Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol40.Issue2.Sep2000:
Mathematical Social Sciences 40 (2000) 157–173
www.elsevier.nl / locate / econbase
Economies with a measure space of agents and a separable
commodity space
Mitsunori Noguchi*
Department of Economics, Faculty of Commerce, Meijo University, 1 -501 Shiogamaguchi Tenpaku-ku,
Nagoya 468 -8502, Japan
Received January 1999; received in revised form June 1999; accepted September 1999
Abstract
We prove the existence of an equilibrium in an economy with a measure space of agents and a
separable Banach commodity space whose positive cone admits an interior point. We follow the
truncation argument given at the end of Yannelis [Yannelis, N.C., 1987. Equilibria in noncooperative models of competition, J. Econ. Theory 41, 96–111] and the abstract economy
approach as in Shafer [Shafer, W., 1976. Equilibrium in economies without ordered preferences or
free disposal, J. Math. Econ. 3, 135–137] and Khan and Vohra [Khan, M.A., Vohra, R., 1984.
Equilibrium in abstract economies without ordered preferences and with a measure space of
agents, J. Math. Econ. 13, 133–142], which allows preferences to be interdependent. Our result
may be viewed as an extension of the result in Kahn and Yannelis [Khan, M.A., Yannelis, N.C.,
1991. Equilibria in markets with a continuum of agents and commodities. In: Khan, M.A.,
Yannelis, N.C. (Eds.), Equilibrium Theory in Infinite Dimensional Spaces, Springer-Verlag, Tokyo,
pp. 233–248] employing production and allowing preferences to be interdependent. We utilize
Mazur’s lemma at the crucial point in the truncation argument. We assume that the preference
correspondence is representable by an interdependent utility function. The method in the present
paper does not rely on the usual weak openness assumption on the lower sections of the preference
correspondence. 2000 Elsevier Science B.V. All rights reserved.
Keywords: General equilibrium; Infinite dimensional commodity space; Measure space of agents; Separable
Banach space; Fixed point theorem
JEL classification: D51
*Tel.: 181-56-138-8514; fax: 181-52-833-4767 (office), 181-56-138-8514 (home).
E-mail address: [email protected] (M. Noguchi)
0165-4896 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 99 )00046-3
158
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
1. Introduction
The author has proven the existence of equilibria for economies with a measure space
of agents (T, T, m ), a separable Banach commodity space - whose positive cone admits
an interior point, and an interdependent preference correspondence P: T 3 L1 ( m,
X) → -, where L1 ( m, X) is the set of Bochner integrable selections of the consumption
correspondence X: T → - (Noguchi, 1997a,b). In order to utilize a fixed point argument,
the author requires X to be weakly compact valued so that L1 ( m, X) becomes weakly
compact in L1 ( m, - ), where L1 ( m, - ) is the Banach space of Bochner integrable
functions f : T → - (Yannelis, 1991, Theorem 3.1, p. 7). In both Noguchi (1997a) and
Noguchi (1997b) the author assumes that for almost all t [ T, P(t, ? ): L1 ( m, X) → admits weakly open lower sections. In the present paper we consider the preference
correspondence P having the representation P(t, x) 5 h j [ X(t): u(t, j , x) . u(t, r (x)(t),
x)j, where u is an interdependent utility function, and r is a choice function selecting a
representative from each equivalence class x [ L1 ( m, - ). For notational simplicity we
omit r in the sequel. It may appear that such P is already covered by the author’s
previous papers, but it is not immediately clear whether the lower sections of P(t, ? ) are,
in general, weakly open or not. The difficulty is present even in the simplified case P(t,
x) 5 h j [ X(t): u(t, j ) . u(t, x(t))j, where u(t, j ) is assumed to be norm continuous and
quasi-concave in j . For example, one might try to make use of Mazur’s lemma [we
assume (T, T, m ) to be separable so that the weak topology on L1 ( m, X) becomes
metrizable] in order to show that for almost all t [ T, hx [ L1 ( m, X): u(t, j ) # u(t, x(t))j
is weakly closed. Then one would proceed as follows: For each t [ T, define E(t) 5 hh [
X(t): u(t, j ) # u(t, h )j, which is, under the assumptions, closed and convex. We must find
an exceptional set T 0 [ T with m (T 0 ) 5 0 such that for every t [ T \T 0 , if x n converges
¯ [ E(t). We have x¯ [ cl cohx n j and
weakly to x¯ in L1 ( m, X) and x n (t) [ E(t), then x(t)
¯ [ cl cohx n (t)j , E(t) for all t [ T \T x¯ for some m -null subset
consequently, we obtain x(t)
T x¯ [ T which depends on x¯ (x n may be a subsequence of the original sequence). Let
L1 ( m, E) denote the set of (equivalence classes of) integrable selections of t → E(t),
which is weakly compact in L1 ( m, - ) for the same reason as L1 ( m, X) is. We may wish
to define T 0 5 < x¯ [L 1 ( m ,C ) T x¯ , but we do not have any knowledge about the measure
theoretic nature of the set T 0 .
As in Noguchi (1997a), we assume that - is an ordered separable Banach space in
which the positive cone -1 admits an interior point a and is closed, proper. The interior
* of -1 by
point a allows us to define a price simplex D in the dual cone - 1
* : k p, al 5 1j, where k ? , ? l is the natural coupling on - * 3 -. The
D5hp [- 1
properness and closedness of -1 guarantee the ‘right property’ of D: Let j [ -. if k p,
j l $ 0 for all p [ D, then j [ -1 .
Observe that the coupling ( p, x) → eT k p, x(t)l is not, generally, jointly continuous
with respect to the weak* topology on D and the weak topology on L1 ( m, X), and this
fact causes an insurmountable problem in the use of a Shafer–Sonnenschein type
argument, which we rely on for showing the existence of maximal elements of the
correspondences defined in terms of such coupling. However, it can be shown that with
the additional assumption of X being norm-compact valued, the above coupling becomes
jointly continuous, and henceforth, the standard argument becomes applicable. Needless
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
159
to say, the above condition of every consumption set X(t) being norm-compact is too
stringent to be acceptable, and in order to remedy this problem, we consider the set of
norm-compact valued sub-correspondences c (in fact, the set of polytope valued
sub-correspondences) of the weakly compact valued consumption correspondence X.
Such c with some additional technical properties form a non-empty directed set with the
obvious inclusion as an ordering. For the sake of simplicity, let us ignore production for
the time being. For each c, we have a sub-economy % c of the original economy %,
where we can utilize some well-known methods in the literature for obtaining an
‘equilibrium’ (xc , pc ) [ L1 ( m, c ) 3 D.
Now we have a net of equilibria (xc , pc ) [ L1 ( m, X) 3 D, where L1 ( m, X) 3 D
becomes a compact metric space with respect to the weak topology on L1 ( m, X) and the
weak* topology on D. Based on the compactness, we can extract a convergent subnet
¯ p¯ ). We must show that (x,
¯ p¯ ) gives rise to an equilibrium for the
(xc (m) , pc (m) ) → (x,
original economy %. To this end, we need the following crucial proposition: There exists
a m -null set T 0 [ T such that for every consumer t lying outside T 0 , if j [ X(t) is
¯
preferred to x(t),
then the price of j is greater than or equal to the price of the initial
endowment e(t) of t. Once this is established, the rest is simply a routine matter.
The heart of the problem is verification of the following statement: There exists a
¯
m -null exceptional set T 0 [ T such that for every t [ T \T 0 , if j [ P(t, x¯ ), then kp,
¯
j l $ kp, e(t)l. Once we have the above statement justified, the rest follows from the
¯ by such j and pass to the limit
standard argument. For example, we can approximate x(t)
¯
¯
¯
¯
to obtain kp, x(t)l $ kp, e(t)l, which actually becomes the equality showing that x(t)
¯
satisfies the budget constraint. We can also show that for almost all consumers t [ T, x(t)
¯
maximizes u(t, ? , x ) over the budget set.
If P(t, ? ) had open lower sections with respect to the weak topology on L1 ( m, X), we
would have that j [ P(t, xc ) for sufficiently large c. We can also arrange the directed set
c
hc j so that j [ c (t) for sufficiently large c, and consequently, j [ P (t, xc ) 5 P(t,
xc ) > c (t) for sufficiently large c. Then by utilizing the property of (xc , pc ) being an
c
‘equilibrium’ in % with some m -null exceptional set Tc , we conclude that k pc ,
j l $ k pc , e(t)l for t [ T \Tc . Thereafter, we hope to be able to modify hc j so that
¯ we obtain kp,
¯
< c Tc [ T with m ( < c Tc ) 5 0. Now by passing to the limit pc → p,
¯ e(t)l m -a.e.
j l $ kp,
Coming back to the present setting, since the weak topology on L1 ( m, X) and the
weak* topology on D are both metrizable, we can extract a convergent sequence (x n ,
¯ p¯ ). Then, as a consequence of Mazur’s lemma, we have
pn ) 5 (xc (m(n)) , pc (m(n)) ) → (x,
¯ [ cl cohx n (t)j outside some m -null set T a . This does not quite say that x n (t) → x(t)
¯
x(t)
outside T a , but we can find a subsequence x n i such that j [ P(t, x n i ) for sufficiently large
i. Note that prior to the extraction of sequence (x n , pn ), we could have arranged the
directed set hmj so that j [ c (m) for all m, and hence j [ c (m(n i )) for all i. Now we
have j [ P c (m(n i )) (t, x n i ) for sufficiently large i, for every t [ T \T a . We stress the fact that
the choice of sequence (x n , pn ) depends on j and hence T a < ( < n`51 Tc (m(n)) ) also
`
depends on j , so we should write Tj 5 T a < ( < n51
Tc (m(n)) ). Repeating the same
argument as in the case that P(t, ? ) was assumed to admit open lower sections, we
¯ j l $ kp,
¯ e(t)l. This is
obtain the following statement: If j [ P(t, x¯ ) and t [ T \Tj , then kp,
not exactly what we intended to derive since the exceptional set depends on j .
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M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
In order to overcome the above difficulty, we consider a countable collection of
measurable selections hsk j of X such that hsk (t)j is dense in X(t) for all t [ T. This
allows us to work on a measurable selection s instead of j since j can be approximated
by sk . Now let c0 be an element of hc j and consider the convex hull c1 5 cohs, c0 j,
which clearly lies in hc j. Note that there exists m 0 such that for all m $ m 0 , we have
c (m) $ c1 . Then we restrict hmj to those elements that are larger than m 0 and obtain a
directed set with the property we need. Repeating the previous argument again with sk
¯ sk (t)l $ kp,
¯ e(t)l.
instead of j , we obtain that if sk (t) [ P(t, x¯ ) and t [ T \Tsk , then kp,
¯ sk (t)l $ kp,
¯
Note that this means that for every t [ T \( < `k 51 Tsk ), if sk (t) [ P(t, x¯ ), kp,
`
e(t)l. Now define T 0 5 < k51
Tsk . Let t [ T \T 0 . If j [ P(t, x¯ ), we can choose a
convergent subsequence sk i (t) → j . Since P(t, x¯ ) is open, sk i (t) [ P(t, x¯ ) for sufficiently
¯ j l $ kp,
¯ e(t)l.
large i. Then by passing to the limit i → `, we finally obtain kp,
Work relevant to the present paper was previously done by Yannelis (1987), Bewley
(1991), Khan and Yannelis (1991), and Podczeck (1997). Yannelis (1987) discusses the
existence of an equilibrium in an abstract economy with a continuum of agents and a
separable Banach strategy space (Yannelis, 1987, Remark 6.4, p. 108), explains how the
abstract economy approach can be utilized to show the existence of an equilibrium in the
exchange economy with norm compact consumption sets, and discusses the type of
truncation argument which is used in the present paper (Yannelis, 1987, Remark 6.5, p.
108). Bewley (1991) proves the existence of an equilibrium in an economy with
commodity-price paring (l ` , l 1 ) and a continuum of agents. Khan and Yannelis (1991)
and Podczeck (1997) prove the existence of an equilibrium in the exchange economy
with a continuum of agents and a separable Banach commodity space. They follow the
excess demand approach, which allows them, with some measure theoretic restrictions
on the measure space of agents, to dispense with the convexity assumption on
preferences by making use of an infinite dimensional version of Lyapunov’s theorem
(see Rustichini and Yannelis, 1991; Podczeck, 1997).
