DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA. CALIFORNIA 91125
EQUILIBRIA IN MARKETS WITH A RIESZ SPACE OF COMMODITIES
Charalambos D. Aliprantis California Institute of Technology Indiana University and Purdue University at Indianapolis
and �c,1\lUTEOF Donald J. Brown California Institute of Technology � 1:, � Yale University _,, � t:::0 :5 �,.� ,.ct'� � l� /d1�\ i /'l\ \ lf1: ...� � Slf�LL IA"'�\.
SOCIAL SCIENCE WORKING PAPER 427
June 1982 ABSTRACT
Using the theory of Riesz spaces, we present a new proof of the existence of competitive equilibria for an economy having a Riesz space of commodities. 2
EQUILIBRIA IN .MARKETS WITH A RIESZ SPACE OF COMMODITIES production. In proving existence, n�wley considers Lm(µ) with the sup
norm topology, and in this case the dual space is the vector space of
1 • INTRODUCTION all bounded additive functionals on Lm(µ). n In the Arrow-Debreu model of a Walrasian economy, [3] and [7], The spaces 1R and Lm(µ) in addition to being ordered linear the commodity space is :mn and the price space is :m:. where n is the vector spaces are also Riesz spaces or vector lattices. In fact, number of commodities. Agent's characteristics such as consumption considered as Banach spaces under the sup norm, they belong to the sets, production sets, utility functions, the price simplex, excess special class of Banach lattices, i. e. to the class of normed Riesz n demand functions, etc. are introduced in terms of subsets of 1R or spaces which are complete under their norms. In this paper, we
1R: or functions on 1Rn or 1R:. As is well known, the Arrow-Debreu exploit the Riesz space structure of the commodity spaces to give a model allows consumption and production to be contingent on time and new proof of the existence of Walrasian equilibrium. The relevant the state of the world, when there is a finite number of periods and a facts about Riesz spaces may be found in the appendix. finite number of states. In this work the space of commodities will be a Riesz space
In a world of uncertainty where there are an infinite number and the price simplex will be a suitable subset of the order dual. We of states or an intertemporal economy having an infinite number of define an excess demand function as a mapping from the price simplex time periods (e. g. , an infinite horizon), the appropriate model for into the commodity space which is weakly continuous and satisfies a the space of commodities is an infinite dimensional vector space. In boundary condition. If A is the price simplex and E is the excess particular, ordered locally convex topological vector spaces are the demand function with domain D !; A, then we define the revealed most frequently used models of infinite dimensional commodity spaces. preference relation > on D by saying that p > q, whenever p, q & D and
In such models, the price space is the cone of positive continuous p(E ) > The definition of > was suggested by Nikaido's discussion q 0. linear functionals in the dual space. See [5], [12], and [14]. of economies with gross substitutability in [15, Section 18, p. 197].
Our main result is that whenever E satisfies Walras' law, then The paradigmatic example being the dual pair
element q of > is an equilibrium price, i.e. E The existence the space of commodities; L;(µ) is the space of prices. This model q O. was introduced by Bewley in [4], where he proved the existence of a proof uses Ricsz space arguments along with an infinite dimensional
Walrasian equilibrium in both exchange economies and economies with version of Sonnenschein's theorem that an irreflexive, convex-valued, 3 4
n .. upper semicontinuous binary relation on a compact convex subset of :m. 1. Density Condition: D is a w -dense convex subset of 11;
has a maximal element. 2. Continuity Condition: There exists a locally convex topology t It is demonstrated in the paper that our results encompass all on L compatible with the dual system (L, L'> (i.e., of the finite dimensional cases, say in [8), Bewley's result for l. in • a(L, L') !;;;; t !;; i;(L, L')) such that E: (D, w ) -1 (L, t) is continuous. the pure exchange case [4), and several new infinite dimensional w• 3. Boundary Condition: If a net D satisfies p -1 D, markets. [pa)� a q e t.. -
then lim p(E ) > 0 holds for some p e D; and Pa
2. EQUILIBRIUM THEOREMS Walras' Law: p(E = 0 holds ror all p e D. 4. p ) Let
set (-•, p) [q E D p > q) and (p, •) [q e D q > p).
