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, Nagoya468-8502, 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 non-cooperative 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)
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: T3L (1 m,
X )→-, where L (m, X ) is the set of Bochner integrable selections of the consumption 1
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 L (1 m, X ) becomes weakly compact in L (m, -), where L (m, -) is the Banach space of Bochner integrable
1 1
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, ?): L (m, X )→
-1
admits weakly open lower sections. In the present paper we consider the preference correspondence P having the representation P(t, x)5hj[X(t): u(t, j, x).u(t, r(x)(t),
x)j, where u is an interdependent utility function, andr is a choice function selecting a representative from each equivalence class x[L (m, -). For notational simplicity we
1
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)5hj[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 L (m, X ) becomes
1
metrizable] in order to show that for almost all t[T,hx[L (1 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)5hh[
X(t): u(t,j)#u(t,h)j, which is, under the assumptions, closed and convex. We must find an exceptional set T [T withm(T )50 such that for every t[T \T , if x converges
T [T which depends on x (x may be a subsequence of the original sequence). Let
¯x n
L (1 m, E ) denote the set of (equivalence classes of) integrable selections of t→E(t),
which is weakly compact in L (m, -) for the same reason as L (m, X ) is. We may wish
1 1
to define T05<¯x[L (1m,C )T , but we do not have any knowledge about the measure¯x
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- admits an interior point a and is closed, proper. The interior
1
properness and closedness of- guarantee the ‘right property’ of D: Letj[-. if kp, 1
jl$0 for all p[D, then j[- . 1
to say, the above condition of every consumption set X(t) being norm-compact is too stringent to be acceptable, and in order to remedy this problem, we consider the set of norm-compact valued sub-correspondences c (in fact, the set of polytope valued sub-correspondences) of the weakly compact valued consumption correspondence X. Suchc with some additional technical properties form a non-empty directed set with the obvious inclusion as an ordering. For the sake of simplicity, let us ignore production for
c
the time being. For each c, we have a sub-economy % of the original economy %, where we can utilize some well-known methods in the literature for obtaining an ‘equilibrium’ (x , p )c c [L (1 m, c)3D.
Now we have a net of equilibria (x , p )c c [L (1 m, X )3D, where L (1 m, X )3D
becomes a compact metric space with respect to the weak topology on L (1 m, X ) and the weak* topology on D. Based on the compactness, we can extract a convergent subnet
¯ ¯ ¯ ¯
(xc(m), pc(m))→(x, p ). We must show that (x, p ) gives rise to an equilibrium for the original economy%. To this end, we need the following crucial proposition: There exists a m-null set T0[T such that for every consumer t lying outside T , if0 j[X(t) is
¯
preferred to x(t), then the price of j is greater than or equal to the price of the initial endowment e(t) of t. Once this is established, the rest is simply a routine matter.
The heart of the problem is verification of the following statement: There exists a
¯ ¯
m-null exceptional set T0[T such that for every t[T \T , if0 j[P(t, x ), then kp, ¯
jl$kp, e(t)l. Once we have the above statement justified, the rest follows from the
¯
standard argument. For example, we can approximate x(t) by suchjand pass to the limit
¯ ¯ ¯ ¯
to obtain kp, x(t)l$kp, e(t)l, which actually becomes the equality showing that x(t)
¯
satisfies the budget constraint. We can also show that for almost all consumers t[T, x(t) ¯
maximizes u(t, ?, x ) over the budget set.
If P(t, ?) had open lower sections with respect to the weak topology on L (1 m, X ), we would have thatj[P(t, x ) for sufficiently largec c. We can also arrange the directed set
c
hcj so that j[c(t) for sufficiently large c, and consequently, j[P (t, x )c 5P(t, x )c >c(t) for sufficiently large c. Then by utilizing the property of (x , p ) being anc c
c
‘equilibrium’ in % with some m-null exceptional set T , we conclude thatc kp ,c
jl$kp , e(t)c l for t[T \T . Thereafter, we hope to be able to modifyc hcj so that
¯ ¯
<cTc[T with m(<cT )c 50. Now by passing to the limit pc→p, we obtain kp, ¯
jl$kp, e(t)lm-a.e.
