A Note on Woodford's Conjecture: Constructing Stationary Sunspot Equilibria in a Continuous Time Model(Nonlinear Analysis and Mathematical Economics)

ANote

on

Woodford’s Conjecture: ConstructingStationary Sunspot Equilibria in

a

Continuous TimeModel*

TadashiShigoka

KyotoInstitute of EconomicResearch,

Kyoto University,YoshidamachiSakyoku Kyoto

606

Japan

京都大学 経済研究所

新後閑 禎

Abstract

We showhowtoconstruct stationarysunspotequilibria in

a

continuous timemodel,

where equilibriumisindeterninate

near

either

a

steady state

or a

closedorbit.Woodford’s

conjecture thattheindeterminacyof equilibrium implies theexistenceofstationary sunspotequilibriaremains valid in

a

continuous timemodel.

Introduction

If forgivenequilibrium dynamics thereexist

a

continuum ofnon-stationaryperfect

foresightequilibria allconvergingasymptoticaUy to

a

steadystate (adeterministiccycle

resp.),

we say

theequilibrium dynamicsis indeterminate

near

the steadystate (the

deterministiccycleresp.). Suppose that thefundamental characteristicsof

an

economy

are

deterministic,butthateconomicagentsbelieve nevertheless that equihibrium dynamics is affectedbyrandom factors apparentlyirrelevanttothe fundamental characteristics

(sunspots). Thisprophecy couldbe self-fulfilling, and

one

willget

a

sunspotequilibrium,

ifthe resulting equilibrium dynamicsis subjectto

a

nontrivial stochastic

process

and

confirns the agents’ belief. SeeShell [19],andCass-Shell [3].

Woodford[23] suggestedthat there exists

a

closerelation between the indetenninacy of

equilibrium

near

a

deterministic steadystateand theexistenceofstationarysunspot

equilibriaintheimmediatevicinityofit. SeealsoAzariadis[1].We mightsummarize

Woodford’sconjecture

as

whatfollows: “Let $\overline{x}$ be

a

steadystateof

a

deterministic model

which has

a

continuumof

non

stationaIy perfect foresight equilibria allconverging

asymptoticaUytothe steady state.Thengiven

any

neighborhood $U(\overline{x})$ ofit, thereexist

stationarysunspotequilibria with

a

supportin $U(\overline{x})$

.

“

Azariadis [1],Farmer-Woodford[9],Grandmont [10],Guesnerie [11], Woodford[24],

andPeck[18] _{have shown that the}conjectureholds good invariouskindsof models.

Woodford [25], Spear-Srivastatva-WMford[22],_{and Chiappori-Geoffard-Guesnerie} [4]

investigatetheconnectionbetweenthe local indeterminacy of equilibria and theexistenceof

local stationarysunspotequilibriathoroughly andshow theconjectureholds good in

extremely generalsituations.However theexisting results supporting Woodford’s

conjecture

are

allderived Rom discrete time models. SeeChiappori-Guesnerie [5] and

Guesnerie-Woodford[12] forthorough

surveys

on

theexistingsunspotliterature. The

purpose

of thisnoteistoshowthatWoodford’sconjectureextendsto

a

continuous time

state(aclosedorbitresp.)in

a

continuous timemodel,whereequilibriumisindeterminate

near

thesteady state (the_{closed orbit}resp.).One

can use

our

methodtoshow thereexist

stationary sunspotequilibriain such models

as

$\alpha ea\ddagger ed$byHowitt-McAfee[15],Hammour

[13],Diamond-Fudenberg[6],_{Benhabib-Farmer}[2], andDrazen [8]. _{The models}$\alpha ea\iota ed$

by[13,6, 8] _include

a

_{stable limit cycle, where equilibriumis indeterminatearound the}

stable limit cycle.One

can

use our

methodto$consm\iota ct$stationarysunspotequilibria around

the stable limit cycle in these models.

Earlier results

on

theexistenceofsunspotequilibria

are

based

on

the overlapping

generationsmodel,where fluctuations exhibited all

occur on

timescaletoolong compared

tothelife times ofagents. However,

as

shown by Woodford[24], _Spear[21],and

Kehoe-Levine-Romer[17],$ffic\dot{\mathfrak{a}}ons$likecash-in-advanceconstraints, externalities, and

proportionaltaxation

can

generatemarketdynamics amenabletotheconstructionofsunspot

equilibriain otherwise well-behaved mMels havingfinitely-many infmite-livedagents. See

Kehoe-Levine-Romer [16]_{for the well-behaved}

_case.