In the present paper, we follow the abstract economy approach as in Shafer (1976)
and Khan and Vohra (1984), which allows preferences to be interdependent. Our result
may be viewed as an extension of the Kahn–Yannelis result (Khan and Yannelis, 1991)
allowing preferences to be interdependent by the use of Mazur’s lemma and employing
production. We also present a concrete example in which our Main Theorem is
applicable. This example demonstrates under some additional conditions that if the size
of initial endowments e(t) and production possibility sets is small relative to the
universal bound on consumption sets X(t), for almost all consumers t, the equilibrium
¯ actually avoids the satiation points due to the ‘boundary’ of X(t).
consumption x(t)
1.1. Definitions
For a linear space - and a subset A , -, co A denotes the convex hull of A. For a
topological space - and a subset A , -, cl A denotes the closure of A and int A denotes
the interior of A. Let -, = be any sets and f : - → = a correspondence. When = is a
linear space, co f : - → = denotes the correspondence defined by the relation co
f (x) ; co [f (x)] for each x [ -, and when = is endowed with a topology cl f : - → =
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
161
denotes the correspondence defined by cl f (x) ; cl [f (x)] for each x [ -. Gf denotes
the graph of f, i.e. Gf 5 h(x, y) [ - 3 = : y [ f (x)j. Let - and = be paired linear
spaces. Following the literature, s ( -, = ) denote the weak topology on -. For a Banach
space -, - * denotes the dual of -, and i ? i - the norm of -. When = is the dual of
some Banach space, = w * denotes = endowed with the weak* topology, and for a subset
A , =, Aw * denotes the set A endowed with the topology induced by the weak*
topology on =. Let -, = be any topological spaces. A correspondence f : - → = is
said to be upper semi-continuous (u.s.c.) at x [ - if for every neighborhood V of f (x),
there exists a neighborhood U of x such that f (x9) ,V for every x9 [ U. The
correspondence f : - → = is said to be upper semi-continuous (u.s.c.) if it is u.s.c. at
every x [ -. A correspondence f : - → = is said to be lower semi-continuous (l.s.c.) at
x [ - if for every open set V in = with f (x) >V ± 5, there exists a neighborhood U of
x such that f (x9) >V ± 5 for every x9 [ U. The correspondence f : - → = is said to be
lower semi-continuous (l.s.c.) if it is l.s.c. at every x [ -. A correspondence f : - → =
is said to be continuous (at x [ - ) if it is both u.s.c. and l.s.c. (at x [ - ). A
correspondence f : - → = is said to have an open graph if Gf is an open subset of
- 3 =. Let (T, T) be a measurable space. A correspondence f : T → = is said to admit
a measurable graph if Gf [ T ^ @ ( = ), where @ ( = ) denotes the Borel sigma algebra
on =. For sigma algebras t1 and t2 , t1 ^ t2 denotes the product sigma algebra. A
correspondence f : T → = is said to be lower measurable if the set ht [ T : f (t) >V ±
5j belongs to T for every open set V , =. For measurable spaces (T, T), (S, S), and a
function f : T → S, f is said to be (T, S)-measurable whenever it is measurable with
respect to the sigma algebras T and S. Let (T, T, m ) be a finite measure space and - a
Banach space. L1 ( m, - ) denotes the space of equivalence classes of - -valued Bochner
integrable functions f : T → - normed by i f i 5 eT i f(t)i - (for details see Diestel and
Uhl, 1977). For a Banach space - and its dual - *, k ? , ? l denotes the natural paring.
For a Banach space - and a correspondence f : T → -, define L1 ( m, f ) 5 h f [ L1 ( m,
- ): f(t) [ f (t) m -a.e.j. Let R denotes the set of real numbers. A correspondence f :
T → - is said to be integrably bounded if there exists a function h [ L1 ( m, R) such that
suphi j i - : j [ f (t)j # h(t) m -a.e. For an ordered vector space - with order # , we write
for a, b [ -, [a, b] 5 h j [ - : a # j # bj, (a, b] 5 h j [ - : a # j # b, a ± j j, etc.
2. Main theorem
Throughout this paper, all measures are assumed to be positive, and any version of
definitions of measurability and integrability of functions can be adopted as long as they
are compatible with those of Tulcea and Tulcea (1969). Our commodity space is an
ordered separable Banach space - whose positive cone -1 is closed and has an interior
* is non-trivial. [If -1 is proper, then
point a. We further assume that the dual cone - 1
* ± (0) (see Jameson, 1970, p. 231).] Recall that L1 ( m, - ) is a set of equivalence
-1
classes of functions. Let r be a choice function selection one representative function
from each class x [ L1 ( m, - ). When there is no fear of confusion, we simply write x for
r (x).
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M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
An economy % is a tuple [(T, T, m ), X, u, e, (S, S, p ), Y, u ] which is characterized
by:
1.
2.
3.
4.
5.
6.
7.
a measure space of consumers (T, T, m );
a consumption correspondence X: T → - ;
an interdependent utility function u: GX 3 L1 ( m, X) → R;
an initial endowment e: T → -, where e(t) [ X(t) for all t [ T;
a measure space of producers (S, S, p );
a production correspondence Y: S → - ;
a share u : T 3 S → R, where u (t, s) denotes the share of consumer t in the profit of
producer s.
Let ‘\’ denote the set theoretic subtraction. Let -1 be the usual positive cone of -, and
- *1 the positive cone induced by -1 on - *. Denote the budget set of consumer t at
price p [ - *
1 \(0) and production plan y [ L 1 (p, Y) by B(t, p, y) 5 h j [ X(t):k p, j l # k p,
eS u (t, s)y(s)l 1 k p, e(t)lj. An attainable state is a pair (x, y) [ L1 ( m, X) 3 L1 (p, Y) such
that eT x(t) 2 eS y(s) 2 eT e(t) [ 2 -1 . Denote the set of attainable states by F.
A competitive equilibrium for % is a price p¯ [ - *
1 \(0) together with an attainable
¯ y¯ ) [ F satisfying:
state (x,
¯ [ B(t, p,
¯ y¯ ) and P(t, x¯ ) > B(t, p,
¯ y¯ ) 5 5, where P(t,
1. For almost all t [ T, x(t)
¯
x¯ ) 5 h j [ X(t): u(t, j , x¯ ) . u(t, x(t),
x¯ )j, and 5 denotes the empty set. [Note that this
¯
condition is well-defined regardless of the particular representative chosen for x.]
¯ y(s)l
¯
¯ h l for all h [ Y(s).
2. For almost all s [ S, kp,
$ kp,
Define for each t [ T, a (t) 5 heS u (t, s)z(s) [ - : z [ L1 (p, Y)j. Let Xˆ 5 hx [ L1 ( m, X):
(x, y) [ F for some y [ L1 (p, Y)j.
We now state the set of assumptions needed for the proof of our Main Theorem.
(A.1) (T, T, m ) is a complete finite separable measure space.
(A.2) (S, S, p ) is a complete finite separable measure space.
(A.3) X: T → - is an integrably bounded, weakly compact, convex, non-empty
valued correspondence such that GX [ T ^ @ ( - ).
(A.4) Y: S → - is an integrably bounded, weakly compact, convex, non-empty
valued correspondence such that GY [ S ^ @ ( - ).
(A.5) u: GX 3 L1 ( m, X)w → R is a function such that:
(a) For every t [ T, u(t, ? , ? ) is jointly continuous on X(t) 3 L1 ( m, X)w .
(b) For every t [ T and x [ L1 ( m, X), if j 1 and j 2 are two points of X(t) such that
u(t, j 1 , x) , u(t, j 2 , x), and l is a real number in (0,1], then u(t, j 1 , x) , u(t,
(1 2 l)j 1 1 lj 2 , x). (As Debreu, 1982, p. 705, pointed out, this condition implies
that u(t, ? , x) is quasi-concave.)
(c) If o ni 51 li x i is a convex combination of x i [ L1 ( m, X), i 5 1, . . . , n, there
exists x i 0 [ hx i j, i 5 1, . . . ,n, such that for almost all t [ T, we have u(t, x i 0 (t),
x i 0 ) # u(t, o in51 li x i (t), x i 0 ).
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
163
ˆ either u(t, j , x) . u(t, x(t), x) for some j [ X(t) or x(t) 2
(d) For each x [ X,
a (t) 2 e(t) , -1 holds for almost all t [ T. (Note that this condition is welldefined independent of the choice of a representative for x.)
(e) For every x [ L1 ( m, X), u( ? , ? , x) is a measurable function on GX [
T ^ @ ( - ).
(A.6) u is a T ^ S-measurable function such that for every s [ S, eT u (t, s) 5 1 and
for every (t, s) [ T 3 S, 0 # u (t, s) # 1.
(A.7) e is (T, @ ( - ))-measurable.
(A.8) There exists a measurable selection w of X such that e(t) 2 w(t) [ int -1 for all
t [ T.
(A.9) 0 [ Y(s) for almost all s [ S.
Remark 1. The condition in (A.5)(b) is a little stronger than the usual quasi-concavity
condition. Quasi-concavity is needed to assure the convexity of individual demand sets.
We also need concavity of u(t, ? , x) on X(t) in order to prove Proposition 4, which is the
main proposition in our truncation argument. To be more specific, let X(t) be a
non-empty, weakly compact, convex consumption set for consumer t, where preference
relation s is defined by u(t, ? , x). In our truncation argument, we approximate X(t) by
subsets C(t) , X(t) which are non-empty, norm compact, convex. What is required for
the proof of Proposition 4 is the following statement: If j [ C(t) is not a satiation point,
there exists j 9 [ C(t) which is arbitrarily close to j and j 9 s j . It may appear to be true
that if the upper contour set of s on X(t) is convex and h s j on C(t), then
(1 2 a)h 1 aj s j for all a [ (0,1], but this may not be the case even in twodimensional Euclidean space (see Noguchi, 1997a, p. 20, for a counterexample).
(A.5)(b) certainly serves our purpose.
Remark 2. (A.1)–(A.4) imply that both L1 ( m, X)w , L1 (p, Y)w are non-empty, convex,
and compact metrizable; the non-emptiness of L1 ( m, X) follows from the fact that X
admits a measurable selection f : T → - (Castaing and Valadier, 1977, Theorem III.30,
p. 80), which clearly lies in L1 ( m, X). The identical argument applies to L1 (p, Y). The
convexity is trivial, and the compactness follows from Yannelis (1991, Theorem 3.1, p.
7). The metrizability follows from Kolmogorov and Fomin (1970, p. 381) and Dunford
and Schwartz (1958, Theorem 3, p. 434).
We state our Main Theorem as follows:
Main Theorem. Let % 5 [(T, T, m ), X, u, e, (S, S, p ), Y, u ] be an economy satisfying
(A.1)–(A.9). Then % has an equilibrium.
Example. For the sake of simplicity, we construct an example in which u is constant in
x [ L1 ( m, X). Let - 5 C(M) be the family of all real valued continuous functions on a
compact metric space, M, endowed with the usual sup norm. We consider the obvious
order in C(M). Let K be a non-empty, convex, weakly compact subset in C(M). We
assume that K contains 0 as a lower bound and 1 as an upper bound such that for all
j [ K except 1, 1 2 j [ int C(M) 1 . Such K can be constructed as follows: let K0 be a
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M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
convex subset containing 0, which is weak compact in C(M). Since K0 is bounded, we
have K0 , [2l, l] for some l [ R. If l is sufficiently large, l 2 K0 , int C(M) 1 . Now
we can let K 5 coh1, l 21 K0 j > C(M) 1 . Define X by X(t) 5 K for all t [ T. X satisfies
(A.3). Let Y be any correspondence as in (A.4) and (A.9) such that Y(s) # ]14 for all s [ S.
Let u be any function satisfying (A.6). Let e be any function as in (A.7) such that
e(t) [ int C(M) 1 and e(t) # ]14 for all t [ T. For (A.8), we may choose w(t) 5 0 for all
t [ T. Observe that by Corollary 2 in Yoshida (1968, p. 134), a (t) # ]14 for all t [ T. Let
u: T 3 K → R be measurable in t [ T, and continuous and concave in j [ K in the sense
of (A.5)(b). Note that the quasi-concavity of u implies (A.5)(c). We assume that u is
strictly increasing in j in the sense that u(t, j 1 ) , u(t, j 2 ) whenever j 2 2 j 1 [ int
C(M) 1 . Note that u(t, j ) . u(t, x(t)) for some j [ X(t) if and only if x(t) ± 1. We
construct an example of such u in the following manner: let (M, @ (M), i ) be a measure
space, where i is a positive Borel measure. Let U : T 3 M 3 R → R be a function such
that: (1) for every (t, m) [ T 3 M, U(t, m, ? ): R → R is continuous, concave, and strictly
increasing; (2) for every y [ R, U( ? , ? , y): T 3 M → R is T ^ @ (M) measurable; and
(3) for every (t, y) [ T 3 R, U(t, ? , y) [ L1 (M, @ (M), i ). Define u(t, j ) 5 eM U(t, m,
j (m)) di. Note that since K is bounded, for all j [ K, we have sup m [M u j (m)u # L for
some L. It follows from (2) that for every j [ K, U(t, m, j (m)) is jointly measurable in
(t, m). Since U(t, m, y) is increasing in y, we have uU(t, m, j (m))u # uU(t, m, 2 L)u 1 uU(t,
m, L)u, and by (3), U(t, m, j (m)) is integrable for all t. Thus by Fubini’s theorem, u( ? , j )
is T measurable. We next show that u(t, j ) is norm-continuous in j . Let j n → j [ K be
a convergent sequence. Then j n (m) → j (m) for all m [ M, and hence U(t, m,
j n (m)) → U(t, m, j (m)). Since uU(t, m, j n (m))u # uU(t, m, 2 L)u 1 uU(t, m,L)u, (3) and the
dominating convergence theorem imply that u(t, j n ) → u(t, j ). It is trivial to check that
u(t, j ) is strictly increasing in j . Since 1 2 e(t) 2 a (t) $ 0 for all t [ T, and if j 1 and j 2
are two points of K such that u(t, j 1 ) , u(t, j 2 ), and l is a real number in (0, 1], then u(t,
j 1 ) , u(t, (1 2 l)j 1 1 lj 2 ), (A.5)(b),(d) are satisfied. In our construction, 1 is a satiation
point for u for each t [ T and sets the upper limit for each commodity available for
consumption. The present example demonstrates that if the size of initial endowments
and production possibility sets is small relative to the size of the upper limit, the
¯ occurs at the satiation point only for those consumers lying
equilibrium consumption x(t)
¯ x(t)l
¯
¯
in a m -null subset. This assertion follows from the fact that kp,
5 eS u (t, s)kp,
¯
¯ e(t)l holds for (p,
¯ x,
¯ y¯ ) [see (5.2) in the proof of Proposition 5], and if x¯ 5 1
y(s)l
1 kp,
on some measurable subset T¯ , T with m (T¯ ) . 0, then the integration of the former
equality over T¯ gives rise to ip¯ i C(M )* m (T¯ ) # ]12 ip¯ i C(M )* m (T¯ ), which is absurd. This
establishes the claim.
