Download NOVEMBRE 1990 MARKETS WITH COUNTABLY MANY PERIODS

NOVEMBRE 1990
.
MARKETS WITH COUNTABLY MANY PERIODS
•
Mon~que FLORENZANO
CNRS-CEPREMAP
N°
9019
1
MARCHES SUR UNE INFINITE DENOMBRABLE DE PERIODES
Résumé
•
Un des premiers résultats d'existence de l'equilibre pour une économie
avec une infinité de biens est celui où Peleg et Yaari (1970) considèrent
une économie d'échange définie sur l'espace des suites réelles ; les
consommateurs y sont supposés avoir des préférences totalement préordonnées, convexes, continues et strictement monotones, tandis que l'offre
totale de ressources est supposée stictement positive.
Une abondante littérature a depuis été consacrée aux économies définies
sur un espace de biens de di~ension infinie. La théorie des espaces de
Riesz et une hypothèse dite de "propreté uniforme" des préférences et/ou de
la production se sont avérées jouer un rôle crucial dans les théorèmes
d'existence.
L'objectif du papier est de trouver l'hypothèse cachée de propreté uniforme à l'oeuvre dans Peleg et Yaari et, grâce aux espaces de Riesz, de généraliser les résultats aux économies intertemporelles admettant l'incertitude à chaque période.
Espaces de
clés
Mots
-
symétrique
Economie
Riesz
localement
-
d'échange
convexes
Equilibre
-
Systéme de Riesz dual
intertemporel
-
Préférences
uniformément propres .
•
MARKETS WITH COUNTABLY MANY PERIODS
Abstract
The existence of equilibria is proved in a stochastic
finite number of agents and countably many periods.
words
Key
Riesz
dual
:
Locally convex-solid
systems
Exchange
Uniformly proper preferences.
JEL
:
021
market with a
topological vector lattices
economy
Intertemporal
-
Symmetric
equilibrium -
2
MARKETS WITH COUNTABLY MANY PERIODS
Monique FLORENZANO
CNRS-CEPREMAP, 75013 Paris, France
The
existence
stochastic market with a
of equilibria is proved in a
finite
number of agents and countably many periods.
1. Introductio n
This note is intended to reconsider, at the light of some recents works,
an equilibrium existence result obtained by Peleg and Yaari (1970) for an
exchange economy with IR'° (the space of all real sequences) as commodity
space, IR'° as price space and no explicit commodity- price duality.
We first restate the topological vector lattice properties of the
commodity space to be considered in this model. A limited attempt in this
direction is made by Besada et al. (1988) who introduce Kothe perfect
spaces
as commodity spaces which arise in a natural way in connection with
go further and do a similar
constructio n in the case where, at each period, the consumption choice of
each agent may be stochastic. The resulting commodity- price duality is
described by what is called a symmetric Riesz dual system by Aliprantis et
initial
the
endowments
the
of
agents.
We
al. (1989),
We
then look
establish
the relevance
pointed
satisfied in a
it
is
at the
out
by
hidden uniform properness assumptions which would
of this vector lattice theoretic constructio n. As
Aliprantis et al. (1989), uniform properness is
subspace of the commodity space,conve niently topologize d;
in this subspace, an equilibrium existence theorem can be deduced from MasColell's theorem (1986). The main result of this note is to show that under
to
assumptions
similar
equilibrium
is associated
with prices in its dual.
the
ones
used
an equilibrium
by
Peleg
in the
and
Yaari,
to
this
initial commodity space
3
2. The economic model
and Burkinshaw
Aliprantis
period.
time
the
denotes
used
terminology
and
notations
The
in
et al. (1989). The
(1978), Aliprantis
duality
commodity-price
The
note are borrowed from
this
at
index t
period t is
represented by a symmetric Riesz dual system <Et, E;>.If xtE Et and PtE E;,
we writè Pt·xt for <xt, Pt>.
We consider m infinitely lived agents i = 1, ... ,m, each of which has at
m
oo
L
oo
wi E nE;
initial endowment Wi= (w;) E nE;. Let w =
t=l
i=l
t=l
be the total endowment.
its
disposal an
00
00
P
= {p E
nEt
L lpt l .wt
t=l
t=l
<
+oo }
and
00
00
!\(P) = {x E n Et
will
The
I
L lpt 1. lxt 1 <
+oo
for all p
E
P}
t=l
t=l
be respectively the price space and the commodity space of our model.
commodity-price duality is represented by the bilinear form: <x,p> =
00
L
Pt .xt. If x E /\(P) and p E P, p.x is the value of x reckoned at
t=l
the prices p = (pt).