E s {p < n 11 P » O). The binary relation > is said to be: • is w -dense in t..; and 1. irreflexive, whenever p f p holds for all p e D;
• b) The cone generated by S (i.e., the set u A.S) is w -dense in 2. convex e A. L o valued, whenever (p, •) is convex for each p D; and , L+. 3. i;- upper semicontinuous, whenever ( -•, p) is ,;-open in D for
Now let t.. be a price simplex for a Riesz dual system
An excess demand function E for /1 is a mapping p I-> E , from a convex p An element p e D is said to be a maximal element for ), whenever q 1 p subset D (=dom E) of L+ into L, satisfying the following four holds for all & D. properties: q 5 6
The following theorem will be of fundamental importance. contr,u, ;l, the irreflexivity of). Now according to [ll, Lemma 1, p.
21] we have n F(p) f �. as desired. • peD TIIEOREM 2. lt Let (E, 'C) be a Hausdorff topological vector For the rest of the discussion in this section space, and let D be a non-empty. convex, and "C-compact subset of E. = (( L,L'),A,E) will be a fixed ""onomy . The economy defines a If ) is an irreflexive. convex valued, and upper semicontinuous binary E E binary relation on D (which will referred to as the revealed relation on D, then the set of all maximal elements of ) is non-empty > he preference relation [15, Section as follows: and "C-compact. .lii.3])
If p, q e D, then we say that p dominates q (in symbols, PR OOF. Let F(p) = D - (-m,p). Since p � (--<0,p), it follows p) q), whenever p(E ) > 0 holds. that F(p) is non-empty for all p e A. Also, by the upper q
semicontinuity of), we see that each F(p) is "C-closed, and hence, TIIEOREM 2 .2 . The revealed preference relation for an
n F(p) is "C-compact. Now it is a routine matter to verify that arbitrary economy is irreflexive, convex valued, and w•-upper peD semicontinuous . 0 F(p) is precisely the set of all maximal elements of >. Thus, it peD remains to be shown that F(p) is non-empty . 0 PROOF. Since p(E = 0 hol.h for all p e D, we see that p I P peD p)
£ To this end, let p •···• £ D. Then we claim that for each p D, and so,) is irreflexive. i Pn n n To see that > is convex valued, let q ,q e m 1 2 (p, ) and co{p ,••• ,p } !: U F(p.) holds. Indeed, if q = \ ap . is a convex 1 n f.; 1. 1 i=l t 1 0 is upper semicontinuous. To
this end, let q & (-m, p). This means that p(E Since p is a q )) O. t The argument in this proposition is due to H. F. Sonnenschein [18], t-continuous linear functional on L, it follows that the function who used it to prove Theorem 2. 1. for a finite dimensional topological r p ( )' from w*) into L, is continuous. Consequently , there vector space. We thank Kim Border for recalling Sonnenschein's I -7 E r (D, theorem to our attention. exists a w*-neighborhood of q such that r e D implies p(E V n V r) > 0
(i. e. , p) r). Therefore, q is an interior point of ( -en, p), and
hence, (-w, p) is an open subset of (D, w*). • 7 8
finite subsets of D. For each a s let D denote the convex hull of An element p e D is said to be a free disposal equilibrium A, a
a Clearly , each D is w•-compact, and U D = D holds. Also, the price, whenever E i 0 holds. . a a p aeA The free disposal equilibrium prices are characteriz ed as collection {D l is directed upwards by inclusion. Now let a e A be a follows. fixed. By Theorem 2.2 the revealed preference relation > restricted
to D is irreflective, convex valued, and w•-upper semicontinuous. THEOREM 2. 3. For an element p s D the following statements a Hence, by Theorem 2. 1, there exists some p e D that is maximal for ) are equivalent: a a
on D (i. e., p( E ) i 0 holds for each p s D ). Next consider the net 1. p is a free disposal equilibrium price. a p a a
{p s where is indexed by the inclusion Since is w•- 2. p is a maximal element for the revealed preference relation ). a:a Al, A 2. A 3. q( E ) 0 holds for all q compact, we can assume (by passing to a subnet if necessary) that p i e D. w• p � q holds in !J.. PROOF. (1) = > (2 ) Let E i 0. Then q( E ) i 0 holds for all a P p We claim that q e D. InJeed, if q e /J. - D holds, then by our q e D, and so, q f p for all q s D. That is, p is a maximal element boundary condition there exists some e D with lim p( E ) O. On for ). p ) Pa
(2 ) = > (3) Obvious. the other hand, must lie in some D ; and since {D l is directed p a a
= • upwards, there exists some p so that p s D holds for all a p. (3) > (1) Since D is w -dense inA, statement (3) implies that e A a ;J But then for each a p we have p( E ) 0, and so, lim p( E ) 0 q( E ) i 0 holds for all q e A. From this it follows easily that ;J i p i p Pa a 1 q( E ) i 0 holds for all q e L • But then Theorem (in the p + 4.4 holds, which is a contradiction. Thus, q e D. appendix) shows that E Thus, p is a free disposal equilibrium p i O. We now establish that q is a free disposal equilibrium price. price, and the proof of the theorem is finished. • To this end, let p Since the function, r I� p( E ) from (D, w•) e D. r into JR is continuous, it follows that p( E w• q ) -lim p( E ). As We now come to one of the main results of this paper. Pa above, we see that there exists some p satisfy ing p( E ) i 0 for e A p • THEOREM 2.4. Every economy E has a non-empty w -compact set a all a p, and so, p( E ) 0 holds for all p e D. By Theorem 2. 3, q of free disposal equilibrium prices. 2 q i is a free disposal equilibrium price.
