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What
is
Perfect Competition?
A Ido Rustichini
Northwestern University
Department of Economics
Nicholas
C
Yannelis
University of Illinois
Department of Economics
\Ud Library of the
APR
University
i
1991
c
of yriwria-Charopalgr:
•
Bureau of Economic and Business Research
College of
Commerce and Business
University of Illinois
at
Administration
Urbana-Champaign
Digitized by the Internet Archive
in
2011 with funding from
University of
Illinois
Urbana-Champaign
http://www.archive.org/details/whatisperfectcom1715rust
BEBR
FACULTY WORKING PAPER NO. 90-1715
College of
Commerce and
University of
Business Administration
Illinois at
Urbana-Champaign
December 1990
What
is
Perfect Competition?
Aldo Rustichini
Northwestern University
and
Nicholas C. Yannelis
University of Illinois at
Urbana-Champaign
Department of Economics
ABSTRACT
We
provide a mathematical formulation of the idea of perfect competition for any economy with
many agents and commodities. We conclude that in the presence of infinitely many
commodities the Aumann ( 1 964, 1 966) measure space of agents, i.e., the interval [0, 1 endowed with
infinitely
]
model the idea of perfect competition and we provide a
characterization of the "appropriate" measure space of agents in an infinite dimensional commodity
space setting. The latter is achieved by modeling precisely the idea of an economy with "many more"
Lebesgue measure,
is
not appropriate to
agents than commodities. For such an
economy
The convexity assumption on preferences
is
the existence of a competitive equilibrium
not needed
in
the existence proof.
is
proved.
What
is
Perfect Competition?
Aldo Rustichini and Nicholas C. Yannelis
Abstract.
We
provide a mathematical formulation of the idea of perfect cominfinitely many agents and commodities. We conclude that in the presence of infinitely many commodities the Aumann (1964,
1966) measure space of agents, i.e., the interval [0,1] endowed with Lebesgue
measure, is not appropriate to model the idea of perfect competition and we
provide a characterization of the "appropriate" measure space of agents in an
infinite dimensional commodity space setting. The latter is achieved by modeling precisely the idea of an economy with "many more" agents than commodities. For such an economy the existence of a competitive equilibrium is proved.
The convexity assumption on preferences is not needed in the existence proof.
petition for an
economy with
Introduction
1.
Perfect competition prevails in an
economy
fluence the price at which goods are bought
In order to
mann
is
no individual can
if
and
in-
sold.
model rigorously the idea of perfect competition, Au-
(1964, 1965, 1966) assumed that the set of agents in the
economy
an atomless measure space. As a consequence of the non-atomicity
sumption, each agent in the economy
is
negligible
and therefore
will
as-
take
prices as given.
A
special feature of the
modities in the economy
his
model
is
Aumann model
is finite.
number
that the
is
In particular the
of com-
commodity space
the positive cone of the Euclidean space
R
n
.
This
is
quite
important because given the fact that the measure space of agents
atomless and the dimension of the commodity space
that the convexity assumption on preferences
is
is finite, it
is
used to convexify the aggregate
fixed point
set
Lyapunov
theo-
and make the standard
is
quite different
when
the
commodity space
not finite dimensional. Indeed, the Lyapunov theorem
dimensional spaces
[see for
important to note that
main
all
fails in infinite
instance Diestel-Uhl (1977)] and one loses
the nice convexifying effect on the aggregate
stitute the
turns out
argument applicable.
However, the situation
is
demand
is
not needed to prove the
existence of a competitive equilibrium. In particular, the
rem
in
the basic results of
technical tools to
demand
Aumann
set.
(Here,
it
is
(1965) which con-
model the idea of perfect competition
23
fail in infinite
dimensional spaces
[see for
instance Rustichini (1989) or
Yannelis (1990)].)
Moreover, as
it is
presence of finitely
known from
the work of
many commodities,
Aumann
(1964), in the
core allocation characterize com-
Aumann
petitive equilibrium allocations. However, contrary to the
core
equivalence theorem, in infinite dimensional commodity spaces in general,
core allocations do not characterize competitive equilibrium allocations.
In particular, Rustichini- Yannelis (1991) showed that even
sure space of agents
the mea-
atomless, preferences are (weakly) continuous,
is
convex, monotone, and
strictly
if
initial
core equivalence theorem ceases to be
endowments
Does
true.
strictly positive, the
this
then suggest that
the nonatomicity assumption
may
commodity spaces
model the idea of perfect competition?
It
may
in order to
not be enough in infinite dimensional
be useful, before we proceed, to put aside for a
mathematical nature of the problem, and look more closely
strictly
economic significance of the nonatomicity condition, and
in
the case where the commodity space
tion,
moment
is finite
any subset of the space of agents, that
positive
measure
as a "critical
On
is
any
coalition,
We may
at the
implications
By
dimensional.
in order not to be insignificant.
mass" condition on any
its
the
defini-
must have
think of this
coalition.
the other hand, thanks to the nonatomicity condition, any coali-
tion of nonzero
measure
positive measure) which
will
is
have a
set of possible subcoalitions (still of
so large that
it
makes any
collusive behavior
arduous.