Coming back to the present setting, since the weak topology on L (1 m, X ) and the weak* topology on D are both metrizable, we can extract a convergent sequence (x ,n
¯ ¯
p )n 5(xc(m(n)), pc(m(n)))→(x, p ). Then, as a consequence of Mazur’s lemma, we have
¯ ¯
x(t)[cl cohx (t)n j outside somem-null set T . This does not quite say that x (t)a n →x(t)
outside T , but we can find a subsequence xa ni such thatj[P(t, x ) for sufficiently largeni
i. Note that prior to the extraction of sequence (x , p ), we could have arranged then n
directed set hmj so thatj[c(m) for all m, and hence j[c(m(n )) for all i. Now wei argument as in the case that P(t, ?) was assumed to admit open lower sections, we
¯ ¯ ¯
In order to overcome the above difficulty, we consider a countable collection of measurable selections hskj of X such that hsk(t)j is dense in X(t) for all t[T. This
allows us to work on a measurable selections instead ofj sincej can be approximated by sk. Now letc0 be an element of hcj and consider the convex hull c 51 cohs,c0j, which clearly lies in hcj. Note that there exists m such that for all m0 $m , we have0
c(m)$c1. Then we restricthmj to those elements that are larger than m and obtain a0 directed set with the property we need. Repeating the previous argument again withsk
¯ ¯ ¯
instead of j, we obtain that ifsk(t)[P(t, x ) and t[T \T , thensk kp, sk(t)l$kp, e(t)l. `
¯ ¯ ¯
Note that this means that for every t[T \(<k51 T ), ifsk sk(t)[P(t, x ),kp,sk(t)l$kp,
`
¯
e(t)l. Now define T05 <k51 T . Let tsk [T \T . If0 j[P(t, x ), we can choose a
¯ ¯
convergent subsequenceski(t)→j. Since P(t, x ) is open,ski(t)[P(t, x ) for sufficiently
¯ ¯
large i. Then by passing to the limit i→`, we finally obtain kp, jl$kp, e(t)l. 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 ) and a continuum of agents. Khan and Yannelis (1991)` 1 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
¯
consumption x(t) actually avoids the satiation points due to the ‘boundary’ of 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 andf:-→= a correspondence. When= is a linear space, co f: -→= denotes the correspondence defined by the relation co
denotes the correspondence defined by cl f(x);cl [f(x)] for each x[-. G denotes f
the graph of f, i.e. G 5h(x, y)[-3=: y[f(x)j. Let - and = be paired linear f
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
w
*
some Banach space,= denotes=endowed with the weak* topology, and for a subset
w
*
A,=, A 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 off(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 correspondencef:-→= is said to be lower semi-continuous (l.s.c.) at Banach space. L (m,-) denotes the space of equivalence classes of--valued Bochner
1
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- is closed and has an interior
1 *
point a. We further assume that the dual cone - is non-trivial. [If- is proper, then
1 1
*
- ±(0) (see Jameson, 1970, p. 231).] Recall that L (m, -) is a set of equivalence
1 1
classes of functions. Let r be a choice function selection one representative function from each class x[L (m,-). When there is no fear of confusion, we simply write x for
1
An economy % is a tuple [(T, T, m), X, u, e, (S,S,p), Y,u] which is characterized by:
1. a measure space of consumers (T, T, m); 2. a consumption correspondence X: T→-;
3. an interdependent utility function u: G 3L (m, X )→R;
X 1
4. an initial endowment e: T→-, where e(t)[X(t) for all t[T;
5. a measure space of producers (S,S,p); 6. a production correspondence Y: S→-;
7. a shareu: T3S→R, whereu(t, s) denotes the share of consumer t in the profit of producer s.
