JOURINALOF
Mathematical
ELSEVJER
Journal of Mathematical Economics 26 ( 1996) 63-84
ECDNOMICS
The existence of security market equilibrium
non-atomic
Andreu Mas-Cole11a, William R. Zame b*cv
*
a Department of Economics, Harvurd Institute of Economic Research. Harvard University, Cambridge,
MA, USA
b Depurtment of Economics, The Johns Hopkins University, Bultimore, MD, USA
’ Depurtment of Economics, University of California Los Angeles, CA 90024, USA
Submitted November 1991; accepted March 1993
Abstract
We considera finite horizon economywith incompletemarketsand, at eachperiod,a
non-atomiccontinuumof statesof nature.We combinebackwardrecursionmethodswith a
reformulationof theproblem in termsof first-orderconditionsto establishthe existenceof
a Radnerequilibrium.Oneof our hypotheses,
not requiredin thecaseof a finite numberof
states,saysthat in anydateand statethe positionreachedafterany possiblepreviousasset
trades,but beforespot trades,constitutesa feasibleconsumption.An exampleshowsthat
this hypothesiscannotbe dispensedwith (evenin the caseof countablemany states).
JEL clussificrrtion: D52
Keywords: Incomplete markets; Infinite state space
1. Introduction
Generalequilibriummodelswith incompleteassetmarketshaveprovedto beof
increasingimportancesince their introductionby Arrow (1964) and Radner
* Corresponding author: Professor W. Zame, University of California, Los Angeles, Departmen! of
Economics, Bunche Hall, 4@5Hilgard Avenue, Los Angeles, CA 90024, USA.
’ Hart (1975) provided an example to show that, without such a short sale constraint, equilibrium
may fail to exist.
03044068/96/$15.00
0 1996 Elsevier Science S.A.
i All rights reserved
SSDI 0304-4068(95)007431
64
A. Mas-Colell, W.R. Zame / Journal of Mathematical Economics 26 (1996) 63-84
(1972). For finite statespaces,existenceof equilibriain suchmodelshas been
established
in quite generalcircumstances
(by Radner,1972;Cass,1984;Duffie,
1985;Werner, 1986;Geanakoplos
and Polemarchakis,
1986;Duffie and Shafer,
1985,1986;seeMagi11
andShafer,1991,for a survey).For countablestatespaces,
existenceof equilibriumhas been establishedby Green and Spear(1989) and
Zame(1988). For relatedresultsfor infinite horizoneconomies,seeHemandez
(1988),Levine (1989),Hemandezand Santos(1991), Levineand Zame (1996),
andMagi11andQuinzii( 1992).
In many applications,
however,particularlythosearisingfrom continuoustime
financl>l models(see,for instance,Black and Scholes,1973; Breeden,1979;
Kreps, 1981, 1982;Duffie, 1988),the statespaceis neitherfinite nor countable,
but rather is a continuum.Unfortunately,the methodsused to establishthe
existenceof equilibriumfor countablestatespacesdependcruciallyon a limiting
argumenton the numberof states,and suchan argumentis not availablewith a
continuumof states.(The essentialdifficulty is that the wealthmappingis not
jointly continuous;seeMas-Cole11
and Zame, 1991; and Mas-Colell,1991,for
moreextensivediscussion.)
In this paperwe presentan equilibriumexistenceresult for a model with a
continuumof states.Thekey fact - or assumption,if you prefer- that underlies
our work is that utility functionsare separable(acrossstatesof naturebut not
necessarily
acrosstime).We treat real securities(that is, securitiesdenominated
in
physicalcommodities),andallow for multipletradingdates;in the spirit of Radner
(1972),we imposean a priori lower boundon short sales.’
The ideaunderlyingour methodis that separabilityof preferences
allows us to
solvethe equilibriumproblemat eachstateat the terminaldateas a function of
earlierchoices.We maythenrepresentactionsat the terminaldateby a correspondence that embodiesthe first-order characteristics(marginal utilities) of the
equilibrium.Becausethis correspondence
hasa finite-dimensional
range,we may
applyLyapunov’sconvexitytheorem(in the form assertingthat the integralof a
correspondence
is convex and compact)and familiar fixed point arguments
(althoughin a somewhatunfamiliarguise).Becauseof the multipletradingdates,
this is not yet the end.However,an inductivetigument allowsus to successively
replaceeachtradingdatein turn by an appropriatecorrespondence,
leadingagain
to a solutionof the first-order equilibriumproblem.In eachcase,convexityof
preferences
allowsus to showthat a selutionto the first-orderproblemis actually
anequilibrium.
A methodologybasedon combiningrecursivityand Lyapunov’stheoremhas
alsobeenusedby Mellwig(1996). Simultaneously
with the presentwork, MasCole11
andMonteiro(1996)havealsoobtainedanexistenceresultfor a continuum
of statesandtwo tradingdates.That resultdependson a regularityassumptionthat
allowsthe constructionof selectionsof conditionalequilibriahavinga continuity
property strongerthanwhat canbe obtainedin general(and,in particular,stronger
thanwhat is obtainedhere).It appearshowever,that this regularitymethodology
A. Mas-Colell, W.R. Zame/Journal of Mathematical Economics26 (199ti,! 63-84
65
doesnot generalizeto multiple tradingdates.Monteiro(I 996)providesa different
argumentfor the caseof two tradingdates.
We setour frameworkin Section2. SecGons
3 and4 isolatethe abstractresults
that are the heart of our approach;the proof uf the main existenceresult is in
Section5. Section6 presentsa counterexample
establishingthe indispensability
of
oneof our hypotheses:
we requirethat no possibleassettradingplan ever leaves
ex post endowmentsoutsidethe consumptionset. (That is, we requirethat any
possibleshortsalesof assetscanbe coveredby initial holdingsof commodities.)It
might well be possibleto weakenthis assumption,but it is interestingthat it
cannotbe dispensed
with entirely.(It is not neededin the casestudiedby Radner,
of a finite number of states.) Finally, the Appendix containsthe proof of a
technicalmeasurabilitypropertyof equilibrium.
We emphasizethat a great deal remainsto be done on this problem.In
particular,applicationsto finance pull in two directions.The first is toward
extensionsto an infinite horizon and to continuoustime; the secondis toward
computationallytractablemodels.For a financemodelwith incompletemackets,
we refer the readerto Karatzaset al. (1989).
2. The security market model
In this sectiorlwe describea modelof a securitymarketwith a continuumof
statesandmultipletradingdates.
Thereare T + 1 tradingdates,labeled0, 1,. . . , T; 0 is the initial date and7’ is
the terminal date. Write D = (0,. . . , T}, D-(t) = (0,. . . , t) and D+(t) = (t +
1, . . . , T) for the setsof all tradingdates,of dates(weakly) beforedate t, andof
dates(strictly) after date t, respectively.