_{The models treated by}[15, 13, 6, 2,

8]_{also include infinite-lived agents, and,inspiteof}this, generateindeterminate equilibria

throughvariouskinds of market imperfections.

Thenote iscomposedof foursections. Section 1 presents

our

model. Section2 describes

deterministic equilibrium dynamics. Section

3

specifies

a

Markov

process

whichgenerates

sunspotvariables. Section4

proves

theexistenceofstationarysunspotequilibria.

1 TheModel

Let $\{\begin{array}{l}\dot{K}_{\prime}(E_{t}dq_{t})/dt\end{array}\}=F(K_{t},q_{t})\in R^{2}$ (1)

be

a

frst order condition of

some

intertemporaloptimizationproblem withmarket

equilibrium conditions incorporated.$F$isassumedtobe

a

continuouslydifferentiable

function(i.e.

a

$C^{1}$ function). $K_{t}$is

a

predetermined variable.

$q_{t}$is

a

forwardlooking

an

economy

such

as

preferences,technologies, andendowments

are

deterministic. Inother

words,thereis

no

inninsic$uncer\ddagger a\dot{m}$ty.Any random factoris irrelevanttothefundamentals (sunspot).Thatis,the only uncertainty isextrinsic.

Our equilibriumstochastic

process

is describedby

$\{\begin{array}{l}\dot{K}_{t}dtdq_{t}\end{array}\}=F(K_{t},q_{t})dt+s\{\begin{array}{l}0d\epsilon_{t}\end{array}\}$, (2)

where

we

assume

$\lim_{harrow+O}E(\epsilon_{t+h}-\epsilon_{t}1\epsilon_{s}, s\leq t)/h$is well defned and equalto$0$

so

that $(E_{t}d\epsilon_{t})/dt=0$

.

$s\in(-\underline{\eta}, \overline{\eta}),$ $\underline{\eta}$ and

fi

are

sufficiently smallpositivenumbers. $dt$is

a

Lebesgue

measure.

$dq_{t}$and$d\epsilon_{t}$

are

Lebesgue-Stieltjes signed

measures

withrespectto$t$

.

We

assume

$d\epsilon_{t}$is

a

“singular” signed

measure

of$t$relativetothe Lebesgue

measure

$dt$

.

$\int_{l}d\epsilon_{s}=\epsilon_{t+h}-\epsilon_{\iota}$and $\epsilon_{t}$is

a

randomvariableirrelevanttofundamentals (i.e.

a

sunspot

variable).For$s=0$,the system isdetenninistic,whereasit is stochastic for$s\neq 0$

.

Wedefine

a

sunspotequilibrium

as

follows. A sunspotequilibriumis

a

stochastic

process

$\{(K_{t}, q_{t}, \epsilon_{t})\}_{i\geq 0}$ with

a

compact support suchthat $\{(K_{t}, q_{t})\}_{i\geq 0}$is

a

solution ofthe

stochastic differentialequation(2) with$s\neq 0$

.

If thesunspotequilibriumis

a

stationary

stochastic

process, we

callit

a

stationarysunspotequilibrium.

2.

Deterninistic Dynamics

We

assume

the detenninisticequilibrium dynamics,where sunspotsdonotmatter,

satisfies the following condition.

Assumption 1. $(\dot{K},\dot{q})=F(K, q)$ is

a

$C^{1}$ vector field defined

on

an

open

subset$W$

on

$R^{2}$

.

$W$includes

a

compact

convex

subset$D$withnonemptyinteriorpoints such that thevector

Under Assumption 1thedifferential equationhas

a

uniqueforward solution for

any

initial condition located

on

$D$

.

Let$x_{t}=\phi(t,x)$be

a

solution of$\dot{x}=F(x)$with

an

initial condition

$x_{0}=x\in D$

.

$\phi:[0, +\infty$)$\cross Darrow D$is

a

welldefmedcontinuouslydifferentiablefunction.

If

thereexist$x\in D$and

a

monotonicallyincreasing

sequence

$t_{n}arrow\infty,$$n=1,2,$$\ldots$ such

that$\lim_{narrow\infty}\mu t_{n},x$)$=y\in D,$$y$is called

an

wlimit pointof$x$

.