The construction of /\(P) makes it natural to consider the topology T
p.x =
defined on /\{P) by the Riesz seminorms
00
L
lpt 1. lxt 1 , p E P •
t=l
Similarly, r'is the topology defined on P by the Riesz seminorms
Pp (x) =
00
L
lpt 1. lxt 1 , x E /\{P) .
t=l
As it is proved in the propositions 1 and 2 of the appendix, <A(P),P> is a
d ual pair; T ( resp. T t) is a Hausdorff locally convex-solid topology on
Px(p) =
!\(P)
(resp. P), consistent with
the duality.
Moreover, let a(/\(P),P) and
a{P,/\{P)) be the weak topologies associated with the duality; every orderinterval of /\(P) (resp.P) is a(/\(P),P){resp. a(P,/\{P)))-compact. In other
words, <A(P),P> is a symmetric Riesz dual system.
As Peleg and Yaari (1970), Besada et al. (1988), we consider an exchange
economy, to be called i:E. A commodity bundle is an element of /\(P)+ .Each
agent
i
has
indifference
reflexive,
an
initial
relation
~i
transitive and
bundle
on
wi
> 0
/\(P)+
complete. In
in
The
/\{P)+
relation
and
a preference or
~i
is assumed to be
addition, the initial bundles and
preference relations are required to satisfy the following assomptions :
4
X
For i = 1, ... ,m, if x,y
monotonicity
1) Strict
E
A(P)+ and x >y, then
>iy.
{ y
i = l, ... ,m,
For
2) Convexity
x
each
for
E
A(P) +
the
set
A{P) + 1 Y ~ix } is convex.
E
the sets
For i = 1, ... ,m, for each X E A(P) + •
r- continuity
{ y E A(P}+ I y ~ix } and { z E A(P}+ 1 X ~iz } are both r-closed in A(P}+.
The r-continuity assumption is a myopy assumption on preferences. A
fourth assumption corresponds to the positivity of totai supply with an
additional requirement which is automatically satisfied if each Etis an
3)
Euclidean space:
4) For every period t, wt is an order-unit of Et and Et= (Et)~ (the order-
continuous dual of Et).
yi{p} = { x E A(P)+ 1 p.x ~ p.wi}
c5i{p} = { x E A(P)+ 1 p.x < p.wi }.
Let
(resp. a quasi-equilibrium) for economy c8 is an
A competitive equilibrium
-m -
-1
-
-
=
(x 1 •••• ,xm))
(m+l)-tuple (x , ... ,x ,p} such that p E P, p
m
m .
a)
L xi= L wi
i=l
b)
budget set of i and set
be the
(attainability of
X
~
0 and
i=l
and y >i xi implies
each i = 1, ... ,m
xi E yi(p) for
y$ yi(p} {resp.
y$oi{p}).
3. Existence theorem
A{P} is Dedekind-complete. Let Aw be the principal ideal of A{P} generated by w,
Aw = { x E A(P) 1 there exists À> 0 with lxl ~ ÀW }.
Recall that under the Riesz norm
llxll 00 = inf{
.
Aw is an
by
.
> 0 1 lx 1
À
~ ÀW }
'
AM-space with unit w. The norm dual of (Aw, 11.11cc) will be denoted
AC:,.
According
to
Mas-Colell
(1986),
an
economy
is
said to satisfy the
closedness condition whenever for every sequence of attainable allocations
(x 1 ·v, ... ,xm,v) which satisfies xi,v+l ~i xi,v for all v and all i=l, ... ,m,
there exists another attainable allocation (x 1 ,
•••
,xm) satisfying xi~ xi,v
i=l, ... m. If viand viare respectively a vector and a 0neighborhood of the commodity space L, ~i is said to be (vi, Vi)-uniformly
proper if for all x EL+, x + ÀVi + z EL+, À> 0 and z E ÀVi imply
for
all v and all
x + ÀVi + z >i x.
w is
said
desirable
for every i = 1, ... ,m, if x EL+
5
x +
implies
>i x for all œ > O. In a topological vector lattice, if an
a!.()
condition,
the closedness
satisfies
endowment
total
desirable
a
and
preferences
proper
uniformly
continuous,
monotone,
convex,
with
economy
exchange
has a quasi-equilibrium with prices in the topological
then it
dual of the commodity space.
In order to apply Mas- Colell's existence result, we will first consider
©w, the same economy than ©, but restricted to the commodity space Aw. It
is worth noticing that © and ©w have the same attainable allocations.
p)
quasi-equilibriwn. (x,
Proof.
It
the asswrrptions
Under
1.