There is uncertaintyabout the stateof nature at date T; this uncertaisryis
graduallyresolvedbetweensuccessivetrading dates.We model uncertaintyby
specifyinga (regular,Borel) probabilitymeasurep on the spacea= [O,llT of
oT) E J2 for a typical state.If t s T,
statesof nature.* We write o=(w,,...,
then a t-history is an element h E H(t) = 0,
0,. If o is a state,then
w[t]=(o,,...,
wI) is the history beforedate t + 1; we usea similarnotationfor
the truncationof histories.We write
X
fi(h)=(h)x~,+,x
l
l
l
l
-- x&.=(ww[t]==h),
for the set of statesthat are possiblefollowing the hiTtory h, and
w+ I)Pl= PI x or+1=(hEH(t+
l):h[t]=h},
’ The assumption that L? has a product structure has real bite. However, the assumption that each
factor is the unit interval is innocuous, since eveby (complete, separable) probability space is
isc>morphicto a probability space on the unit interval.
66
A. Mas-Colell, W.R. Zame/ Journal of Mathematical Economics 26 (1996) 63-84
for the set of t + l-historiesthat are possiblefollowing the history h. Note that
thereis a uniqueO-history(it is empty) andthat n(O) = 0.
Let 9 be the Bore1sigmafield of a, andlet To be the sigmafield of null
events.For each t ,< T, write .Tl for the sub-sigmafield of 9 generatedby 9$
and the Bore1subsetsof fl that are independentof the last T - t coordinates.
Note that (Fl} is an increasingfiltration andthat FT = SC We saythat a function
f definedon D X 0 is progressively measurable if
) is measurablewith
respectto Ft, for each t. (Equivalently:f( t, o) dependsonly on the first t
coordinates
of 0.) If f is progressively
measurable,
then it is sometimesusefulto
abusethe notationandview
) asa functionon t-histories,or asa functionon
statesthat dependsonly on thefirst t coordinates
of the state;this shouldcauseno
confusion.We shall abuseterminology and refer to a function defined on
D - ( t) X 6! as progressively measurable if f( t’ , ) is measurable
with respectto
Fit, for each t’ I t.
For eacht I T, disintegrationof measures
yieldsa measurep-cIon the spaceof
historiesH(t) and,for eachh E H(t), a measure( ph) definedon fl( h) suchthat,
for eachmeasurable
function g : C2--) IF&
fit,
fit,
l
l
l
For eachhistory h, definethe measureph by ph( E) = )uh(E X [0, l]r- ‘+ I). We
shallassumethroughoutthat eachof the measures
( ph) is non-atomic.3
Thereare I(t) 2 0 commoditiesavailablefor tradeat eachhistoryat date t; we
insistthat I(t) 2 1, but we allow for the possibilitythat I(t) = 0 for t < T (in
which case consumptiontakes place only at the terminal date).4 Write I =
max I(t); without loss of generality,we view (W’(‘) as the subspaceof IR’
consistingof vectors x suchthat x, = 0 for 1> I(t). A consumption plan is a
bounded,progressivelymeasurablefunction x : D X J’2+ IF!?with the property
that x( t, 0) E R ‘W for each t. A commodity price process is a progressively
measurable
function p : D X a --) rW$ suchthat p( t. U) E if?!$‘)for eacht. (Keep
in mind that requiringconsumptionplansandpricesto be progressivelymeasurablemeansthat x( t, o) andp( t, o) dependonly on o[ t].) If x is a consumption
planand p is a commodityprice process,the associatedwealth process p q x is
definedby
po x(t, 0) =p(t,
o)‘x(t,
0) =CP/(tv
“)X,(h
4
’ For example, supposethat p is a product: JL= A, X . . . X AT. Then for each history h E H(t), the
measure plr reduces to A,, so non-atomicity of each p” reduces to the requirement that each of the
factor measures A, be non-atomic. Allowing for the possibility that p is not a product measure means
that we allow for the possibility that current histories affect future probability distributions.
4 There would be only notational difficulties in allowing for the possibility that some commodities
arc not available for trade following some histories.
A. Mus-Colell, W.R. Zume / Journal of Mathematical Economics26 (1996) 63-84
67
In the usualformulation,a security is a claim to a dividendstreamof real
commodities;in general,a securitymay live for manyperiodsand be tradedat
manydates.However,we adopta formulationin which securitiesrepresentclaims
to realcommoditiesandother securities,but live for a singleperiodandaretraded
only once.As we shallseevery shortly,thesetwo formulationsareequivalent;Lhe
onewe adoptis simplymore convenientfor our purposes.
Thereis a set J of securities;of these,the subsetJ(t) C J existsat date r, and
the subsetJ * (t) C J(t) is tradedat date t. 5 Thereareno securitiesavailableat
the terminaldate,so J * (T) = J(T) = (6.6 Securityj E J is availableat date t(j);
its payoff at eachstateat date t(j) + 1 is a non-negativevectorof commodities
and securitiesin J( t( j) + 1); tiat is, Ai
= ( A;(o), with A;( 0)) E R’(‘(j)+ I’
x lRJ(‘(j)+? Of course,we requirethat Aj be measurable
with respectto am+ 1.
A simple exampleshould suffice to demonstratethat our formulation is
equivalentto the usualone. Considera securitythat is tradedonly at date0 but
yieldscommoditydividendsat everyintermediatedate.In our formulation,sucha
securitycanbeexpressed
as: a securityA, tradedat date0, payingat date 1 in real
commodities,anda securityA, that is not tradedat date 1, but paysat date2 in
realcommodities,and a securityA, that is not traded,etc.
A trading plan is a progressivelymeasurable
function z : D( T - 1) X 6! + IF!j;
&, o) is the trade of securitiesat date t in the state o. A trading plan is
admissible if z(t, o) E RJ l (?
If z is a tradingplan,we write div z for the associated
dividendprocess;thus:
div z( t, o) =
C Zj(t, w)Aj(m)*
jur- I)
We follow the conventionthat securitytradestake placeex dividend(so that
dividendsat datet accrueto theholderof the securityat datet - 1. As before,we
write (div ’z, div*z) to distinguishthe payoffi in commoditiesand in other
securities.
A security price process is a progressively
measurable
functionq : D(T - 1) X
0 -+ RJ suchthat q( t, o) E R J l (‘) for eacht; here,qj(t, o) is the price of the
tradablesecurityj at datet in the statecr).If z is an admissible
tradingplanandq
is a securitypriceprocess,the associatednet expenditureprocess q c] ( z - div* z)
is definedby
qO(z-div*
z)(t, o) =
C qj(t, “)(zj(t,
jEJ ‘(1)
0) -div* ~)(t, 0).