For$t>0,x\in D$,define

$\mu-r,x)\in D$

as an

inverseimage$z$ofx$=\mu t,$ $z$),ifthe latter is well defined. Suppose

$\mu-t,x)\in D$iswell defined for$\forall t>0$for

some

_{$x\in D$}

.

Ifthereexists

a

monotonically

increasing

sequence

$t_{n}arrow\infty,$ $n=1,2,$ $\ldots$such that$\lim_{narrow\infty}\mu-t_{n},x$) $=y\in D$,

we

call$y$

an

a-limitpoint of$x$

.

A limitsetof$D$isdefined

as

a

setofall points in$D$ such that each of

themis either

an

$\omega-$-limit

or an

a-limit pointof

some

$x$in$D$respectively. The$s\alpha ucture$of

a

limitsetof

a

planar dynamicalsystemis

very

simple. The limitsetiscomposedof steady

states (Figure2),closedorbits (Figure3),andtrajectories joining steadystates (Figure4).

If

a

steadystate(aclosed orbitresp.)is stable,the equilibrium isindeterminate

near

the

steady state(theclosedorbitresp.). (SeeFigures2and3.)

As shownbelow,thestochasticdifferentialequation(2)generates

a

familyof

perturbations of the deterministic equilibrium dynamics $(\dot{K},\dot{q})=F(K, q)$.Totalk about

“perturbation”precisely,

we

introduce the following functional

space

endowed with the$C^{1_{-}}$

topology. $C(W)=$

{

$g:g:Warrow R^{2}$

.

$g$ is

a

$C^{1}$

function.}

Note that$F\in C(W)$

.

A

perturbation of$F$is

an

elementof

some

neighborhoodof$F$in$C(W)$ withrespecttothe$C^{1_{-}}$

topology.The followingproposition is

an

obvious

consequence

of thesffuctural stability

(Hirsch-Smale [14,Theorem 16.3.2]), _whereint$X$and$X$denote

a

setofallinterior

points and the boundaryof

some

closedset$X$,respectively.

Proposition 1. (1) (Figure 1)Thereis

a

neighborhoodV(W) $\subset C(W)$ of$F$such thatfor

$\forall g\in V(W),\dot{x}=g(x)$points inward

on

$\partial D$

.

(2)(Figure 2)Supposethelimit setof$D$iscomposed of

a

unique stable steady state $\overline{x}$,

$M(W)\subset C(W)$of$F$with the followingproperty. For$\forall g\in M(W)$,thelimit setof the

dynamicalsystem $\dot{x}=g(x)$

on

$D$is composed of

a

unique stable steadystate $\overline{x}(g)$ such

that $\overline{x}(g)\in U(\overline{x})$,andthereexists

a

compactsubset $X(\overline{x})$ of $U(\overline{x})$ such thatfor $\forall g\in$ $M(W),\overline{x}(g)\in$ Int $X(\overline{x})$ and $\dot{x}=g(x)$ pointsinward

on

$\partial X(\overline{x})$

.

(3) (Figure3)Supposethe limitsetof$D$iscomposed of

a

uniqueunstable steady state $\overline{x}$

and

a

uniquestable limit cycle $\gamma$,and fix

some

open

neighborhood $U(\gamma)\subset D$ of$\gamma$ Then

thereexists

a

neighborhood$N(W)\subset C(W7$of$F$with the followingproperty. For$\forall g\in$

$N(W)$_,the limit_setof the dynamical_system $x=g(x)$

on

$D$iscomposedof

a

unique

unstablesteadystate $\overline{x}(g)$and

a

uniquestable limit cycle $\gamma\langle g$) such that$Xg$) $\subset U(\gamma)$, and

thereexists

a

compactsubset $X(\gamma)$ of $U(\gamma)$ such that for$\forall g\in N(W),$ $\chi g$)$\subset$ Int $X(\gamma)$

and $\dot{x}=g(x)$pointsinward

on

$\partial X(\gamma)$

.

3.

Stochastic Process

We specify

a

stochastic

process

$\{\mathfrak{g}\}_{t3\}}$generatingsunspotvariables in

a way

consistent

with theformulationin theequations(1) and(2).

We

assume

the sunspot

process

takes

a

finitenumber ofvalues andis subjectto

a

continuous timeMarkov

process

with

a

stationary transitionmatrix. Let$Z$bedefined

as

$Z=$

$\{z_{1}, z_{2}, z_{N}\},where-\underline{\epsilon}\leq z_{1}<Z_{2}<\ldots Z_{N}\leq\overline{\epsilon}$with sufficiently small positiveconstants $\underline{\epsilon}$

and $\overline{\epsilon}$, andwith

a

positive butfiniteinteger$N$

.

