Crises and prices : information aggregation, multiplicity and volatility

Digitized

by

the

Internet

Archive

in

2011

with

funding

from

Boston

Library

Consortium

Member

Libraries

HB31

J1415

DEWEY

Massachusetts

Institute

of

Technology

Department

of

Economics

Working

Paper

Series

CRISES

AND

PRICES

Information

Aggregation,

Multiplicity

and

Volatility

George-Marios

Angeletos

Ivan

Werning

Working

Paper

04-43

December

15,

2004

Room

E52-251

50

Memorial

Drive

Cambridge,

MA

02142

Thispapercan bedownloadedwithout charge fromthe

SocialScience Research Network PaperCollectionat

MASSACHUSETTSINSTITUTEj

OFTECHNOLOGY

FEB

1 2005

Crises

and

Prices

Information

Aggregation,

Multiplicity

and

Volatility*

George-Marios

Angeletos

Ivan

Werning

MIT

and

NBER

MIT,

NBER

and

UTDT

angeletOmit.edu iwerningOmit.edu

First draft:

January

2004.

This

version:

December

2004.

Abstract

Many

argue that crises- suchascurrency attacks, bankrunsandriots- can be

describedastimesofnon-fundamental volatility.

We

argue thatcrises arealsotimes

when

endogenoussources ofinformation arecloselymonitoredandthusan important

part ofthephenomena.

We

studytherole ofendogenousinformationin generating

volatilitybyintroducing afinancialmarketina coordinationgamewhere agentshave heterogeneousinformationaboutthefundamentals. Theequilibrium priceaggregates

information withoutrestoring

common

knowledge. In contrast tothe casewith

ex-ogenousinformation, wefindthatuniqueness

may

notbe obtainedas aperturbation

from

common

knowledge: multiplicityisensuredwhenindividualsobserve

fundamen-talswithsmall idiosyncraticnoise. Multiplicity

may

emergealso inthefinancial price.

When

the equilibriumisunique,itbecomes moresensitivetonon-fundamentalshocks

asprivate noiseisreduced.

JEL

Codes: D8,E5, F3,Gl.

Keywords: Multipleequilibria, coordination, global games, speculative attacks,

cur-rencycrises,bankruns,financialcrashes,rationalexpectations.

*Forusefulcommentsandsuggestions,wethankDaronAcemoglu,ManuelAmador,V.V.Chari,Christian Hellwig,PatrickKehoe, BalazsSzentes,andseminarparticipantsatMIT, Stanford,UCLA,

UC

Berkeley,

Duke, the Minneapolis and ChicagoFederal Reserve Banks, theMinnesota SummerWorkshop, andthe

1

Introduction

It'sa love-hate relationship,economistsare atonce fascinatedand uncomfortable with

mul-tiple equilibria.

On

the onehand, economic and political crises involve large and abrupt

changesin outcomes, but oftenlack obviouscomparable changes in fundamentals.

Com-mentatorsattributeanimportantroleto

more

orlessarbitraryshiftsin'market sentiments'

or'animal spirits',and modelswith multipleequilibriaformalizetheseideas.

On

the other hand,modelswith multipleequilibriacanalsobe viewedasincompletetheoriesthat should ultimatelybe extendedalong

some

dimensionto resolvetheindeterminacy.

The

first view is represented by a large empirical and theoretical literature.

On

the empiricalside,

Kaminsky

(1999), forexample, documentsthat thehkelihoodofeconomic

crises is affected by observable fundamentals, but that a significant

amount

of volatility

remainsunexplained.

On

thetheoreticalside,modelsfeaturingmultipleequilibriaattemptto

address suchnon-fundamentalvolatility.

Bank

runs, currencyattacks,debtcrises,financial

crashes, riotsandpoliticalregimechangesaremodeledasacoordinationgame: attackinga

regime -suchasa currencypegorthebanking system

-

isworthwhileifandonlyifenough

agentsare alsoexpectedto attack.1

Morris and Shin (1998, 2000) contributeto the secondviewby showingthat a unique equilibriumsurvives insuch coordinationgames

when

individualsobservefundamentalswith smallenoughprivate noise.2

The

resultismoststriking

when

seenasa perturbationaround

the original common-knowledgemodel, which isridden withequilibria.

Most

importantly,

their contributionhighlights theroleoftheinformation structureindeterminingand

char-acterizingthe equilibriumset.

The

aim of thispaper istounderstand the roleofinformationin crises.

We

focuson

two distinct forms ofnon-fundamental volatility: the existence ofmultiple equilibria and

the sensitivity of a unique equilibrium to non-fundamental disturbances.

We

argue that

consideringendogenousinformationiscrucialforthesequestions.

Information istypically treatedexogenouslyin coordinationmodels, butis largely

en-dogenous in most situations of interest. Financial prices and macroeconomic indicators

convey information regarding others actions and theirinformation. During times ofcrises

suchindicators aremonitoredintenselyand appeartobe an importantpartofthe

phenom-ena.

As

an example, consider the Argentine 2001-2002crisis, whichincluded devaluation

ofthe peso, sovereign-debt default, andsuspension ofbank payments. Leadingup tothe

crisisthroughout 2001,bankdepositsand the peso-forwardratedeterioratedsteadily. Such

1

See.forexample, Diamondand Dybvig (1983), Obstfeld (1986,1996), Velasco (1996), Calvo (1988),

CooperandJohn(1988),ColeandKehoe(1996).

2

variableswere widely reportedby news media andinvestor reports,andwerecloselywatched bypeoplemakingcrucialeconomicdecisions.

These observations lead usto introduceendogenous sourcesof public information in a coordinationgame. In ourbaselinemodel, individualsobservetheir privatesignalsandthe

price of afinancialasset.

The

rational-expectations equilibrium price aggregates disperse

privateinformation, but avoidsperfectrevelationbyintroducingenoughnoise inthe

aggre-gationprocess, as in

Grossman

andStiglitz (1976). Thus,noneofourresultsaredrivenby

restoring common-knowledge.3

The

maininsight toemergeisthat theprecision ofthe endogenous publicinformation

increaseswith theprecision ofthe exogenousprivateinformation.

When

privateinformation

is

more

precise, individuals' asset

demands

are

more

sensitiveto theirinformation.

As

a

result,the equilibriumpricereacts

more

sensitivelytofundamentalvariables,thusconveying

more

preciseinformation.

This resulthas important implicationsforthedeterminacyofequilibria,asa horse-race

between private and public information emerges.

An

increase in the precision ofprivate

information makes coordination

more

difficult as individuals relymore on their

own

dis-tinct information. However, theconsequent increase intheprecision ofendogenous public

informationfacilitates coordination. Thisindirect effect typicallydominates, reversingthe

limiting result: multiplicity is ensured

when

individualsobserve fundamentals with small

enoughprivate noise.

Uniqueness therefore can not be attained as a small perturbation around

common-knowledge.

To

illustrate thispointFigure 1 displays the regions ofuniqueness and

multi-plicityinthespaceofexogenouslevelsofpublic andprivate noise (<7Eandax, respectively).

Multiplicityisensured

when

eithernoiseissufficientlysmall. Inthissense,public and

pri-vatenoise actsymmetrically. Incontrast,

when

informationisexogenous,less public noise

facilitates multiplicity, but less private noise contributes towards uniqueness. Moreover, with endogenousinformation, equilibriaoutcomes arecontinuous withrespect to

informa-tionparameters. In contrast,withexogenousinformation,thelimiting resultdiscussedabove

illustratesasharpdiscontinuity.

Interestingly, multiplicity

may

emerge, not only with respecttotheprobability ofregime

change, but also in the asset price. This occurs

when

the asset's dividend depends on theendogenous aggregate activity in the coordinationgame, which one can interpret also

as ashort ofinvestmentgame. Different equilibriumprices arethen sustained bydifferent

expectationsaboutthe aggregatesizeofattack/investment,andthereforeaboutthe dividend

3_Atkeson

(2000)hasalreadypointedoutthat perfectly revealing assetmarketswouldrestore equilibrium

uniqueness

multiplicity

Figure 1:

a

x measures the exogenous noise in private information and <r £ the exogenous publicnoise intheaggregationofinformation.

ofthe asset,which inturnare possible onlythanks tothecoordinatingroleprices playin

thefirstplace.