Thereare N < OQtraders, definedb;, consumptionsets,endowments,andutility
functions.Tradern’s consumption set C” is definedby constraintsat eachdate
5 As we shall see, allowing for securities that are not traded gives us the flexibility we require.
6 Again, there would be only notational difficulties in allowing for the possibility that the set of
tradable securities depends on the history, not just on the date.
68
A. Mas-Colell. W.R. Zame/.iournal of Mathematical Economics 26 (1996) 63-84
and state; we require that consumptionplans of trader n satisfy xn(t, o) E
Cn(t, o) for eacht, w, where C”(t, o) C F!?‘) is an upper set (i.e. C”(t, w) is
closed,convex, boundedbelow, has a non-empty interior, and { y) + rW$'k
Cn( t, o) for eachy E C”(t, o)), andthe correspondence
o + C”( t, 0): J2--$+
OX’(‘)is Yt measurable.Endowments e” are arbitrary elementsof C”. 7 Utility
finctions Un : C” + IF!areassumedto be separable
acrossstates(but not necessarily acrosstime), so that
U”(x)=~d’[x”(O,
u), x”(l,w)
,..., x”(T, w); o] dp((rl),
where~“(x~,..., xr; o) is a smooth(continuouslydifferentiable),strictly monotone, concavefunction of x0,. . . , xT, and is measurablein cr).We assume,in
addition,that the partialderivativesof u” are,on eachboundedsetof R’, bounded
aboveandboundedawayfrom 0, and that theseboundsare independent
of o. 8
Tradern’s tradingplanszn areconstrained
to lie in the feasibletradingset Z”,
definedby restrictionson tradesat eachdateandstate: z”( t, o) E Z”( t, U) where
Z”(t, o> c RJ(‘) is an upperset, 0 E Int Z”( t, 01, and the correspondence
ct)+
Zn(t, (0): a + --) RJ(‘) is Ft measurable.
g
Giventhe endowmenten,thecommoditypriceprocessp andthe securityprice
processq, trader n’s budgetset B”(e”, p, q) consistsof consumptionplans x”
andadmissibletradingplans z” suchthat
pOx”+qO(z’-div*
z’“)~;pOe”.
Whenthis inequalityis satisfied,we saythat Z” cfinanccsxn.
Given the abovedata, a Radner equilibrium consistsof a commodityprice
processp, a securitypriceprocessq, andconsumptionandtradingplans x”, z”
for eachtrader n such that
Cx” = Zen,
n
n
(1)
c zn= 0,
n
( x”, z”) ‘5 B”( e”, p,
q),
for all 12,
If (xl”, 2”‘) EB”( e”, p, q),
thenI/“( xl”) s U”( x”),
for all n. (4)
Thatis, commoditymarketsclear,securitymarketsclear(andsecuritiesarein zero
net supply),andconsumption/tradingplansarebudgetfeasibleandoptimal.
7 But we impose an addition.4 assumption on endowments below.
* Differentiability of u” is not actually necessary;it would be possible - although cumbersome - to
work with subdifferentials rather than with differentials.
9 In imposing lower bounds on short sales, we follow Radner ( 1972).
A. Mas-Colell, W.R. Zame / Journal of Mathematical Economics 26 (1996) 63-84
69
To establishthe existenceof Radnerequilibrium,we requirefive additional
assumptions.
We havealreadyimposedlower boundson eachconsumptionand
net tradeset:the first two additionalassumptions
requireuniformlower bounds;as
follows
(A.l) there is a constant B > 0 such that y+ -B for all y E Cn(t, w), all
6 n, t, 0,
(A.2) there is a constantB’ > 0 such that Yj2 -B’ for all y E Z”(t,
j, 12,t, 0.
w), all
We have alreadyrequired that 0 be interior to each trade set; the next
assumptionrequires,first, that the samebe true of the assetstrading set and,
second,that the interiority be uniform (in particular,for tradableassetsa fixed
amountof short sellingis alwayspermitted):
(A.3) thereis a constantc > 0 suchthatif zn E 2” and yj - (div,?z”Xt, w) > -c
for each j E .I( t), then y E Zn(t, 0).
The fourth assumptionis crucial;it requiresthat the commoditycomponentof
short salescanalwaysbe coveredfrom dividendsandendowments,beforetrades
takeplace:
(A.4) thereis a constantc’ < 0 suchthat if zn E Z” is an admissibletradingplan
and y E IF!?(‘)is such that y/r c’ + (div ’z;)( t, o) + e$ t, w) for all J’E I(t)
then, y E C”(t, 0).
We havealreadyassumedthat dividendsare non-negative;
the fifth assumption
requiresthat dividendsbe uniformly boundedaboveand that expecteddividends
be uniformly positive:
(AS) there is a constant d > 0 such that 11Ai
11< l/d
dp,$ o) 2 d for everyj andeveryhistory h E H( t( j)).
and / II Aj( 0) II
Our centralresultis as follows.
Main Theorem.Underthe assumptions
(A.I)-(AS), a Radnerequilibriumexists.
With the exceptionof (A.41, all our assumptionsare either standardor
relativelyinnocuous.Assumption(A-4), however,is quitestringent.It is alsoa bit
surprising,becausenothinglike it is requiredin the completemarketscase,or in
Radner’s(1972)modelwith a finite numberof states.It is neitherjust a matterof
70
A. bias-Colell,
W.R. &me/
Journul
oj’Matkemutica1
Economics 26 (1996) 63-54
convenience
nor an artifact of the proof, however; in Section6 we presenta
counterexample
establishing
its indispensability.
We makethreemoreobservations
concerning(A.4):
If we wereto imposemild conditionson the structureof securityreturns,(A.4)
wouldimply (A.2).
0 Supposethat consumptionsets are alwaysthe non-negativeorthantsof the
corresponding
spaces.Thenin (A.4) we cantake c’ = 0 providedthat for every
n, o and t, the endowmente”( t, o) includesa (uniformly)positiveamountof
some good not appearingin the assetreturns. This follows without any
modificationof the proofs.
0 An alternativeto (A.4) is to let consumptionsetsC(t, o) equalthe entirespace
IW’(‘)andto requirethat at-least-as-good
setssatisfyappropriateboundedbelow
and uniformity conditions.This is the approachtaken in Mas-Cole11
and
Monteiro(1996).
The idea of the proof of the Main theoremis to reducethe concept of
equilibriumto a first-ordersequentialproblem(this will be donein this section),
which is then solvedinductivelyby the applicationof an abstractstatic equilibrium result(which will be formulatedand provedin Section3). The key stepis
thenin Section4, whichessentiallyshowshow the abstractresultcan be usedto
establishthe caseT = 1. Finally,Section5 completesthe inductionby establishing
theMain theoremfor arbitrary 7’.