Let_{$[\{\epsilon_{t}(\omega)\}_{t\geq 0}, (\Omega, B_{\Omega}, P)]$}be

a

continuous time

stochastic

process,

where$\omega\in\Omega,$$B_{\Omega}$is

a

$\sigma$-fieldin $\Omega,$$P$is

a

probability

measure, and$\epsilon_{t}()$

:

$\Omegaarrow Z$ is

a

random variable for$\forall r\geq 0$

.

Let_{$P(h)=[\rho_{ij}(h)]_{1\leq ij<}\ovalbox{\tt\small REJECT},$} $h\geq$

$0$,denote

an

$N\cross N$ stationary transition probabilitymatrix,where$p_{ij}(h)$is the conditional

probability that $\epsilon_{t}(\omega)$

moves

from$\epsilon_{t}(\omega)=z_{i}$to$\epsilon_{t+h}(\omega)=z_{j}$through the length oftime$h$

under the condition$\epsilon_{t}(\omega)=z_{i}$

.

$\sum_{i=1}^{N}p_{jj}(h)=1$ for$\forall i,$$\forall h\geq 0$

.

We

assume:

Assumption

2.

(1) $\{\mathfrak{g}(\omega)\}_{i\geq 0}$is

a

continuous time Markov

process

with

a

stationary

(2)Thetransitionmatrix satisfies the followingcontinuitycondition.

Iim$harrow+0p_{ij}(h)=1$,for $i=j$, and$=0$_, for $i\neq j$

.

(3)The stochastic

process

$\{\epsilon\sqrt{}\omega)\}_{f\geq 0}$ is “separable“.

SeeDoob[7] for theconcept of separability. Under Assumption2,

we

havethe

followingtwoobservations about thesunspot

process,

where $\epsilon_{t}(\omega)=\epsilon(t, \omega)$

.

Observation 1. (Doob [7,Theorem 6.1.2])

(1)The limit $\lim_{tarrow+O}\frac{1-p_{\ddot{u}}(t)}{t}=q_{i}<+\infty$existsforall$i$

.

(2)$P$

{

$\epsilon(t,$ $\omega)\equiv z_{i}$,for all $t_{0}\leq r\leq t_{0}+\alpha 1\epsilon(r_{0},$ $\omega)=z_{i}$

}

_$=e$ $\iota$

and if$\epsilon(t_{0}, r)=z_{i},$ $\epsilon(r, on)=$

$z_{i}$in

some

neighborhood of$t_{0}$ (whose size depends

on

on.) withprobability

one.

A function$g()$will be caUed

a

_{step function,}ifit has only finitely

many

pointsof

discontinuity in

every

fmiteclosedinterval,ifit isidenticallyconstantin

every open

interval

ofcontinuity points andif,when$t_{0}$is

a

pointofdiscontinuity,

$g(t_{0}-)\leq g(t_{0})\leq g(t_{0}+)$,

or

$g(t_{0}-)\geq g(t_{0})\geq g(t_{0}+)$

.

Afunction$g()$willbe saidtohaveajumpat

a

point$t_{0}$,ifit is discontinuousthere,andif

theonesided limits$g(t_{0}-)$ and$g(t_{0}+)$existand satisfy

one

of thetwopreceding inequalities.

Observation

2.

(Doob [7,Theorems 6.1.3, and 6.1.4])

(1)Thelimit $\lim_{\iotaarrow+O}\frac{p_{ij}(t)}{t}=q_{ij}$ _{$i\neq l$} exists,and

$\sum_{j*i}q_{ij}=q_{i}*$

(2) If$q_{i}>0$andif$\epsilon(t, \omega)=z_{i}$, thereiswith probability 1

a

samplefunction discontinuity,

whichis ajump; if$0<\alpha\leq\infty$,the probability that if thereis

a

discontinuity in the interval

$[t_{0}, t_{0}+\alpha)$ the firstjumpis ajumpto$z_{j}$is $q_{i_{\dot{J}}}/q_{i}$

.

Observation

2-3

implies for arbitrarily large but finite$T>0$, the sample paths $\epsilon=\epsilon_{t}(\omega)$

are

stepfunctions and include only finitely

many

discontinuous jumps

over

[$o,$ $\eta$with

probability

one.