* : k p, al 5 1j be the price simplex for %.
Let D 5 h p [ - 1
Remark 3. Since D is weak* compact (see Jameson, 1970, Theorem 3.8.6, p. 123) and
- is separable, the weak* topology on D is metrizable by a translation invariant metric
on - * (Dunford and Schwartz, 1958, Theorem 1, p. 426). Furthermore, D is bounded in
- * (Dunford and Schwartz, 1958, Corollary 3, p. 424). Note that by the separating
hyperplane theorem, if q(x) $ 0 for all q [ D, then x [ -1 (see Remarks 1 and 2 in
Noguchi, 1997a, p. 5).
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
165
Let ( be the collection of all correspondences c : T < S → - such that c u T 5
coh < in51 fi j for some measurable selections fi of X, c u S 5 coh < jl 51 g j j for some
measurable selections g j of Y, e and w are measurable selections of c u T , and 0 is a
measurable selection of c u S . As in Noguchi (1997a, p. 17), we can show that ( forms a
non-empty directed set under the obvious inclusion.
Observe that for each c [ (, c u T admits a measurable graph. This follows immediately from Himmelberg (1975, p. 69) and Theorem III.30 in Castaing and Valadier
(1977, p. 80), and the same holds also for c u S .
For (t, p) [ T 3 D, define:
E
A 1 (t, p) 5 h j [ c (t): k p, j l # u (t, s)P (s, p) 1 k p, e(t)lj
S
and for (x, y, p) [ L1 ( m, c u T ) 3 L1 (p, c u S ) 3 D, define:
P2 ( y, p) 5 hz [ L1 (p, c u S ):
E k p, z(s)l .E k p, y(s)lj
S
P3 (x, y, p) 5 hq [ D: kq 2 p,
S
E x(t) 2E y(s) 2E e(t)l . 0j
T
S
T
where P (s, p) ; suph [ c (s) k p, h l.
Remark 4. Note that the coupling ( p, v) → eT k p, v(t)l and ( p, v) → eS k p, v(t)l are
jointly continuous on D w * 3 L1 ( m, c u T )w , D w * 3 L1 (p, c u S )w , respectively (see Noguchi,
1997a, Lemma 1, p. 8), and it follows that P2 and P3 admit an open graph. Note also
that P (s, p) is non-negative [implied by (A.9)], bounded in (s, p), measurable in s, and
weak* continuous in p (see Remark 4 in Noguchi, 1997a, p. 7). A standard argument
such as Noguchi (1997a, Proposition 3, p. 9) shows that under (A.8), A 1 (t, ? ):
D w * → c (t) is a continuous correspondence with non-empty compact convex values.
We define pseudo-utility functions u 2 : (L1 (p, c u S )w 3 D w * ) 3 L1 (p, c u S )w → R and
u 3 : (L1 ( m, c u T )w 3 L1 (p, c u S )w 3 D w * ) 3 D w * → R by:
u 2 (( y, p), z) 5 dist[(( y, p), z), ((L1 (p, c u S )w 3 D w * ) 3 L1 (p, c u S )w)\GP 2 ]
u 3 ((x, y, p), q) 5 dist[((x, y, p),q),((L1 ( m, c u T )w 3 L1 (p, c u S )w 3 D w * ) 3 D w * )\GP 3 ]
where dist denotes the obvious distance function.
We define a correspondence H1 : T 3 L1 ( m, X)w 3 D w * → - by:
H1 (t, x, p) 5 Arg Maxh u(t, j , x): j [ A 1 (t, p)j
We next define correspondences H2 : L1 (p, c u S )w 3 D w * → L1 (p, c u S )w , H3 : L1 ( m,
c u T )w 3 L1 (p, c u S )w 3 D w * → D w * by:
166
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
H2 ( y, p) 5 Arg Maxhu 2 (( y, p), z): z [ L1 (p, c u S )w j
H3 (x, y, p) 5 Arg Maxhu 3 ((x, y, p), q): q [ D w * j
The standard argument as in Berge (1963) (see Noguchi, 1997a, Proposition 4, p. 11)
shows that H1 (t, ? , ? ), Hi (i 5 2, 3) are non-empty compact valued u.s.c. correspondences. [We need (A.5)(a) for showing that H1 (t, ? , ? ) is u.s.c.]
We focus on the properties of H1 . Note that H1 is convex valued.
Proposition 1. H1 ( ? , x, p): T → - admits a measurable graph.
See Appendix A for the proof.
Let & 5 L1 ( m, c u T )w 3 L1 (p, c u S )w 3 D w * . In the light of Proposition 1, we can
define a non-empty convex valued correspondence f1 (x, p) 5 hw [ L1 ( m, c u T )w : w(t) [
H1 (t, x, p) m -a.e.j. Following Yannelis (1991, Theorem 5.5, p. 19), we can prove the
closedness of f1 : L1 ( m, c u T )w 3 D w * → L1 ( m, c u T )w in the following manner: let (x k ,
pk , w k ) [ L1 ( m, c u T )w 3 D w * 3 L1 ( m, c u T )w be a sequence such that w k [ f1 (x k , pk ),
¯ pk → p,
¯ and w k → w.
¯ Now by the weak sequential convergence theorem of
x k → x,
¯ [ cl co Lshw k (t)j on T \T a for some T a [ T with
Yannelis (1989), we have w(t)
m (T a ) 5 0, where Ls is taken with respect to the weak topology on -. Note that
w k (t) [ H1 (t, x k , pk ) on T \T k for some T k [ T with m (T k ) 5 0. Then, for t [ T \[T a <
¯ p¯ ).
( < `k 51 T k )], if j [ Ls hw k (t)j, there exists a subsequence (w k i (t), x k i , pk i ) → ( j , x,
Recall that H1 (t, ? , ? ) is u.s.c. and closed valued, and since X(t) is a regular topological
¯ p¯ ), and hence Ls
space, it follows that H1 (t, ? , ? ) is closed. Consequently, j [ H1 (t, x,
¯ p¯ ). Since H1 (t, x,
¯ p¯ ) is closed convex, we obtain w(t)
¯ [ H1 (t, x,
¯ p¯ ) for
hw k (t)j , H1 (t, x,
¯ p¯ ).
all t [ T \[T a < ( < k`51 T k )], and therefore, w¯ [ f1 (x,
We also define f2 5 cl co H2 , f3 5 cl co H3 .
Proposition 2. Let - be a Hausdorff locally convex space and let C , - be a compact
convex subset. If V , - is an open neighborhood of C, then there exists an open convex
subset V 9 of - such that C ,V 9 , cl V 9 ,V.
See Appendix A for the proof.
Proposition 2 implies that f2 , f3 are a non-empty compact convex valued u.s.c.
correspondence, and in particular, are closed (cf. Noguchi, 1997a, Proposition 6, p. 12).
We apply the fixed point theorem (Fan, 1952, Theorem 1, p. 122) to f 5 f1 3 f2 3 f3 :
& → &, we obtain a fixed point (xc , yc , pc ). By the standard Shafer–Sonnenschein
argument with little care given to the treatment of f2 , f3 (see Noguchi, 1997a, p. 15),
(xc , yc , pc ) is seen to have the following properties (for a detailed argument, see
Appendix B):
1. for almost all t [ T, xc (t) [ Arg Maxhu(t, j , xc ): j [ A 1 (t, pc )j;
2. eS k pc , yc (s)l $ eS k pc , z(s)l for all z [ L1 (p, c u S );
3. k pc , (eT xc (t) 2 eS yc (s) 2 eT e(t)l $ kq, (eT xc (t) 2 eS yc (s) 2 eT e(t)l for all q [ D.
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
167
Let Tc [ T be the exceptional set in (1) with respect to xc . We have the following
proposition whose proof appears in Noguchi (1997a, Proposition 12, p. 16):
Proposition 3. For almost all s [ S, pc ( yc (s)) $ pc (h ) for all h [ c (s).
Proposition 3 implies that:
P (s, pc ) 5 k pc , yc (s)l
(3.1)
for almost all s [ S. It follows that A 1 (t, pc ) 5 B c (t, pc , yc ), where B c (t, p, y) 5 B(t, p,
y) > c (t) for (t, p, y) [ T 3 D 3 L1 (p, c u S ). Eq. (3.1) implies that for all t [ T \Tc :
E
k pc , xc (t)l # u (t, s)k pc , yc (s)l 1 k pc , e(t)l
(3.2)
S
and also,
P c (t, xc ) > B c (t, pc , yc ) 5 5
where P c (t, x) 5 h j [ c (t): u(t, j , x) . u(t, x(t), x)j for x [ L1 ( m, c u T ). We integrate both
sides of (3.2) with respect to t. Observe that by (A.6), we can exchange the order of
integration and obtain k pc , eT xc (t)l # k pc , (eS yc (s) 1 eT e(t))l. Combining this with (3)
above, we have:
Kq, SE x (t) 2E y (s) 2E e(t)DL # 0
c
T
c
S
(3.3)
T
for all q [ D.
Consider a net (xc , yc , pc ) [ L1 ( m, X)w 3 L1 (p, Y)w 3 D w * . We can extract a
¯ yc (m) → y,
¯ and, pc (m) → p.
¯
convergent subnet (xc (m) , yc (m) , pc (m) ) such that xc (m) → x,
¯ ¯ ) [ F.
By Remarks 3 and 4, and (3.3), we obtain (x,y
Recall that P(t, x) 5 h j [ X(t): u(t, j , x) . u(t, x(t), x)j.
Proposition 4. Let z [ L1 (p, Y). There exists T z [ T with m (T z ) 5 0 such that for all
¯ j l $ kp,
¯ eS u (t, s)z(s)l 1 kp,
¯ e(t)l.
t [ T \T z , j [ P(t, x¯ ) implies kp,
See Appendix A for the proof.
Proposition 5. Let (x, y, p) [ L1 ( m, X) 3 L1 (p, Y) 3 D such that (x, y) [ F. Let
z [ L1 (p, Y). If there exists T z [ T with m (T z ) 5 0 such that for all t [ T \T z , j [ P(t, x)
implies k p, j l $ k p, eS u (t, s)z(s)l 1 k p, e(t)l, then (x, y, p) is an equilibrium for %.
See Appendix A for the proof.
Proposition 5 clearly completes the proof of the Main Theorem.
168
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
3. Concluding remarks
We remark that our approach extends to economies with commodity-price paring (l ` ,
l 1 ) in which u is not interdependent and u(t, ? ) is Mackey-continuous (weak*
continuous) on X(t) , K, where K is a closed ball in l ` , and convex in the sense of
(A.5)(b). At this point, we are uncertain about the possibility of extending our approach
to cover economies with an interdependent utility function and the commodity-price
paring (l ` , l 1 ), or to cover economies with a continuum of commodities (M, M, i ),
where (M, M, i ) can be assumed to be separable and totally s -finite.
Acknowledgements
The author is indebted to anonymous referees of MASS for helpful comments and
suggestions. As a matter of fact, the use of Mazur’s lemma, which enabled us to extend
the earlier result in the original draft to cover interdependent preferences was suggested
by one of the referees. The author also wishes to thank Prof. K. Urai for reading the
original draft.