In the remainderof this section,we formulatethe equilibriumconceptas a
first-orderproblem.
A sufficientconditionfor p, 4, X, z to constitutea Radnerequilibriumis th
parts (l), (2), (3) of the definition be satisfiedand that there be progressive
measurable
functions o(t, o) and p( tw), taking valuesin ([W$~X ll’J$:))“’
respectively,
suchthat, for all n, t, h E H(t), &“E I(t), we have
V+,
@(t, h) =~~d’(x”(O,
A
o),...,Y’(T,
0); 0) dp/@),
for all t s T, h E H(t), /‘E I(t),
vjyt,
h) =/#(t+
forall t<T,
( P”( P* (I) - v”)
1, 0) -A#+
hEH(t),
l
((C
(5)
1, 0) dE.Lh(
o),
jEJ(t).
Zln) - (x”, zn))( t, 0) 2 0,
wheneverx’~( t, o) E C”,
z’~(t, U) E Z”
and
( Z‘n - div*z”)( t, o) E RJ‘(I).
(6)
Thisis straightforwardto verify if we recallthat un is concaveandwe interpret
v,?(t, o) (resp., u$ t, w)) as the marginalutility of commodity/’ (resp.,assetj)
at (t, o), and p”( t, o) as the marginalutility of incomeat (t, 0). As a matterof
A. Mas-Colel’, W.R. Zame / Journal of Mathematical Economics26 (1996) 63-84
71
terminology, if fct any feasible consumptionx we have functions v( t, W)
satisfying(5) we say that v(t, 0) are marginal valuation fi4nctions for x, 2.
It is helpful to think of v” and p” as part and parcelof the definition of
equilibrium.Thus!.we can view an equilibriumas beinga six-tuple( p, q, x, z,
v, PI*
Without lossof generalitywe normalize:
C p/+(t9m)+
/El(t)
E
jEJ
qj(t9 0)=1*
*(r)
3. Abstract equilibrium
In this sectionwe present an abstractequilibrittm theoremthat is key to
establishingthe mainresult.
Thedatafor our abstractresultarea finite collectionr ‘, . . . , r N of subsetsof
R K (we write r= n nr “) and a correspondence
V : r + + Rpt, enjoyingthe
following properties:
(a) for eachn, P is an upperset,
(b) for eachn, 0 E Int r n,
(c) for each g E r, V(g) is non-empty,convexandcompact,
(d) V is an upper-hemicontinuous
correspondence.
It is convenientto view elementsof IRNKas N X K matrices;we write V”(g) for
the nth row of the matrix V(g), andVt( g) for the k entry of the nth row.
The interpretationwe have in mind is that K standsfor the total numberof
consumptiongoodsand assets,P is the set of feasibleportfolio andconsumption net tradesfor trader n, and any v” E V “(9) is a vectorof marginalutilities.
Write AK for the unit price simplexin IRf and A:+ for the set of strictly
posit,iveprices.By an abstract equilibrium we mean a four-tuple (q, g, v, A)
consistingof prices q E AK, a vector of choices g E r, a vector of marginal
utilities v E V(g), anda vector A E RT+ of marginalutilitiesof money,suchthat
(1) C,g”
= 0;
0 for each n;
(2) q*g”=
(3) ( A”q - un) (f” - g “) 2 0 for eachi andf” E r n Note that, in particular,
if vn*fn>vn=gn,then q*f”>q-g”=O.
Our basicresultis that an abstractequilibriumalwaysexists.
l
Theorem 3.1. With the assumptions above, an abstract equilibrium exists.
Proof. We considerfirst the specialcasein which eachV(g) is a singleton(so
that V is a continuousfunction). For eachn andeach g E r, define
P”(g)=lf”El-”
: V”( g) l f”
> V”(g)
-g#‘).
72
of Mathemutical Economics 26 (1996) 63-84
A. Mus-Colell, W.R. Zame/Journal
Thisis anh-reflexive,strictlymonotone,convex-valued
(non-transitive)preference
relationwith an opengraph.Hence,by the equilibriumexistencetheoremof Gale
and Mas-Cole11
(1975),the economydefinedby the consumptionssets r ‘, the
preferencesP ‘, and the endowments0 E r “, has a Walrasian(competitive)
equilibrium(q, g), with q E A:,. Thus:
(1) C,g” = 0;
(2) 4.8” = 0 for eachn;
(3’)if 2+fn>ungn,then
q.f”>q*g”=O.
We now derivethe sameconclusions
for the caseof a generalV by meansof a
standardapproximationargument.For each integer m, choosea continuous
function(9m. r + R Tt suchthat dist(graphq,,,, graphV ) < 1/m; this is possible
becauseV is upperhemicontinuous,
convexandnon-empty-valued.
If we replace
V by (9m,we arein the specialcaseconsidered
above;let ( gm, qm) be corresponding Walrasianequilibria.Note that (q,$, (g,), (p,,$g,)) are boundedsequences
(because:q, E AT +; each r n is boundedbelow; and upper hemicontinuity
entailsthat the restrictionof V to boundedsubsetsof r has boundedrange).
Hence,passingto a subsequence
if necessary,we may assumethat qm -+ q E
AK, g, + g E r, Q,,,(
g,) --) v E V(g). This yields:
(1’) C,g” = 0;
(2’) q g” = 0 for eachn;
(3”)if vn~fn>vn~gn,then qmf”Zq.g”=O.
To seethat thelast inequalityin (3”) is, in fact, strict, andhencethat (3’) holds,
supposenot.Thenvn~fn>vn~gn but q~f”=q*g”=O.SinceOEInt
P and
P is open and convex,there is a f’“E.P
such that vn*fn>un~g”
and
q 0f’” < 0; this contradicts(3”) andestablishes
(3’), as desired.
It remainsto obtain h E RT+ and to establish(3) in full. This can be done
separately
for eachagent.Fix n andconsiderthe convexsets:
l
l
A=((f”,
cY)Er”XR:tYsu”*f”),
Then A n int B = fl and ( g n, u” g “) E A n B. Therefore, by the separating
hyperplane
theoremthereis a non-zero(q”, - pn) E R K+ I suchthat qn f n - p”ar
2 q” . f’” - $un~gn wheneverf”EP,
f’“ERK,
asun*fn
and qef’nsqe
g“. From this it is easilyseenthat q” = p “q for somep n > 0. Hencep “4 9f n pnon*fnrpnq~gnCL”u” g n for anyf n E r n, which canbe rewritten as
l
l
l
( 1l
P”
Fq-vn
(f”-g”)rO,
for anyf n E P. Thus,An= pn/$
will do. 0
In additionto the existenceof equilibrium,we shall alsorequireestimatesof
the equilibrium prices and of the vectors A; moreover,we shall need these
A. Mas-Coiell, W.R. Zame / Journal of Mathematical Economics 26 (1996) 63-84
73
estimatesto dependonly on the distancesfrom the origin to the complementof
P, on the lower boundson r ‘, andon the rangeof the restrictionof V to the
feasibleset. To be precise,we assumethe existenceof real numbersE> 0, p > 0
anda compactset V * C WY+ satisfying:
(e) if 11
f 11< E, then fE P, for all 12,
(f) if g” E P, then g” > -PI (where1 =(l, l,..., l)), for all n,
(g)if g~r andC,g”=O,thenV(g)cV*.