Proposition
E (A:)m x A~+ with
assumption
from
follows
2)
p (w)
= 1.
w is
that
1)
©w has a
3),
and
1),
desirable for every
also well known that, under assumption 1), every point of
i=l, ... ,m .It is
1
int (A:) ( the Il. 11-interior of A:), in particular - w, is a properness vector
m
every ~i in ©w. As it is pointed out by Aliprantis et al. (1989), the
for
the order-convergence
11.11 00- convergence implies
the order-convergence
topology,
continuity
of
satisfies
the
it
a Lebesgue
is
So the 11.11 00-
assumption 3), Finally, ©w
follows
the
from
easily
3) and the (a(A(P}, P))m-compactness of the set of all
2) and
assumptions
from
as
condition
closedness
T
r-convergence.
implies the
©w follows
in
~i
each
; since
attainable allocations. The conclusion follows from Mas-Colell's theorem.
D
Coroiiary.
Let
p be the normal component of
p
(x,
p)
is a quasi-equili-
briwn. of ©w and p(w) > O.
pE
(A~)+ c (A;)+ ,so by the Riesz decomposition theorem, there is a
the disjoint
unique decomposition of p, p E (Aw);: and q E ((Aw);:)d
=
complement of (Aw);:, p ~ 0, q ~ 0, such that p = p + q. We first remark
that (Aw);: separates the points of Aw; indeed, as it is shown immediately
after the Proposition 2 of the appendix, P c (A{P}};; c (Aw);: and P
separates the points of Aw • Then let Nq= { x E Aw I lq 1 (x) = 0 } be the
Proof.
null
q in
ideal of
Aw.It
follows from
Theorem 2.2 in Aliprantis et al.
(1989) that Nq is order dense.
Then
that
suppose
for
some
xi >i
i,
xi
0
in
. -i
(xia) c Nq, xia ~xi.For large enough œ, xia >i x
(p
+
i
such
©w· There exists a net
-
-
.
.
and p(xia) = p(xia)
~
q)(wi) ~ p(wi}, By passing to limit, p(xi) ~ p(wi). Since there exists
that
(p
+
q)(wi) > 0,
it
is
easily
deduced
from
the
strict
6
monotonicity
of preferences
~
that p
O.
From the strict monotonicity of
.
m .
we get that for each i, p(:i=1') 2:: p(wi). Finally, since
=
Ix'I,
preferences,
i=l
Lm wi,
= -i
p(x ) = p(wi) holds for all i.
i=l
Recall now that
Theorem 6.4
(see
Aw is a Banach lattice and thus that A~ is a band of A:
in Aliprantis
w is
p E A~; since
in
the
and Burkinshaw
li.li- interior
of
(1978)). It
follows that
A_;,, p(w) > 0 and p can be
re-normalized so that p(w) = 1.
D
Proposition
(1989),
1 is
nothing else
reproved there
construction
(1972).
To
used in
go
in order
than Theorem
to keep
5.6 in Aliprantis et al.
this note
self-contained. The
the proof of the corollary is classical since Bewley
further,
we
need
to
exploit Assumption 4), through the
proposition 3 of the appendix.
Proposition 2.
Under
the
asswrrptions
1),
2),
3)
and 4), (x,
p)
is an
equitibrium of <5 and pis a strictiy positive eiement of P.
Proof.
From
Proposition 3 in the appendix, we know that Aw is order dense
(x, p) is a
quasi-equilibrium of <5w as in the corollary of Proposition 1, p E P.
Suppose now that for some i, xi >i xi in <5. There exists a net
(Xia) C Aw, Xia: xi. For large enough œ, Xia >i xi and p(xia) = p .Xia 2::
p(wi) = p.wi. By passing to limit, p.xi 2:: p.wi, which shows that (x, p} is
a quasi equilibrium of <5.
Since p. w > 0, there exists i such that p. wi > O. For all z > 0 in
and
thus
r- dense
in
À(P)
and
that
P = (Aw);.
Then if
-i
-i
À(P), by the strict monotonicity of preferences, X + z >i X and by the T-i
-i
=
=
We get
continuity
of
preferences,
p=
X
(x + z) > p . wi =
p
:i
p .z > 0, which shows that pis strictly positive. Hence p= . wi > 0 for all
. .
.
i and
(x, p) is an equilibrium.
D
Proposition
of
2 is
the existence
(1988).
a straight generalization (as to the commodity space)
statements in
Assumptions 1-4
Peleg and
Yaari (1970)
or Besada et al.
on the economic model are exactly the assumptions
made in these two papers. By adding an explicit desirability assumption for
the
total
assumption
endowment
w, it would
on preferences.
be
easy
to
The quasi-equilibrium
weaken
the monotonicity
price would not need be
7
assumption would be necessary to
an irreducibility
positive and
strictly
guarantee the existence of an equilibrium.
assumption
4) allow
A(P)
space
commodity
together
the
with
to consider
at each period a commodity-price duality
the idea
of a stochastic choice at each period. A
<L00, L1 > which captures
the case where at each period, the commodity space
application is
simpler
the
of
construction
The
is an Euclidean space, of which the dimension may depend on time.