Inregionswherethe equilibrium isunique,

we

are able toperform comparativestatics.

We

findthata reductioninexogenousnoiseincreasesthesensitivity oftheregimeoutcome

tonon-fundamentaldisturbances.

When

theasset'sdividendisendogenous, a reductionin

noise

may

alsoincrease the non-fundamentalvolatility in thefinancial price. Since these

results parallel the determinants ofequilibrium multiplicity,

we

conclude that lower noise

increases volatility forboth formsofvolatility.

We

alsostudy a separatemodelmotivatedbythe

bank

runandriotsapplications. Inthis

modelthereisnofinancialmarket. However,informationisendogenousbecause individuals

observe a public noisysignal ofothersactions.

An

advantageisthatthismodelallowsusto

studyendogenous informationwith aminimal modification oftheMorris-Shinframework.

The

modelshares the

main

insights and results forequilibrium multiplicity obtainedwith

thefinancial pricemodel.

Thispapercontributestothe real-expectationsliteratureby examiningthecoordinating

role of financial prices anditsimplicationsfor volatility. In

Grossman

andStiglitz (1981),

wherethedividendoftheassetisexogenous andtherearenocomplementaritiesin real

in-vestmentchoices,the equilibriumpriceisuniqueandislessvolatilethe lower thenoise inthe informationstructure. Thisisbecauseinthatframeworkprices merelyconveyinformation abouttheunderlying fundamentals,whichhelpsevery individual figureouthis bestchoice without regardto

what

other agents are choosing. In the presence ofcomplementarities,

instead, financialpricesserve alsoa coordinatingrole: they help agents coordinatetheir real

investment choicesbyimproving theability ofeach agent topredict whatother agents do.

And

it isthiscoordinatingrolethatexplains

why

multipleequilibriaandhighvolatility

may

emergealso intheprices.4

Our

workbuilds on Morris-Shin andunderscores theirgeneraltheme thatmultiplicity

or uniqueness depends onthe information structure. Within such a framework, our aim

is to understand the effect ofendogenous sources of public informationsuch as financial

pricesandotherindicators ofaggregateactivity. Angeletos,

HeUwig

andPavan (2003,2004)

alsoendogenize information,but notthroughinformation aggregationas here. Instead,they considerthesignalingeffectsofpolicy interventionsandtheflow ofinformationinadynamic

globalgame.

RelatedisMukherji, TsyvinskiandHellwig(2004),

who

consider acurrency-crises

game

and allow financial prices to directlyaffect the coordination outcome. In particular, they

focuson

how

multiplicity oruniqueness dependson whetherthecentralbank'sdecision to

devalue istriggeredbylargereserve losses or high interestrates. Tarashev (2003) also

en-dogenizesinterest rates inacurrency-crisesmodel, but does notinvestigatethe implications

forequilibriummultiplicity.

Finally,

we

sharewith Chariand

Kehoe

(2003) the motivationinexaminingtheroleof

information in generating non-fundmentalvolatility, but

we

focus on coordination rather

thanherding. Indeed,

we

bring closerthe global-games and social-learning approachesto

crisesbyallowingendogenouspublicsignalsaboutothers' activity,butavoid informational cascadesby assumingthat noagentis "big". Building againonMorris-Shin,

we

also

show

how

coordination

may

helpunderstand highnon-fundamentalvolatilityeven with aunique

equilibrium.

The

restofthepaper isorganizedas follows.

We

describethebasic modelinSection2

andreview theexogenousinformationbenchmark. Section3incorporates thefinancialasset

and examinesmultiplicity of equilibria. Section 4examinesthe determinantsof volatilityas

acomparative staticalong auniqueequilibrium. Section5studiesthemodelwith adirect

signal ofaggregateactivity. Section6concludes.

BarlevyandVeronessi (2004) consideraGrossman-Stiglitz-likeeconomywhich admitsmultiple

rational-expectationsequilibria,butthe source of multiplicityisverydifferent. In theirmodel,there areno

comple-mentaritiesinrealinvestmentchoices (thedividendofthe assetisactuallyexogenous) andthe pricedoes

not playanycoordinatingrole like inour framework. Multiplicity instead originatesinthe non-linearity of the inferenceproblemfacedby uninformedtraderswhentheyinteract with informedandless risk-averse agents.

2

The

Basic

Model: Exogenous

Information

Beforeendogenizing public information,

we

review thebasicmodelwithexogenous

informa-tion.

Actions

and

Payoffs. Thereisastatus quo andameasure-one continuum of agents,

indexedbyi £ [0, 1].

Each

agenticanchoosebetweentwoactions, eitherattackthestatus

quo(oj

=

1)ornot attack(a2

=

1).

The

payofffromnot attackingisnormalizedto zero.

The

payofffromattackingis 1

—

c

>

ifthestatus quois abandoned and

—

cotherwise, where

c 6 (0,1) parametrizes thecost of attacking.

The

statusquois inturn abandonedifand onlyifyl

>

6,where

A

denotes themassofagents attackingand9representstheexogenous fundamentals,namelythe strengthofthestatusquo. Itfollowsthatthe payoffofagentiis

U(a

i,A,6)

=

al

(R(A,6)-c)

(1)

where

R

(A,9) denotes theregimeoutcome,with

R =

1 if

A

>

9and

R

=

otherwise. Interpretations. Inmodelsofself-fulfillingcurrencycrises(Obstfeld, 1986, 1996;Morris

and Shin, 1998), a "regime change" occurs

when

a sufficiently large mass ofspeculators

attacksthecurrency, forcingthecentralbanktoabandonthepeg; 9then parametrizes the

amount

offoreignreserves or

more

generallytheabilityandwillingness ofthecentral

bank

to

maintain thepeg. Inmodels ofself-fullingbankruns,onthe otherhand, a "regime change" occurs once a sufficiently large

number

of depositors decide to withdraw their deposits,

forcingthe

bank

tosuspenditspayments; 9then parametrizes theliquidresourcesavailable

tothebankingsystem.5

Throughoutthe paper,

we

focusoncrisesasthe

main

interpretation ofthemodel. Other

applications, however, are also possible. For example, one could interpret the model as

an

economy

with investment complementarities, inwhicha;

=

1 represents undertake an

investment,

A

theaggregatelevelofinvestment, and9theexogenousproductivity.

In short, the key property ofthe payoff structure

we

consider is that it introduces a

coordination motive:

U

(1,A,9)

—

U

(0,A,9) increaseswith A,meaning thattheindividual

incentive totakeoneactionincreaseswith the

mass

ofotheragentstaking the

same

action.

Indeed,

when

9is

common

knowledge, both

A =

I and

A =

is an equilibriumwhenever

9 € (#, #]

—

(0,1].

The

interval (9,6] thus represents the set of"criticalfundamentals" for

which agentscancoordinateonmultiple coursesofactionunder

common

knowledge. Information. Following Morris-Shin,

we

assume 6 is not

common

knowledge. Inthe

Otherrelated interpretations include debtcrises andfinancial crashes (Calvo, 1988; Coleand Kehoe,

1996;Morris andShin, 2003, 2004). Finally,Atkeson (2000) interprets themodelasdescribingriots: the potentialriotersmayormaynotoverwhelmthe police forceinchargeofcontainingsocialunrestdepending

beginning of the game, nature draws 6 from a given distribution, which constitutes the

agents'

common

prior about 9. For simplicity, this prior is assumed to be an (improper)

uniform over the entire real line.6 Agent i then receives a private signal Xj

=

9-f-0"x^, whereo~xisapositive scalarand

_{^ ~}

AA(0,1)isidiosyncratic noise,i.i.d. acrossagentsand independentof9. Agentsalsoobserveanexogenous publicsignalz

=

9

+

a.v,where az isa

positive scalarandv

~

Af(0,1)is

common

noise,distributedindependentlyofboth9and£.

The

information structureisthusparametrized parsimoniouslybyaxand az,thestandard

deviations ofthetwo noises; or equivalently by

a

x

=

o~

2 _and

a

z

=

o~~ 2

, theprecisions of

privateandpublic information.

Equilibrium

Analysis.

We

focus on monotone equilibria, that is, equilibria inwhich

an agent attacksifand onlyifhis private signal is sufficiently low.'