Let ‘\’ denote the set theoretic subtraction. Let- be the usual positive cone of-, and 1
*
- the positive cone induced by - on -*. Denote the budget set of consumer t at
1 1
A competitive equilibrium for % is a price p[- \(0) together with an attainable 1
condition is well-defined regardless of the particular representative chosen for x.]
¯ ¯ ¯
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 G [T^@(-).
X
ˆ
(d) For each x[X, either u(t, j, x).u(t, x(t), x) for some j[X(t) or x(t)2 a(t)2e(t),-1 holds for almost all t[T. (Note that this condition is
well-defined independent of the choice of a representative for x.)
(e) For every x[L (1 m, X ), u(?, ?, x) is a measurable function on GX[
T^@(-).
(A.6)u is aT^S-measurable function such that for every s[S,eT u(t, s)51 and for every (t, s)[T3S, 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)2w(t)[int- for all 1
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: Ifj[C(t) is not a satiation point, there existsj 9[C(t) which is arbitrarily close tojandj 9sj. It may appear to be true
that if the upper contour set of s on X(t) is convex and hsj on C(t), then
(12a)h 1ajsj for all a[(0,1], but this may not be the case even in two-dimensional Euclidean space (see Noguchi, 1997a, p. 20, for a counterexample).
(A.5)(b) certainly serves our purpose.
w w
Remark 2. (A.1)–(A.4) imply that both L (m, X ) , L (p, Y ) are non-empty, convex,
1 1
and compact metrizable; the non-emptiness of L (1 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 L (1 m, X ). The identical argument applies to L (1 p, Y ). The
convexity is trivial, and the compactness follows from Yannelis (1991, Theorem 3.1, p.
7). The metrizability follows from Kolmogorovand 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[L (m, X ). Let-5C(M ) be the family of all real valued continuous functions on a
1
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
convex subset containing 0, which is weak compact in C(M ). Since K is bounded, we0
Let u be any function satisfying (A.6). Let e be any function as in (A.7) such that 1
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, j1),u(t, j2) whenever j 2 j2 1[int C(M ) . Note that u(t, j).u(t, x(t)) for some j[X(t) if and only if x(t)±1. We
1
construct an example of such u in the following manner: let (M, @(M ),i) be a measure space, wherei is a positive Borel measure. Let U : T3M3R→Rbe a function such dominating convergence theorem imply that u(t, jn)→u(t, j). It is trivial to check that
u(t, j) is strictly increasing inj. Since 12e(t)2a(t)$0 for all t[T, and ifj1andj2
are two points of K such that u(t,j1),u(t,j2), andlis a real number in (0, 1], then u(t,
j1),u(t, (12l)j 1 lj1 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
¯
equilibrium consumption x(t) occurs at the satiation point only for those consumers lying
¯ ¯ ¯
in a m-null subset. This assertion follows from the fact that kp, x(t)l5eS u(t, s)kp,
¯ ¯ ¯ ¯ ¯ ¯
y(s)l1kp, e(t)lholds for (p, x, y ) [see (5.2) in the proof of Proposition 5], and if x51
¯ ¯
on some measurable subset T,T with m(T ).0, then the integration of the former 1
Remark 3. SinceDis weak* compact (see Jameson, 1970, Theorem 3.8.6, p. 123) and
-is separable, the weak* topology onDis metrizable by a translation invariant metric on-* (Dunford and Schwartz, 1958, Theorem 1, p. 426). Furthermore,Dis 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[- (see Remarks 1 and 2 in 1
Let ( be the collection of all correspondences c: T<S→- such that cuT5
n l
coh<i51 fij for some measurable selections f of X,i cuS5coh<j51 gjj for some measurable selections g of Y, e and w are measurable selections ofj cuT, and 0 is a measurable selection ofcu . As in Noguchi (1997a, p. 17), we can show that( forms a
S
non-empty directed set under the obvious inclusion.
Observe that for each c[(,cuT admits a measurable graph. This follows immedi-ately from Himmelberg (1975, p. 69) and Theorem III.30 in Castaing and Valadier (1977, p. 80), and the same holds also for cuS.