The previous
discussion suggests that one should not look for a char-
acterization of perfect competition (or
more generally of a
strategic situ-
ation where each single player has an insignificant influence) in the space
of agents or players by
itself,
but in the relation of the dimension of the
measure space of agents and the dimension of the commodity (or strategy)
space.
The concept
of dimension, of course, has to be given a rigorous
formulation. Simply and informally stated, the characterization of perfect
how
with how
competition that we introduce in this paper does not just specify
many agents there
many commodities
tion which
is
are,
but
how many
agents (or players) deal
(or strategy choices).
introduced in this paper
is
To put
it
that the
differently, the
economy needs
assumpto have
"many more" agents than commodities. Such an assumption (see Sec24
tions 3
and 6
for
a rigorous definition)
stronger than the nonatomicity
is
condition of the measure space of agents in an infinite dimensional com-
modity space setting and
in finite
equivalent to the nonatomicity assumption
dimensional commodity spaces.
assumption
The
it is
in the
In essence, this
Aumann model which
drives his results.
main contribution of this paper
lation of the idea of perfect competition,
to those of
Aumann
for separable
is
to provide a rigorous formu-
space.
As
it
measure space of agents,
measure
is
may
i.e.,
satisfy the condition that the
rest of the
definitions
main
interval
[0, 1]
Aumann
(1964, 1966)
endowed with Lebesgue
not "large enough" to model the idea of perfect competition
in the presence of infinitely
The
bigger than that of the
is
be expected, the
the
results
Banach spaces. Moreover, we charac-
dimensionality of the measure space of agents
commodity
the hidden
and derive the analogous
measure space of agents which
terize the
is
1
paper
many commodities.
is
organized as follows:
Section 2 contains
and some mathematical preliminaries. Section 3 contains the
result, i.e., the integral of
a Banach- valued correspondence
is
convex
provided that the dimensionality of the measure space
is
dimensionality of the Banach space.
interpreted as the
convexifying effect on the aggregate
This result
demand
the measure space whose dimensionality
space of commodities.
Some important
collected in Section 5. Section 6 uses the
is
set.
is
greater than the
Section 4 characterizes
bigger than that of the
corollaries of the
main
main
result of this
Banach
result are
paper to prove
the existence of the competitive equilibrium. Finally, Section 7 contains
an application of our main theorem to the problem of the existence of a
pure strategy Nash equilibrium in games with a continuum of players and
with an infinite dimensional strategy space.
It
should be noted that several authors have already
made remarks
of this
example, Mertens (1990), Mas-Colell (1975), Gretsky-Ostroy
Ostroy-Zame (1988)], but they did not provide a precise mathematical
modeling of the idea of "many more" agents than commodities. The only
exemption is the work of Lewis (1977) which uses nonstandard analysis. In
particular, she makes use of a nonstandard analog of the Lyapunov theorem
nature
[see for
(1985),
proved by Loeb (1971).
25
2.
and Preliminaries
Definitions
The space
X
is
a Banach space over the
The norm
R
field
assumed
to be separable.
by
Unless otherwise specified, the topology on
||x||.
of real numbers;
it is
X will be denoted
X will be the norm
of an element x G
topology.
A special importance will be assumed by a fixed weak compact subset
K of X. We recall that (Dunford-Schwartz [1958], V. 6.3) the weak
topology of K is a metric topology. As usual for a set A C X we denote by
co A and coA the convex hull and the closed convex hull of A, respectively.
For a subset A C X we denote by P{A) or 2 A the set of all nonempty
subsets of A, and PAA) the set of all closed subsets of A.
,
The Banach space topology on X induces a natural structure of
measure space on it if we denote by P(X) the set of norm Borel subsets
of
X
(and for any
A
G 0(X), /3(A) are the Borel subsets of A).
denotes the Borel cr-algebra generated by the weak topology of
from Masani
We now
[1978],
Theorem
T
proceed to describe our measure space.
to be complete and finite.
/z)
such condition
will
X
,
then
we have (3(X) = P W (X).
2.5(b),
able space, with a a-algebra r, and a measure
(T, r,
W {X)
If
We do
we
fi;
be a measur-
will
shall always
assume
not assume nonatomicity, since
be contained in an assumption which we
will
introduce
later.
//^(/i)
is
the space of real valued, measurable, essentially bounded
functions defined on T.
is
For any
naturally defined; and so
is
E
f, the
G
L^
the space
measure space (E,te ,h e )
E (fj.) = {/ E — R, / is
:
te measurable and /z^-essentially bounded}.