Thus

suppose that,

givena realization z ofthe public signal, an agent attacksifand only iftherealizationx

of hisprivate signalislessthan a thresholdx*(z).

The

sizeofthe attackisthen A(9,z)

=

<1>

(

v

/q^(x*(z)

—

9)) andisdecreasingin6. Itfollowsthatregimechangeoccursifandonly

if9

<

9*{z),where9*(z) istheuniquesolution toA(9,z)

—

9, orequivalently

x*(z)

=

9*{z)

+

^-\9*

{£)). (2)

The

posterior ofthe agent about 9 is

Normal

with

mean

_aa*

x

+

-ar a z and precision ctx

+

oc z, where

a

x

—

a~ 2 _and

_a

z

=

a~

2

denote, respectively, theprecision ofprivate and publicinformation. Itfollowsthattheagentfindsitoptimaltoattackifandonlyifx

<

x*(z)

,

wherex*(z) solvetheindifferenceconditionPr[9

<

6*(z)\x,z]

=

c, orequivalently

$

[V^+cZ

(e*(z)

-

^

zx*{z)

-

^-z))

=

c. (3)

Substituting(2) into(3) givesa singleequationin 6*(z) :

Hence, anequilibriumissimplyidentifiedwith asolution to (4).

A

solution to (4) alwaysexist andisuniquefor all z ifand onlyif

a

z/^/ax

~

<

V2n, or

equivalently o~x

<

a

2

z^/2n. (See

Appendix

for adetailed derivation.)

We

conclude that the

Thisassumptioniswithoutanyseriouslossof generality:bylettingthecommonpriorbeuninformative,

webiasthe results againstmultiplicity.

7

This is without seriouslossofgenerality fortwo reasons. First, when themonotoneequilibrium is

uniqueandtheinformationisexogenous,astandardargumentof iterrated deletion of stronglydominated

strategiescanbe usedtoshowthat there arenoothernon-monotoneequilibriaeither. Second,toprove the existence ofmultiple equilibriaandtheconvergencetocommon-knowledgeoutcomes witheitherexogenous

Figure2:

a

_x and

a

z parametrize thenoise in private andpublicinformation; uniquenessis

ensuredforaxsmallenough.

equilibriumisuniqueifand onlyiftheprivate noiseissufficientlysmall.

Proposition

1 (Morris-Shin) Inthe

game

withexogenous information, the equilibriumis

uniqueifandonlyifax

<

a^y/2/K.

This result is illustrated in Figure 2, which depicts the regions of (o~x,az) for which

the equilibrium isunique. For any az

>

0, uniqueness in ensuredby letting o~x

>

suf-ficiently small. In the limit as ax

—

> 0, the incomplete-information

game

approaches the

common-knowledge

game,inwhichtherearemultipleequilibria;yet,the equilibriumofthe

incomplete-information

game

remainsuniqueand actuallyasymptotestoasituationwhere the regime outcome becomes independent of e. In other words, not only sunspots cease

tomatter,but all non-fundamentalvolatility-thevolatility of

A

or

R

conditional on9 -vanishes.

Corollary 1 Inthe limit aso~x

—

>0, there is a unique equilibrium,inwhich

A

((9,z)

—

> 1 if

<

6 and A{9,z) -* if6

>

0, where6

=

1

-

c.

Thisresultisintriguing,asitmanifests asharpdiscontinuity ofthe equilibriumsetaround

o~_x

=

: a tiny perturbation

away

fromperfect information obtains aunique equilibrium.

The

key intuition is that disperse private information serves as an anchor for individual

behavior: itlimitstheabilityto forecast eachothers'actions andthereby tocoordinateon multipleequilibria.

Indeed,

when

allagentsshare the

same

informationabouttheunderlying fundamentals,

they canperfectly forecast each others' actions inequilibriumand cantherefore perfectly

coordinateoneithereverybodyor

nobody

attacking

when

6G (6,6}.

But when

agentshave

heterogeneous informationabout the fundamentals, eachagentfacesuncertainty regarding otheragents'beliefsabout9andtheir actions. Foranygivenprecision ofpublicinformation, the higher theprecision oftheagents' privateinformation,the

more

heavilyagents condition

their actionsontheir

own

privateinformation,andthus theharderitisforother agentsto

predict their actions.

When

a

xis sufficiently smallrelative to a., thisanchoring effectof

privateinformationis strong enoughthat the abilityto coordinateonmultiple coursesof

action totallybrakes

down

andauniqueequilibriumsurvives. Finally, as

a

x

—

> 0,theprivate

signal

become

infinitely precise relative to the publicsignal, inwhichcaseagents ceaseto

conditiononthe publicsignaland byimplicationthesensitivityofequilibriumoutcomesto

theshocke vanishes.

3

Endogenous

Information

I:

Financial

Prices

The

resultsreviewedabovepresumethattheprecisionofpublicinformationremains

invari-ant whilechanging theprecision ofprivateinformation. This, however,

may

notbepossible

when

the public information consists of financial prices that endogenously aggregate the

disperse privateinformationinthe population.

To

capture the role of prices as achannel ofinformation aggregation,

we

modify the environmentas follows. Thereare

now

twostages. Instage1,agents tradea financial asset

whosedividenddependsontheunderlyingexogenousfundamentals and/ortheendogenous outcomeofStage2. Stage2inturnislikethe

benchmark

regime-change

game

oftheprevious

section, exceptforthat the publicsignal is

now

thepricethat clearedtheasset market in

stage 1.

Thisframework opens

up

new

modelingchoicesregardingthespecificationofthe asset's

dividendandthepreferencesoverrisky payoffs.

We

guide our choiceswithaneye towards

tractability: to isolatetheeffectsofinformation aggregationfrom any directpayofflinkages

between thetwo stages,

we

assumethat preferences are separable between thefirst-stage

portfolio choiceandthe second-stage attacking decision;8 _{and topreservenormality ofthe}

informationstructure,

we

model stage1 along the

CARA-normal

framework of

Grossman

and Stiglitz (1976, 1980)andHellwig (1980).

3.1

Model

Set-up

Timing,

information,

and

payoffs.

The

game

startsagainwith naturedrawing 8 from

an improper uniformoverR.

Each

agentihas a given

endowment

ofwealth Wiandreceives

an exogenousprivate signal Xi

=

9+-cXx^.

The

distribution ofw-iisarbitraryandthenoise

£;is_{A/"(0, 1)},i.i.d. across agents,andindependentof9.

In stage 1,agents caninvest theirwealtheither inarisk-lessasset orarisky asset.

The

riskless asset isin infinitelyelastic supply, it costs 1 inthe first stage, and itdelivers 1 in

thesecondstage.

The

risky asset,onthe otherhand,costsp inthefirststageanddelivers

/inthe second.

The

agent enjoys utility from final-stage consumption and has constant absolute risk

aversion

(CARA).

The

payofffromtheportfoliochoiceisthus

V

(c)

=

—

exp(—7c) /7,where

c

=

w —

pk

+

fk

is consumption, k denotes investment in the risky asset, and 7

>

is the coefficient ofabsoluterisk aversion.

The

net supply ofthe risky asset is given by

an unobserved

random

variable

K

(e)

=

cr

se,where £

~

jV(0,1) is independent of both the fundamentals and the private noise and a£ parametrizes the exogenous noise in the

aggregation process.

The

unobserved shock e can also be interpreted as the

demand

of

'noisetraders' and its role is to introduce noise in the information revealed by financial

pricesaboutfundamentals.

In stage 2, agents chose whether to attack or not.

The

the status quo is abandoned

(R

=

1)ifandonlyif

A >

9andtheagent'spayofffromstage2isU(a,A,9)

=

a(R

(A, 6)

—

c)

,

as inthe

benchmark

model.

The

regime outcome, theasset'sdividend,andtheagents'

pay-offsare realized attheendofstage2.9

Asset.

What

remains to closethe modelis thespecification oftheasset's dividend/.

We

are interested intwoalternatives:the case that/isdeterminedmerelybytheunderlying

economic fundamentals9; andthe case that / isafunctionoftheendogenousactivity

A

of

the coordinationstage.

What

we

wishto capturewith the latter case isenvironments wherethe coordination

game

is

some

shortofaninvestment

game

whichalsodeterminesassetreturns. For example, imagine agents have the optionto invest in a

new

sector (or adopt a

new

technology), the dividend oftheasset dependsontheprofitability of this sector, andthe

new

sectoris

profitable onlyifenoughagentsinvest.