(Theassumptions
on feasibletradingsetsmadein Section2 aredesignedprecisely
to ensurethat theseconditionswill be met when we translateto the abstract
setting.)
Theorem 3.2. With the assumptions above including (e)-(g), there are constants
Q=Q(e, /I, V*)>O,
A+=A+(e,
/3, V*)>O,
and A-= A-(E, /3, V*)>O
such that, if (q, g, v, A) is an equilibrium for the abstract economy then
(1) Q I qk I 1 - Q for each k, and
(2) A-I A* n I A+ for each n.
Proof. We write supV * for the supremumof all componentsof all vectorsin
V * , and inf V * for the infimum of all componentsof all vectorsin V * . Set
Q=(~/2K)((infV*)/[(su~V*)1[(~/2)+~~Pl).
Supposethat qL < Q for someF’. Becausepricessumto 1, thereis a k suchthat
qk 2 l/K. We write
A= ((inf V*)/[(sup
V*)][(E/~)
+HKp])-’
(so that Q = (e/2 K)/A). We write 8, for the vector whosekth entry is 1, and
whoseother entriesare0 (and similarlyfor s/>. We fix an arbitrary n. Assumption (e), togetherwith the fact that P 1san upperset, guarantees
that the vector
However,
assumption
(f)
guarantees
that
f= ,4&- (e/2& belongsto r “.
g” 1~H/3 1. Direct calculationnow showsthat q gf< 0 and v lf> v . (HP 1) 2 v
g”, contradictingthe equilibriumconditions.We concludethat qr< Q for eachk,
asasserted.
Since( A”q - vn) (fn - g “) 2 0 for everyf” E P, and r n is an upperset,it
follows, in particular, that ( A”q - vn) 2 0. Since q E AK, we concludethat
A” 2 v[ for every k, so A” 2 inf V * . Let us fix k’. Since -(e/2)& E r n, it
follows that ( A”q - vn) (-(e/2)&g “) 2 0. Direct calculation,usingthe fact
that g n51HP 1, showsthat
l
l
l
( A”q) 9[ ( W) 4-1 5 (SUP V * ) [( l /2) + HP 13
andhencethat
A” 5 (sup V* )[(~/2) +HP]/(
This completesthe proof. 0
d2)Q.
74
A. Mus-Colell. W.R. Zame/ Journal of Muthemutica! Economics 26 (1996) 63-84
For future reference,we notethat Theorem3.2 appliesto competitiveequilibrium pricesin an Arrow-Debreueconomy:simplytake I$“( g) to be n’s marginal
utility of consumptionfor the kth commodityat consumptionlevel g”.
4. First-order equilibrium
In this sectionwe posea first-order equilibriumproblemcorrespondingto a
securitymarketwith two tradingdates,andusethe abstractresultof the previous
sectionto find a solution.This is a genericstepin a recursionargumentcompleted
in Section5.
We considera securitymarketeconomywith two tradingdates,i.e. T = 1. The
economyis as in Section2 except that utility functions are left unspecified.
Instead,we assumethat we aregivena measurable,
closed-valued
correspondence:
@: rp(O)
x I-p x [o, l] + RT’!O)XRTT’!
n
n
We assumethat thereis a compactsubsetL c R y’!o’x RN’(‘) suchthat C, x”(0) ,(
&P(O), C, znI 0, x” E Cn(0) and znE 2” implies @(x(O), z, o) c L for
almostall 0 E [0, 11.
The interpretationwe have in mind is as follows: for x”(0) E C”(O), Z” E
2”(0)=Zn, n= l,..., N, we have (M, E) E @(x(O), z, o) if, when date 0
consumptionsare given by x(0) and date 1 endowmentsby e(1, o) +
div’ z”( 1, o), then at the equilibriumreachedat (1, o> we havethat M,” (resp.
ET) is trader n’s marginalutility for the consumptionof commodity/’ at date0
(resp.at date 1).
Given x(0) and z let pV(o,,zbe the restrictionof the correspondence
@ to
(x(0)) X (z) X [0, 11.A selection from @$(oJ
7 is a measurablemapping cp=
( MP’ EJ: (0, 1) -+ R!‘!O’X R,N$” such that i(o) E @(x(O), z, o) for (almost)
all o E [0, l]. For such a selection cp, we define a vector u = v( (9)=
(U,(Q), U,(Q)) E @‘!“X R:: by
Vz”j(Q)=jE~(o)*A:(o)dCL(W),
jEJ.
Giventhe abovedata,a first-order equilibrium consistsof a price q E A$‘)” l,
a consumptionvector x(O),a tradevector z, a marginalutility vector v E R T’(“)X
RyJ, and a vector of marginalutilities of incomeat t = 0, A E R y+, such that
(recallthat J * denotesthe tradableassets):
(1) C,x”(O) s &e”(O), x” E C”(O).
(2) &zn = 0, Z” E Z”(O) and ~7= 0, if j 6EJ * .
(3) q W(o)
- e’(O), z”) = 0 for eachn.
l
A. Mas-Colell, W.R. Zume/.Iournal of Muthemutical Economics 26 (1996) 63-84
(4) ( A”q - u”) ((x”,
l
75
z’“)) - (x”, z”)) > 0 for every n and( x’~, z”‘) E C”(O)
with zi” =0 for jEJ*.
(5) Thereis a selectionq from 4p,(0)
.c_ suchthat v = v( cp).
X2"
Theorem.4.1. A first-order equilibrium exists.
Proof. For each n set r n = (C”(O)- {e”(o)])X (2” n RJ‘) CR'(') X R’ l . Note
that for each n, r ’ is an upper set and0 E Int T ‘. We write r= I-l, r ‘. For
g = (x(0) - e(O),z) E r defineV( 2) = { v( cp): - cpis a selectionfrom @$(,,,,,c
R(‘(‘)? It follows from the resultsof Aumann(1965)on measurable
correspondencesthat V(g) is compact,convrzxandnon-empty,andthat the correspondence
v: r-, pm+J) is upper hemiob>ntinuous.