-
In this paper a set-valued function (or correspondence)
to be a
map from T
denote the graph of
to a set of all
F
by
GF -
the
nonempty subsets
{(t,x) G
T
x
X
correspondence F, the new correspondence co
(co F)(t)
=
F
:
of
X
F
,
is
defined
P{X).
We
x G F(t)}. For a given
and co.F are defined by
co(F(t)), and analogously for co.F.
Various notions of measurability are discussed in Himmelberg [1957].
In the case where the measure space
T
ditions which are satisfied in our case)
and
out to be equivalent [Himmelberg [1957],
following:
26
complete,
is
F
is
X
is
Souslin (con-
closed valued they turn
Theorem
3.5].
We
adopt the
Definition. The correspondence
if
the graph of F,
T
{(t,x) G
product measure space r
K
C{K)
this last space is (isometrically
of
all
space (in the weak topology).
known [Dunford-Schwartz
well
It is
an element of the
9.1 J.
K
be the space of continuous functions on
space.
,s
operation preserve measurability
77ie co
now a compact Hausdorff
is
x G F(t)}
:
said to be measurable
is
:
x A'
® /?( X).
[Himmelberg [1915], Theorem
F T — P(X)
and C(K)
,
Theorem
[1957]
m
its
Let
dual
IV. 6. 3] that
isomorphic to) the space rca(A'), the space
regular, countably additive real valued set functions, defined
on the
a-field of the Borel (norm) subsets of K\ rca + (ii
r
) are the nonnegative
For any Borel subset A, rca(A) and rca + (A) are
elements of rca(A').
A
defined in the natural way. For any Borel subset
of rca + (>l); that
the unit ball
MA =
If fi(A)
=
1,
we
refer to
/x
:
at
DA
,
is
the subset of
(6 a ,<f>)
For any
m
G
=
MA
<f>(a)
rca(ii'), its
1}.
we
set of
for every
norm
Dirac measures on A,
the set of
of elements S a
\m\
MK
.
More
<f>
with a G A, such that
,
G C{K).
the total variation.
is
precisely
M^-valued functions / which
we introduce
satisfy:
(i)
G
j|/||
L^
Ionescu-Tulcea, A. and
L£faMF
=
C,
(1969), p. 99].
Also we
L£(n,MK ) u(t)
L£(p D) = {ie L£(n,M K K0
)
{7 e
:
)
t
L£(p,D F = {je L£(vl,D)
)
K
:
u(t)
e
Also for u G
M, F
where \f
tne indicator function of the set F.
IS
a closed subset of
27
,
we
T
We
<f>
shall
and with
(/i,M /<-), the
L°°(fi); (ii)
a measurable (real valued) function for every
(f(t),<f>) is
Among
single out those having unit
consider on the following measurable functions defined on
values in the space
MA
one point:
Definition. For any Borel subset A,
denoted
<
y.(A)
as usual as a probability measure.
the probability measures over a Borel set
mass concentrated
K, we denote
is:
G rca + (A)
{fji
of
G C(K);
t
t-+
[see
let
G
M
e
D
F(t) /i-a.e.}
^-a.e.}
D F{t)
define (v,
If
/x-a.e.}.
F) = jK xxp>(x)dv(x)
A is a Banach space,
we
will
denote by L x (n,X) the space of equivalence classes of X-valued
Bochner integrable functions / :T
ll/H
The
=
X
—
jf ||/(0II^(0-
F T—
integral of the set-valued function
/ F(t)dfjL(t)
=
2
:
x(t)dfi(t)
I
normed by
:
X
is
x(t)
£ F(t)
or
Fdfi.
as usual:
/J-a.e. I
.
J
We
will
F
denote the above integral by J
/
The Main Theorem
3.
(Convexifying Effect on Aggregation)
The
Lemma
lemmata
following simple
MA
1. Let v £
DA
\
.
be used
will
Then
later.
there exist two measures
v± which
satisfy:
v±
Remark.
=
v
±
Note v2
Proof.
Since v
A, 5 and
B
is
v2 £
is
MA
^
v2
with:
not an element of
0,
MA
v2 £ rca(A).
.
not Dirac, there exist two disjoint Borel subsets of
'»
say, such that
<
i/(5)
< u(B) <
1.
(1)
We
denote with ^(5), v B the restriction of u to the two set 5, B. Define
now 6 = u(S)/i/(B): from (1) above follows that 6 G (0, 1]; also define:
v2
= — v$ +
^B
(clearly
£/
2
7^
computation to check that u ± G
Lemma
MF
r
t
\
\
2. Let
DF
/
t)
E
^
an d ^±
MA
^
t
—
"
i
u2-
^
1S
now an
eas y
.
£ r with /z(£) >
for every
v2 {i) G rca(F(0),
0);
and v G L£((*,MF ) with
0,
£ E. Then there
an d v ± v2 £
28
exists a
MF ^,
fi-a.e.
i/(t)
G
measurable function
Proof.
The
function v2 will result as a measurable selection from a
We
correspondence.