Equilibrium. Because ofthe

CARA-normal

specification, the

demand

for the risky

assetisindependentofwealth.

The

individual's

demand

kin stage1 andactionainstage

9_Given

the separability of payoffbetweenthetwostages,onecould reinterpret themodelasiftheagents

whomoveinstage1weredifferent thanthe agentswhomoveinstage2, providedofcourse thatthelatter

observe thepricethat clearedtheassetmarketinstage1

2 arethus functionsof(x,p) alone. Since thereisacontinuumofagents,the corresponding aggregatesarefunctionsof (9,p).

The

equilibriumprice,ontheotherhand, isa functionof

theexogenousstate(0,e)~.

We

thusdefine

Definition 1

A

rational- expectations equilibriumisapricefunction P(9,e),individual

strate-giesk(x,p) anda(x,p) for investmentandattacking, and correspondingaggregates K{Q,p) andA(9,p), suchthat: (i) instage1,

k(x,p)

=

argmaxE

[V((f

—

p) k) \ x,p] (5)

K(9,p)

=

I k(x,p)d$

(^—)

(6)

K(9,P(9,e))

=

K(e)

(7)

(ii) instage2,

a(x, p)

=

arg

max

E

[ U(a, A(9,p),6) \ x,p] (8)

A(0,p)=

fa(x,p)d$(—

-)

(9)

The

interpretation ofthese conditionsisstraightforward. Condition(5) and (9) requires

that an agent's investment take into account the information contained in their private

informationand prices,whereascondition (7)imposes marketclearing inthe assetmarket. Note that (5)- (7) define astandard rational-expectations equilibriumfor stage 1, whereas

(8)-(9) defineastandardPerfect Bayesian equilibriuminstage2. Finally, with

some

abuse

ofnotation,

we

let

R

(9,e)

= R

(9,

A

(9,

P

(9,£))) denote the equilibriumregimeoutcomeas

afunctionoftheexogenousstate(9,e)

.

3.2

Exogenous

dividend

We

consider first the case that the dividend / depends onlyon the exogenous economic

fundamentals9.

To

preserveNormality,

we

let /

=

/(9)

=

9.

We

start the characterization of the equilibrium with stage 1 (the financial market).

Thankstothe

CARA-Normal

specification,theexpectedutilityoftheagent reducesto

E{V(c)\x,p}

=

V((E[f\x,p}-p)k-lV

&r[f\x,p}k2)

The

FOC,

whichisbothnecessaryandsufficient, impliesthattheoptimal

demand

forthe

assetis

E[f\x,p]-p

k(x,p)

=

r, 10

iVax\f\x,p]

where /

=

9.

We

proposethattheposterior of 9conditionalon x,phas

mean

Sx

+

(1

—

S)p

and precision a, for

some

8 G (0,1) and

a

>

0. It fohowsthat k(x,p)

=

(Sa/j)(x

—

_p)

and therefore K(8,p)

=

(8a/j)(9

—

p). In equilibrium, K(9,p)

=

K

(e),which gives the

market-clearingprice asp

=

9

—

(Sa/j)~

a

E e, orequivalently

P(9,e)

=

9-o

_pe (11)

where

a

_p

=

(5a/-y)~1

_a

E isthestandard deviationofthepriceconditionalon9.

The

observa-tion of

p

isthereforeequivalenttothe observationofa

Normal

signalabout 9withprecision

a

_p

=

a~

2

. Moreover, x and

p

areindependent and therefore9\x,p

~

JV(Sx

+

(1

—

5)p, a)

,

£

=

&

x/a, and

a

=

^

-(-

a

p, whichverifiesourinitialguess. Solvingfora,8, and

a

p,

we

get

a

=

a

x (l

+

a

x

a

E/^y2),8

=

1/(1

+

a

x

a

e/j

2

), and

a

p

=

a

ea^/7 2

.Thatis,theprecision ofthe

publicinformation revealedbythepriceisanincreasing functionofboththeprecisionthe

aggregation processandtheagents'primitive privateinformation. Equivalently,

a

p

=

ja

eal, (12)

whichtogetherwith (11) completes the characterizationofstage 1.

Next, consider stage 2 (the coordinationstage). Given (11), thecontinuation

game

in

stage2isisomorphictothe

benchmark

game

ofSection2,exceptforthefactthatthe public

signalisz

=

p

=

P

(9,e) andhence

a

z

=

a

p.

The

following arethusimmediateimplications

of our earlier analysis in Section 2: first, the agent's strategy is a(x,p)

=

1 if

and

only

ifx

<

x*_(p), the aggregate attack is

A

(6,p)

=

$

(a/o^(x* (p)

—

6)), andthe regime is

abandonedifand onlyif9

<

9*(p);second,the thresholdsx*(p) and9*(p) aregivenbythe

solutionto(2)and(4)once

we

replacezwithp anda,with

a

p; third,thesolutionisunique

ifandonlyifandonlyifax

<

a_p^/2n.Using then (11),

we

conclude that the equilibriumis

uniqueifandonlyif<J2 e

u

x

>

j

2

(2ir)~1^2

.

Proposition 2 Inthe economywith _f

=

_f(9), there aremultiple equilibria ifand onlyif

o~2.o\

<

7

2(27r)-1/2

. There arethen multiplestrategies, a(x,p), attacksA(9,p) and regime

outcomes,

R

(9,p),forthecoordinationstage, but the asset demands, k(x,p) and

K

(9,p),

andtheprice function, P(9,e), arealways uniquely determined.

Note

how

thisresultcontrastswith Proposition1:

when

publicinformationisexogenous,

asufficientlylow

a

x ensures uniqueness; but

when

the public information isendogenous as

Figure3:

With

endogenouspublicinformation,as

a

xdecreases, o~zalsodecreases;

multiplic-ityisthereforeensuredforsufficientlysmall

a

x.

inthe presentmodel, asufficientlylow

a

x ensuresmultiplicity. Thisisbecausemoreprecise

privateinformation

now

implies

more

precisepublicinformation,indeedataratehighenough

that theagents' ability topredicteachothers'actions increases as theirinformation improves. Figure3 illustratestheresult inthe (ax,a.) space: unlikeFigure2, as

a

x decreases,

a

z also

decreasesandfastenoughthatthe

economy

necessarily entersthe regionof multiplicity.

Uniqueness canthereforenotbeseenasa smallperturbation

away

from

common

knowl-edge.

As

it was earlier illustrated in Figure 1,

when

either

a

x or ae is small enough, the

precision oftheendogenouspublicinformationissufficientlyhigh that multiple coursesof

ac-tioncanbesustainedinthe coordinationstage. Moreover, thecommon-knowledge outcomes

canactuallyberecoveredas as eithertypeofnoise vanishes.

Corollary2 Considerthe limit asa_x

—

> for giveno~ E, orthe limit as o~ s

—

> for given

o~_x. There exists an equilibrium in which R(9,s)

—

» whenever9 £ {9,6), as well as an

equilibrium in which

R

(9,e)

—

»1 whenever9€ (0,9). In every equilibrium, P(9,e)

—

9for

all (9,e).

3.3

Endogenous

dividend

We

now

consider the case thatthe payoffofthe assetisdetermined bytheoutcome ofthe coordination stage.

To

preserve Normality,

we

now

let /

=

f (A)

=

—^~

1(A).

The

asset

thuspays

more

the lower thesizeoftheattack,which

we

couldinterpret asasituationwhere

"attack"

means

refrainingfrom

some

shortofinvestment.

We

startagain the equilibriumanalysiswith stage1. Guessing (andlaterverifying)that agentsuse

monotone

strategies instage2 such that a(x,p)

=

1 ifand onlyifx

<

x*_(p) for

some

threshold x*(p) ,

we

havethat

A

(8,p)

=

$

(y^fx*

(p)

—

#]) and therefore /

=

s/a^[9

—

x*(p)]. Hence, ifthe agentsposterior about 9 is

Normal

(which

we

verify later),

so is /. It follows that the individual optimal

demand

for the asset is againas in (10).