We

assume:

Assumption

3.

(1) _For$\forall i,j=1,2,\ldots.,N,$$q_{i}>0$, and$q_{ij}>0$, where $q_{i}$ and$q_{i_{\dot{J}}}$

are

specified

as

in Observations 1,and

2.

(2)Thesample paths

are

continuous

on

the rightateachjumpdiscontinuitywithprobability

one.

A typical sample path ofthe sunspot

process

$\{\epsilon_{t}(\omega)\}_{t\geq 0}$is depicted in Figure

5.

4 StationarySunspot Equilibria

We have sufficientpreparationto

prove

that themodel specified insection 1 has

stationary

sunspotequilibria underAssumptions 1, 2, and

3.

4-1.

OntheSolution of(2)

We

use

Assumptions2and 3in thepresentsubsection explicitly, and show howto

consmlct

a

solution of the stochastic differentialequation(2).

We

can

rewrite (2)

as

$\{\begin{array}{l}\dot{K}dtdq-sd\epsilon\end{array}\}=F(K, q)dt,$ $s\neq 0$

.

Inwhat follows,$s\in(-\underline{\eta}, \overline{\eta})$ is

a

fixedparameterand$s\neq 0$,unless stated otherwise.Let

$y$be defined

as

$y\equiv q-s\epsilon$

.

Then

we

have

Let$G$ be defined

as

$G(K, y, \epsilon:s)\equiv F(K,y+s\epsilon)$

.

Then

we

have

$\{\begin{array}{l}\dot{K}dtdy\end{array}\}=G(K,y, \epsilon:s)dt$

.

(3)

Weinterpretthe stochasticdifferential equation(3)

as an

ordinarydifferential$\eta ua\dot{u}on$for

each fixed value $\epsilon$

.

$\{\begin{array}{l}\dot{K}\dot{y}\end{array}\}=G(K,y, \epsilon:s)$

.

(4)

Since$\epsilon_{t}(\omega)$ takes$N$values,_{$z_{1,2,\ldots N}zz$}, this generates

a

family of$N$ordinarydifferential

equations,

$\{\begin{array}{l}\dot{K}\dot{y}\end{array}\}=G(K,y, z_{i};s),$ $s\neq 0,$$i=1,2,\ldots,$N. (5)

The familyconstitutes

a

setof$C^{1}$-perturbations of $(\dot{K},\dot{q})=F(K, q),$$s=0$

.

Weconstructthe solution of thestochastic differential equation (5)

as

follows. Under

Assumptions2and3,for almost

every

$\omega$,the sample path of$\epsilon_{t}(\omega)=\epsilon(t, \omega)$ is

a

righthand

continuous step function,for $t\in[0, +\infty$),

assumes

only$N$different values, _{$z_{1,2,\ldots N}zz$},

andfor arbitrarily large but finite$T$,includesatmostfinitely

many

discontinuousjumps

over

$[0, T$)withprobability

one.

Fix $\omega=\mathfrak{U}$,and

suppose

the sample path of$\epsilon_{t}(ab)=$ $\epsilon(t, w)$, for$t\in[t_{1}, t_{2}$), includes

one

and only

one

discontinuous jumpat $t_{1}+h$from$z_{i}$to $z_{\dot{j}}$,where$z_{i}\neq z_{j}$

.

SeeFigure

6.

Then the systemis subjecttothe differentialequation,

$(\dot{K},\dot{y})=G(K, y, z_{j}:s)$ during $t\in[t_{1}, t_{1}+h$),and then subject to $(\dot{K},\dot{y})=G(K, y, z_{j};s)$

equation, $\dot{x}=G(x, z_{i}:s)$,with

an

initialcondition$x=x_{t}$,where$u$isthe length oftime

during which$x$

moves

from$x_{t}$to$x_{t+u}$, and$z$;is fixed. Let$xn=(K(t_{1}), y(t_{1}))$

.

If $\dot{x}=$

$G(x, z_{i}:s)$has

a

solution for

an

initial value$x=(K(t_{1}), y(t_{1}))$, which implies$x=$

$\phi(h, K(t_{1}),$$y(t_{1}),$ _$z;:s$) exists, andif $\dot{x}=G(x, z_{j}:s)$has

a

solutionfor the initialvalue,

$x=\phi(h, K(t_{1}),$$y(t_{1}),$$z_{j}:s$), then forthe $W$ fixed above,

we

have $(K_{t}, y_{i})=$

$\phi(t-t_{1}, K(t_{1}),$$y(t_{1}),$ $z_{i};s$), during $t\in[t_{1}, t_{1}+h$), and $(K_{t}, y_{t})=$

$\phi(t-(t_{1}+h), \phi(h, K(t_{1}), y(t_{1}), z_{i}:s), z_{j}:s)$, during $t\in[t_{1}+h, t_{2}$).