1997a, Lemma 1, p. 8), and it follows that P and P admit an open graph. Note also2 3
thatP(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 (t,1 ?):
w
*
D →c(t) is a continuous correspondence with non-empty compact convex values.
w
H ( y, p)2 5Arg Maxhu (( y, p), z): z2 [L (1p,cuS) j
w*
H (x, y, p)3 5Arg Maxhu ((x, y, p), q): q3 [D j
The standard argument as in Berge (1963) (see Noguchi, 1997a, Proposition 4, p. 11) shows that H (t,1 ?, ?), H (ii 52, 3) are non-empty compact valued u.s.c. corre-spondences. [We need (A.5)(a) for showing that H (t,1 ?, ?) is u.s.c.]
We focus on the properties of H . Note that H is convex valued.1 1
Proposition 1. H (?, x, p): T→- admits a measurable graph.
H (t, x, p)1 m-a.e.j. Following Yannelis (1991, Theorem 5.5, p. 19), we can prove the
w w w Recall that H (t,1 ?, ?) is u.s.c. and closed valued, and since X(t) is a regular topological
¯ ¯
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 V9of - such that C,V9,cl V9,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) tof 5 f 3 f 3 f1 2 3: &→&, we obtain a fixed point (x , y , p ). By the standard Shafer–Sonnenschein
c c c
argument with little care given to the treatment off2,f3 (see Noguchi, 1997a, p. 15), (x , y , p ) is seen to have the following properties (for a detailed argument, seec c c Appendix B):
1. for almost all t[T, x (t)c [Arg Maxhu(t, j, x ):c j[A (t, p )1 c j; 2. e kS p , y (s)c c l$e kS p , z(s)c l for all z[L (1p,cuS);
Let T [T be the exceptional set in (1) with respect to x . We have the following
c c
proposition whose proof appears in Noguchi (1997a, Proposition 12, p. 16):
Proposition 3. For almost all s[S, p ( y (s))c c $p (ch) for all h[c(s).
Proposition 3 implies that:
P(s, p )c 5kp , y (s)c c l (3.1)
c c
for almost all s[S. It follows that A (t, p )1 c 5B (t, p , y ), where B (t, p, y)c c 5B(t, p, y)>c(t) for (t, p, y)[T3D 3L (1p, cuS). Eq. (3.1) implies that for all t[T \T :c
kp , x (t)c c l#
E
u(t, s)kp , y (s)c c l1kp , e(t)c l (3.2)S
and also,
c c
P (t, x )c >B (t, p , y )c c 55
c
where P (t, x)5hj[c(t): u(t,j, x).u(t, x(t), x)jfor x[L (1 m,cuT). We integrate both sides of (3.2) with respect to t. Observe that by (A.6), we can exchange the order of integration and obtainkp ,c eT x (t)c l#kp , (c eS y (s)c 1eT e(t))l. Combining this with (3) above, we have:
q,
E
x (t)2E
y (s)2E
e(t) #0 (3.3)K
S
c cD
L
T S T
for all q[D.
w w w
*
Consider a net (x , y , p )c c c [L (1 m, X ) 3L (1p, Y ) 3D . We can extract a
¯ ¯ ¯
convergent subnet (xc(m), yc(m), pc(m)) such that xc(m)→x, yc(m)→y, and, pc(m)→p. ¯ ¯
By Remarks 3 and 4, and (3.3), we obtain (x,y )[F.
Recall that P(t, x)5hj[X(t): u(t, j, x).u(t, x(t), x)j.
Proposition 4. Let z[L (1p, Y ). There exists Tz[T with m(T )z 50 such that for all
¯ ¯ ¯ ¯
t[T \T ,z j[P(t, x ) implies kp, jl$kp, eS u(t, s)z(s)l1kp, e(t)l. See Appendix A for the proof.
Proposition 5. Let (x, y, p)[L (1 m, X )3L (1 p, Y )3D such that (x, y)[F. Let z[L (p, Y ). If there exists T [Twithm(T )50 such that for all t[T \T , j[P(t, x)
1 z z z
implies kp, jl$kp, e u(t, s)z(s)l1kp, e(t)l, then (x, y, p) is an equilibrium for %.