Appendix
Let <A(P),P>,
and
T
T'
be defined as in the beginning of Section 2.
resp. T') is a Hausdorff locally
Proposition 1; <A(P),P> is a dual pair.T
convex-solid topology consistent with this duality.
00
00
Proof. Let cp (resp. 4>' )denote the set of all elements of nEt (resp. nE;)
t=l
t=l
and Pare
with only finitely many non-zero coordinates. Obviously A{P)
00
00
nEt
spaces, more precisely ideals of
Riesz
and
nEt
containing 4> et 4>' •
t=l
t=l
00
properties
separation
The
<x, p> = p.x =
bilinear form
of the
L Pt .xt
t=l
follow from the last remark.
00
Each
i.e.
a
Px(P)
~
L lptl· lxtl
seminorm Pp (x) (resp.px{p)) =
seminorm
Riesz
Px(q) ) ;thus r
lxl
( resp.
~
t=l
~
lyl
r')is
Pp(x)
~
is obviously monotone,
Pp(Y)
(resp. lpl
~
lql
~
a locally convex-solid topology on
A(P){resp. P).
clear that P c (A{P),r)', the topological dual of A{P).To
should be
It
the converse
prove
with
coincides
seminorms : xt
Pt E E;
~
the
inclusion, let
absolute
weak
that on
us remark
each factor
Et, r
lal(Et,Et) generated by the
topology
lptl· lxtl· Hence, if f E (A(P),r)' ,for each t there exists
such that
f(O, ... ,O,xt,O, ... ) = Pt·xt.
Since for
all x in A{P),
00
(x 1 ,x2
that
,xt,O, ... ) r-converges to x, f(x) = LPt·xt· It remains to prove
t=l
p = (Pt) E P. lfl {0, ... , 0, xt, 0, ... ) = lpt l .xt, so that If 1{w) =
, •••
00
L lpt i .wt < +oo. From a similar proof, it follows that A(P) = (P ,r')'.
t=l
Finally, if a(A(P) ,P) and a{P,A{p)) are the weak topologies associated with
8
duality <A(P),P>, remark that a(A(P),P) c
the
proves that
T
and a(P,A(P)) cr', which
and r' are Hausdorff.
T
D
Proposition
2. Each
order-interval of A(P) (resp. P) is a(A(P) ,P) (resp.
a(P,A(P))-compact.
Proof.
é
If u > 0 in A(P), let [-u,+u] be an order-interval of A(P). For any
> 0, Pu(P) <
and x E [-u,+u] imply
E
00
L Pt .xt
This
shows
compactness
now
00
L
L
lpt .xt 1 ::;;
lpt 1. lxt 1 ::;; Pu(P) < E.
t=1
t=1
r'- equicontinuity of [-u,+u]. The a(A(P),P)-relative
t=l
the
1 ::;;
of [-u,+u]
that each
00
follows,from
the Alaoglu-Bourbaki theorem. Recall
<Et,Et> is a symmetric Riesz dual system. Each [-ut,+ut] is
a(Et,E;)-compact. It follows easily that I-u,+u] is a(A(P},P)-closed.
The a(P,A(p)}-compactness of order-intervals of Pis proved in a similar
way.
D
',,
Applying
Theorem 22.1
of Aliprantis
and Burkinshaw(1978}, we see that
A(P) (resp.P) is Dedekind-cornplete and, (resp. ,') is a Lebesgue topology.
If
(A(P)};; (resp.
P;;)
linear
order-bounded
denotes
forms
the
on
collection
A(P}
(resp.
of all order-continuous,
P),
P
c (A(P));;
(resp.
A(P) c P;;).
Assume now, as in assumption 4), that for every t, wtis an order-unit of
Et and that E; = (Et)~.Recall that Aw is the principal ideal generated by w
in A(P).
Proposition
3. Under
assumptions 4),
Awis
order dense in A(P) and
(Aw)~ CP C (A(P))~ C (Aw)~.
Proof.
From the
first part of the assumption, it follows that each factor
Et can be considered as a Riesz subspace of Aw. For all x in A(P},
(x 1 ,x2 , ••• ,xt,O, ... ) î x. This proves the first part of the statement.
To
second
prove the
part of
first inclusion,
the assumption,
let f be any element of (Aw)~. From the
one deduces
that there
exists PtE (Et)~
00
= E~ such that j(x) = LPt·xt and that p = (pt} E P.
t=l
The second inclusion was proved above; the third one is obvious.
D
~
9
References
economies
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C.D.,D.J.
Aliprantis,
with
countable
a
number
Analysis and Applications 142,
and
C.D.
Aliprantis,
Burkinshaw,1978,
O.
New York and London).
Besada,
Esteves and
M.
countably many commodities,
Bewley,
T.F.,
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Këthe,
1969,
G.,
Mas-Colell,
lattices,
Peleg,
B.
A.,
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Herves,
M.
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Solid
Riesz
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514-540.
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