£ E; also 6
t
Now
MF
= max{0 >
Define now: 6
for every
recall
is
:
t
B{v(t),6)
ric).
The
=
the correspondence N(t)
MK
C
M
F(t) }.
From Lemma
B(~}(t)),6(t))
111-41]
result follows
d
>
is
measurable
[see
with complete values, and takes
a separable space (because this space
,
1,
.
a measurable function.
Castaing and Valadier, (1977),
values in
MK
a u;*-compact, convex subset of
is
\
,
compact and met-
is
from the previous lemma and the Kuratowski and
Ryll-Nardzewski (1962) measurable selection theorem.
now
Recall
(Hamel) basis
we denote,
that for any vector space over the real field an algebraic
exists.
The
cardinality of
dimX
any vector space X,
for
any Hamel basis
is
the same, and
the cardinality of any of
its
bases.
We
introduce the following condition.
(Al) For the pair ((T, r,/x),X),
E
if
belongs to r, ^{E)
>
0,
then
dim L E )00 (/x) > dimX.
Remark
This
1.
is
the condition that there are
"many more" agents
than commodities. In section 4 we shall characterize the measure spaces
that satisfy the above condition.
Remark
LE
ooifi) to
This
is
The assumption
2.
X
map T from
G X
T(f) =
implies that for any linear
there exists a function
/ such that / ^
0,
the condition which will be actually used in the proof of the
theorem and
main
its corollaries.
X
is finite
measure space (T,r,fi)
is
non-atomic implies the assumption (Al) above.
Of course, the reverse
is
also true.
Remark
When
3.
dimensional, the assumption that the
Hence the Main Theorem below, as
well as the corollaries of the section 5 imply the analogous results in
Aumann
(1965).
Remark
We
4.
and so (since every
empty
interior)
Hamel
basis.
it
On
recall here that
finite
a Banach space
is
dimensional proper subspace
cannot have,
if it is
infinite
of second category
is
closed
and with
dimensional, a countable
the other hand such a space, as being a separable metric
space, has cardinality at most K
hypothesis) d'imX
=
N
.
We
.
29
conclude (under the continuum
Remark
Readers familiar with the proof by Lindenstrauss (1966) of
5.
Lyapunov's convexity theorem, or Knowles' theorem
or Kluvanek- Knowles (1975)] will recognize the
Main Theorem.
and let F T —* 2 K
:
sume
Let
K
[see
Knowles (1974)
main idea of the
be a weakly compact,
proof.
nonempty subset of
X
be a measurable, closed valued correspondence. As-
Then
that the pair ((T, r,fi),X) satisfy (Al).
l'-l coF.
Proof.
Fix
t
€ T. Define
7(M F(t) ) =
{x G
X
:
x
=
/
with
adfi(a),
//
G
MF^}.
Clearly
coF(t)Cl(M F{t) ).
Also, since
linear,
MF
,
\
t
is
iu*-compact convex, and the
and w* to tu-continuous,
I(MF
)
map
/x
— fk adfj.(a)
is
closed and convex, and therefore
is
coF C I{M F ).
Clearly
/ Fdfji=
{xeX :x=
Jt
Obviously,
sion.
fT Fdfi C JT
We
shall
now prove
this will
,
t
\) it
coFdfiC
We
F(t))dfi(t),
shall
now prove
follows that
J I(MF(t) )dfi(t).
that
J
and
coFd}i.
From coF(t) C I{MF
J
f (6(t),
Jt
m
I(M
)dn(t)
C
J
Fd/x,
complete the proof of the theorem.
30
6e D F }.
the converse inclu-
For any x £ JT Fdfi, define
Hx
The
set
shall
now prove
by ext# z ) are contained
Fdp., this will
for
)
:
J
(v,F)dfi
=
x}.
Hx is convex, and tu*-compact [see Castaing and
We
V-2].
= {ue L^(n,M F
DF
g"
From Lemma
,
2
in Z™"(/z,
DF
).
From
v>
2
t
t j,
the representation formula
we
;
i.e.,
that for a set
E
derive the existence of a rca( if) valued r-measur-
sucn tnat
^ ^(0
KO
(fv2l F)dn(t)
Now
extend / to a function / £
=
0;
^<x>(fi)
0.
||/|| LjSiOo0l)
>
0.
settm g /(0
=
0» for
Now
every
t
from
£ E.
conclude:
vx
so that
i/j
is
±
/z/ 2
£ #,
i/j
+
/i^ 2
^
/j/2
is
X is the commodity space.
be a separable metric space denoting the price space. Denote by
D(t,p) the demand
from
-
Suppose that (T,T,fi)
result tells us the following:
interpreted as the measure space of agents and
P
i/j
no£ an extreme points of 77, a contradiction.