Appendix A. Proofs
A.1. Proof of Proposition 1
Fix p [ D, and define a(t) 5 A 1 (t, p). We first show that a: T → - admits a
measurable graph. Define g(t, j ) 5 k p, j l 2 eS u (t, s)P (s, p) 2 k p, e(t)l. Recall that (S,
S, p ) is complete, P bounded, and P ( ? , p) S-measurable. It follows that Fubini’s
theorem, (A.6), and (A.7) imply that g( ? , j ) is T-measurable. Note also that since g(t,
? ) is continuous on -, Castaing and Valadier (1977, Lemma III.14, p. 70) implies that g
is T ^ @ ( - ) measurable. Recall that Gc u T [ T ^ @ ( - ). Since Ga 5 Gc u T > h(t, j ) [
T 3 - : g(t, j ) # 0j, we obtain Ga [ T ^ @ ( - ).
Recall from (A.5)(e) that for every x [ L1 ( m, X), u( ? , ? , x) is measurable on
¯ ? , ? , x) be an extension of u( ? , ? , x) to the entire T 3 - as a
GX [ T ^ @ ( - ). Let u(
T ^ @ ( - ) measurable function. Since - is Suslin, (T, T, m ) complete, and H1
non-empty valued, Castaing and Valadier (1977, Lemma III.39, p. 86) is applicable to
u¯ ( ? , ? , x) and A 1 , and we obtain GH 1 (?,x, p) [ T ^ @ ( - ). h
A.2. Proof of Proposition 2
´ (1966, p. 145), we can find a neighborhood U of zero such
By Lemma 2 in Horvath
that (C 1 U ) > (V c 1 U ) 5 5. Since - is locally convex, we may assume that U is an
open convex neighborhood of zero. Then V 9 5 C 1 U has the required properties. h
A.3. Proof of Proposition 4
We first establish the following claim:
Claim 1. Let s be a measurable selection of X and let z [ L1 (p, Y). There exists a
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
169
m -null subset Tz, s , T such that if t lies in the set ht [ T : s (t) [ P(t, x¯ )j > (T \Tz, s ), then:
E
¯ s (t)l $ u (t, s)kp,
¯ z(s)l 1 kp,
¯ e(t)l
kp,
S
Proof of Claim 1. Note that:
c5
H
cohe(t) < w(t) < s (t)j
coh0 < z(s)j
for t [ T
for s [ S
lies in (, and hence there exists m 1 [ } such that c (m 1 ) $ c. Thus, for every m $ m 1 ,
we obtain:
1. s is a measurable selection of c (m)u T ;
2. z is a measurable selection of c (m)u S .
Consider the obvious convergent subnet obtained by restricting } to } 9 5 hm [ } :
¯ y,
¯ p¯ ). Observe that since L1 ( m,
m $ m 1 j and denote it also by (xc (m) , yc (m) , pc (m) ) → (x,
X)w 3 L1 (p, Y)w 3 D w * is metrizable, we can extract a convergent sequence (xc (m(n)) ,
¯ y,
¯ p¯ ). Recall that for each n, xc (m(n)) [ B c (m(n)) (t, pc (m(n)) , yc (m(n)) )
yc (m(n)) , pc (m(n)) ) → (x,
c (m(n))
and P
(t, xc (m(n)) ) > B c (m(n)) (t, pc (m(n)) , yc (m(n)) ) 5 5 outside some m -null subset
T n 5 Tc (m(n)) , T. Observe that the construction of the sequence (xc (m(n)) , yc (m(n)) ,
pc (m(n)) ) depends upon the choice of z and s. For the sake of simplicity, we write (x n , y n ,
¯ r) be an open ball centered at x¯ with radius r.
pn ) 5 (xc (m(n)) , yc (m(n)) , pc (m(n)) ). Let B(x;
By Mazur’s lemma we can choose a convex combination A 1 [ co hx n : 1 # n # n 1 j >
¯ 1) for some n 1 . Repeating the same argument for the weakly convergent sequence
B(x;
¯ we can choose a convex combination A 2 [ cohx n : n 1 1 1 # n # n 2 j > B(x;
¯ ]12 )
x n 1n 1 → x,
for some n 2 . Inductively, we can construct a sequence A i such that A i → x¯ in the norm
topology and the terms appearing in A i are all strictly greater than those appearing in
¯ in X(t) for all t [ T \T a for some T a [ T with
A i 21 . We may assume that A i (t) → x(t)
m (T a ) 5 0. Let T 5c be the exceptional set in (A.5)(c) for all A i . Define T z, s 5 < `n 51 T n <
T a < T 5c . Let t be a point in the set ht [ T : s (t) [ P(t, x¯ )j > (T \T z, s ). Note that by
(A.5)(c), we have for every A i , u(t, x n i (t), x n i ) # u(t, A i (t), x n i ) for some x n i appearing in
¯
the convex combination A i . If s (t) [ P(t, x¯ ), or u(t, s (t), x¯ ) . u(t, x(t),
x¯ ), then since
¯
(A i (t), x n i ) → (x(t),
x¯ ) in X(t) 3 L1 ( m, X)w , by (A.5)(a) we have u(t, s (t), x n i ) . u(t, A i (t),
x n i ) $ u(t, x n i (t), x n i ), or s (t) [ P(t, x n i ) for all i $ i 0 for some i 0 . Thus, for i $ i 0 , we
have:
1. s (t) [ P(t, x n i );
2. s is a measurable selection of c (m(n i ))u T ;
3. z is a measurable selection of c (m(n i ))u S .
Combining 1. and 2. we have:
s (t) [ P c (m(n i )) (t, xn i )
(4.1)
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
170
for i $ i 0 . Since Proposition 3 implies that pn i ( y n i (s)) $ pn i (h ) for all h [ c (m(n i ))(s), for
almost all s [ S, we have, by 3. pn i ( y n i (s)) $ pn i (z(s)) for almost all s [ S, and hence:
E u(t, s)k p , y (s)l $E u(t, s) k p , z(s)l
ni
ni
S
ni
(4.2)
S
On the other hand, (4.1) implies that:
k pn i , s (t)l . k pn i , x n i (t)l
(4.3)
(Recall that t [ T \T n i .) Let j l 5 (1 2 l)x n i (t) 1 ls (t). If it were true that:
E
k pn i , x n i (t)l , u (t, s) k pn i , y n i (s)l 1 k pn i , e(t)l
S
there would exist l¯ [ (0, 1] such that:
E
k pn i , j l¯ l , u (t, s)k pn i , y n i (s)l 1 k pn i , e(t)l
S
c (m(n i ))
i.e. j l¯ [ B
(t, pn i , y n i ). By (A.5)(b), we have j l¯ [ P c (m(n i )) (t, x n i ), contradicting
t[
⁄ Tn i . Thus, we have:
E
k pn i , x n i (t)l $ u (t, s)k pn i , y n i (s)l 1 k pn i , e(t)l
S
for all i $ i 0 , and combining this with (4.2) and (4.3), we obtain
E
k pn i , s (t)l . u (t, s) k pn i , z(s)l 1 k pn i , e(t)l
S
for all i $ i 0 . Since pn i → p¯ in D w * , and noting that eS u (t, s)k p, z(s)l 5 k p, eS u (t, s)z(s)l
for all p [ D, we deduce that:
E
¯ s (t)l $ u (t, s)kp,
¯ z(s)l 1 kp,
¯ e(t)l h
kp,
S
Proof of Proposition 4 (continued). We have X(t) 5 clhsk (t)j for all t, where sk are
measurable selections of X. Applying Claim 1 to each sk , we can define T z 5 < k`51
T z, sk . Let t [ T \T z and j [ P(t, x¯ ). We can choose a convergent subsequence sk i (t) → j ,
and since P(t, x¯ ) is norm-open in X(t), sk i (t) [ P(t, x¯ ) for sufficiently large i. Since t lies
in the set ht [ T : sk i (t) [ P(t, x¯ )j > (T \T z, sk ) for sufficiently large i, our result
i
follows. h
Remark 5. The author is indebted to an anonymous referee of MASS, who pointed out
that the use of Mazur’s lemma is possible for producing hx n i j at the crucial step in the
above proof. In the original draft, u was not assumed to be interdependent, and the
weak sequential convergence theorem of Yannelis (1989) was used. Mazur’s lemma
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
171
allowed us to generalize the early results to the case in which u is interdependent. Note
that Mazur’s lemma does not have a weak* counterpart, but the weak sequential
convergence theorem of Yannelis (1989) does and can be used for treating economies
with commodity-price paring (l ` , l 1 ) in our framework.
Proof of Proposition 5. Define T 9 5 ht [ T : u(t, j , x) . u(t, x(t), x) for some j [ X(t)j.
If t [ T 9 > (T \T z ), we can extract a sequence hi [ P(t, x) such that hi → x(t). Then we
have for every t [ T 9 > (T \T z ):
K E u(t, s) z(s)L 1 k p, e(t)l
k p, x(t)l $ p,
(5.1)
S
In fact, by (A.5)(d), (5.1) holds for all t [ T \(T z < T 5d ), where T 5d is the exceptional set
in (A.5)(d) with respect to x. Let Z 5 ht [ T \(T y < T 5d ): k p, x(t)l . k p, eS u (t, s)y(s)l 1
k p, e(t)lj. Clearly, Z [ T. If m (Z) . 0, we have eT k p, x(t) 2 eS u (t, s)y(s) 2 e(t)l $ eZ k p,
x(t) 2 eS u (t, s)y(s) 2 e(t)l . 0, which contradicts (x, y) [ F. Hence, m (Z) 5 0, and
consequently:
K E u(t, s)y(s)L 1 k p, e(t)l
k p, x(t)l 5 p,
(5.2)
S
for all t [ T \(T y < T 5d < Z), where m (T y < T 5d < Z) 5 0. In particular, x(t) [ B(t, p, y)
for all t [ T \(T y < T 5d < Z). Combining (5.1) and (5.2), we obtain:
k p,
E u(t, s) y(s)L $ K p, E u(t, s) z(s)L
S
(5.3)
S
for all t [ T \(T y < T z < T 5d < Z). We integrate both sides of (5.3) with respect to t and
apply Proposition 12 in Noguchi (1997a, p. 16) with ( p, y) instead of ( p* , y * ). We then
have:
k p, y(s)l $ k p, h l
(5.4)
for all h [ Y(s), for almost all s [ S.
We next show that for t [ T \(T y < T 5d ), if j [ X(t) satisfies:
E
k p, j l # u (t, s) k p, y(s)l 1 k p, e(t)l
S
then j [
⁄ P(t, x). We may assume that t [ T 9 > (T \(Ty < T 5d )) since P(t, x) 5 5 otherwise.
Since from (5.4), we have k p, y(s)l $ 0 for almost all s [ S, (A.8) implies that
k p,w(t)l , eS u (t, s)k p, y(s)l 1 k p, e(t)l. Let j l 5 (1 2 l)w(t) 1 lj . Then, for each
l [ (0, 1), we have k p, j l l , eS u (t, s)k p, y(s)l 1 k p, e(t)l, which means that j l [
⁄ P(t, x)
for all l [ (0, 1). Since P(t, x) is open, we deduce that j [
⁄ P(t, x). This and (5.4)
complete the proof since (T, T, m ) is assumed to be complete [(A.1)]. h
172
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
Appendix B. Properties of (xc , yc , pc )
We have yc [ f2 ( yc , pc ) 5 cl co H2 ( yc , pc ). We claim that P2 ( yc , pc ) 5 5; assume
the contrary, that there exists a w [ P2 ( yc , pc ). Then, u 2 (( yc , pc ), w) . 0, and
consequently, u 2 (( yc , pc ),w9) . 0 for all w9 [ H2 ( yc , pc ). Thus, we have H2 ( yc ,
pc ) , P2 ( yc , pc ). Note that P2 ( yc , pc ) is an open convex subset of L1 (p, c u S )w . We have
the following lemma:
Lemma 1. Let - be a Hausdorff locally convex space and let A , - be a compact
convex subset. Let C , A be a compact subset and let V be an open convex
neighborhood of C in A. Then we have cl co C ,V.
Proof of Lemma 1. Choose an open subset V˜ , - such that V 5 V˜ > A. By Lemma 2 in
´ (1966, p. 145), we can find a neighborhood U of zero such that (C 1 U ) >
Horvath
c
(V˜ 1 U ) 5 5. Since - is locally convex, we can assume that U is a closed convex
˜ Since C is compact, we have
neighborhood of zero. Then we have C 1 U , V.
N
C , < i 51 (x i 1 U ), where x i [ C, i 5 1, . . . , N. Let Ci 5 (x i 1 U ) > A. Note that Ci is a
compact convex subset of -. By Jameson (1970, p. 208), co( < Ni 51 Ci ) is compact, and
in particular, closed. Therefore, cl co C , cl co ( < iN51 Ci ) 5 co( < Ni 51 Ci ). Note that
˜ and co( < N C ) , co A 5 A. Thus cl co
co( < iN51 Ci ) , co(C 1 U ) , co V˜ 5 V,
i 51
i
C , V˜ > A 5V. h
By Lemma 1, we have cl co H2 ( yc , pc ) , P2 ( yc , pc ), and consequently, yc [ P2 ( yc ,
pc ), which is a contradiction. The same argument applies to f3 also.