The sameis true of the correspondenceV’ : r+ RN(‘(o)+J’)obtaine,dby replacingevery v by its projectionv’ on
R’lo)+J*. Hence,Theorem3.1 yieldsan abstractequilibrium(q, (x(O), z), v, A).
The definition of V providesa selection<psuch that u = ZJ(
cp), and v’ is the
projectionof v on R’(“)+J*. The!-efore,the list q, x(O), z, v, A is a first-order
equilibrium,asdesired. 0
Supposethat y > 0 is suchthat, for all n:
(1) y E C”(0) whever II y II < y,
(2) y E 2” whez%‘~~~~
II y II < y, and
(3) c(o) 2 (l/yh
P2 (l/y)l.
Thencombiningthe proof of Theorem4.1 with Theorem3.2 we get
Theorem 4.2. The set of first-order equilibria is compact. Moreover, there is a
‘(“)xJ’ X ~,,C”(O) X n,,2” X (RN’(‘) X RN”) X RT,, dependcompact set L’ c A+
ing only on y and L such that (q, x(O), z, v, h) E L’ for any first-order equilibrium .
5. Existenceof a Radner equilibrium
We denote our base economy by Z. Let us fix a vector x of feasible
consumptionplans,a vector z of admissibletradingplans,a period t 5 T anda
history h E H(t). We define a conditionaleconomy Z’( x, z, h) beginningin
period t andendingin periodT by fixing consumptions
at periodst’ < t at x(h),
defininginitial endowmentsat (t, h) to be e(t, h) + (div ’ z”)( t, h), anddefining
trading sets at (t, h) to be Z”( t, h) - (div * z”)( t, h). Note that the economy
a( X, z, h) dependson X, z only throughx(h), z(h).
For any X, z, t, h as above,we say that a feasibleconsumptionplan E is
compatible with (x, z, t, h) if X is the vector of consumptionplans of an
equilibriumof the conditionaleconomya( X, z, h) and X(t’, h) = x( t’) for t’ I t.
A. Mus-Colell, W.R. Zume/ Journul ofMathemuticu1 Economics 26 (1996) 63-84
76
We define
v(x,
=
2, h)
v E r-pyx
t’s t
qy
: : thereis an E compatiblewith ( x, z, t , h)
andmarginalvaluationfunctionsE for X suchthat v = V( h) .
>
Note that V( x, z, h) dependon x, z only through x(h), x(h); we sometimes
abusenotationandwrite V( x( h), z(h), h).
We find it convenientto use an inductiveargument;to this end,we introduce
the following:
Induction hypothesis. For all t 2 ? there is a compact L c n ,,s , R fy”) X RTJ!“)
such that for any feasible x and z the set V(x, z, h) is a non-empty, compact
subset of L for all h E H(t). Moreover, the dependenceon h is m-;asurable and
the dependenceon x(h), z(h) is upper hemicontinuous.
Recall that there is a uniqueO-history,which is empty. If x is a vector of
feasibleconsumptionplansand z is a vectorof admissibletradingplans, then
~(0) and ~(0) are empty. Note that V(@, x, z) = V(fl, #,@> (in our abuseof
notation)is the set of equilibriumallocationsfor our baseeconomy8. Since
non-emptiness
of V(@,0,(b) is just the validity of the inductionhypothesisfor
7= 0, establishingtheMain theoremreducesto establishing
the inductionhypothesisfor I= T, T- 1,. . . , 1,0.
Proof. We first verify theinductionhypothesis
for 3 = T. Note that, for t = T, the
conditionaleconomy8(x, z, h) isjust anordinaryfinite-dimensional
Arrow-De-
breu economy in which the data (endowmentsand utility functions) depend
continuouslyon x(h) and z(h), andmeasurably
on h. Note alsothat V( x, z, h) is
just the collectionof vectorsof marginalutilities at equilibrium.More precisely,
we write I = CTXoI( t ); then V( x, z, h) consistsof all Mtuples
(gradientu’( R’), . . . ,gradientuN( RN)),
where
in= (~~(0, h), x”( 1, h) ,..., x”(T-
1, h), X”(T, h)),
and E(T, h) is an Arrow-Debreu equilibriumallocationfor 8( ;w z, h). By
Theorem4.2, V( x, z, h) liesin a compactsubsetof RN’. Standardarguments(see
Hildenbrand,1974, for example)show that the set of equilibrmmallocations
dependsupperhemicontinuously
on x(h) and z(h), whenceV( x, z, h) doesalso.
(Keep in mind that utility functions have been assumedto be continuously
A. Mas-Colell, W.R. Zame/ Journal of Mathematical Economics26 (19%) 63-84
77
differentiable.)We deferdiscussionof the measurabledependence
of V on h to
the appendix.This establishes
the inductionhypothesisfor 7= T.
It remainsonly to show that validity of the inductionhypothesisfor t = s + 1
z 1 impliesits validity for t = s 2 0. To seethis, we reducethe problemto the
two-periodcasesolvedin the last section.Fix X, z and h E y(s); we showfirst
that an equilibrium exists for the conditionaleconomy SY(x,z, h), whence
V( X, z, h) is not empty.To this end,definea two-periodfirst-ordereconomyin
the following way. For X’(S), z’(s), w E [0, l] given,define
@(x’(s)9 z’(s), 0) =v*(x(1),...,x(sl), x’(s),
z(l) 9.•*9z(s- I), Z’(S), (h, a)),
where(h, o) E H( s + l), andan asteriskindicatesthat we neglectall termsof v
corresponding
to periodst < t’ (of course,we take I( s + 1) U .I( s + 1) as the set
I( 1) of date 1 consumptiongoods, Z(s) as the set I(0) of date0 consumption
goods,etc.). Accordingto Theorem4.1, this two-periodeconomyhasan equilibrium, and this providesthe desiredequilibrium for the conditionaleconomy
2%x, z, h).
Theorem4.2 yields compactnessof V( x, z, h), and upper hemicontinuity
follows as before.For measurability,seethe appendix.This completesthe proof.
cl
6. A counterexample
In this sectionwe show,by meansof anexample,that the keyassumption(A.4)
cannotbe dispensed
with.
Thereare two periods,i.e. T = 1, andtwo traders.In period0 thereis a single
consumptiongood,while in period1 therearetwo consumption
goods.The states
of the world are o c [0, l] with p Lebesguemeasure.
The utility functionsof the two tradersareof the form
u”( x) = x”(0) + I;“(
x”( 1, 0)) dp( o)
l
Consumptionsetsarethe non-negative
orthants.Note that the utility functionsare
stateindependent.