The above
Let
of //-measure
£ E.
h
We
(denoted
prove our claim. Let v £ extH we assume, arguing
€ ^F(t) an(^ u2 7^
(Al) we deduce the existence of an / £ L E ^(p) such that
able function
Hx
that the set of extreme points of
fT
by contradiction that v £ L^£(n,D F ),
nonzero: u(t)
Valadier, (1977),
T
x
P
to
set of
K.
If
agent
(Al)
is
at prices p,
t
demand
the aggregate
despite the fact that
D is a set-valued function
satisfied (i.e., if there are
agents than commodities), then
in Section 5). In other
i.e.,
it
follows
"many more"
from our Main Theorem that
J T D(t,p)dfi(t) is convex (see also Corollary 1
words there is a convexifying effect on aggregation
set
we have
infinitely
many commodities [compare
with
Khan-Yannelis (1990) or Ostroy-Zame (1988)].
4.
We now
of agents.
Models of Agents' Spaces
take a closer look at the possible variety of measure spaces
Our main purpose
is
to describe those spaces for which the
31
assumption (Al) holds, when A'
an
is
dimensional) separable
(infinite
Banach space.
In the following
1,2, are isomorphic
A
between them.
such that <MTX
i
=
1,2,
E
every
<f>
£ rx
modulo the
;
we
t
isomorphism
we
a one to one mapping
is
\ <f>(E),
said to
), i
—
:
rx
—*
i
t
t2
for
for
consider here the equivalence classes of measurable sets
ideal of sets of
Consider
t
^((JSi E ) = USi <K^i) if E $ T x
be measure preserving if /i 2 (0(£')) = p~ 1 (E)
\E) = T2
is
,
there exists a measure preserving set isomorphism
if
set
two measure spaces (T r-,^
shall say that
now
measure
zero.
the unit interval /
=
[0,1],
with
/?
the Borel subsets
number a we now
of /, and A the Lebesgue measure. For every ordinal
define that product
TQ —
The measure
n 7 <Q ^-y'
A7
all
for every
different
with IQ
,
=
I for every a.
where
Recall that a measurable rectangle
Ay =
G P otherwise. The
clan of
7Q
theoretic product space, a complete finite measure space,
denoted (T a ,r Q ,/x Q ).
form
I
I
I for
all
but a
<r-algebra r Q
is
number
finite
/z
Q
-
(for details, see for instance
the measure
set of the
of indices, and
satisfies
measurable rectangle where the sides {A 7J
When a < u
a
the <r-algebra generated by the
measurable rectangles. The measure
from /
is
is
|
i
=
1,
.
.
.
,m}
are
von Neumann (1950)).
space (T Q ,r Q ,/x Q )
is
(isomorphic to) the
unit interval with Borel subsets and Lebesgue measure. Note that by the
isomorphism theorem of Halmos and von Neumann (1948) every separable
and nonatomic measure space of
total
measure one
For cardinals of cardinality higher than
of the measure spaces
haram
which
A
=
(1942)].
is
is
possible thanks to
For a a-algebra
the power of a basis of r.
f for any principal ideal A
r, let f
The
lj
is
isomorphic to
it.
classification
Maharam's Theorem [Ma-
be the least cardinal number
cr-algebra
< r which
is
a complete
is
called
homogeneous
if
not the null ideal. Maharam's
theorem now states that every homogeneous measure algebra of total one
32
is
isomorphic to a (T Q ,T a
a.
We may
a
,
/j,
),
some
as previously described for
cardinal
therefore restrict our attention to such product spaces.
For any ordinal number 7, 7
<
and every
a,
AG/3 we
set
define the
function:
ifx,
1
x« i A'
if
where x^
is
eA;
the coordinate of the point x in the measure space which
corresponds to the ordinal 7. Clearly, every such function
(note {/^
The
>
1/2}
=
fl 7 <^
set of functions
D^xAx [[ K ^ <a D y
{/7 } 7<a
have proved
Theorem
and
essentially
bounded.
a set of linearly independent functions,
is
and has cardinality K(a). Setting a = u^
We
)
measurable
is
clearly
is
enough to
satisfy (Al).
the following result:
4.1. For every separable Banach space
sure space (T-pTj,
x )
X
,
there exists a mea-
such that the assumption (Al)
This
is satisfied.
measure space can be taken to be the product measure space with factors
(/,/?, A)
The
interval
mulated
and a =
u>
1
.
conclusion to be drawn from the above result
endowed with Lebesgue measure
his
model]
is
in infinite dimensional
[as
Aumann
is
that the
[0, 1]
(1964, 1966) for-
not "large enough" to model perfect competition
commodity
spaces.
Some
Corollaries
5.
F
Suppose that the correspondence
:
T —
2 K satisfies
all
Then we can deduce the
sumptions of the Main Theorem.
the as-
following
Corollaries.
Corollary
Proof.
1.
fT Fdfi
is
convex.
Directly from the fact that
jT coFdfi is convex because it
{y e L l (fj.,X) y(t) G coF(t)
:
Corollary 2 (Fatou's
2K
(n
—
is
JT Fdfi = JT coFdfi, and that
the linear image of the convex set
/x-a.e.}.