PROOF. We first show that the set of free disposal Finally, let us show that the set of all free disposal equilibrium prices is non-empty . Let A denote the collection of all equilibrium prices is w•-compact. It is enough to show that this set 9 10
is w•-closed in So, let {p } be a net of free disposal equilibrium Finally, we note in passing that if the excess demand function D. a w• E satisfies Samuelson's weak axiom of revealed preference, i.e., prices satisfying p � q e If q e - D, then by our boundary a A. A p, q e D and p(E ) i 0 implies q(E ) ) 0, then the economy has £ q p E condition there exists some p D with lim p(E ) ) O. On the other Pa precisely one free disposal equilibrium price. Indeed, note first hand, since each p is a maximal element, p(E ) i 0 holds for each a, a p a that Samuelson's axiom is equivalent to the statement : If p, q £ D,
) e which contradicts lim p(E ) O. Thus, q D. From the continuity satisfy p f q, then q ) p holds. Thus, if p, q e are two distinct Pa D of the function r I-� p(E ) we get p(E ) i 0 for all p e D, and so, free disposal equilibrium prices, then by the maximality of p and q we r q by Theorem 2. 3, q is a free disposal equilibrium price. The proof of have pf q and qf p, which contradicts Samuelson's axiom. the theorem is now complete. •
3• EXAMPLES OF ECONOMIES A strictly positive price p e D is said to be an equilibrium First, we present examples of price simplices. Our examples price for the economy E, whenever E = 0 holds. p will be related one way or another to AM-spaces with unit. A Banach The following fundamental theorem guarantees equilibrium lattice L is said to be an AM-space (abstract M-space), whenever for prices. + each f, g e L we have
TIIEOREM 2.5. Every economy E with D S has a non-empty 11fvg11 MaxCl lfl I. I lg l IJ. w•-compact set of equilibrium prices.
(The letter M stands for maximum.) The element e ) 0 in an AM-space L PROOF. If D = S holds, then we claim that p & S satisfies is said to be the unit of L, whenever the order interval [-e, e] E i 0 if and only if E O. Indeed, if E i 0 holds, then either p p p coincides with the closed unit ball of L, i.e., whenever (a) E < 0 or (b) E = O. If (a) is true, then (since p e S) we have p p = l [-e, e] {f e L : I lf I i 1} holds. Here are some examples of p(E ) < 0, which contradicts Walras' law. Consequently, (b) is true, p AM-spaces with units: i.e., E = 0. p
& D 1. The Riesz space C bounded real valued Thus, an element p is a free disposal equilibrium price if b(D) of all continuous and only if p is an equilibrium price. The conclusion now follows functions on a topological space D, with the sup norm from Theorem 2.4. • 11f11"' sup{lf(wll: w & D}. 11 12
The unit e is the constant function one on n. s {p e < p » 0 and p(e) 1}.