Substituting /

=

y/a^[9

—

x*_(p)] intothelatterandletting7

=

7^/0^ and

p

+

x*(p) (13)

we

canrewritetheoptimal

demand

ask

=

(E[9\x,p]

—

_{p) / (^fVax[9\x,p\).} Likeinthe previous

section,

we

then propose E[#|x,p]

=

Sx

+

(1

—

5)p andVar[0|x,p]

=

a. It follows that

K

(9,p)

=

(5a/7

))(9

—

p) andthereforemarketclearingimplies

p

=

9-

_T

^-=e.

(14)

Hence, provided that thereis a one-to-one

mapping

between p andp, the observationof

_p

isequivalent tothe observation ofa

Normal

signalabout 9withprecision

a

_p

=

(5a/^)2

a

e

andtherefore 9\x,p~J\f(8x

+

(1

—

5)p,a) with5

=

a

x/(qx

+

a

p) and

a

=

a

x

+

a

p. Solving

ford,a,and

a

_pgives

a

p

=

a

E

a

x/^f

=

a

e

a

x /'j

2

,orequivalently

a_p

=

ja^l-

(15)

Next, consider stage2.

The

threshold9*(p) solves 9

— A

(9,p); equivalently,

x*(p)

=

8*(p)

+

^-\9*(p)).

(16)

The

thresholdx*(p),onthe otherhand,solves Pr[9

<

9*(p)\x,p]

=

c; equivalently,

$

(v^T^

(>(P)

-

_5fe^(p)

"

_sfep))

=

c

(17)

Theseequationsaretheanaloguesof(2) and(4) inthe

benchmark

game. However,once

we

substitute (13) and (16) into (17),

we

get auniquesolution for 9*(p) :

*(P)

-

*

(y^

$_1C1

-

c)

-

_S=fe?)

> (18)

inwhichcase(16) implies alsoauniquex*(p). Hence,unlike eitherthebenchmark

game

or

thecase ofthe previoussection,the strategyofthe agentsinthe coordination

game

andthe

regimeoutcomearealways uniquelydeterminedasfunctionsoftherealized price.

To

complete theanalysis,

we

needtodetermine the

mapping

between p andp, that is,

the equilibriumprice

mapping

p

=

P(9,e). Using(13), the aggregate

demand

fortheasset

is

K(9,p)

=

j{6-^=p-x*(p)).

(19)

Sincex*_(p) isuniquely determined,

K

(9,p) is alsouniquely determined (andsimilarly for

k(x,p)). Moreover, for any9, K(9,p) iscontinuous inp, K(9,p)

—

> oo asp

—

*

—

oo, and K{9,p)

—

*

—

ooasp

—

>-foo. Itfollowsthat

K

(9,p)

—

e,orequivalently (13),alwaysadmits

asolution; thatis,a market-clearing pricep

=

P

(9,e) exists forany9 ande. However,the

market-clearingpriceneednot alwaysbeunique. Notethat 9*(p) andx*(p) aredecreasing

inp, whichinturnimplies theasset'sdividend, /

=

yfaZ\9

—

x*(p)], isitselfincreasinginp.

As

aresult,the

demand

fortheassetcanbenon-monotonic. Indeed, (16)and (19)imply

*»{^}--^-,(.-<0)}

andtherefore

K

(9,p)isgloballydecreasinginpifandonlyif^fa^jav

>

y2ft, orequivalently

a

_la_x

>

72/\/27r. Ifinstead o\(j\

<

j

2/\/2x, the aggregate

demand

for the asset is non-monotonicandthereisanon-emptyinterval(p\,p~2)such that(13) admits asinglesolution

whenever p ^ (pi,p2)butthreesolutionswhenever

p

E (pi,p%)

.Different selectionsfromthese

solutionsthensustain differentmappings between p andp,thatis,differentequilibriumprice

functions

P

(9,e)

.

Proposition 3 In the economy with_f

—

_f(A), there are multiple equilibria ifandonlyif

o\a\

<

72(27r)-1/ 2

. There are thenmultiple pricefunctions P(9,e), but the assetdemands

k(x,p),

K

_(9,p),the strategya (x,p), andtheattack

A

(9,p) arealways uniquely determined.

Inthemodelofthe previoussection,the dividendwasa functionof 9aloneandtherefore

thepriceplayed theroleofasignalabouttheexogenousfundamental.

As

aresult,

indeter-minacy emergedforthestrategies andthesizeofthe attackinthe coordinationstage, but

not fortheprice clearingthe assetmarket. Here, instead, the dividenddepends on

A

and

therefore thepriceplaystherole ofan anticipatory signalabout theendogenoussizeofthe

attack.

A

particular price realizationcoupledwithan agent's privateinformationthenpins

down

auniqueposterioraboutthesizeoftheattack, inwhichcasetheagent'sbest response

isofcourse unique. This explains

why

the strategies instage2areuniquelydetermined.

In stage 1, onthe other hand, differentlevelsofthepriceare associatedwithdifferent

expectationsaboutthesize ofthe attackand thereforeaboutthedividend ofthe asset: in

equilibrium,a higherprice signalsa higher dividend.

When

thiseffectisstrongenough,itcan

offsetthe usual negative payoffeffectoftheprice- namelythatahigherprice

means

ahigher

cost ofobtaining theasset- and

may

thereforeinduce the

demand

fortheassettoincrease

with the price. This non-monotonicity ofthe asset

demand

introduces thepossibility for

multiple market-clearingprices. Finally,thepositiveeffectofthepriceisstronger

when

a

_p

ishigher, fora higher

a

pincreasesthesensitivity oftheagents'actions instage2totheprice

andtherefore ofthe dividendtothe price. This explains

why

multiple equilibrium prices

exist for forsmalllevelsof noise.

To

further understand

why

multiplicity emerges in the price and not in the strategy

ofthe agents, it helps to consider fora

moment

the no-noise

game

(ax

=

a

£

=

0).

The

exogenoussignalisthenx

=

9andthe equilibriumprice isp

— —^~

1

(A) andthereforean

agentlearns perfectly9byobservingx and

A

byobservingp. Clearly,the agentthenfinds

itoptimaltoattackifandonlyif

A

>

9, orequivalently x

< $(—

p),which

means

thathis

strategyisuniquelydeterminedasa functionofx andp. However,the equilibrium valuesof

p

and

A

arenot uniquely determined. Instead, forevery9£ (9,9],both(p,A)

=

(oo, 0) and

(p,A)

=

(—oo,1) canbesustainedin equilibrium.

In conclusion, small noise

now

ensures multiplicity inboth theasset priceand the

co-ordinationoutcome: different equilibriumpricefunctions resultto different realizations of

theprice and thesizeofthe attackforthe

same

9 ande and therefore to differentregime

outcomesaswell.

What

ismore, the no-noise (orcommon-knowledge) outcomes are once againobtainedasthenoise vanishes.

Corollary 3 Considerthe limit aso~x

—

> for given_{<x £}, or the limit as

a

_e

—

> for given

ax. There is anequilibriuminwhichR(9,e)

—

*1 andP(9,e)

—

>

—

oo whenever96 (9, 9), as wellasanequilibriuminwhichR(9,e)

—

>1 andP(9,e)

—

>

-co

whenever9£ (9,9).

3.4

Discussion

Although the property that theprecision ofendogenous publicinformation increases with

theprecision of exogenousprivate informationis likelytobevery robust,

how

strongthis

effect is and whether it can overturn the direct effect of private information

may

depend

onthedetails ofthechannelthrough whichprivate informationis aggregated. Indeed,the

resultthatmultiplicityisensuredfor smallax relies ontheprecision ofpublicinformation

increasing ataratehigherthanthe squareroot oftheprecision ofpublicinformation. That

wasthe case inthe twospecifications

we

considerabove (as evidentin (12) and (15)) but

neednotbetrueingeneral.

We

now

provideanexample whichillustrates thispossibility.

To

this aim,

we

modify stage 1 as follows.

The

agent is assumed to be risk neutral andface a quadraticliquidity/transaction cost for investing intheriskyasset.

The

indirect

utilityfromtheportfolio choiceisthus givenby

V(c)

=

c

=

w-pk-^k

2

+

fk (20)

where thescalark

>

parametrizes the liquidity/transaction cost.

We

consider againtwo

cases forthe dividend,exogenousand endogenous.