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www.elsevier.nl / locate / econbase
Economies with a measure space of agents and a separable
commodity space
Mitsunori Noguchi*
Department of Economics, Faculty of Commerce, Meijo University, 1 -501 Shiogamaguchi Tenpaku-ku,
Nagoya 468 -8502, Japan
Received January 1999; received in revised form June 1999; accepted September 1999
Abstract
We prove the existence of an equilibrium in an economy with a measure space of agents and a
separable Banach commodity space whose positive cone admits an interior point. We follow the
truncation argument given at the end of Yannelis [Yannelis, N.C., 1987. Equilibria in noncooperative models of competition, J. Econ. Theory 41, 96–111] and the abstract economy
approach as in Shafer [Shafer, W., 1976. Equilibrium in economies without ordered preferences or
free disposal, J. Math. Econ. 3, 135–137] and Khan and Vohra [Khan, M.A., Vohra, R., 1984.
Equilibrium in abstract economies without ordered preferences and with a measure space of
agents, J. Math. Econ. 13, 133–142], which allows preferences to be interdependent. Our result
may be viewed as an extension of the result in Kahn and Yannelis [Khan, M.A., Yannelis, N.C.,
1991. Equilibria in markets with a continuum of agents and commodities. In: Khan, M.A.,
Yannelis, N.C. (Eds.), Equilibrium Theory in Infinite Dimensional Spaces, Springer-Verlag, Tokyo,
pp. 233–248] employing production and allowing preferences to be interdependent. We utilize
Mazur’s lemma at the crucial point in the truncation argument. We assume that the preference
correspondence is representable by an interdependent utility function. The method in the present
paper does not rely on the usual weak openness assumption on the lower sections of the preference
correspondence. 2000 Elsevier Science B.V. All rights reserved.
Keywords: General equilibrium; Infinite dimensional commodity space; Measure space of agents; Separable
Banach space; Fixed point theorem
JEL classification: D51
*Tel.: 181-56-138-8514; fax: 181-52-833-4767 (office), 181-56-138-8514 (home).
E-mail address: [email protected] (M. Noguchi)
0165-4896 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 99 )00046-3
158
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
1. Introduction
The author has proven the existence of equilibria for economies with a measure space
of agents (T, T, m ), a separable Banach commodity space - whose positive cone admits
an interior point, and an interdependent preference correspondence P: T 3 L1 ( m,
X) → -, where L1 ( m, X) is the set of Bochner integrable selections of the consumption
correspondence X: T → - (Noguchi, 1997a,b). In order to utilize a fixed point argument,
the author requires X to be weakly compact valued so that L1 ( m, X) becomes weakly
compact in L1 ( m, - ), where L1 ( m, - ) is the Banach space of Bochner integrable
functions f : T → - (Yannelis, 1991, Theorem 3.1, p. 7). In both Noguchi (1997a) and
Noguchi (1997b) the author assumes that for almost all t [ T, P(t, ? ): L1 ( m, X) → admits weakly open lower sections. In the present paper we consider the preference
correspondence P having the representation P(t, x) 5 h j [ X(t): u(t, j , x) . u(t, r (x)(t),
x)j, where u is an interdependent utility function, and r is a choice function selecting a
representative from each equivalence class x [ L1 ( m, - ). For notational simplicity we
omit r in the sequel. It may appear that such P is already covered by the author’s
previous papers, but it is not immediately clear whether the lower sections of P(t, ? ) are,
in general, weakly open or not. The difficulty is present even in the simplified case P(t,
x) 5 h j [ X(t): u(t, j ) . u(t, x(t))j, where u(t, j ) is assumed to be norm continuous and
quasi-concave in j . For example, one might try to make use of Mazur’s lemma [we
assume (T, T, m ) to be separable so that the weak topology on L1 ( m, X) becomes
metrizable] in order to show that for almost all t [ T, hx [ L1 ( m, X): u(t, j ) # u(t, x(t))j
is weakly closed. Then one would proceed as follows: For each t [ T, define E(t) 5 hh [
X(t): u(t, j ) # u(t, h )j, which is, under the assumptions, closed and convex. We must find
an exceptional set T 0 [ T with m (T 0 ) 5 0 such that for every t [ T \T 0 , if x n converges
¯ [ E(t). We have x¯ [ cl cohx n j and
weakly to x¯ in L1 ( m, X) and x n (t) [ E(t), then x(t)
¯ [ cl cohx n (t)j , E(t) for all t [ T \T x¯ for some m -null subset
consequently, we obtain x(t)
T x¯ [ T which depends on x¯ (x n may be a subsequence of the original sequence). Let
L1 ( m, E) denote the set of (equivalence classes of) integrable selections of t → E(t),
which is weakly compact in L1 ( m, - ) for the same reason as L1 ( m, X) is. We may wish
to define T 0 5 < x¯ [L 1 ( m ,C ) T x¯ , but we do not have any knowledge about the measure
theoretic nature of the set T 0 .
As in Noguchi (1997a), we assume that - is an ordered separable Banach space in
which the positive cone -1 admits an interior point a and is closed, proper. The interior
* of -1 by
point a allows us to define a price simplex D in the dual cone - 1
* : k p, al 5 1j, where k ? , ? l is the natural coupling on - * 3 -. The
D5hp [- 1
properness and closedness of -1 guarantee the ‘right property’ of D: Let j [ -. if k p,
j l $ 0 for all p [ D, then j [ -1 .
Observe that the coupling ( p, x) → eT k p, x(t)l is not, generally, jointly continuous
with respect to the weak* topology on D and the weak topology on L1 ( m, X), and this
fact causes an insurmountable problem in the use of a Shafer–Sonnenschein type
argument, which we rely on for showing the existence of maximal elements of the
correspondences defined in terms of such coupling. However, it can be shown that with
the additional assumption of X being norm-compact valued, the above coupling becomes
jointly continuous, and henceforth, the standard argument becomes applicable. Needless
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
159
to say, the above condition of every consumption set X(t) being norm-compact is too
stringent to be acceptable, and in order to remedy this problem, we consider the set of
norm-compact valued sub-correspondences c (in fact, the set of polytope valued
sub-correspondences) of the weakly compact valued consumption correspondence X.
Such c with some additional technical properties form a non-empty directed set with the
obvious inclusion as an ordering. For the sake of simplicity, let us ignore production for
the time being. For each c, we have a sub-economy % c of the original economy %,
where we can utilize some well-known methods in the literature for obtaining an
‘equilibrium’ (xc , pc ) [ L1 ( m, c ) 3 D.
Now we have a net of equilibria (xc , pc ) [ L1 ( m, X) 3 D, where L1 ( m, X) 3 D
becomes a compact metric space with respect to the weak topology on L1 ( m, X) and the
weak* topology on D. Based on the compactness, we can extract a convergent subnet
¯ p¯ ). We must show that (x,
¯ p¯ ) gives rise to an equilibrium for the
(xc (m) , pc (m) ) → (x,
original economy %. To this end, we need the following crucial proposition: There exists
a m -null set T 0 [ T such that for every consumer t lying outside T 0 , if j [ X(t) is
¯
preferred to x(t),
then the price of j is greater than or equal to the price of the initial
endowment e(t) of t. Once this is established, the rest is simply a routine matter.
The heart of the problem is verification of the following statement: There exists a
¯
m -null exceptional set T 0 [ T such that for every t [ T \T 0 , if j [ P(t, x¯ ), then kp,
¯
j l $ kp, e(t)l. Once we have the above statement justified, the rest follows from the
¯ by such j and pass to the limit
standard argument. For example, we can approximate x(t)
¯
¯
¯
¯
to obtain kp, x(t)l $ kp, e(t)l, which actually becomes the equality showing that x(t)
¯
satisfies the budget constraint. We can also show that for almost all consumers t [ T, x(t)
¯
maximizes u(t, ? , x ) over the budget set.
If P(t, ? ) had open lower sections with respect to the weak topology on L1 ( m, X), we
would have that j [ P(t, xc ) for sufficiently large c. We can also arrange the directed set
c
hc j so that j [ c (t) for sufficiently large c, and consequently, j [ P (t, xc ) 5 P(t,
xc ) > c (t) for sufficiently large c. Then by utilizing the property of (xc , pc ) being an
c
‘equilibrium’ in % with some m -null exceptional set Tc , we conclude that k pc ,
j l $ k pc , e(t)l for t [ T \Tc . Thereafter, we hope to be able to modify hc j so that
¯ we obtain kp,
¯
< c Tc [ T with m ( < c Tc ) 5 0. Now by passing to the limit pc → p,
¯ e(t)l m -a.e.
j l $ kp,
Coming back to the present setting, since the weak topology on L1 ( m, X) and the
weak* topology on D are both metrizable, we can extract a convergent sequence (x n ,
¯ p¯ ). Then, as a consequence of Mazur’s lemma, we have
pn ) 5 (xc (m(n)) , pc (m(n)) ) → (x,
¯ [ cl cohx n (t)j outside some m -null set T a . This does not quite say that x n (t) → x(t)
¯
x(t)
outside T a , but we can find a subsequence x n i such that j [ P(t, x n i ) for sufficiently large
i. Note that prior to the extraction of sequence (x n , pn ), we could have arranged the
directed set hmj so that j [ c (m) for all m, and hence j [ c (m(n i )) for all i. Now we
have j [ P c (m(n i )) (t, x n i ) for sufficiently large i, for every t [ T \T a . We stress the fact that
the choice of sequence (x n , pn ) depends on j and hence T a < ( < n`51 Tc (m(n)) ) also
`
depends on j , so we should write Tj 5 T a < ( < n51
Tc (m(n)) ). Repeating the same
argument as in the case that P(t, ? ) was assumed to admit open lower sections, we
¯ j l $ kp,
¯ e(t)l. This is
obtain the following statement: If j [ P(t, x¯ ) and t [ T \Tj , then kp,
not exactly what we intended to derive since the exceptional set depends on j .
160
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
In order to overcome the above difficulty, we consider a countable collection of
measurable selections hsk j of X such that hsk (t)j is dense in X(t) for all t [ T. This
allows us to work on a measurable selection s instead of j since j can be approximated
by sk . Now let c0 be an element of hc j and consider the convex hull c1 5 cohs, c0 j,
which clearly lies in hc j. Note that there exists m 0 such that for all m $ m 0 , we have
c (m) $ c1 . Then we restrict hmj to those elements that are larger than m 0 and obtain a
directed set with the property we need. Repeating the previous argument again with sk
¯ sk (t)l $ kp,
¯ e(t)l.
instead of j , we obtain that if sk (t) [ P(t, x¯ ) and t [ T \Tsk , then kp,
¯ sk (t)l $ kp,
¯
Note that this means that for every t [ T \( < `k 51 Tsk ), if sk (t) [ P(t, x¯ ), kp,
`
e(t)l. Now define T 0 5 < k51
Tsk . Let t [ T \T 0 . If j [ P(t, x¯ ), we can choose a
convergent subsequence sk i (t) → j . Since P(t, x¯ ) is open, sk i (t) [ P(t, x¯ ) for sufficiently
¯ j l $ kp,
¯ e(t)l.
large i. Then by passing to the limit i → `, we finally obtain kp,
Work relevant to the present paper was previously done by Yannelis (1987), Bewley
(1991), Khan and Yannelis (1991), and Podczeck (1997). Yannelis (1987) discusses the
existence of an equilibrium in an abstract economy with a continuum of agents and a
separable Banach strategy space (Yannelis, 1987, Remark 6.4, p. 108), explains how the
abstract economy approach can be utilized to show the existence of an equilibrium in the
exchange economy with norm compact consumption sets, and discusses the type of
truncation argument which is used in the present paper (Yannelis, 1987, Remark 6.5, p.
108). Bewley (1991) proves the existence of an equilibrium in an economy with
commodity-price paring (l ` , l 1 ) and a continuum of agents. Khan and Yannelis (1991)
and Podczeck (1997) prove the existence of an equilibrium in the exchange economy
with a continuum of agents and a separable Banach commodity space. They follow the
excess demand approach, which allows them, with some measure theoretic restrictions
on the measure space of agents, to dispense with the convexity assumption on
preferences by making use of an infinite dimensional version of Lyapunov’s theorem
(see Rustichini and Yannelis, 1991; Podczeck, 1997).