2. The Riesz space Lm(µ) of all essentially bounded measurable This implies that S is a convex set. Also, since S is a subset of the
functions on a measure space (X,L,µ), with the essential sup norm closed unit ball of L*, Alaoglu's theorem shows that the w•-closure
A of S in L* is convex and w•-compact. Clearly, I Im= I lfl ess sup fl. • {p e L : p(e) = 1}. A£ + , The constant function one is the unit. Now assume that �. Fix some p e Then for each q e L S I S. +
with q(e) 1 we have ap + (1 - a)q e S for all 0 < a ( 1, and so, 3. The Riesz space lm of all order bounded real sequences with the from w* - lim [ap + (1 - a)q] = q we see that U AS is w•-dense in sup norm. Again, the constant sequence one is the unit. a�O+ x 2 0 , (and that q e Thus, is a price simplex for THEOREM 3.1 • Let AM-space with unit. If the convex set all special cases of the preceding theorem. The finite dimensional case comes first. {p e L' p » 0 and I p 1} s + I 11 EXAMPLE 3. 2. The convex set is non-empty, then the w*-closure A of S in L* is a price simplex for , . n (L, L'>. In addition, if L is w*-dense in L ' then n + + (p (p ,• ,p ) e p 0 for i 1, ••. ,n and p 1} A l • . n :m. i [ i 2 1=1 * A e L IIp 11 1). {p + n is a price simplex for the Riesz d!!al system • rest follows from Theorem 3. 1 by observing that if p = Cp , ••• ,p) 0 get p( f) = 0 for all p B , from which it follows that f This 1 n 2 e + o . n n implies that p is strictly positive on • holds in the norm dual of (JR , � L. ll·llm>• then llpll = p( e) = Pi ' Also, note that Now consider the Banach lattice = for some measure L L1(µ) space (X, µ). It is a fact that Lis an ideal in its second norm n Z, n {p (p p 0 for i l,. .. ,n and p 1}•• s l'• .. ,pn)e JR i > [ i dual; see [l, 9. 2, p. 61]. If, in addition, (X, is 1=1 Thm Z, µ) • a-finite, then holds (see, for example [2, Thm 27. 10, L1Cµ) Lm(µ) This next case is that of C( O) with 0 a compact metric space. p. 234]), and so, in the a-finite case is a Riesz dual the conyex set details follow. {p * I I l} EXAMPLE 3.4. If (X, µ) is a a -finite measure space, then A e L+ lpl Z, the conyex set is a price simplex for the Riesz dual system To see this, note that according to Theorem 3. 1 it is enough is a price simplex for the Riesz dual system To this end, start by observing that since is a compact 0 + s {feL Cµ>na f » 0} metric space, C( O) is a separable Banach lattice; see, for instance 1 • • [16, Prop, 7. 5, p. 105]. This implies that B = {pe L : llpll.�l} = {f : f >> 0 on and llfll =1). + + e L1(µ) Lm(µ) 1 with the w•-topology is a compact metric space [10, Thm 1, p. 426], We show next that is non-empty. To this end, pick a • • S and in particular, that (B ,w) is separable. Fix a countable + m disjoint sequence {A } of satisfying A = X and µ( A ) m for • -n • n Z u n n < w•-dense subset {p ,p ,••• } of B ' and put p = \ 2 p L · Then n=l 1 2 + � n e + P 1 -n each n. Next for each n choose 0 A ) and < n < 1 with Anµ( An < 2 • is strictly positive on L. m then let g = A 'X. Now the linear functional For if p( f) = 0 holds for some f 0, then p (f) = 0 likewise [ n A e L1 (µ). 2 n n=l n • holds for all n, and so, by the w*-denseness of {p ,••. } in B we 1,p2 + 15 16 = e norm dual of C O) with the sup norm), and hence, An important special case of the previous example is the Riesz The proof goes as follows: By [19, Thm 14, p. 192 and Thm 18, • dual system p p , •••,} is a countable w• - (0}, then Now let us consider a completely regular Hausdorff topological ( 1, 2 -dense subset of M; m p -n space n. In [17] F. D. Sentilles defined a notion of a topology P on p = 2 • e M is strictly positive on C (see the [ n----11p ; b (0) n=l n C (O) which extends the notion of the strict topology introduced by R. b corresponding proof in Example 3.3). The rest follows from Theorem C. Buck [6] for locally compact n. It was shown in [17] that the 3.1. • strict topology P has the following properties: An interesting special case of the preceding example is when a) is the finest locally convex topology on C (O) for which Dini's p b n C[O,l] with the topology generated by the sup norm. theorem holds (i.e., (f } � (0) and f (w) i 0 for all w e 0 imply a a Finally, we close this