Lemma,
exact version).
1,2,.. ) are measurable correspondences, then
w-Ls /
JT
S^ F =
Fn (t)dfi, C
33
/
JT
vt-Ls
Fn (t)dfji.
If
Fn (-) T —
:
Proof.
The proof
the following chain of inclusions:
is
w-Ls /
Jt
Fn (t)dfxC
cl
w-Ls
/
Jt
=
co w-Ls
/
Jt
—
/
Jt
The
first
inclusion
is
dimensional spaces
theorem [Khan
Fn (t)dp.
Fn (t)dfj.
w-Ls Fn (t)dfi.
the approximate version of Fatou's
Yannelis (1988)], the second
[see
lemma in
is
the
(1985)]. Finally the third inclusion follows
infinite
Datko-Khan
from our Main
Theorem.
Corollary 3 (integration preserves u.s.c).
space and F A X T — 2 K be a correspondence such
Let
€ T, F(-
,t) is
weakly
Proof.
Since for each
w-LsF(p,
We
C
then w-Ls F(p n ,t)
to p,
show that fT F(-
,
(w-u.s.c),
u.s.c.
t
t)
i.e.,
if
the sequence
F(p,t). Then
fT F(-
G T, F(-
w-u.s.c,
,
t)
C F(pQ ,t),
t)dfi(t) is
is
i.e., if
p n converges
,t)dp. is w-u.s.c.
we have that
whenever
w-u.s.c,
be a metric
that for each fixed
:
t
A
pn
—
p
—
p
then
pn
*
.
(5.1)
w-Ls / F(p n ,t)dn{t) C / F{p Q ,t)dii(t).
Jt
jt
It
follows from Corollary 2 that
w-Ls / F{p n ,t)dfi(t) C / w-Ls F(p n
Jt
Jt
From
(5.1)
it
Combining now
(5.2)
w-Ls
J F(-
t)dfi(t).
(5.2)
follows that
/ w-Ls F(pni t)dfi(t)
i.e.,
,
and
J
(5.3)
F{P n
,
C
/
F(pQ ,t)dti(t).
we can conclude
t)dfi(t)
,t)dp,(t) is w-u.s.c.
34
C [ F{p
that
,t)dn(t),
(5.3)
Note that Corollary
Theorem
Aumann
in
1
1
the infinite dimensional generalization of
is
(1965) or Debreu (1967, p. 369)
Lyapunov-Richter theorem. Corollary 2
Lemma
dimensional spaces.
in infinite
sequence of set-valued functions
latter
is
Fn (-) need
also true for Corollary 3,
not be convex valued contrary
and Rustichini (1989).
no convexity assumption
i.e.,
needed to show that integration preserves w-u.s.c.
that
as the
worth pointing out that the
It is
to the related results obtained in Yannelis (1988)
The
known
an exact version of the Fatou
is
It is
is
worth pointing out
the above Corollaries are false in infinite dimensional spaces [see
all
for instance Rustichini (1989) or Yannelis (1990) for a counterexample]
even
the measure space
if
results
is
What seems
atomless.
to drive the above
condition (Al).
is
The Existence
6.
of a Competitive
Equilibrium
X
Let
ble
denote the commodity space where
Banach space whose
economy £
is
(T,t,h)
is
(2)
Y T—
2
(3)
> C
(4) e
T—
:
/x-a.e.,
an ordered, separa-
is
has an interior point
u.
An
where
e]
the measure space of agents;
X+
is
the consumption set-valued function of each agent;
Y(t) x Y(t)
t
X+
a quadruple [(T, r,/i),y, >,
(1)
:
positive cone
X
A"+
and
is
is
the preference relation of agent
the initial
for all
t
£ T,
endowment
e{i)
of agent
belongs to a
t,
t;
where
e(t)
€ Y(t)
norm compact subset
of
Y(t);
(5) the pair [(T,r,fi),X] satisfies the
Denote the budget
p x < p
B(t,p)
:
•
x
The demand
e(Z)}.
>
t
y for
pair (p, /), p e
X+
(0 /(0 £ D{t,p)
(ii)
set of agent
all
l
We now
(a.l)
at prices
set of agent
y £ B(t,p)}.
\ {0},
/ £
/z-a.e.,
and
iT f{t)d i{t)<ST
t
assumption (Al).
A
t
p by B(t,p) = {x £ Y{i)
at prices
p
is
D(t,p)
competitive equilibrium for £
L^^X^)
such that:
e{i)d li{t).
introduce the following assumptions:
(T,t,h)
is
a complete
finite
separable measure space.
35
=
:
{x 6
is
a
The correspondence Y
(a.2)
T — 2*+
:
is
integrably bounded, closed,
convex, nonempty, weakly compact valued, and
able graph,
(a.3)(a) For each
t
i.e.,
G
T
GY
G t
it
has a measur-
P(X + ).