In the present paper, we follow the abstract economy approach as in Shafer (1976)
and Khan and Vohra (1984), which allows preferences to be interdependent. Our result
may be viewed as an extension of the Kahn–Yannelis result (Khan and Yannelis, 1991)
allowing preferences to be interdependent by the use of Mazur’s lemma and employing
production. We also present a concrete example in which our Main Theorem is
applicable. This example demonstrates under some additional conditions that if the size
of initial endowments e(t) and production possibility sets is small relative to the
universal bound on consumption sets X(t), for almost all consumers t, the equilibrium
¯ actually avoids the satiation points due to the ‘boundary’ of X(t).
consumption x(t)
1.1. Definitions
For a linear space - and a subset A , -, co A denotes the convex hull of A. For a
topological space - and a subset A , -, cl A denotes the closure of A and int A denotes
the interior of A. Let -, = be any sets and f : - → = a correspondence. When = is a
linear space, co f : - → = denotes the correspondence defined by the relation co
f (x) ; co [f (x)] for each x [ -, and when = is endowed with a topology cl f : - → =
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
161
denotes the correspondence defined by cl f (x) ; cl [f (x)] for each x [ -. Gf denotes
the graph of f, i.e. Gf 5 h(x, y) [ - 3 = : y [ f (x)j. Let - and = be paired linear
spaces. Following the literature, s ( -, = ) denote the weak topology on -. For a Banach
space -, - * denotes the dual of -, and i ? i - the norm of -. When = is the dual of
some Banach space, = w * denotes = endowed with the weak* topology, and for a subset
A , =, Aw * denotes the set A endowed with the topology induced by the weak*
topology on =. Let -, = be any topological spaces. A correspondence f : - → = is
said to be upper semi-continuous (u.s.c.) at x [ - if for every neighborhood V of f (x),
there exists a neighborhood U of x such that f (x9) ,V for every x9 [ U. The
correspondence f : - → = is said to be upper semi-continuous (u.s.c.) if it is u.s.c. at
every x [ -. A correspondence f : - → = is said to be lower semi-continuous (l.s.c.) at
x [ - if for every open set V in = with f (x) >V ± 5, there exists a neighborhood U of
x such that f (x9) >V ± 5 for every x9 [ U. The correspondence f : - → = is said to be
lower semi-continuous (l.s.c.) if it is l.s.c. at every x [ -. A correspondence f : - → =
is said to be continuous (at x [ - ) if it is both u.s.c. and l.s.c. (at x [ - ). A
correspondence f : - → = is said to have an open graph if Gf is an open subset of
- 3 =. Let (T, T) be a measurable space. A correspondence f : T → = is said to admit
a measurable graph if Gf [ T ^ @ ( = ), where @ ( = ) denotes the Borel sigma algebra
on =. For sigma algebras t1 and t2 , t1 ^ t2 denotes the product sigma algebra. A
correspondence f : T → = is said to be lower measurable if the set ht [ T : f (t) >V ±
5j belongs to T for every open set V , =. For measurable spaces (T, T), (S, S), and a
function f : T → S, f is said to be (T, S)-measurable whenever it is measurable with
respect to the sigma algebras T and S. Let (T, T, m ) be a finite measure space and - a
Banach space. L1 ( m, - ) denotes the space of equivalence classes of - -valued Bochner
integrable functions f : T → - normed by i f i 5 eT i f(t)i - (for details see Diestel and
Uhl, 1977). For a Banach space - and its dual - *, k ? , ? l denotes the natural paring.
For a Banach space - and a correspondence f : T → -, define L1 ( m, f ) 5 h f [ L1 ( m,
- ): f(t) [ f (t) m -a.e.j. Let R denotes the set of real numbers. A correspondence f :
T → - is said to be integrably bounded if there exists a function h [ L1 ( m, R) such that
suphi j i - : j [ f (t)j # h(t) m -a.e. For an ordered vector space - with order # , we write
for a, b [ -, [a, b] 5 h j [ - : a # j # bj, (a, b] 5 h j [ - : a # j # b, a ± j j, etc.
2. Main theorem
Throughout this paper, all measures are assumed to be positive, and any version of
definitions of measurability and integrability of functions can be adopted as long as they
are compatible with those of Tulcea and Tulcea (1969). Our commodity space is an
ordered separable Banach space - whose positive cone -1 is closed and has an interior
* is non-trivial. [If -1 is proper, then
point a. We further assume that the dual cone - 1
* ± (0) (see Jameson, 1970, p. 231).] Recall that L1 ( m, - ) is a set of equivalence
-1
classes of functions. Let r be a choice function selection one representative function
from each class x [ L1 ( m, - ). When there is no fear of confusion, we simply write x for
r (x).
162
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
An economy % is a tuple [(T, T, m ), X, u, e, (S, S, p ), Y, u ] which is characterized
by:
1.
2.
3.
4.
5.
6.
7.
a measure space of consumers (T, T, m );
a consumption correspondence X: T → - ;
an interdependent utility function u: GX 3 L1 ( m, X) → R;
an initial endowment e: T → -, where e(t) [ X(t) for all t [ T;
a measure space of producers (S, S, p );
a production correspondence Y: S → - ;
a share u : T 3 S → R, where u (t, s) denotes the share of consumer t in the profit of
producer s.
Let ‘\’ denote the set theoretic subtraction. Let -1 be the usual positive cone of -, and
- *1 the positive cone induced by -1 on - *. Denote the budget set of consumer t at
price p [ - *
1 \(0) and production plan y [ L 1 (p, Y) by B(t, p, y) 5 h j [ X(t):k p, j l # k p,
eS u (t, s)y(s)l 1 k p, e(t)lj. An attainable state is a pair (x, y) [ L1 ( m, X) 3 L1 (p, Y) such
that eT x(t) 2 eS y(s) 2 eT e(t) [ 2 -1 . Denote the set of attainable states by F.
A competitive equilibrium for % is a price p¯ [ - *
1 \(0) together with an attainable
¯ y¯ ) [ F satisfying:
state (x,
¯ [ B(t, p,
¯ y¯ ) and P(t, x¯ ) > B(t, p,
¯ y¯ ) 5 5, where P(t,
1. For almost all t [ T, x(t)
¯
x¯ ) 5 h j [ X(t): u(t, j , x¯ ) . u(t, x(t),
x¯ )j, and 5 denotes the empty set. [Note that this
¯
condition is well-defined regardless of the particular representative chosen for x.]
¯ y(s)l
¯
¯ h l for all h [ Y(s).
2. For almost all s [ S, kp,
$ kp,
Define for each t [ T, a (t) 5 heS u (t, s)z(s) [ - : z [ L1 (p, Y)j. Let Xˆ 5 hx [ L1 ( m, X):
(x, y) [ F for some y [ L1 (p, Y)j.
We now state the set of assumptions needed for the proof of our Main Theorem.
(A.1) (T, T, m ) is a complete finite separable measure space.
(A.2) (S, S, p ) is a complete finite separable measure space.
(A.3) X: T → - is an integrably bounded, weakly compact, convex, non-empty
valued correspondence such that GX [ T ^ @ ( - ).
(A.4) Y: S → - is an integrably bounded, weakly compact, convex, non-empty
valued correspondence such that GY [ S ^ @ ( - ).
(A.5) u: GX 3 L1 ( m, X)w → R is a function such that:
(a) For every t [ T, u(t, ? , ? ) is jointly continuous on X(t) 3 L1 ( m, X)w .
(b) For every t [ T and x [ L1 ( m, X), if j 1 and j 2 are two points of X(t) such that
u(t, j 1 , x) , u(t, j 2 , x), and l is a real number in (0,1], then u(t, j 1 , x) , u(t,
(1 2 l)j 1 1 lj 2 , x). (As Debreu, 1982, p. 705, pointed out, this condition implies
that u(t, ? , x) is quasi-concave.)
(c) If o ni 51 li x i is a convex combination of x i [ L1 ( m, X), i 5 1, . . . , n, there
exists x i 0 [ hx i j, i 5 1, . . . ,n, such that for almost all t [ T, we have u(t, x i 0 (t),
x i 0 ) # u(t, o in51 li x i (t), x i 0 ).
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
163
ˆ either u(t, j , x) . u(t, x(t), x) for some j [ X(t) or x(t) 2
(d) For each x [ X,
a (t) 2 e(t) , -1 holds for almost all t [ T. (Note that this condition is welldefined independent of the choice of a representative for x.)
(e) For every x [ L1 ( m, X), u( ? , ? , x) is a measurable function on GX [
T ^ @ ( - ).
(A.6) u is a T ^ S-measurable function such that for every s [ S, eT u (t, s) 5 1 and
for every (t, s) [ T 3 S, 0 # u (t, s) # 1.
(A.7) e is (T, @ ( - ))-measurable.
(A.8) There exists a measurable selection w of X such that e(t) 2 w(t) [ int -1 for all
t [ T.
(A.9) 0 [ Y(s) for almost all s [ S.
Remark 1. The condition in (A.5)(b) is a little stronger than the usual quasi-concavity
condition. Quasi-concavity is needed to assure the convexity of individual demand sets.
We also need concavity of u(t, ? , x) on X(t) in order to prove Proposition 4, which is the
main proposition in our truncation argument. To be more specific, let X(t) be a
non-empty, weakly compact, convex consumption set for consumer t, where preference
relation s is defined by u(t, ? , x). In our truncation argument, we approximate X(t) by
subsets C(t) , X(t) which are non-empty, norm compact, convex. What is required for
the proof of Proposition 4 is the following statement: If j [ C(t) is not a satiation point,
there exists j 9 [ C(t) which is arbitrarily close to j and j 9 s j . It may appear to be true
that if the upper contour set of s on X(t) is convex and h s j on C(t), then
(1 2 a)h 1 aj s j for all a [ (0,1], but this may not be the case even in twodimensional Euclidean space (see Noguchi, 1997a, p. 20, for a counterexample).
(A.5)(b) certainly serves our purpose.
Remark 2. (A.1)–(A.4) imply that both L1 ( m, X)w , L1 (p, Y)w are non-empty, convex,
and compact metrizable; the non-emptiness of L1 ( m, X) follows from the fact that X
admits a measurable selection f : T → - (Castaing and Valadier, 1977, Theorem III.30,
p. 80), which clearly lies in L1 ( m, X). The identical argument applies to L1 (p, Y). The
convexity is trivial, and the compactness follows from Yannelis (1991, Theorem 3.1, p.
7). The metrizability follows from Kolmogorov and Fomin (1970, p. 381) and Dunford
and Schwartz (1958, Theorem 3, p. 434).
We state our Main Theorem as follows:
Main Theorem. Let % 5 [(T, T, m ), X, u, e, (S, S, p ), Y, u ] be an economy satisfying
(A.1)–(A.9). Then % has an equilibrium.
Example. For the sake of simplicity, we construct an example in which u is constant in
x [ L1 ( m, X). Let - 5 C(M) be the family of all real valued continuous functions on a
compact metric space, M, endowed with the usual sup norm. We consider the obvious
order in C(M). Let K be a non-empty, convex, weakly compact subset in C(M). We
assume that K contains 0 as a lower bound and 1 as an upper bound such that for all
j [ K except 1, 1 2 j [ int C(M) 1 . Such K can be constructed as follows: let K0 be a
164
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
convex subset containing 0, which is weak compact in C(M). Since K0 is bounded, we
have K0 , [2l, l] for some l [ R. If l is sufficiently large, l 2 K0 , int C(M) 1 . Now
we can let K 5 coh1, l 21 K0 j > C(M) 1 . Define X by X(t) 5 K for all t [ T. X satisfies
(A.3). Let Y be any correspondence as in (A.4) and (A.9) such that Y(s) # ]14 for all s [ S.
Let u be any function satisfying (A.6). Let e be any function as in (A.7) such that
e(t) [ int C(M) 1 and e(t) # ]14 for all t [ T. For (A.8), we may choose w(t) 5 0 for all
t [ T. Observe that by Corollary 2 in Yoshida (1968, p. 134), a (t) # ]14 for all t [ T. Let
u: T 3 K → R be measurable in t [ T, and continuous and concave in j [ K in the sense
of (A.5)(b). Note that the quasi-concavity of u implies (A.5)(c). We assume that u is
strictly increasing in j in the sense that u(t, j 1 ) , u(t, j 2 ) whenever j 2 2 j 1 [ int
C(M) 1 . Note that u(t, j ) . u(t, x(t)) for some j [ X(t) if and only if x(t) ± 1. We
construct an example of such u in the following manner: let (M, @ (M), i ) be a measure
space, where i is a positive Borel measure. Let U : T 3 M 3 R → R be a function such
that: (1) for every (t, m) [ T 3 M, U(t, m, ? ): R → R is continuous, concave, and strictly
increasing; (2) for every y [ R, U( ? , ? , y): T 3 M → R is T ^ @ (M) measurable; and
(3) for every (t, y) [ T 3 R, U(t, ? , y) [ L1 (M, @ (M), i ). Define u(t, j ) 5 eM U(t, m,
j (m)) di. Note that since K is bounded, for all j [ K, we have sup m [M u j (m)u # L for
some L. It follows from (2) that for every j [ K, U(t, m, j (m)) is jointly measurable in
(t, m). Since U(t, m, y) is increasing in y, we have uU(t, m, j (m))u # uU(t, m, 2 L)u 1 uU(t,
m, L)u, and by (3), U(t, m, j (m)) is integrable for all t. Thus by Fubini’s theorem, u( ? , j )
is T measurable. We next show that u(t, j ) is norm-continuous in j . Let j n → j [ K be
a convergent sequence. Then j n (m) → j (m) for all m [ M, and hence U(t, m,
j n (m)) → U(t, m, j (m)). Since uU(t, m, j n (m))u # uU(t, m, 2 L)u 1 uU(t, m,L)u, (3) and the
dominating convergence theorem imply that u(t, j n ) → u(t, j ). It is trivial to check that
u(t, j ) is strictly increasing in j . Since 1 2 e(t) 2 a (t) $ 0 for all t [ T, and if j 1 and j 2
are two points of K such that u(t, j 1 ) , u(t, j 2 ), and l is a real number in (0, 1], then u(t,
j 1 ) , u(t, (1 2 l)j 1 1 lj 2 ), (A.5)(b),(d) are satisfied. In our construction, 1 is a satiation
point for u for each t [ T and sets the upper limit for each commodity available for
consumption. The present example demonstrates that if the size of initial endowments
and production possibility sets is small relative to the size of the upper limit, the
¯ occurs at the satiation point only for those consumers lying
equilibrium consumption x(t)
¯ x(t)l
¯
¯
in a m -null subset. This assertion follows from the fact that kp,
5 eS u (t, s)kp,
¯
¯ e(t)l holds for (p,
¯ x,
¯ y¯ ) [see (5.2) in the proof of Proposition 5], and if x¯ 5 1
y(s)l
1 kp,
on some measurable subset T¯ , T with m (T¯ ) . 0, then the integration of the former
equality over T¯ gives rise to ip¯ i C(M )* m (T¯ ) # ]12 ip¯ i C(M )* m (T¯ ), which is absurd. This
establishes the claim.