Suppose that the$N$differential equations(5) have

a

common

compact support$X$, and

that the$N$vectorfields allpoint inward

on

theboundary$\mathfrak{X}$of$X$

.

Then $\phi(t, x, z;:s)$is

well defined, and belongs to$X$, for

any

$x\in X$_, for

any

$i=1,2,\ldots,N$, andfor

any

$t\geq 0$

.

Hence,it isthe

case

thatfor almost

every

$\omega$, andfor

any

fmed initial condition$x\in X$,

we

can

construct

a

solution of thestochastic differential equation(4)with$s\neq 0$by

means

of

successiveapplications of the abovemethod,sincefor almost

every

$\omega$the sample path of

$\epsilon_{t}(\omega)=\epsilon(r, \omega)$is

a

stepfunction of$t\in[0, +\infty$). If $\{\epsilon_{t}\}_{t\in[0.T)}$includes $m(>0)$

discontinuities at $t=t_{1},$ $t_{2},\ldots t_{m}(0<t_{1}<t_{2}\ldots<t_{m}<T)$, then

we

have $xr=\phi(T-t_{m}, \phi(t_{m}-t_{m-1}, \phi(\ldots\phi(t_{1}, x_{0}, \epsilon_{0}:s),\ldots..),\epsilon_{bn-1}: s),\epsilon_{bn}:s)$,

where $x_{0}=(K_{0}, y_{0})$, and$xr=(K_{T}, y_{T})$

.

Let $\{(x_{t}, \epsilon_{t})\}_{t\geq 0}=\{(K_{t}, y" \epsilon_{t})\}_{i\geq 0}$ be

a

constructed solution of(4) together with

$\{\epsilon_{t}\}_{i\geq 0}$, for

some

fixed$x=x0=(K_{0}, y_{0})\in X$, and for

some

fixed $\omega$

.

For the fixed $\omega$and

the fixed $(K_{0}, q_{0})=(K_{0}, y_{0}+sq_{I})$,

we

have $\{(K_{t}, q_{t})\}_{t\geq 0}=\{(K_{t}, y_{t}+s\epsilon_{t})\}_{t\geq 0}$

as a

solution

ofthestochasticdifferential equation (2).Note that if$\{(x_{t}(\omega), \epsilon_{t}(\omega))\}_{t\geq 0}=$

$\{(K_{t}(\omega), y_{t}(\omega), \epsilon_{t}(\omega))\}_{i\geq 0}$is stationary, then $\{(K_{t}(\omega), q_{i}(\omega), \epsilon_{t}(\omega))\}_{t\geq 0}=$ [$(K_{t}(\omega), y_{t}(\omega)+s\epsilon_{t}(\omega),$$\epsilon_{t}(\omega))\}_{t\geq 0}$ is also stationary.

We

can

show that together withsunspot variables,theconstructed solution $\{(x_{t}, \epsilon_{t})\}_{t\geq 0}$is

subjectto

a

Markov

process

with

a

stationary$\alpha ansition$probabilityand

a

compact support

$X\cross Z$ (Shigoka [20,Proposition4]). We

can

also show the Markovoperatorassociated

some

continuousfunction

on

it([20,_Proposition6])._Thereforeby Yoshida [26,_Theorem

13.4.1],thereexists

an

invariantprobability

measure on

$X\cross Z$ suchthatif

we

assign this

measure

to $X\cross Z$

as an

initialprobabilitymeasure,thenthe resulting Markov

process

$\{(x_{t}(\omega), \epsilon_{t}(\omega))\}_{t\geq 0}$is stationary.Indeeditcouldbe ergodic ([26,Theorem 13.4.3]).

4-2.

Existence of Stationary SunspotEquilibria.

We

use

Assumptions 1, 2, and

3

in thepresentsubsection explicitly. We

can

establish

a

mainresult.