section with several examples of p f � 0) [17, p. 32 8]; a economies. b) is locally solid [17, 6.1, p. 32 7]; and n n p Thm EXAMPLE 3 .6. Consider the Riesz dual system OR , m. > of Example 3.2 with the price simplex c) when n is either a-compact or complete separable metric, then the n topological dual of (C (O), p) is precisely the vector space Mt b (p (p ,... ,p ) p l} . !:,. l n 6 m.: [ i functionals on C (O) that are representable as an 1=1 of all linear b integral with respect to a unique compact-regular Borel measure D In Section 3 of [8], Debreu derives the excess demand [17. Thm 9 .1 , p. 332 ). correspondence for an economy having a finite number of profit maximizing firms and a finite number of utility maximizing consumers. Thus, according to Theorem 4.3 of the appendix, if D is a • If, in addition to his assumptions, we assume that production sets are complete separable metric space, then M is an ideal of C (D) (the t b 17 18 strictly convex and that utility functions are strictly quasi-concave, EXAMPLE 3 .8. Let <1..,, 11> be the special Riesz dual system of • then the excess demand correspondence is a function, which we denote Example 3.4 with the price simplex A= {p £ 1.., : p 2 0 and llpll = 1}. by E. Debreu shows that the domain of E is A; E is continuous; and E e 1}. Note that S= {p 11 : p )) 0 on 1.., and llpll1= n n satisfies Walras' law. Hence, (OR , JR. ), A, E) is an economy in our Consider 1.., with the Macke y topology �(1..,. 11> as the sense, and so, by Theorem 2.4 there exists some p e A with E and commodity space of an exchange economy having a finite number of p � O p(E consumers. Each consumer has with the relative �<1..,, )-topology p)= O. This proves Theorem 6 of [8] for the special case where 1: 11 the excess demand correspondence is a function. • as his consumption set. Endowments are in 1: - (0). Agents maximize strictly quasi-concave, monotone, and �<1..,. 11)-continuous EXAMPLE 3. 7. Again consider the Riesz dual system S; that E is continuous; and that it satisfies Walras' law. Moreover, For this economy, the "excess demand function E" is well defined, with n E is bounded from below (i.e., there exists some x £ JR satisfying p £ 0 domain D= ( 11 : p 2 and llpll1=1). However, since E need not x i E for all p £ I "' p ) S satisfies be continuous on D, we cannot invoke our existence Theorem 2. 4 p S), and IE p 11 -? whenever ( n � n directly. Rather we shall use the argument in the proof of Theorem p p £ A - S. It is easy to see that the latter two properties of n -? 2. 4 to demonstrate the existence of an equilibrium price for this E imply our boundary condition: (p S and p p e A - S imply n} !!;; n -? example. lim p(E ) ) 0 for some p e S. Hence, by Theorem 2.5 there exists Pn Theorem 2. 4, let D : a£ As in the proof of ( a Al be the family some p e S with E = 0, and this is the conclusion of Theorem 8.3 of P of all finite dimensional simplices contained in D. It is easy to [9]•• show that E: (D ' w•) �u . 1 ) is continuous, using a -? Cl..,, .., 1> 19 20 This means that there exists some i with q ) > O. Put arguments--say in Debreu (8). Then the revealed preference relation c(i restricted to D is irreflexive, convex valued, and w•-upper A. (min{q.. (X ), llwll }) 2q (i), and note that y = A.e. + X satisfies a q "' / c 1 q semicontinuous (see the proof of Theorem 2.2). Thus, by Theorem 2.1, y ) X for each t. By the ... 1 )-continuity of the ), there exists q •<1 1 t t > has a maximal element of D ' say q . Now consider the nets a a some n so that y = •••,y ,O,O, •••) satisfies X for all t n (y1, n Yn � q • {q } and {E } [-w, w]. Since is a<1..,, 1.. )-compact and a �A q � A a and q(y ) = q (y ) < q(X ) = q(w). Hence, for some t we have D C D q v v [-w, w] is a<1... 1 )-compact, by passing to two subnets (if necessary) 1 q (y ) < q (w ) and > X (t) for all sufficiently large a, a a n a t y il q • t a aU ...1..,> aU ... 