* : k p, al 5 1j be the price simplex for %.
Let D 5 h p [ - 1
Remark 3. Since D is weak* compact (see Jameson, 1970, Theorem 3.8.6, p. 123) and
- is separable, the weak* topology on D is metrizable by a translation invariant metric
on - * (Dunford and Schwartz, 1958, Theorem 1, p. 426). Furthermore, D is bounded in
- * (Dunford and Schwartz, 1958, Corollary 3, p. 424). Note that by the separating
hyperplane theorem, if q(x) $ 0 for all q [ D, then x [ -1 (see Remarks 1 and 2 in
Noguchi, 1997a, p. 5).
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
165
Let ( be the collection of all correspondences c : T < S → - such that c u T 5
coh < in51 fi j for some measurable selections fi of X, c u S 5 coh < jl 51 g j j for some
measurable selections g j of Y, e and w are measurable selections of c u T , and 0 is a
measurable selection of c u S . As in Noguchi (1997a, p. 17), we can show that ( forms a
non-empty directed set under the obvious inclusion.
Observe that for each c [ (, c u T admits a measurable graph. This follows immediately from Himmelberg (1975, p. 69) and Theorem III.30 in Castaing and Valadier
(1977, p. 80), and the same holds also for c u S .
For (t, p) [ T 3 D, define:
E
A 1 (t, p) 5 h j [ c (t): k p, j l # u (t, s)P (s, p) 1 k p, e(t)lj
S
and for (x, y, p) [ L1 ( m, c u T ) 3 L1 (p, c u S ) 3 D, define:
P2 ( y, p) 5 hz [ L1 (p, c u S ):
E k p, z(s)l .E k p, y(s)lj
S
P3 (x, y, p) 5 hq [ D: kq 2 p,
S
E x(t) 2E y(s) 2E e(t)l . 0j
T
S
T
where P (s, p) ; suph [ c (s) k p, h l.
Remark 4. Note that the coupling ( p, v) → eT k p, v(t)l and ( p, v) → eS k p, v(t)l are
jointly continuous on D w * 3 L1 ( m, c u T )w , D w * 3 L1 (p, c u S )w , respectively (see Noguchi,
1997a, Lemma 1, p. 8), and it follows that P2 and P3 admit an open graph. Note also
that P (s, p) is non-negative [implied by (A.9)], bounded in (s, p), measurable in s, and
weak* continuous in p (see Remark 4 in Noguchi, 1997a, p. 7). A standard argument
such as Noguchi (1997a, Proposition 3, p. 9) shows that under (A.8), A 1 (t, ? ):
D w * → c (t) is a continuous correspondence with non-empty compact convex values.
We define pseudo-utility functions u 2 : (L1 (p, c u S )w 3 D w * ) 3 L1 (p, c u S )w → R and
u 3 : (L1 ( m, c u T )w 3 L1 (p, c u S )w 3 D w * ) 3 D w * → R by:
u 2 (( y, p), z) 5 dist[(( y, p), z), ((L1 (p, c u S )w 3 D w * ) 3 L1 (p, c u S )w)\GP 2 ]
u 3 ((x, y, p), q) 5 dist[((x, y, p),q),((L1 ( m, c u T )w 3 L1 (p, c u S )w 3 D w * ) 3 D w * )\GP 3 ]
where dist denotes the obvious distance function.
We define a correspondence H1 : T 3 L1 ( m, X)w 3 D w * → - by:
H1 (t, x, p) 5 Arg Maxh u(t, j , x): j [ A 1 (t, p)j
We next define correspondences H2 : L1 (p, c u S )w 3 D w * → L1 (p, c u S )w , H3 : L1 ( m,
c u T )w 3 L1 (p, c u S )w 3 D w * → D w * by:
166
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
H2 ( y, p) 5 Arg Maxhu 2 (( y, p), z): z [ L1 (p, c u S )w j
H3 (x, y, p) 5 Arg Maxhu 3 ((x, y, p), q): q [ D w * j
The standard argument as in Berge (1963) (see Noguchi, 1997a, Proposition 4, p. 11)
shows that H1 (t, ? , ? ), Hi (i 5 2, 3) are non-empty compact valued u.s.c. correspondences. [We need (A.5)(a) for showing that H1 (t, ? , ? ) is u.s.c.]
We focus on the properties of H1 . Note that H1 is convex valued.
Proposition 1. H1 ( ? , x, p): T → - admits a measurable graph.
See Appendix A for the proof.
Let & 5 L1 ( m, c u T )w 3 L1 (p, c u S )w 3 D w * . In the light of Proposition 1, we can
define a non-empty convex valued correspondence f1 (x, p) 5 hw [ L1 ( m, c u T )w : w(t) [
H1 (t, x, p) m -a.e.j. Following Yannelis (1991, Theorem 5.5, p. 19), we can prove the
closedness of f1 : L1 ( m, c u T )w 3 D w * → L1 ( m, c u T )w in the following manner: let (x k ,
pk , w k ) [ L1 ( m, c u T )w 3 D w * 3 L1 ( m, c u T )w be a sequence such that w k [ f1 (x k , pk ),
¯ pk → p,
¯ and w k → w.
¯ Now by the weak sequential convergence theorem of
x k → x,
¯ [ cl co Lshw k (t)j on T \T a for some T a [ T with
Yannelis (1989), we have w(t)
m (T a ) 5 0, where Ls is taken with respect to the weak topology on -. Note that
w k (t) [ H1 (t, x k , pk ) on T \T k for some T k [ T with m (T k ) 5 0. Then, for t [ T \[T a <
¯ p¯ ).
( < `k 51 T k )], if j [ Ls hw k (t)j, there exists a subsequence (w k i (t), x k i , pk i ) → ( j , x,
Recall that H1 (t, ? , ? ) is u.s.c. and closed valued, and since X(t) is a regular topological
¯ p¯ ), and hence Ls
space, it follows that H1 (t, ? , ? ) is closed. Consequently, j [ H1 (t, x,
¯ p¯ ). Since H1 (t, x,
¯ p¯ ) is closed convex, we obtain w(t)
¯ [ H1 (t, x,
¯ p¯ ) for
hw k (t)j , H1 (t, x,
¯ p¯ ).
all t [ T \[T a < ( < k`51 T k )], and therefore, w¯ [ f1 (x,
We also define f2 5 cl co H2 , f3 5 cl co H3 .
Proposition 2. Let - be a Hausdorff locally convex space and let C , - be a compact
convex subset. If V , - is an open neighborhood of C, then there exists an open convex
subset V 9 of - such that C ,V 9 , cl V 9 ,V.
See Appendix A for the proof.
Proposition 2 implies that f2 , f3 are a non-empty compact convex valued u.s.c.
correspondence, and in particular, are closed (cf. Noguchi, 1997a, Proposition 6, p. 12).
We apply the fixed point theorem (Fan, 1952, Theorem 1, p. 122) to f 5 f1 3 f2 3 f3 :
& → &, we obtain a fixed point (xc , yc , pc ). By the standard Shafer–Sonnenschein
argument with little care given to the treatment of f2 , f3 (see Noguchi, 1997a, p. 15),
(xc , yc , pc ) is seen to have the following properties (for a detailed argument, see
Appendix B):
1. for almost all t [ T, xc (t) [ Arg Maxhu(t, j , xc ): j [ A 1 (t, pc )j;
2. eS k pc , yc (s)l $ eS k pc , z(s)l for all z [ L1 (p, c u S );
3. k pc , (eT xc (t) 2 eS yc (s) 2 eT e(t)l $ kq, (eT xc (t) 2 eS yc (s) 2 eT e(t)l for all q [ D.
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
167
Let Tc [ T be the exceptional set in (1) with respect to xc . We have the following
proposition whose proof appears in Noguchi (1997a, Proposition 12, p. 16):
Proposition 3. For almost all s [ S, pc ( yc (s)) $ pc (h ) for all h [ c (s).
Proposition 3 implies that:
P (s, pc ) 5 k pc , yc (s)l
(3.1)
for almost all s [ S. It follows that A 1 (t, pc ) 5 B c (t, pc , yc ), where B c (t, p, y) 5 B(t, p,
y) > c (t) for (t, p, y) [ T 3 D 3 L1 (p, c u S ). Eq. (3.1) implies that for all t [ T \Tc :
E
k pc , xc (t)l # u (t, s)k pc , yc (s)l 1 k pc , e(t)l
(3.2)
S
and also,
P c (t, xc ) > B c (t, pc , yc ) 5 5
where P c (t, x) 5 h j [ c (t): u(t, j , x) . u(t, x(t), x)j for x [ L1 ( m, c u T ). We integrate both
sides of (3.2) with respect to t. Observe that by (A.6), we can exchange the order of
integration and obtain k pc , eT xc (t)l # k pc , (eS yc (s) 1 eT e(t))l. Combining this with (3)
above, we have:
Kq, SE x (t) 2E y (s) 2E e(t)DL # 0
c
T
c
S
(3.3)
T
for all q [ D.
Consider a net (xc , yc , pc ) [ L1 ( m, X)w 3 L1 (p, Y)w 3 D w * . We can extract a
¯ yc (m) → y,
¯ and, pc (m) → p.
¯
convergent subnet (xc (m) , yc (m) , pc (m) ) such that xc (m) → x,
¯ ¯ ) [ F.
By Remarks 3 and 4, and (3.3), we obtain (x,y
Recall that P(t, x) 5 h j [ X(t): u(t, j , x) . u(t, x(t), x)j.
Proposition 4. Let z [ L1 (p, Y). There exists T z [ T with m (T z ) 5 0 such that for all
¯ j l $ kp,
¯ eS u (t, s)z(s)l 1 kp,
¯ e(t)l.
t [ T \T z , j [ P(t, x¯ ) implies kp,
See Appendix A for the proof.
Proposition 5. Let (x, y, p) [ L1 ( m, X) 3 L1 (p, Y) 3 D such that (x, y) [ F. Let
z [ L1 (p, Y). If there exists T z [ T with m (T z ) 5 0 such that for all t [ T \T z , j [ P(t, x)
implies k p, j l $ k p, eS u (t, s)z(s)l 1 k p, e(t)l, then (x, y, p) is an equilibrium for %.
See Appendix A for the proof.
Proposition 5 clearly completes the proof of the Main Theorem.
168
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
3. Concluding remarks
We remark that our approach extends to economies with commodity-price paring (l ` ,
l 1 ) in which u is not interdependent and u(t, ? ) is Mackey-continuous (weak*
continuous) on X(t) , K, where K is a closed ball in l ` , and convex in the sense of
(A.5)(b). At this point, we are uncertain about the possibility of extending our approach
to cover economies with an interdependent utility function and the commodity-price
paring (l ` , l 1 ), or to cover economies with a continuum of commodities (M, M, i ),
where (M, M, i ) can be assumed to be separable and totally s -finite.
Acknowledgements
The author is indebted to anonymous referees of MASS for helpful comments and
suggestions. As a matter of fact, the use of Mazur’s lemma, which enabled us to extend
the earlier result in the original draft to cover interdependent preferences was suggested
by one of the referees. The author also wishes to thank Prof. K. Urai for reading the
original draft.