Proposition

4

Suppose

V

isgivenby (20).

(i)

When

_f

—

9, the equilibrium pricefunction

P

is always unique, whereas the

equilib-riumregimeoutcome

R

isuniqueifandonlyifax is either sufficientlysmall orsufficiently

highrelative to a£.

(ii)

When

_f

=

—

$

-1(v4), there are multiple equilibriaifand only ifo~x and/or ae are

sufficiently small. Multiplicity then emerges in both the regime outcome

R

and the price

functionP.

Here,unliketheearlier

CARA-normal

cases, theincrease intheprecision ofpublic

infor-mationgeneratedby anincrease intheprecision of privateinformationisnot always strong enoughtooffset thedirect effectoftheprivateinformation

when

the payoffofthe assetis

exogenous. Thisisbecause the slopeof individual

demand

on theexpected dividend is

in-dependentoftherisk inthe dividend,whichimpliesthattheprecision ofpublicinformation

increaselesswithprivateinformationthan

when

theslopeitselfincreaseswithmoreprecise

information(lessrisk).

When

thedividenddependsonlyon9,theanchoringeffectofprivate

information dominates, thus ensuring uniquenessfor axsmallenough.

When,

however, the dividendoftheassetdepends onthesizeoftheattack,thefactthat

thesensitivity ofthesizeofthe attackandtherefore ofthe dividenditselftothefundamental

increases astheagents'informationimproves compensatesforthelack ofsuch asensitivity

inthe

demand

fortheasset.

As

aresult,coordinationonce againbecomeseasier asprivate

informationbecomes morepreciseandmultiplicityreemergesforsmallenoughnoise.

These exampleshighlightthatthedetailsofthe processofinformation aggregation

mat-ter.

They

also suggestthat the coordinating effect of pricesislikely tobe stronger

when

agents are risk averse and

when

the dividend ofthe traded assets dependdirectly onthe coordinationoutcome. InSection5,

we

willconsideranalternative channelofinformation

aggregation,namelyadirectsignalaboutothers'activity.

We

willfindthatthe coordinating

effect of this endogenoussignal isonce again strongto deliver multiplicity

when

eithero~x

oro~z aresmall enough. Inwhatfollows,

we

continue to focusonthe

more

interestingcase

that theprecision ofpublicinformationincreasesfastenoughwith theprecision of private

information.

4

Noise

and

Volatility

We

now

investigatetheroleoftheinformationstructure fornon-fundamentalvolatility,that

is, thevolatilityofthe regimeoutcome

R

(orthesizeofthe attackA)conditional

on

9.

In general, thereare twosources ofnon-fundamentalvolatility: volatilitygenerated by

the shock e

when

the equilibrium is unique; and volatility generated by payoff-irrelevant

variables (sunspots)

when

there are multiple equilibria and agents use these sunspots to

coordinatetheirbehavior.

With

exogenousinformation, multiplicitydisappears

when

agents observethe

fundamen-tals withsmall idiosyncratic noise.

By

implication, thereisnosunspotvolatility

when

ax

is small enough.

What

ismore, as o~x

—

> 0, the size ofthe attack and by implication the regimeoutcome

become

independent ofe.Hence,allnon-fundamentalvolatilityvanishesas

a

x

—

>0.

On

the other hand,non-fundamentalvolatilityismaximized

when

az—* forgiven

o~x, as in thiscasethe

common-knowledge

outcomesareobtained.

With

endogenousinformation, theimpactof private noiseonvolatilityisquitedifferent.

A

sufficientlylargereductionineithero~xoro~ecanincrease volatilitybyensuringmultiplicity

and therefore introducing

more

sunspot volatility. Corollaries 2 and3 indeed imply that

sunspot volatility is maximized

when

either noise vanishes: as ax

—

> or

a

_£

—

> 0, the regime can either collapse or survivefor anygiven 9 € (9, 9], purely as a functionofthe

sunspot.

What

ismore, sunspotvolatilitycan

show up

in prices aswell:

when

thedividend

isendogenous, the equilibriumpricecan bearbitrarilylowor arbitrarily highforanygiven

9e

{9,9}.

The

propertythat, withendogenous information,less noise

may

increase volatilitydoes notrely onthe existenceofmultiple equilibria.

As we show

next,

when

the equilibriumis

unique,areductionin eithero~xoro~ s

may

increasethesensitivity oftheregimeoutcome and

theassetpricetotheexogenousshockeand

may

therefore resulttohighernon-fundamental

volatility.

4.1

Regime

Volatility

In equilibrium, the regime is abandoned

(R

=

1) ifand only if9

<

9*

[p), but pin turn

depends on9, sincep

=

P

(6,e).

To

expressthe equilibrium regime outcome asa function ofthe exogenousvariables 9ande, note that, as long asthe equilibrium isunique, 9*

(p)

iscontinuously decreasingin

p

and

P

(9,e) is continuouslyincreasing in 9. Itfollowsthat

R

(9, e)

=

1 ifandonlyif9

<

9(s),where9(e) istheuniquesolution to

e

=

6*(P(9,e)).

We

can thusexaminethenon-fundamentalvolatility ofthe regime outcome by examining

thesensitivityof9(e) toe.

Considerfirstthecasethedividendisexogenous(/

=

/(9)). Usingtheresults ofSection

3.2,

we

get

e(£)

=

$(ip

+

-2—

£

}

(2i)

whereip

=

y/l

+

1/

(^a^)^

1

(1

-

c). Itfollowsthat

39

_{_}

_<t>{§-\9))

de -ya£a2x

(22)

and therefore, for anygiven 9, a reduction in a£ or ax increases the slope of 9(e) with

respect to e.

By

implication, 9 satisfies asingle-crossing property with respect to ax or

a

£: let £o be the unique value of e for which

d9/da

e

=

or equivalently 89/dax

=

0;

forany s\ and e^ such that e\

<

Eq

<

£2,

we

still have that d\9(s2)

—

9(e{)\/a£

<

and

similarly 3|^(e2)

—

9(ei)\/ge

<

0. Inthis sense, a reductionineithertype ofnoiseincreases

non-fundamentalvolatility.

A

similar result holds

when

the dividend isendogenous (/

=

/{A)). Indeed, (21) and

(22) continuetoholdif

we

replace

7

with7

=

ja

x.

We

conclude that

Proposition 5 Less noise implies

more

non-fundamental volatility even

when

the

equilib-rium is unique: for anygiven 9, a reduction in

a

e or

a

x increases the slope of 9(e) with

respecttoe.

Our

earlier resultsregarding equilibriummultiplicitycanthusbe viewed as an extreme reincarnationoftheaboveresult.

When

thenoiseissufficientlysmall,volatilitycan behigh,

not only because theoutcomeisverysensitivetotheexogenousnoise, butalsobecause the outcome can depend onarbitrarysunspots.

4.2

Price

Volatility

We

next examine the comparative statics of the volatility of prices.

To

emphasize the implicationsforvolatilitythat donot derivefrommultiplicity,

we

firstfocus againoncase that the equilibriumisuniqueorthat therearenosunspots.

When

thedividendis exogenous,

we

have/

=

9,

p

=

/

—

a_pe, and

a

p

—

^a

za\.

When

instead the dividendisendogenous,

we

have /

=

"3?

-1

[A)

=

(9

—

x*)/ax,p

=

f

"—

(o-p/ax)e,

and a_p

=

7cr_£0"^.Ineither case,

we

canwritetheequilibriumprice asthe

sum

ofthedividend andthesupplyshockappropriatelyweighted:

V

=

/

-

(l<yeo\)e (23)

Keeping /constant,thevolatilityoftheprice clearlydecreases witha reductionineither

a

x

or

a

6. But

what

aboutthevolatilityofthedividenditself?

When

the dividend is exogenous, / is independent of e.

The

impact ofnoise is then exactlylike in Grossman-Stiglitz: a reductionin eithero~_x oraeimplies lowervolatility in

equilibriumprices.

When

insteadthe dividend is endogenous, / depends on e, because agents' actions in

the second stagedepend ontheprice, whichinturndepends ontheshocke. Moreover, for

essentiallythe

same

reason that a reductionin noiseincreasesthe sensitivity oftheregime

outcometotheshocke, a reductioninnoise increasesthesensitivityof/ one.