S
See Appendix A for the proof.
3. Concluding remarks
We remark that our approach extends to economies with commodity-price paring (l ,`
l ) in which u is not interdependent and u(t,1 ?) 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 ), or to cover economies with a continuum of commodities (M,` 1 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)5A (t, p). We first show that a: T→- admits a 1
measurable graph. Define g(t, j)5kp, jl2eSu(t, s)P(s, p) 2kp, 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) isT-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 Gcu [T^@(-). Since Ga5Gcu >h(t, j)[
T T
T3-: g(t, j)#0j, we obtain G [T^@(-).
a
Recall from (A.5)(e) that for every x[L (1 m, X ), u(?, ?, x) is measurable on
¯
G [T^@(-). Let u(?, ?, x) be an extension of u(?, ?, x) to the entire T3- as a
X
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 , and we obtain G [T^@(-). h 1 H (1?,x, p)
A.2. Proof of Proposition 2 ´
By Lemma 2 in Horvath (1966, p. 145), we can find a neighborhood U of zero such
c
that (C1U )>(V 1U )55. Since- is locally convex, we may assume that U is an open convex neighborhood of zero. Then V9 5C1U has the required properties. h
A.3. Proof of Proposition 4
We first establish the following claim:
¯
Consider the obvious convergent subnet obtained by restricting } to }9 5hm[}:
¯ ¯ ¯
B(x; 1) for some n . Repeating the same argument for the weakly convergent sequence1 1
¯ ¯ ]
xn1n1→x, we can choose a convex combination A2[cohx : nn 111#n#n2j>B(x; 2)
¯
for some n . Inductively, we can construct a sequence A such that A2 i i→x in the norm
topology and the terms appearing in A are all strictly greater than those appearing ini
for i$i . Since Proposition 3 implies that p ( y (s))0 ni ni $p (nih) for allh[c(m(n ))(s), fori
almost all s[S, we have, by 3. p ( y (s))ni ni $p (z(s)) for almost all sni [S, and hence:
E
u(t, s)kp , y (s)ni ni l$E
u(t, s)kp , z(s)ni l (4.2)S S
On the other hand, (4.1) implies that:
kp ,ni s(t)l.kp , x (t)ni ni l (4.3)
(Recall that t[T \T .) Letni j 5l (12l)x (t)ni 1ls(t). If it were true that: kp , x (t)ni ni l,
E
u(t, s)kp , y (s)ni ni l1kp , e(t)ni lS
¯
there would exist l[(0, 1] such that:
kp ,ni jl¯l,
E
u(t, s)kp , y (s)ni ni l1kp , e(t)ni lS
c(m(n ))i c(m(n ))i
i.e. jl¯[B (t, p , y ). By (A.5)(b), we haveni ni jl¯[P (t, x ), contradictingni
t[⁄ T . Thus, we have:ni
kp , x (t)ni ni l$
E
u(t, s)kp , y (s)ni ni l1kp , e(t)ni lS
for all i$i , and combining this with (4.2) and (4.3), we obtain0
kp ,ni s(t)l.