Appendix A. Proofs
A.1. Proof of Proposition 1
Fix p [ D, and define a(t) 5 A 1 (t, p). We first show that a: T → - admits a
measurable graph. Define g(t, j ) 5 k p, j l 2 eS u (t, s)P (s, p) 2 k p, e(t)l. Recall that (S,
S, p ) is complete, P bounded, and P ( ? , p) S-measurable. It follows that Fubini’s
theorem, (A.6), and (A.7) imply that g( ? , j ) is T-measurable. Note also that since g(t,
? ) is continuous on -, Castaing and Valadier (1977, Lemma III.14, p. 70) implies that g
is T ^ @ ( - ) measurable. Recall that Gc u T [ T ^ @ ( - ). Since Ga 5 Gc u T > h(t, j ) [
T 3 - : g(t, j ) # 0j, we obtain Ga [ T ^ @ ( - ).
Recall from (A.5)(e) that for every x [ L1 ( m, X), u( ? , ? , x) is measurable on
¯ ? , ? , x) be an extension of u( ? , ? , x) to the entire T 3 - as a
GX [ T ^ @ ( - ). Let u(
T ^ @ ( - ) measurable function. Since - is Suslin, (T, T, m ) complete, and H1
non-empty valued, Castaing and Valadier (1977, Lemma III.39, p. 86) is applicable to
u¯ ( ? , ? , x) and A 1 , and we obtain GH 1 (?,x, p) [ T ^ @ ( - ). h
A.2. Proof of Proposition 2
´ (1966, p. 145), we can find a neighborhood U of zero such
By Lemma 2 in Horvath
that (C 1 U ) > (V c 1 U ) 5 5. Since - is locally convex, we may assume that U is an
open convex neighborhood of zero. Then V 9 5 C 1 U has the required properties. h
A.3. Proof of Proposition 4
We first establish the following claim:
Claim 1. Let s be a measurable selection of X and let z [ L1 (p, Y). There exists a
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
169
m -null subset Tz, s , T such that if t lies in the set ht [ T : s (t) [ P(t, x¯ )j > (T \Tz, s ), then:
E
¯ s (t)l $ u (t, s)kp,
¯ z(s)l 1 kp,
¯ e(t)l
kp,
S
Proof of Claim 1. Note that:
c5
H
cohe(t) < w(t) < s (t)j
coh0 < z(s)j
for t [ T
for s [ S
lies in (, and hence there exists m 1 [ } such that c (m 1 ) $ c. Thus, for every m $ m 1 ,
we obtain:
1. s is a measurable selection of c (m)u T ;
2. z is a measurable selection of c (m)u S .
Consider the obvious convergent subnet obtained by restricting } to } 9 5 hm [ } :
¯ y,
¯ p¯ ). Observe that since L1 ( m,
m $ m 1 j and denote it also by (xc (m) , yc (m) , pc (m) ) → (x,
X)w 3 L1 (p, Y)w 3 D w * is metrizable, we can extract a convergent sequence (xc (m(n)) ,
¯ y,
¯ p¯ ). Recall that for each n, xc (m(n)) [ B c (m(n)) (t, pc (m(n)) , yc (m(n)) )
yc (m(n)) , pc (m(n)) ) → (x,
c (m(n))
and P
(t, xc (m(n)) ) > B c (m(n)) (t, pc (m(n)) , yc (m(n)) ) 5 5 outside some m -null subset
T n 5 Tc (m(n)) , T. Observe that the construction of the sequence (xc (m(n)) , yc (m(n)) ,
pc (m(n)) ) depends upon the choice of z and s. For the sake of simplicity, we write (x n , y n ,
¯ r) be an open ball centered at x¯ with radius r.
pn ) 5 (xc (m(n)) , yc (m(n)) , pc (m(n)) ). Let B(x;
By Mazur’s lemma we can choose a convex combination A 1 [ co hx n : 1 # n # n 1 j >
¯ 1) for some n 1 . Repeating the same argument for the weakly convergent sequence
B(x;
¯ we can choose a convex combination A 2 [ cohx n : n 1 1 1 # n # n 2 j > B(x;
¯ ]12 )
x n 1n 1 → x,
for some n 2 . Inductively, we can construct a sequence A i such that A i → x¯ in the norm
topology and the terms appearing in A i are all strictly greater than those appearing in
¯ in X(t) for all t [ T \T a for some T a [ T with
A i 21 . We may assume that A i (t) → x(t)
m (T a ) 5 0. Let T 5c be the exceptional set in (A.5)(c) for all A i . Define T z, s 5 < `n 51 T n <
T a < T 5c . Let t be a point in the set ht [ T : s (t) [ P(t, x¯ )j > (T \T z, s ). Note that by
(A.5)(c), we have for every A i , u(t, x n i (t), x n i ) # u(t, A i (t), x n i ) for some x n i appearing in
¯
the convex combination A i . If s (t) [ P(t, x¯ ), or u(t, s (t), x¯ ) . u(t, x(t),
x¯ ), then since
¯
(A i (t), x n i ) → (x(t),
x¯ ) in X(t) 3 L1 ( m, X)w , by (A.5)(a) we have u(t, s (t), x n i ) . u(t, A i (t),
x n i ) $ u(t, x n i (t), x n i ), or s (t) [ P(t, x n i ) for all i $ i 0 for some i 0 . Thus, for i $ i 0 , we
have:
1. s (t) [ P(t, x n i );
2. s is a measurable selection of c (m(n i ))u T ;
3. z is a measurable selection of c (m(n i ))u S .
Combining 1. and 2. we have:
s (t) [ P c (m(n i )) (t, xn i )
(4.1)
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
170
for i $ i 0 . Since Proposition 3 implies that pn i ( y n i (s)) $ pn i (h ) for all h [ c (m(n i ))(s), for
almost all s [ S, we have, by 3. pn i ( y n i (s)) $ pn i (z(s)) for almost all s [ S, and hence:
E u(t, s)k p , y (s)l $E u(t, s) k p , z(s)l
ni
ni
S
ni
(4.2)
S
On the other hand, (4.1) implies that:
k pn i , s (t)l . k pn i , x n i (t)l
(4.3)
(Recall that t [ T \T n i .) Let j l 5 (1 2 l)x n i (t) 1 ls (t). If it were true that:
E
k pn i , x n i (t)l , u (t, s) k pn i , y n i (s)l 1 k pn i , e(t)l
S
there would exist l¯ [ (0, 1] such that:
E
k pn i , j l¯ l , u (t, s)k pn i , y n i (s)l 1 k pn i , e(t)l
S
c (m(n i ))
i.e. j l¯ [ B
(t, pn i , y n i ). By (A.5)(b), we have j l¯ [ P c (m(n i )) (t, x n i ), contradicting
t[
⁄ Tn i . Thus, we have:
E
k pn i , x n i (t)l $ u (t, s)k pn i , y n i (s)l 1 k pn i , e(t)l
S
for all i $ i 0 , and combining this with (4.2) and (4.3), we obtain
E
k pn i , s (t)l . u (t, s) k pn i , z(s)l 1 k pn i , e(t)l
S
for all i $ i 0 . Since pn i → p¯ in D w * , and noting that eS u (t, s)k p, z(s)l 5 k p, eS u (t, s)z(s)l
for all p [ D, we deduce that:
E
¯ s (t)l $ u (t, s)kp,
¯ z(s)l 1 kp,
¯ e(t)l h
kp,
S
Proof of Proposition 4 (continued). We have X(t) 5 clhsk (t)j for all t, where sk are
measurable selections of X. Applying Claim 1 to each sk , we can define T z 5 < k`51
T z, sk . Let t [ T \T z and j [ P(t, x¯ ). We can choose a convergent subsequence sk i (t) → j ,
and since P(t, x¯ ) is norm-open in X(t), sk i (t) [ P(t, x¯ ) for sufficiently large i. Since t lies
in the set ht [ T : sk i (t) [ P(t, x¯ )j > (T \T z, sk ) for sufficiently large i, our result
i
follows. h
Remark 5. The author is indebted to an anonymous referee of MASS, who pointed out
that the use of Mazur’s lemma is possible for producing hx n i j at the crucial step in the
above proof. In the original draft, u was not assumed to be interdependent, and the
weak sequential convergence theorem of Yannelis (1989) was used. Mazur’s lemma
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
171
allowed us to generalize the early results to the case in which u is interdependent. Note
that Mazur’s lemma does not have a weak* counterpart, but the weak sequential
convergence theorem of Yannelis (1989) does and can be used for treating economies
with commodity-price paring (l ` , l 1 ) in our framework.
Proof of Proposition 5. Define T 9 5 ht [ T : u(t, j , x) . u(t, x(t), x) for some j [ X(t)j.
If t [ T 9 > (T \T z ), we can extract a sequence hi [ P(t, x) such that hi → x(t). Then we
have for every t [ T 9 > (T \T z ):
K E u(t, s) z(s)L 1 k p, e(t)l
k p, x(t)l $ p,
(5.1)
S
In fact, by (A.5)(d), (5.1) holds for all t [ T \(T z < T 5d ), where T 5d is the exceptional set
in (A.5)(d) with respect to x. Let Z 5 ht [ T \(T y < T 5d ): k p, x(t)l . k p, eS u (t, s)y(s)l 1
k p, e(t)lj. Clearly, Z [ T. If m (Z) . 0, we have eT k p, x(t) 2 eS u (t, s)y(s) 2 e(t)l $ eZ k p,
x(t) 2 eS u (t, s)y(s) 2 e(t)l . 0, which contradicts (x, y) [ F. Hence, m (Z) 5 0, and
consequently:
K E u(t, s)y(s)L 1 k p, e(t)l
k p, x(t)l 5 p,
(5.2)
S
for all t [ T \(T y < T 5d < Z), where m (T y < T 5d < Z) 5 0. In particular, x(t) [ B(t, p, y)
for all t [ T \(T y < T 5d < Z). Combining (5.1) and (5.2), we obtain:
k p,
E u(t, s) y(s)L $ K p, E u(t, s) z(s)L
S
(5.3)
S
for all t [ T \(T y < T z < T 5d < Z). We integrate both sides of (5.3) with respect to t and
apply Proposition 12 in Noguchi (1997a, p. 16) with ( p, y) instead of ( p* , y * ). We then
have:
k p, y(s)l $ k p, h l
(5.4)
for all h [ Y(s), for almost all s [ S.
We next show that for t [ T \(T y < T 5d ), if j [ X(t) satisfies:
E
k p, j l # u (t, s) k p, y(s)l 1 k p, e(t)l
S
then j [
⁄ P(t, x). We may assume that t [ T 9 > (T \(Ty < T 5d )) since P(t, x) 5 5 otherwise.
Since from (5.4), we have k p, y(s)l $ 0 for almost all s [ S, (A.8) implies that
k p,w(t)l , eS u (t, s)k p, y(s)l 1 k p, e(t)l. Let j l 5 (1 2 l)w(t) 1 lj . Then, for each
l [ (0, 1), we have k p, j l l , eS u (t, s)k p, y(s)l 1 k p, e(t)l, which means that j l [
⁄ P(t, x)
for all l [ (0, 1). Since P(t, x) is open, we deduce that j [
⁄ P(t, x). This and (5.4)
complete the proof since (T, T, m ) is assumed to be complete [(A.1)]. h
172
M. Noguchi / Mathematical Social Sciences 40 (2000) 157 – 173
Appendix B. Properties of (xc , yc , pc )
We have yc [ f2 ( yc , pc ) 5 cl co H2 ( yc , pc ). We claim that P2 ( yc , pc ) 5 5; assume
the contrary, that there exists a w [ P2 ( yc , pc ). Then, u 2 (( yc , pc ), w) . 0, and
consequently, u 2 (( yc , pc ),w9) . 0 for all w9 [ H2 ( yc , pc ). Thus, we have H2 ( yc ,
pc ) , P2 ( yc , pc ). Note that P2 ( yc , pc ) is an open convex subset of L1 (p, c u S )w . We have
the following lemma:
Lemma 1. Let - be a Hausdorff locally convex space and let A , - be a compact
convex subset. Let C , A be a compact subset and let V be an open convex
neighborhood of C in A. Then we have cl co C ,V.
Proof of Lemma 1. Choose an open subset V˜ , - such that V 5 V˜ > A. By Lemma 2 in
´ (1966, p. 145), we can find a neighborhood U of zero such that (C 1 U ) >
Horvath
c
(V˜ 1 U ) 5 5. Since - is locally convex, we can assume that U is a closed convex
˜ Since C is compact, we have
neighborhood of zero. Then we have C 1 U , V.
N
C , < i 51 (x i 1 U ), where x i [ C, i 5 1, . . . , N. Let Ci 5 (x i 1 U ) > A. Note that Ci is a
compact convex subset of -. By Jameson (1970, p. 208), co( < Ni 51 Ci ) is compact, and
in particular, closed. Therefore, cl co C , cl co ( < iN51 Ci ) 5 co( < Ni 51 Ci ). Note that
˜ and co( < N C ) , co A 5 A. Thus cl co
co( < iN51 Ci ) , co(C 1 U ) , co V˜ 5 V,
i 51
i
C , V˜ > A 5V. h
By Lemma 1, we have cl co H2 ( yc , pc ) , P2 ( yc , pc ), and consequently, yc [ P2 ( yc ,
pc ), which is a contradiction. The same argument applies to f3 also.
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