As

aresult,

the overall impactofnoiseonthevolatilityof pricesis

now

ambiguous: ontheonehand,

less noisereducesthevolatilityoftheprice foranygivenvolatilityofthe dividend; onthe

otherhand,lessnoise increasesthevolatilityofthe dividenditself.

The

secondeffectindeeddominatesin

some

cases.

An

exampleisillustrated inFigure??,

whichdepicts

P

(9,e) fordifferentvaluesof eanda given9.

The

solidlinecorrespondstoa

relativelyhigho~_x,thedottedonetoanintermediateax,andthedashed onetoarelatively

lowax. (Inallcases,however,

a

xishighenoughthatthe equilibriuminunique.)

The

effect

of

a

xonthesensitivity of

P

(9,e) to eisnon-monotonic: it ishighest

when

a_xiseitherhigh

or low.

A

similarpictureemerges if

we

considertheimpactofae.

We

thus conclude:

Proposition 6

When

the dividendis exogenous, thevolatilityof the price conditional on9

necessarily decreaseswitha reductionin eitherax oro~ E. But

when

the dividend dependson

the coordinationoutcome, areductionin eithernoise

may

increase pricevolatility.

5

Endogenous

Information

II:

Observable

Actions

In the analysis so far,

we

have assumed that agents observe a signal generated from a

differentstagethanthe coordinationgame.

We

now

examinethecasethattheinformation

is generated within the coordination game. In particular,

we

assume away the financial

marketandletthe agents observe a noisy publicsignalabouttheaggregateactivity ofother agentsinthecoordinationgame.

We

first consideramodel wherethesignalisaboutothers' contemporaneousactions. In

this case, our equilibrium concept isnovel andunavoidably atthe crossroads of

rational-expectationsand

game

theory.

We

later

show

that the

same

resultscan be obtained ina

dynamicvariant of thismodel, inwhichthe populationisdivided intotwogroups: 'early'

movers,

who

have onlyprivateinformationaboutthefundamentals, and'late' movers,

who

can alsoobserve a public signalabouttheearly movers' activity.

The

equilibrium concept

isthen standard game-theoretic.

5.1

Model

set-up

Thereisnoassetmarketand,except,fortheendogenouspublicsignal,the

game

isidentical to

thebenchmarkmodelanalyzedinSection2. Allagents

move

simultaneouslyafterobserving

private signalsaboutthe fundamentalsanda publicsignalaboutthesizeoftheattack.

The

private signals areagainx

—

9

+

o~x£,whereasthe publicsignalis

y

=

s(A,e)

where

s:[0,l]xR-^l

ande isnoise, independentof 9 and £.

To

preserveNormalityof

the information structureand obtain closed-form solution,

we

let s(A,e)

=

§~

l

(A),

+

a_£e

ands

~

jV(0,1).

10

_The

exogenousinformation structureisthen parameterizedbythepair ofstandard deviations (<xx,cr_£).

Sincetheinformation ofeach agent includes asignal about otheragents' actions, the equilibriumconcept

we

defineisahybridofaperfectBayesian equilibriumanda

rational-expectationsequilibrium.

Definition 2

A

rational-expectationsequilibriumco?isistsofan endogenoussignaly

=

Y(9,e),

anindividual attack strategya{x,y), andanaggregateattackA[9,y), thatsatisfy:

a(x,y)

=

arg

max

E

[ U(a,R(9,y)) \ x, y

} (24)

a€[0,l]

A(6,y)

=

fa(x,y)d$(—)

(25)

y

=

s(A(9,y),e) (26)

forall (9,e,x,y) €

M

4, where R{9,y)

=

1 ifA(9,y)

>

9 andR{9,y)

=

otherwise. Thisconvenient specificationwas introduced byDasgupta(2001)inadifferentset-up.

Condition(24)

means

that_{a(x, y)}istheoptimal strategyfortheagent,whereascondition

(25)

means

that A(9,y) is simply the aggregate across agents.

The

fixed-point relation

introducedby(26) isthe rational-expectations feature ofourcontext: thesignaly

must

be generatedbyindividual actions,whichinturn are contingentony.

5.2

Equilibrium

analysis

We

focus againon

monotone

equilibria, inwhich an agent attacksifandonlyifx

<

x*(y)

andthestatus quoisabandonedifandonlyif9

<

9*(y).

A

monotone

equilibrium isthus

identifieswith atriplet offunctionsx*,9*, andY.

Giventhat anagent attacksifand onlyifx

<

x*(y),the aggregate attackisA{9,y)

=

$

(y/a^(x*(y)

—

6)}.Itfollowsthattheregimeisabandonedifandonlyif9

<

9*(y),where

9*{y) solvesA(9,_y)

=

9, orequivalently

x*(y)

=

8*(y)

+

^-

1(9*(y)). (27)

Condition(26),onthe otherhand,impliesthatthesignalmustsatisfyy

=

y/a^[x*(y)

—

9]-a

Ee, orequivalently

x*{y)

-a

xy

=

9-

<rx

a

Ee. ₍₂₈₎

Notethat(28) isa

mapping

betweenyandz

=

9

—

a

x

a

ee. Define the correspondence

y(z)

=

{yeR\x*(y)-a

x

y

=

z}.

(29)

We

will later

show

that y(z) isnon-empty and examine

when

itissingle-or multi-valued.

Now

take any function Y(z) that is a selection from this correspondence

-

that is, such that Y(z) Gy(z) for allz

-

and letY(9,e)

=

Y(9

—

a

x

a

Ee).

Any

suchselectionpreserves normalityoftheinformationstructure.

Indeed, the observation of y

=

Y{9,e) is then equivalent to the observation of z

=

9

—

oxo-s

= Z

(y), where

Z

_(y)

=

x*(y)

—

a_xy. Therefore, itis asifthe agents observe a

Normal

publicsignalabout9withprecision

a

z

=

a

x

a

E,or equivalently

az

—

ax

a

£. ₍₃₀₎

The

precision ofendogenouspublicinformationisthus once again increasingintheprecision

ofexogenousprivateinformation.

For theindividual agentinturnto finditoptimaltoattackifandonlyifx

<

x*(y),it

must bethatx*(y)solvestheindifferenceconditionPr(9

<

9*(y)\x,y)

=

c, orequivalently

$

(v^T^I

(<r(y)

-

^x*

(y)

-

-^-Z

(y)))

=

c. (31)

Using

Z

(y)

=

x*(y)

—

axyandsubstituting x*(y)from (27),

we

get

9*(y)

= $

_(-^r-y

+

J^hr*-

1(i

-

c))

, (32)

\a

x

+

a

z \

a

x

+

a

z J

which together with (27) determines unique 6*(y) and x*(y). Hence, the strategy ofthe agents a_{(x, y)} andtheaggregate attack

A

(9,y) areuniquelydetermined.

We

finallyneedtoexaminethe equilibriumcorrespondence y(z). Recall thatthisisgiven

bythesetofsolutions tox*(y)

—

o~xy

=

z. Using (27) and(32),thisreducesto

F(y)

=

^(-^—y

+

A)+-

l

=(--^—y

+ A\=z,

(33)

where

A

=

\Jaxj(ax-f

a

2)$

_1

(l

—

c). NotethatF{y) iscontinuousiny,and F(y)

—

»

-co

as y

—

> +oo, and F(y)

—

+co

_{as y}

—

> -co, which ensures that 3^(z) is non-empty and

thereforethat anequilibriumalwaysexist. Next, note that

sign {F'(y)}

=

-signI1

-

_-^=<p

_(^^y

+

Aj

>

andthereforeF(y) isgloballymonotonicifandonlyif

a

z/\fa^

<

v

27r,inwhichcasey(z)is

singlevalued. Ifinstead

a

zj'

s

fa

x

~

>

y/2n,thereisanon-emptyinterval (z,z)such that y(z)

takesthree valueswheneverz£ (z, z).Different selectionsthen sustaindifferentequilibrium

signalY.

Using

a

z

=

a

£

a

x,

we

conclude that multiple equihbria surviveaslongas eithersourceof

noiseissufficiently small.

Proposition 7

A

monotone rational-expectations equilibrium existsforall (cr_x,a_£) and

is unique ifand onlyifo\ox

>

l/\/27r. //o\o~ x

<

l/\/27r, the equilibriumstrategya and

aggregate attack

A

remainunique, but there are multiple signalfunctionsY.