11> contradiction. we can assume that q � q £ A and E � Z e [-w, w] . a q q a We shall show that q e S and Z E = 0, i. e. , that q is an b) q = O. q q c = Ilwll .. equilibrium price. This means that q �· Choose A.such that 0 Suppose for some p £ D we have p(Z ) O. Then p(E ) > O and note that y = A.e. + X satisfies y ) X for all agents t, and q > q 1 q q a t < w holds for all sufficiently large a. But for all sufficiently large a moreover y 2 . By the •<1..,, 11)-continuity of the ) there exists t we aiso have p e D ' which contradicts the maximality of q with a a some n so that y = (y •••,y ,O,O, •••) satisfies y X for all t. n 1, n n � q respect to > on D • Hence, p(Z i 0 holds for all p e D, and this a q ) Moreover, q(y ) = 0 and q(w) > 0 (since w is uniformly bounded away n implies Z i 0. We can write Z = X - w, where q q q from zero). Hence, for some agent t we have q q w a(yn) a( t) and X X (for a(1 , 1 )) and X is the aggregate demand at prices q . i -7 q .. 1 a � � y > X (t) for all sufficiently large a, a contradiction. D a t Hence, X i w. A standard argument now shows that q(X ) = q(w). q q Now by the Yosida-Hewitt theorem (see Proposition 2.6 in the CASE II : q.. ) = 0. (Xq � appendix of (4 )) we can write q = q + e 1 and q"" is Note (sinGe w is unifo.nnly bounded away from zero) c �· where qc i that purely finitely additive. Next we claim that q.., = O. To see this, �(w) > 0 holds. But then the relation assume by way of contradiction that q"' > O. We have two cases. q ) ) + = ) c (Xq qc (Xq �(Xq) q(Xq CASE I: q (X ) > 0. q(w) = q w + �(w) > .., q c( ) qc(w), In this case, we have two possibilities. contradicts X i w and q LO. q c Thus, q = q e D holds. Next we claim that q e S. To see c a) q 0. c > 21 22 this, assume that q(i) = 0 holds for some i. Choose some C(il) consisting ot nonz ero functions and define -n 0 < A. < llwll.,, with A.e + X < 2w. If y = A.e. + X, then y > X , and 2 e i q 1 q n q e u lf l. Then it is easy to see that � \ F - e is t n n ll., n p� p n p by arguing as in Case I(a) above, there exists some • an excess demand function (with domain L - {0}) which is homogeneous + ••• ••• � Y = Cy • ,y ,O,O, ) with q(y ) < q(w) and y X for all t. n 1 n n n q of degree 0. • Hence, for some t, q > X (t) hold for all a (y n ) < q a (wt) and y n q t a Our final example uses the method of Example 3.9 to construct sufficiently large a, a contradiction. Therefore, q e S. an excess demand function E on a simplex A where the domain D of E is Finally, a standard argument shows that for each t, a proper subset of A. q(X q(t)) = q(wt) and that Xq(t) is maximal in t's budget set with EXAMPLE 3.10. Consider the Riesz dual system D={pel : llpll l}, u (1,1, •••) = e, Uz (0,1, 1,••• ), ; 1 l The next example presents a method of constructing excess u = (0,0, 1,1, •••) etc. and e = (1,0,0, •••) e (0,1,0,0, •••>. 3 . 1 . 2 demand functions. etc. EXAMPLE 3.9. Let "' "' Now define F: A -1 1: by \ I "' bounded and lu 11 • then put F p(e )u for each p e A.) n < [ n n � P n=l Now it is a routine matter to verify that the function F p [ A.nUp(un)] en. n=l E: (A , w*) (L, II II> defined by E = F - p(F )e, p e A, is an -1 p p p Then it is easy to verify that the functions pl-1 F , from (A, w•) excess demand function (with domain A). p = into .11·11>. and pl-1 p(F ) from w*) into are both Finally, for a specific example. Let L C(Q) for some U "' p (A, JR, • •. e D, compact metric space Fix a countable norm dense subset {f } of continuous. Also, if p = (p ,p , ) then from the inequality il. n 1 2 23 24 a> that 0 < p(F ) < q(F ). This implies A. �[p(u )] A. � p ) ( p p p n n n ( L i 2. A.n � n) ' 1=n q(F ) P a> lim q(E ) = 1 > 0, \ p(F ) - )p > 0. Pa p if follows that p(F ) 2. A. �(pn n Thus, the function p ,f=;1 n • E:(D, w•) -7 (f.,,,11·11> defined by and we are finished. F These examples suggest that the appropriate commodity space E _JL_ - e p p(F ) p for economies are the AM-spaces with unit, which admit a strictly is continuous and satisfies Walras' law. In addition, we claim that positive linear functional. it satisfies our boundary condition, i. e. if a net {p D satisfies a} !;;;; w• p � q e - then lim p(E ) > 0 holds for some p e D. a 11 D, p a w• To this end, let {p } D satisfy p � p e - D. Write a � a 11 = p e and .,, purely finitely additive. P Pc + P.,,• with c {� P CASE I: p O(i. e. , p = ). c = P.,, Clearly, p(F 0 holds, and so, the element e e D satisfies p) = 1 (l) A.l� lim e (E ) lim[ - l] "'· 1 p (F ) Pa a p a CASE II: p > O. c