Choose $\underline{\eta},$ $\overline{\eta},$

$\underline{\epsilon}$ , $\overline{\epsilon},$

$s\in(-\underline{\eta}, \overline{\eta}),$ $s\neq 0,$ $and-\underline{\epsilon}\leq z_{1}<z_{2}<\ldots z_{N}\leq\overline{\epsilon}$ in such

a

way

that $G( z;:s)i=1,2,\ldots,N$all belongto $V(W)$in Proposition 1-1. Then

we

can

take

$D$in Assumption 1

as

$X$in Subsection

4-1.

Hence

we can

constructstationarysunspot

equilibria by

means

of the method describedthere. Byconstruction, $(x_{t}, \epsilon_{t})=(K_{t}, y_{t}, \epsilon_{t})\in$

$D\cross Z$

.

Soif

we

choose _$V(W)$ sufficiently small, then $(K_{t}, q_{t})=(K_{t}, y_{C}+s\epsilon_{t})\in W$, where

$W$is

as

in Assumption 1.

Next,

suppose

that theassumptionin Proposition 1-2 is satisfied. $Ch\infty se\underline{\eta},$ $\overline{\eta},$ $\underline{\epsilon}$ , $\overline{\epsilon},$

$s\in(-\underline{\eta}, \overline{\eta}),$$s\neq 0,$ $and-\underline{\epsilon}\leq z_{1}<z_{2}<\ldots z_{N}\leq\overline{\epsilon}$ in such

a

way

that$G( z;:s)$

$i=1,2,\ldots,N$all belong to$M(W)$ inProposition

1-2.

Then

we can

take $X(\overline{x})$ in Proposition

1-2

as

$X$in Subsection

4-1.

Hence

we can

constructstationary sunspotequilibria by

means

of the method described there. Byconstruction, $(x_{t}, \epsilon_{t})=(K_{t}, y_{t}, \epsilon_{t})\in X(\overline{x})\cross Z$

.

So, if

we

choose $X(\overline{x})$ and$M(W7$ sufficiently small, then $(K_{t}, q_{t})=(K_{t}, y_{t}+s\epsilon_{t})\in U(\overline{x})$, where

$U(\overline{x})$ is

as

in Proposition

1-2.

Finally,

suppose

thatthe assumptionin Proposition

1-3

is satisfied. $Ch\infty se\underline{\eta},$ $\overline{\eta},$ $\underline{\epsilon}$ ,

$\overline{\epsilon},$

$s\in(-\underline{\eta}, \overline{\eta}),$$s\neq 0,$ $and-\underline{\epsilon}\leq z_{1}<z_{2}<\ldots z_{N}\leq\overline{\epsilon}$ in such

a

way

that_{$G( z_{i}:s)$}

$i=1,2,\ldots,N$all belongto$N(W)$ in Proposition

1-3.

Then

we

can

take$X(\gamma)$ in Proposition

1-3

as

$X$in Subsection4-1. We

can

proceed in exactly the

same way

as

above. Then

we

have shown the following theorem. See Shigoka[20] for the application ofthe theoremto

Theorem 1. Let $\{\begin{array}{l}\dot{K}(E_{l}dq)/dt\end{array}\}=F(K,q)$ be

a

first order condition of

some

intertemporaloptimizationproblem withmarket equilibriumconditions incorporated. There

is

no

intrinsic uncertainty,

so

fundamentalcharacteristicsof

an

economy

are

detenninistic.

Extrinsic uncertainty(sunspot),if

any,

alone exists. Suppose the

deterministic equilibrium dynamics $(\dot{K},\dot{q})=F(K, q)$ satisfies Assumption 1.

(1)Global Sunspot Equilibria. We

can

constructstationarysunspotequilibria with

a

supportof the endogenous variable$(K_{t}, q_{t})$ in$W$

.

(2)Local Sunspot Equilibria

near a

SteadyState. SupposetheassumptionofProposition

1-2

is satisfied,

so

equilibriumisindeterminate

near

the steady state $\overline{x}$

.

For

any

neighborhood $U(\overline{x})$ ofit,

we can

construct stationarysunspotequilibria with

a

supportof

the endogenous variable$(K_{t}, q_{t})$in $U(\overline{x})$

.

(3)Local Sunspot Equilibriaaround

a

Limit Cycle. Suppose that the assumption of

Proposition

1-3

is satisfied,

so

equilibrium isindeterminate around the limit cycle $\gamma$For

any

neighborhood$U(\gamma)$ofit,

we

can

constructstationarysunspotequilibria with

a

support

of the endogenous variable $(K_{t}, q_{t})$ in$U(\gamma)$

.