<8>
and each x G Y(t) the
set
R(t,x)
=
{t
G Y(t)
l
y > x} is weakly closed and the set R~ (t,x) = {y G Y(t)
y} is norm closed,
> is measurable in the sense that the set {(t,x,y) G T x
X+ y > x} belongs tor® 0(X+ ) /3(X + ),
> is transitive, complete, and reflexive.
:
t
(b)
t
t
(a. 4)
X+
t
x
<g>
:
(c)
:
x >
t
For
all <
norm
to the
Theorem
G T, there exists z(i) G V(<) such that e(<)
interior of
X,
— z(t)
belongs
.
6.1. Let £ be an economy satisfying (a.l)-(a.4).
Then
a
competitive equilibrium exists in £.
Proof.
It
follows
by combining the main existence
Yannelis (1990) together with the Main Theorem and
Note that Theorem
tence
Theorem
6.1
above
is
Khan-
result in
its corollaries.
the counterpart of the
Main
Exis-
of a competitive equilibrium in Khan-Yannelis (1990).
should be emphasized, that contrary to the Khan-Yannelis (1990)
ting, the
economy that we considered
agents than commodities
[recall (5)
in this
It
set-
paper has "many more"
above] and this allows us to drop the
convexity assumption on preferences.
It is
worth pointing out that
for the core equivalence
theorem
[see for
instance Rustichini-Yannelis (1989, 1991)] the convexity assumption on
preferences
is
also not needed regardless as to
whether the economy has
"many more" agents than commodities. The reason
is
that for the proof
of the core equivalence theorem, one has to separate an open set from the
closure of the "aggregate net trade preferred set."
imate Lyapunov theorem we
1991)]
and
virtue of the approx-
have that the norm closure of the "ag-
will
gregate net trade preferred set"
By
is
convex
this will enable us to apply the
[see
Rustichini-Yannelis (1989,
standard separating hyperplane
argument.
However, despite the
sets coincide
(i.e.,
fact that
without convexity of preferences both
the core and the set of competitive equilibrium alloca-
tions are the same), they
may
be empty unless of course we have "many
more" agents than commodities,
as the
36
above theorem indicates.
Finally
Theorem
is
it
worth pointing out that the preference relation >
in
need not be complete contrary to the Khan-Yannelis (1990)
6.1
existence theorem.
The Existence of a Pure Strategy
Nash Equilibrium
7.
We now indicate how from our
main theorem one can obtain a gener-
alization of Schmeidler's (1973) result on the existence of a pure strategy
Nash equilibrium
for
a game with a continuum of players with a
In particular, in view of the failure of the
dimensional strategy space.
Lyapunov theorem
Khan
in infinite
dimensional spaces, as
it
was shown by
still
is
allowed to
However, by applying our main theorem, we show that one
infinite.
can
Nash
(1986), one can only obtain an approximate pure strategy
equilibrium once the dimensionality of the strategy space
be
finite
obtain an exact pure strategy Nash equilibrium for a
game with
a continuum of players and with an infinite dimensional strategy space,
provided that the game has "many more" players than strategies.
We
outline the details below:
A game G
(1) (T,r,fi)
(2)
X
is
is
a quadruple ((T, t,/j.),X,Y, u) where
a complete
finite
measure space of players,
the strategy space which
is
is
assumed
to be
a separable Banach
space,
(3)
Y
:
player
(4)
u
:
for
2X
T —
T
a measurable function denoting the strategy set of
is
t,
X
X
x
-L
x
(/x,
each fixed x,
Y) —*
u(-
,
•
,
x)
R
is
is
the utility function of player
t,
where
a Borel measurable function on {(t,y)
:
yeY(t)},
(5) the pair ((T,T,fi),X) satisfies the condition (A.l).
Note that the above definition of a game
(1986) with the addition of (5),
assumption
in
Khan
i.e.,
is
identical with that of
Khan
we have replaced the nonatomicity
(1986), with the condition that
we have "many more"
players than strategies.
Denote by extY(-) the extreme points of the
ready to state the following result.
37
set Y(-).
We
are
now
Theorem
7.1. Let
G =
((T,r,p,),X,Y,u) be a game satisfying the
following assumptions:
(7.1)
Y
:
T —
2
X
is
a nonempty, closed, convex, weakly
compact and
integrably bounded correspondence;
(7.3)
u:T
u T
XVien
G
(7.2)
:
x
x
X
X
x Z^/x, Y) -*
X X 1 (/x,y)
R
— R
»
on X;
is
linear
is
weakly continuous on
has a pure strategy Nash equilibrium,
L l (fi,extY) such
u(t,x*(t), ij)
that for almost all
t
= max u(t,y,x^)
X L x {p:,Y).
there exists x"
i.e.,
£
in T,
where
y€K(t)
Proof.
X
x*T
=
/
x*(t)dfj.(t).