Since llp II < llp p II = llp 1 c c II + ll a> ll = , there exists some e D satisfying p < q. Also, note that p.,,(u ) = p.,,(e) = llp.,,ll > 0 q c n holds for each n. Therefore, a> a> F = [A.Up(u )] e [A. U (e)]e > > 0. p n n n 2. n Pm n n=l n=l particular, p ) > 0 holds, and so, from p (F ) p(F ) we see In c(Fp c p p 25 26 ACKNCYNLEDGEMENTS 4. APPENDIX: RI ESZ SPACES As we have said in thP- introduction, this paper utiliz es the The research of the first author was supported in part by NSF theory of Riesz spaces (vector lattices). For this reason, this grant MCS 81-00787 and that of the second author in part by a grant section deals exclusively with the fundamental properties of Riesz from NSF to Yale University. spaces. For details concerning the lattice properties of Riesz spaces This work was done at the California Institute of Technology we refer the reader to [13), and for the topological concepts on Riesz while Charalambos D. Aliprantis was a visiting Professor and Donald J. spaces to [l] and [16). Brown was a Sherman Fairchild Distinguished Scholar. We take this A relation l on a non-empty set X is said to be an order opportunity to thank our colleagues at Caltech for their hospitality. relation, whenever a) x e l x holds for all x X; b) x L y and y 2 x imply x y; and x c) l y and y l z imply x L z. An ordered set is a non-empty set X together with an order i relation l The symbol x y is an alternative notation for y l x. Now let A be a non-empty set in an ordered set X. Then an element y e X is said to be a least upper bound (or a supremum) for A, whenever 1) y is an upper bound for A, i. e. , xi y holds for all x e A; and 2) if x i z holds for all x e A, then y i z. Clearly, a set A can have at most one least upper bound, and if it does have one, then it is denoted by sup A. The greatest lower bound (or infimum) of a set A is defined similarly, and is denoted by inf A. 27 28 A lattice is an ordered set X such that sup{x, y} and inf{x,y} exist L means f( w) 2. 0 for all w e Cl. Here are some examples of function for each pair x, y e X. As usual we write spaces. o xvy sup{x,y} and x/\y inf{x, y}. 1. JR , all real-valued functions on a set Cl; In this paper, all vector spaces are real vector spaces. The 2. C( O), all continuous functions on a topological space Cl; symbol JR will stand for the set of real numbers. An ordered vector C 3. b(Cl), all boundej continuous functions on a topological space Cl; space L is a vector space L together with an order relation 2. which is compatible with the algebraic structure of L in the following 4. B( Cl), all bounded real-valued functions on a set !l; manner: P 5. i (1 i p Let L be a Riesz space. Then for each f e L we put + The set L = [f B L : f 2. 0} is called the positive cone of L, and its + - l l members are called the positive elements of L. An ordered vector f fv 0. f (-f)VO , and f fv (-f). space L which is also a lattice is referred to as a Riesz space (or a + The element f is called the positiye part of f, f- the negative part, vector lattice). and IfI the absolute value of f. The following identities hold: Typical examples of Riesz spaces are provided by the function + - spaces. A function space L is a vector space of real-valued functions f /\ f O; defined on a non-empty set such that for each f, g e L the two !l + f f - C; and functions fvg and f/\g defined by + l l = f f + c. fvg(w) max[f(w), g( w)), and f/\g(w) min{f(w), g( w)) In particular, every element of L can be written as a difference of two positive elements. Every set of the form belong to L. Clearly, every function space L under the ordering f g 2. + [-u,u] = {f e L -u � f � u) (u e L ) is called an order interval whenever f( w) l g( w) for all w e !l, is a Riesz space. Also, f 2. 0 in 29 30 of L. A linear functional ' : L -7 1R is said to be order bounded, to be strictly pos itive, in symbols ' >> 0, whenever f > 0 in L (i. e. , whenever ' maps order intervals of L onto bounded subsets of ll; i. e. , f 2 0 and f I 0) implies '(f) ) O. + whenever '([-u,u]) is a bounded subs et of ll for each u & L . The next res ult is the dual of Theorem 4. 