E
u(t, s)kp , z(s)ni l1kp , e(t)ni lS
w
¯ *
for all i$i . Since p0 ni→p inD , and noting thateSu(t, s)kp, z(s)l5kp,eSu(t, s)z(s)l for all p[D, we deduce that:
¯ ¯ ¯
kp,s(t)l$
E
u(t, s)kp, z(s)l1kp, e(t)l hS
Proof of Proposition 4 (continued). We have X(t)5clhsk(t)j for all t, where sk are ` measurable selections of X. Applying Claim 1 to each sk, we can define Tz5<k51
¯
Tz,sk. Let t[T \T andz j[P(t, x ). We can choose a convergent subsequenceski(t)→j,
¯ ¯
and since P(t, x ) is norm-open in X(t),ski(t)[P(t, x ) for sufficiently large i. Since t lies ¯
in the set ht[T : ski(t)[P(t, x )j>(T \Tz,sk) for sufficiently large i, our result
i
follows. h
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 ) in our framework.` 1
Proof of Proposition 5. Define T9 5ht[T : u(t, j, x).u(t, x(t), x) for somej[X(t)j. If t[T9>(T \T ), we can extract a sequencez hi[P(t, x) such thathi→x(t). Then we
have for every t[T9>(T \T ):z
kp, x(t)l$
K
p,E
u(t, s) z(s)L
1kp, e(t)l (5.1)S
In fact, by (A.5)(d), (5.1) holds for all t[T \(Tz<T ), where T5d 5d is the exceptional set in (A.5)(d) with respect to x. Let Z5ht[T \(Ty<T ):5d kp, x(t)l.kp, eS u(t, s)y(s)l1
kp, e(t)lj. Clearly, Z[T. Ifm(Z ).0, we havee kT p, x(t)2eSu(t, s)y(s)2e(t)l$e kZ p, x(t)2eS u(t, s)y(s)2e(t)l.0, which contradicts (x, y)[F. Hence, m(Z )50, and consequently:
kp, x(t)l5
K
p,E
u(t, s)y(s)L
1kp, e(t)l (5.2)S
for all t[T \(Ty<T5d<Z ), where m(Ty<T5d<Z )50. In particular, x(t)[B(t, p, y)
for all t[T \(Ty<T5d<Z ). Combining (5.1) and (5.2), we obtain:
kp,
E
u(t, s) y(s)L K
$ p,E
u(t, s) z(s)L
(5.3)S S
for all t[T \(Ty<Tz<T5d<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:
kp, y(s)l$kp,hl (5.4)
for allh[Y(s), for almost all s[S.
We next show that for t[T \(Ty<T ), if5d j[X(t) satisfies:
kp,jl#
E
u(t, s)kp, y(s)l1kp, e(t)lS
thenj[⁄ P(t, x). We may assume that t[T9>(T \(Ty<T )) since P(t, x)5d 55otherwise. Since from (5.4), we have kp, y(s)l$0 for almost all s[S, (A.8) implies that
kp,w(t)l,eS u(t, s)kp, y(s)l1kp, e(t)l. Let j 5l (12l)w(t)1lj. Then, for each
Appendix B. Properties of (x , y , p )c c c
We have yc[f2( y , p )c c 5cl co H ( y , p ). We claim that P ( y , p )2 c c 2 c c 55; assume the contrary, that there exists a w[P ( y , p ). Then, u (( y , p ), w)2 c c 2 c c .0, and consequently, u (( y , p ),w2 c c 9).0 for all w9[H ( y , p ). Thus, we have H ( y ,2 c c 2 c
w
p )c ,P ( y , p ). Note that P ( y , p ) is an open convex subset of L (2 c c 2 c c 1p,cuS) . 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 V5V>A. By Lemma 2 in ´
Horvath (1966, p. 145), we can find a neighborhood U of zero such that (C1U )>
c
˜
(V 1U )55. Since - is locally convex, we can assume that U is a closed convex
˜
neighborhood of zero. Then we have C1U,V. Since C is compact, we have
N
C, <i51(xi1U ), where xi[C, i51, . . . , N. Let Ci5(xi1U )>A. Note that C is ai N
compact convex subset of-. By Jameson (1970, p. 208), co(<i51 C ) is compact, andi
N N
in particular, closed. Therefore, cl co C,cl co (<i51 C )i 5co(<i51 C ). Note thati
N ˜ ˜ N
co(<i51 C )i ,co(C1U ),co V5V, and co(<i51 C )i ,co A5A. Thus cl co ˜
C,V>A5V. h
By Lemma 1, we have cl co H ( y , p )2 c c ,P ( y , p ), and consequently, y2 c c c[P ( y ,2 c
p ), which is a contradiction. The same argument applies toc f3 also.
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