7 t6T
Combine Khan's (1986) theorem together with our Main The-
orem.
References
Aumann,
R. J., 1964, "Markets with a Continuum of Traders," Econometrica 32, 39-50.
Aumann, R. J., 1965, "Integrals of Set- Valued Functions," J. Math. Anal.
Appi 12, 1-12.
Aumann, "R. J., 1966, "Existence of a Competitive Equilibrium
a Continuum of Traders," Econometnca 34, 1-17.
in
Markets with
Castaing, C. and Valadier, M., 1977, "Convex Analysis and Measurable Multifunctions," Led. Notes Math. 580, Springer- Verlag, New York.
Debreu, G., 1967, "Integration of Correspondences," Proc. Fifth Berkeley Symp.
Math. Stat. Prob., University of California Press, Berkeley, Vol. II, Part I,
351-372.
Diestel, J. and Uhl, J., 1977, Vector Measures, Mathematical Surveys, No. 15,
American Mathematical Society, Providence, Rhode Island.
Dunford, N. and Schwartz, J. T., 1958, Linear Operators, Part I, Interscience,
New York.
Gretsky, N. E. and Ostroy, J., 1985, "Thick and Thin Market Nonatomic Exchange Economies," in Advances in Equilibrium Theory, C. D. Aliprantis
et al, eds., Springer- Verlag.
Halmos, P. R. and Von Neumann, J., 1941, "Operator Methods in Classical
Mechanics, II," Ann. Math. 43:2.
Himmelberg, C. J., 1975, "Measurable Relations," Fund. Math. 87, 53-72.
Ionescu-Tulcea, A. and C, 1969, Topics in the Theory of Lifting, SpringerVerlag, Berlin.
Khan, M. Ali, 1976, "On the Integration of Set- Valued Mappings in a Nonreflexive Banach Space II," Simon Stevin 59, 257-267.
Khan, M. Ali, 1986, "Equilibrium Points of Nonatomic Games over a Banach
Space," Trans. Amer. Math. Soc. 293:2, 737-749.
38
Khan, M. Ali and Yanelis, N. C, 1990, "Existence of a Competitive Equilibrium
in Markets with a Continuum of Agents and Commodities," this volume.
Kluvanek, I. and Knowles, G., 1975, "Vector Measures and Control System,"
Math. Stud. 20, North Holland.
Knowles, G., 1974, Liapunov Vector Measures, SIAM J. Control 13, 294-303.
Kuratowski, K. and Ryll-Nardzewski, C, 1962, "A General Theorem on Selectors," Bull. Acad. Polon. Sci. Ser. Sci. Marsh. Astronom. Phys., 13,
397-403.
Lewis, L., 1977, Ph.D. Thesis, Yale University.
Loeb, P., 1971, "A Combinatorial Analog of Lyapunov's Theorem for Infinitesimal Generated Atomic Vector Measures," Proc. Amer. Math. Soc. 39,
585-586.
J., 1966, "A Short Proof of Lyapunov's Convexity Theorem,"
Math. Mech. 15, 971-972.
Maharam, D., 1942, "On Homogeneous Measure Algebras," Proc. Natl. Acad.
Sci. 28, 108-111.
Masani, P., 1978, "Measurability and Pettis Integration in Hilbert Spaces,"
J. Reine Angew. Math. 297, 92-135.
Mas-Colell, A., 1975, "A Model of Equilibrium with Differentiated Commodities," J. Math. Econ. 2, 263-295.
Mertens, J.-F., 1990, "An Equivalence Theorem for the Core of an Economy
with Commodity Space Loo — t(Loo, L\)" this volume.
Lindenstrauss,
J.
Ostroy,
J.
and Zame, W.
R., 1988,
"Non-Atomic Exchange Economies and the
Boundaries of Perfect Competition," mimeo.
Robertson, A. P. and Kingman, J. F. C, 1968, "On a Theorem of Lyapunov,"
J. London Math. Soc. 43, 347-351.
Rustichini, A., 1989, "A Counterexample and an Exact Version of Fatou's
Lemma in Infinite Dimensions," Archiv der Mathematic, 52, 357-362.
Rustichini, A. and Yannelis, N. C, 1989, "Commodity Pair Desirability and
the Core Equivalence Theorem," mimeo.
Rustichini, A. and Yannelis, N. C, 1991, "Edgeworth's Conjecture in Economies
with a Continuum of Agents and Commodities," J. Math. Econ., to appear.
Schmeidler, D., 1973, "Equilibrium Points of Non Atomic Games, J. Stat.
Phys. 7:4, 295-300.
Von Neumann, J., 1950, "Functional Operators," Ann. Math. Stud. 22, Princeton University Press.
Yannelis, N. C, 1988, "Fatou's Lemma In Infinite Dimensional Spaces," Proc.
Amer. Math. Soc. 102, 303-310.
Yannelis, N. C, 1990, "Integration of Banach-Valued Correspondences," this
volume.
39
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