As for endowments,we take e”(0) 2 20 for n = 1, 2, e’( 1, UJ)
+ e*(2, o) = (6, 6) for all o, el( 1, o) = e:(l) w) = 3 for all o, 2 < e#, o) < 4
for all o and p( ei( 1, o) > 2 + E) > 0 for all e > 0. In other words, the state
affectsonly the allocationof good1 amongthe two traders,andthe distributionof
endowmentsis spreadover the segmentindicatedin the Edgeworthbox of Fig. 1.
We still needto specifythe utility functions un(x”( 1)). Ratherthan doing so
explicitly, we shallsimplymakethreerequirements:
(a) TheParetosetin the Edgeworthbox of periodi is the diagonal;seeFig. 1.
78
A. Mas-Colell, W.R. Zame/ Journal of Mathematical Economics 26 (1996163-84
Fig. 1.
(b) We indexthe Paretosetby 0 r; s I 6 in the obviousway. Let us denoteby
p(s) the shadowprices of the Pareto-optimalahocation(s, S) normalizedto
p,(s) = 1. Then we ask that p2(s) s 1 for s 2 1 and p,(s)= (s + 1)/(3 - s),
otherwise.The propertiesof this assignment
of valuesare describedin Fig. 1.
(c) We denoteby p”(s) the marginalutility of wealth of trader n at the
price-wealthcorrelationcorresponding
to the optimalallocation(s, s). We require
that 1s p’(s) 5; 2 for all s 2 1, p2(s) 2 4 for all s and p2(s) s 5 for s r; 4.
It is tediousbut straightforwardto exhibit smooth,strictly concave,strictly
monotonefunctionsd”, n = 1, 2, satisfying(a)-(c).
Finally, as for the assetstructure,there is, a single securitythat is simply a
futureon good 1, i.e. the stateindependent
returnof the assetin (0, 1). The lower
boundon the assettradeis - 4.
We claim that this economyhasno Radnerequilibrium.Denoteby (Zj, Z’, Z2)
potentialequilibriumvaluesof the securityprice (the price of the consumption
goodin period0 is normalizedto equal 1j and trades.Note that we must have
q>O (and thereforeZz= -2’) and 2’ 2 - 3 (indeed,if Z’ < - 3, tJaenthere is
no spotequilibriumfor a positivemeasureset of a; seeFig. 1). Althoughthis is
not essentialto the argument,it will facilitateour reasoningif we assumethat
whereverZ’ z -3, z2= - z ‘, then for all o there is at most one spot market
equilibrium(in Fig. 1, for all o the period1 ex post endowmentsare in the line
R > [3) lo the right of point a). Then we can denoteby u’( z*), v2(z’), the
marginalutilities (if defined) of the assetwhen the assettrade is zi 2 - 3,
Z2 = - z’ andperiod 1 spotpricesare in equilibrium.Because
of requirement(c)
above,we shouldhaveu’( I’ ) s - 3 (or undefined)for all z’ 2 - 3 andv2( z ' j 2 4
(or undefined)for all Z’ 2 - 3. SeeFig. 2.
Becausethe holdingsof period0 consumptiongood havebeenchosento be
A. Mas-Colell. W.R. Zamc/ Journal of Mathematical Economics 26 (1996) 63-M
0
-3
79
Z'(' 4)
Fig. 2.
large enough, it is easy to verify that an equilibrium (Zj, Z’, - 2I) must be
characterized
by un(Z‘) = 4 for n = 1, 2. Hence,v’(Z’) = v2(2’). But no sucha
2’ exists.Therefore,no equilibriumsetexists.The readermay want to convince
himself that, with referenceto Fig. 2, (Zj,- 3, 3) is not equilibrium:at this
asset-;\riceagent1 (who hasa uniformlyporjitivelevelof wealthat each stateand
is a price-taker)will wish to sell moreof tde asset.
Strictly speaking,the quasilinearityof the utility functionsun with rcspcctto
the consumptiongoodof period0 preventsu” from beingstrictly concave.But,
clearly, we could replacex”(0) by &C’(O)), where g is strictly concavebut
almostlinear,without alteringthe qualitativefeaturesof the counterexample.
We concludewith someremarks:
(1) Why doesthis existencefailurenot occur in Radner’smodelwith a finite
numberof states(seeRadn\:r,1972)?Supposethat in the exampletherewere M
states with correspondingendowmentsei( 1, m) = 2 + 2m/M, m = 1,. . . , M.
Then the analogof Fig. 2 is represented
in Fig. 3. We can seethat (q’, - 3 (2/M j, 3 -I- 2/M) is now an equilibrium.There is, therefore,a discontinuityin
budgetsets.As long as M < x the spotpriceschosenat the statem = 1 (which is
I
-2
I
0
azl(=4)
Fig. 3.
80
A. Mus-Cole& W.R. Zume / Journul
ofMuthemuticul Economics26 (1996) 63-84
p2= l/3) makesit impossiblefor trader 1 to sell more than three units of the
asset(the wealthat state 1 is zero). However,when M = m, thereis no longera
statewith zerowealthandit becomespossiblefor a singletraderto pZan selling
more than three units of the asset (although this cannot possibly occur at
equilibrium).
(2) While theexampledoesdefinitelyestablishthe indispensability
of A.4, Fig.
3 suggeststhat theremay be room for modificationsof the equilibriumnotion
which,in Fig. 2, wouldhavethe effect of making(q’, - 3, 3) into an equilibrium.
(3) If the assettradecouldbe madecontingenton the state(completemarkets)
thentraderscouldmakesurethat, whateverthe state,they tradeto the point a in
Fig. 1. With spotprices(1, l/3) this is thenan equilibrium.
(4) Observethat for the exampleto work the initial endowmentse” only need
to take,as functionsof o, countablymanyvalues.This impliesthat the example
alsoappliesto the casewith a countablenumberof states.
(5) Note that the examplecan be constructedso that the spotmarket equilibrium is uniquefor any z * 2 - 3( Z* = - z’). Therefore,thereis no room for the
useof (trivial) sunspotsfor the solutionof this problem.
(6) Thereis a non-genericityin the example:at the allocationa of Fig. 1 there
is a continuumof equilibria.It is not obviousat all that this canbe modified.It
maywell be that the counterexample
lacksrobustnessin this dimensionandthat a
genericityhypothesiscould play a role in generatingexistence(genericityhypothesesare, for example,quite helpful in the analysisof Mas-Cole11and
Monteiro, 1996).
Appendix
Herewe providethedetailsof the argument(in the proof of the Main Theorem)
that the correspondences
V( X, Z, h) dependmeasurablyon the history h. We
beginby recordinga generalfact aboutintegralsof measurable
correspondences.