Interestingly,

when

multiplicityarises, itiswithrespect toaggregateoutcomes butnot

withrespect to individualbehavior.

To

understandthis result,consider the no-noiselimit

(o~x

—

aE

=

0), inwhichcasex

=

9 andy

=

^

1

(A). Clearly,the agentfinds itoptimalto

attackifandonlyifx

<

3>(y),whichuniquelydetermines the equilibrium strategya(x, y)for

the agent. However, for every9 G {9,9}, both (y,A)

—

(—oo,0) and (y,A)

—

(+oo,1) can

besustained as equilibria.

When

a

xando~ £ arenon-zero,the

same

natureofindeterminacy remains: thebehaviorofthe agentisuniquelydeterminedfor anygivenx andy,butthere can bemultiple equilibrium value fory and

A

foranygivenrealizationof 9 ande. Inthis

sense,themultiplicityhereissimilartotheone

we

encounteredinSection3.3,

when

agents

traded afinancial assetwhosedividend depended onthesizeoftheattack.

Finally, the

common-knowledge

outcomes are once again obtained as either source of

informationbecomesinfinitely precise, and thereforesunspot volatilityismaximized

when

the noisevanishes.

On

theother, theMorris-Shin limit canbeobtainedifthe endogenous

publicsignalbecomesinfinitelyimprecise, inwhichcaseit isasiftheendogenoussignalwere

notavailable.

Corollary 4 (i) Considerthe limit as either

a

x

—

> or_{o~ e}

—

>0. Thereexistsanequilibrium, in which R(9,e)

—

> whenever9 G (9,9], as well asan equilibriuminwhich R(6,e)

—

> 1

whenever9G (9, 9].

(ii) Considerthe limit as

a

£

—

> oo. Thereis a unique equilibrium inwhich

R

(9, e)

—

>1

whenever9

<

9 and

R

(9,e) —> whenever9

>

9, where9

=

1

—

c.

5.3

Non-simultaneous

signal

The

analysisabovehasassumedthat agentscanconditiontheirdecision toattackon anoisy

indicator ofcontemporaneous aggregate behavior.

We

now show

that ourresults extendto

asimpledynamic modelinwhich noagent has informationabout contemporaneous actions ofotheragentsandthereforeastandard game-theoretic equilibrium concept (namely

PBE)

can beused.

The

populationisdividedintotwogroups,'early' and 'late' agents. Early agents

move

first,onthebasis of theirprivateinformationalone. Late agents

move

second, onthebasis

of their private information as well as a noisy publicsignal about the aggregate activity

ofearly agents. Neither group can observecontemporaneous activity, but lateagentscan

conditiontheirbehaviorontheactivityofearly agents.

Let ji G (0,1) denote the fraction of early agents, Ai the aggregate activity of early

agents, and

A2

the aggregate activity of late agents.

The

signal generatedbyearly agents

andobserved onlybylateagentsisgivenby

y^Q-^AJ+e,

(34)

where e

~

A/"(0,o~l) is independent of 9 and £. Early agents can condition their actions

onlyontheir privateinformation, whereaslateagentscanconditiontheiractions alsoony\.

Finally,theregimechangesifandonlyiffiAx

+

(1

—

ji)A2

>

9.

We

lookforperfectBayesianequilibria inwhichthe strategyofthe agentsinmonotonic

intheir privateinformation. Sincelateagentscanconditiontheirbehavioronyx butearly

agentsnot, amonotoneequilibriumisascalarx\ 6

R

and apair offunctionsx₂ :

K

— R

and9*

:

R

—

> (0,1) suchthat: an early agent attacks ifand only ifx

<

x\; alate agent

attacksifandonlyifx

<

x\(y\)\

and

the regimeisabandonedifandonlyif9

<

0*(y1).

Insuchanequilibrium,the aggregate attackof earlyagentsisA\(9)

= $

(s/q^[x±

—

_#]) .

Itfollowsthat_yx

=

<1>

_1

[Ax (9))+e

=

yfa^[x\

—

9]+e.Hence,inequihbrium, the observation

ofyx isequivalent tothe observationof

1

a,

which is apublic signal with precision

a

z

=

ctE

a

x (equivalently, with standard deviation

az

=

cr

£

a

x),likeinthesimultaneous-signalmodel.

Since yx and z have the

same

informationalcontent,

we

can equivalently express the strategy ofa lateagent as a function of z instead of yx and replace the functions x2(yx)

and 8*(yx) with x2(z) and 9*{z).

The

aggregate attack of late agents is then

A

2(9,z)

=

$

[s/a^[x₂(z)

—

_6]) and the overall attackfrom both groups is

A

(9,z)

=

\iAx(9)

+

(1

—

pi)A2(9,z). It followsthat theregime changesifand onlyif9

<

9*(z), where9*(z) solves

A(9*(z),z)

=

9*(z),orequivalently

M

$

(Vc£[*J

-

9*(z)})

+

(1

-

/.)$(yfe[z*2(z)

-

6*(z)\)

=

9*{z). (35)

Next, considerthe optimal behavior ofthe agents. Since therealization of zis

known

to late agents, their decision problemislike inthe

benchmark

model: the thresholdx₂(z)

solvesPr[9

<

9*(z)\x2(z),z]

=

c,or equivalently

$

{s/^(5x*2(z)

+

(l-S)z-

9*(z)))

=l-c,

(36)

where5

—

a

x/(ax

+

a~) and

a

=

a

x

+

a..Earlyagents, onthe otherhand, donot observez

andtherefore faceadoubleforecastproblem: theyareuncertainaboutboththefundamental

andthesignal

upon

whichlateagentswillconditiontheirbehavior.

The

thresholdx* solves

Pr[9

<

9*(y)\xl\

=

c, orequivalently

f

*

{^T

x [x\

-

9*(z)})^l<t>

(V5T

[x{

-

_A)dz

=

\-

c, (37)

wherea^

=

a

x

a

e/(l

+

a

e).

u

A

monotone

equilibriumis therefore ajoint solution to (35)-(37).

We

can reduce the

dimensionalityofthesystembysolving (35) for x^(z):

x*₂(z)

=

9*(z)

+

-Lsr

1 (V(z)

+

-JL-

{8\z)

- $

(,/S

[x*

-

6T

(*)])}).

Substitutingtheaboveinto (36)

and

using5

=

a

x/(ax

+

a

z) and

a

=

a

x

+

a

z,

we

obtain:

r(r(z),x*)

=

5(z), (38) where r(<9,

xO

=

—%e

+

*-

2 fe

+

-r—

{e-§

(^

[x *

-0])} g(z)

=

yjl

+

Q

z/ax

$

_1

(1

—

c)

—

(az/y/a^)z, anda,

=

Q

£

a

x.

We

conclude thatan

equilib-riumisajointsolution of (37)and (38) fora thresholdx|

G

R

anda function9*:

R

—

»(0,1).

Let

C

denote thesetofpiecewisecontinuousrealfunctionswithrange a subsetof[0, 1].

For anygiven function 9* _{G C, (37) always defines} auniquex\

G

R. For given x\

G

R,on the otherhand, (38)

may

admit a unique or multiple solutions in 9* G C, depending on

(ftu

a

E,fi). Differentsolutions to (38) forgivenx\representdifferentcontinuationequilibria

for the

game

between late agents defined

by

a fixed strategy for the early agents.

The

questionof interest,however,isthedeterminacyofequilibriumintheentiregame.

In theappendix

we show

that,

when

(38) admits aunique solution for every x\ G R,

thefixed pointof (37)-(38) is alsounique.

On

the other hand,

when

(38) admits multiple

solutions for every x\

G

R,

we

prove that (37)-(38) alsoadmits multiplefixed points. This

provides us with the followingsufficient (butnot necessary) conditionsforuniqueness and

multiplicity:

Proposition 8 (i) Thereexistsa unique equilibriumif

i 1 ,

ai<7x

>

, (1

—

u)

(ii) There existmultiple equilibriaif

u

_To

seethis,notethat z

=

8—axe

=

x—£—axe.sothat z\x

~

TV(0,a\

+

cr_xoV). Thatis.conditional

onx, zisdistributednormal withprecisiona.\

=

axae/{\

+

ae ).