References

1.

C.Azariadis, Self-fulfilling prophecies, J. Econ. Theory

25

(1981),

380-396.

2.

J.Benhabib, and R. Farmer, $Indete inacy$and Increasing Returns,“ mimeo.

1991.

3.

$D$

.

_{$Cass,$ $andK$}

.

$Shell,$_{$Dosunspotsmatter?J$}. $Polit$. $Econ$. $91(1983),$ $193- 227$

.

4.

P. A. Chiappori, P. Y. Geoffard, and R. Guesnerie, Sunspot fluctuations around

a

steady state: The

case

ofmultidimensional, one-stepforwardlookingeconomicmodels,

Econometrica

60

(1992), loe7-1126.

5.

P. A.Chiappori, and R. Guesnerie, Sunspot equilibria in sequential marketmodels,

Sonnenschein,Eds.) North-Holland, Amsterdam,

1991.

6.

P. Diamond, andD. Fudenberg,Rationalexpectationsbusiness cycles insearch

equilibrium, J. Polit. Econ.

97

(1989),

606-619.

7. J. L. Doob, “StochasticProcesses,” John Wiley, New York,

1953.

8.

A.Drazen,Endogenousbusiness cycles with self-fulfdlingoptimism: Amodel with

entry, $in$ “CyclesandChaos inEconomic Equilibrium,“ (J. Benhabib,Ed.) Princeton

UniversityPress, New Jersey,

1992.

9.

R.Farmer,andM. Woodford, “Self-Fulfilling Prophecies and the Business Cycle,“

mimeo.

1984.

10.

J. M. Grandmont,Stabilizingcompetitive business cycles, J. Econ. Theory 40

(1986),

57-76.

11. R.Guesnerie, Stationarysunspotequilibria in

an

n-commodity world, J.Econ.

Theory

40

(1986),

103-128.

12. R. Guesnerie, and M.Woodford, Endogenousfluctuations, in “Advances in

Economic Theory, Sixth World Congress,Vol. 2,“ (J.J. Laffont,Ed.) Cambridge

UniversityPress, New York,

_1992.

13.

M. Hammour, “Social Increasing Retums inMacroModelswith ExtemalEffects,“ mimeo.

1989.

14.

M. Hirsch, andS. Smale, “Differential Equations, Dynamical Systems, andLinear

Algebra,” AcademicPress, NewYork,

1974.

15.

P. Howitt,and R. McAfee,_{Stability of equilibria with}externalities, Quart. J. Econ.

103

(1988),

261-277.

16.

T. J.Kehoe,D. K.Levine, and P. M. Romer,Determinacy of equilibria in dynamic

modelswith finitely

many

consumers, J. Econ. Theory

50

(1990), 1-21.

17. T. J. Kehoe, D. K. Levine,and P. M. Romer,Oncharacterizing equilibriaof

economieswithextemalities andtaxes

as

solutionstooptimizationproblems, Econ.

18.

J. Peck,On the existence ofsunspotequilibria in

an

overlappinggenerationsmodel,

J. Econ. Theory 44 (1988),

19-42.

19.

K. Shell, “MonnaieetAllocationIntertemporelle,” mimeo. Seminaire d‘Econometrie

Roy-Malinvaud,Paris,

1977.

20.

T. Shigoka, ttA Note

on

Woodford’sConjecture:ConstructingStationary Sunspot

Equilibria in

a

Continuous Time Model,“ mimeo.

1993.

21. S.E. Spear, Growth, externalities, andsunspots, J. Econ. Theory

54

(1991),

215-223.

22. S. E. Spear, S. Srivastava, and M. Woodford,Indeterminacy of stationary equilibrium

in stochastic overlapping generationsmodels, J. Econ. Theory

50

(1990),

265-284.

23.

M. $W\mathfrak{X}ord,$ $\prime\prime Indete inacy$of Equilibrium in the OverlappingGenerations Model:

A Survey,“ mimeo.

1984.

24.

M.Woodford, Stationarysunspotequilibria in

a

finance constrainedeconomy,

J. Econ. Theory

40

(1986),

128-137.

25.

M. $W\mathfrak{X}ord,$$\prime\prime Stationary$ Sunspot Equilibria: TheCase of Small Fluctuations around

a

Deterministic SteadyState,’t mimeo.

1986.

$F_{1f^{ure}}\vee 5^{-}$

$t$