Lemma. Let( X, F) be a measurable space, (Y, g, p ) a finite measure space,
and F : X X Y + RN a measurable correspondence.For each x E X, let S(x) be
the set of measurable selectionsfrom F(x, 9), and set
G(x)=
P(Y)w(Y):P-(x)
T/ten G : X -+ iI?!N is a measurable correspondence.
Proof. If F has a finite range(that is, there is a finite set 2 C RN such that
F( x, y) C 2 for all x, y), the argumentis entirely straightforward,andleft to the
reader.To derivethe generalcasefrom this one,choosea cubeC c 08N contain-
A. Mas-Cole& W.R. Zame / Jourrlal of Mathematical Economics 26 (1996) 63-84
81
ing the rangeof F; without lossof generality,we may assumeeachsideof C has
length 1. We may alsoassumewithout lossthat the measure/3 hastotal mass1.
For eachintegerk, partition the cubeC into disjoint subcubesC!, eachof side
2-‘, and let c: be the centerof C,? Let 7rk : C 3 C be the function that maps
eachof the subcubesCktto its centercf, anddefinea correspondence
Fk = 7~kF.
It is easily seenthat 1 ’ is a measurable
correspondence.
For each x E X, write
Sk(x) for the setof sectionsof _Fk(x, ) and Gk( x) for the corresponding
setof
integrals.SinceFk evidentlyhasa finite range,the integralcorrespondence
Gk is
measurable.
Note that if 9 E S(x) then 7~kqE Sk(x). Sinceeverypointof C lieswithin 2-k
of its imageunder7~’ and /3 has total mass 1, the integrals/q( y) dp( y) and
j7&p( y) ;ip< y) lie within 2- k of eachother. In particular,G(x) is containedin
the 2-“-neighborhoodof Gk( x). Conversely,it is easyto seethat if $ E Sk(x)
thenthereis a selection9 E S(x) suchthat ?Tkq- +; henceGk(x) is containedin
the 2-k-neighborhoodof G(x). It follows that the correspondence
G is the
intersection(over all k) of the correspondences
Gk; sinceeachGk is measurable,
so is G. 0
l
With this result in hand, we turn to the argumentthat V(x, z, h) depends
measurablyon the history h. We beginat the terminaldate;that is, ?= T. Our
goal is to expressV(x, z, h) in termsof countablymanyother correspondences
that are clearly measurable.To this end, begin by fixing a countabledenseset
vectorsin IF8
T’(T)anda countabledenseset q, , . . . of price
XI ,... of consumption
vectors in II%+
‘v) . For each history h and trader n, write w”(h) = e”(r, h) +
(div ‘z”)(t, h); these are the endowmentsin the Arrow-Debreu economy
8(x, z, h). For eachtriple a, d, c, of positive integersand eachpair E, S of
strictly positiverationalnumbers,considerthe set V(a, a’, c, e, 6, h) of vectors
h E RN’(T)satisfyingthe following conditions:
ICI x:’ - w”(h)]1 < e,
4‘. ( xi - w9 < c,
either qc ( xj - w,)
l
>, e or un(x$> < u”( xl:) + 6,
1h* 9gradientu*( x,“) 1< 6.
(The validity of theseconditionsfor every d saysthat A is almostthe vectorof
marginal utilities of an approximate equilibrium.) It is evident that
V( a, d, c, E, 6, Bt)is a measurable
set anddependsmeasurablyon h. Moreover,
we canexpressV( x, z, h) in termsof the variousV(a, a’, c, E, 8, h):
l
V(x, z, h) = n
U
U (IV(a,
e&J, E<E(),S<8, cl,c J
d, c, E, P, h),
wheretie first intersectionis takenover all strictly positiverational Ed,a,, and
the first union is taken over strictly positive rational E< q,, S < S,. This expressesthe corres>pondence
V( x, z, h) in terms of countablymany measurable
correspondences,
whenceV( x, z, h) is measurable.
82
A. i&s-Colell, W.R. Zame / Journal of
Economics 26 (1996) 63-84
Muthematical
We now use inductionto establishthe measurabilityof V( x, z, h) for 7 = t,
givenmeasurabilityfor ?= t + 1. The basicstructureof the argumentis the same
asabove,but the detailsare slightlydifferent.Fix consumptionandtradingplans
x, z prior to date 1. For eachhistory h and trader‘,!, write w”(h) = e”(t, h) +
(div’z”)(t, h) + d’IV* z”( t, h). For eachhistory h and date t consumptionand
tradingplans x(t), z(t), write X, Z for the consumptionand tradingplans that
agreewith X, z beforedatet andwith x(t), z(t) at datet. Write G( x(t), z(t), h)
for the setof vectorsv constructedfrom the correspondence
V asin the definition
of a first-order equilibrium. The definition, together with the lemma above,
guarantees
that G( x(t), z(t), h) is measurable
andclosedvalued.Fix a countable
denseset x, , . . . of consumptionvectorsin R~‘(T),a countabledenseset of asset
tradingplans &,, . . . in R NJ* (I), a countabledenseset q,, . . . of price vectorsin
R:(r)+J(r), and a countabledense set vl,. . . of marginal utility vectors in
RN(l(f+l)J(r+1).F;or positiveintegersa, d, b, b’, c d, and positiverationalnumbersE, 6, considerthesetV(a, d, 6, b’, c, d, E, S, h) of vectorsh E RN(J(r)+J(r)
suchthat
e ~cx,il-w”I <c,
@ IZs,n I < E9
e Iqr*KXo”,&9-w”II < E,
0 either qc [( xj, s,l> - wnl2 E or Wq - uf> K xz, &9 - (xi, &?I > -c,
l
l
0 vdE S-neighborhood
of G( x0, 56, h),
e &, E &neighborhood
of Z”(t, h).
(The validity of theseconditionsfor every d, 6’ saysthat A is almostthe vector
of marginal utilities of an approximateequilibrium.) It is evident that
V(a, d, b, b’, c, 4 e, 6, h) is a measurable
set,and dependsmeasurablyon the
history h. Moreover,
V( %
Z,
h) = n
u
u
n V(u, d, b, b’, c, d, 6 6, h).
C(),Sf)
e< r&k S” u,h,c,du’,h’
Hence,V( n, z, h) is measurable,
anddependsmeasurablyon the history h. This
completesthe proof. c3
Acknowledgements
We are grateful to Darrell Duffie and Paulo Monteiro for many useful
conversations,
to M. Hellwig who at the heightof SanMiniato first exposedone
of the authorsto this problem,to S. TakekumaandA. Yamazakifor forcing us to
work harderon a measurabilityissue,andto the NationalScienceFoundationfor
financialsupport.
A. Mas-Colell, W.R. Zame / Journal of Mathematical Economics26 (1996) 63-84
83
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