1. For a proof see Clearly, the set of all order bounded linear functionals on L is a [2, Thm 24 .3 , p. 190]. vector space. This vector space is called the order dual of L, and is TIIEOREM 4. 2. Let L be a Ries z space. If' e L ' then for - + denoted by L . In L an ordering 2 is introduced by saying that each f e L we have f + 'l l whenever '(f) 2 (f) holds for all f e L . Under this ordering L becomes an ordered vector space. In actuality, L is a Ries z + f '(f ) sup{ (f) 0 .{• .{fl; space. For the proof of the next res ult of F. Ries z see [l, Thm 3. 3, f p. 20] or [2,Thm 24 .2, p. 189]. f(f-) sup{- (f) 0 .{ l .{ fl; and then An ideal A of a Ries z space L is a vector subs pace of L such that I fI Isl and g & A imply f e + .{ A. ' ( f) sup{f(g) 0.{g.{f}; We now turn our attention to topological concepts on Ries z - •I I f ( f) sup{-f(g) 0 .{ g i fl; and spaces . A seminorm 11 on a Ries·'· space L is said to be a lattice (or a Ries z) seminorm, whenever I' I cf> sup{f(g) -f .{ g i fl. l fl i Isl in L implies llfll .{ llsll. Let L denote the pos itive cone of L , If should be clear + A locally convex-s olid Ries z space (L,T) is a Ries z space L equipped that L cons is ts precis ely of all linear functionals ':L for + -7 1R with a Haus dorff linear topology • that is generated by a family of + which f(f) L 0 holds for all f & L . The members of L are called the + lattice seminorms . Recall that the vector space of all •-continuous pos itive linear functionals on L. By Theorem 4. 1, every order bounded linear functionals on Lis known as the topological dual of (L,T). linear functional on L can be written as a difference of two pos itive Regarding topological duals of locally convex-s olid Ries z linear functionals on L. A positive linear functional ' & L+ is said spaces the following res ult holds . (For a proof see [1, Thm 5. 7, p. 36].) 31 32 , 3 • complete metric space is referred to as a Banach lattice. Some THEOREM 4. The topological dualL of a locally convex- important examples of Banach lattices are: solid Riesz space (L,�) is an ideal of its order dual L , that is, , , , Ill i ITI and T £ L .!l!!P.lx f e L . In particular, is a Riesz space L a) The C( D) spaces, D compact, with the sup norm in its own right. ' llfll = sup{lf( w)I : w £ D}. If L is a Riesz space and L is an ideal of L separating the .,, points of L, then the pair - f(f ) sup{-f(f) T e L and 0 i f i f} sup norm, for example, is the only lattice norm that makes C[O,l] a Banach lattice.) On the other hand, every pos itive linear functional sup{-f(f) , 0 i f} i 0. T £ L and i f [2, 24.10, 192] • on a Banach lattice is norm continuous; see Thm p. - - • f f = 0 Thus, if is a Banach lattice and denotes its norm dual, then On the other hand, (f ) 2 0 holds trivially, and so, (f ) holds L L • • - = holds, and moreover L is a Banach lattice, for all 0 E Since is a f = 0 L i ' L . L Ries z space,, this implies (f ) L f 1 - Now let be a Banach lattice. Then there is a natural for all E L , from which it follows that f O. But then L + - + � .. f = f - f = f 0, as desired. • 2 embedding f I� f of L into its double norm dual L defined by A special class of locally convex-solid Riesz spaces are the 1 'f) 'f( f) for 1 E and e f ( L 'f L • Banach lattices. A Riesz space that under a lattice norm is a The embedding is linear, norm and lattice pres erving. Thus , if L is 33 34 •• sublattice of the Banach lattice L The following property of this REFERENCES identification will be needed. [lJ C, D. ALIPRANTISAND O. BURKINSHAW, Locally Solid Riesz Spaces, + TIIEOREM 4. 5. If L is a Banach lattice, then L is Academic Press, New York-London, 1978. •• • •• a( L ,L ) -dense in L + [2] C. D. ALIPRANTIS AND O. BURKINSHAW, Principles of Real Analysis, + •• • + •• PROOF. Let L be the a( L ,L )-closure of L in L North-Holland, New York-Oxford, 1981. • + [3J K. J. ARROW AND F, H. HAHN, General Competitiye Analysis, Clearly, L !; L: . Assume by way of contradiction that there exists Holden-Day, San Francisco, 1971. •• + 0 < e L - L • Then by the Hahn-Banach theorem there exist c e f + JR •• • •• and a a( L ,L )-continuous linear functional on L (i. e. , some [4 ] T. F. BEWLEY, Existence of equilibria in economies with • + infinitely many commodities,J, Econom. Theory 4( 1972), 514-540 . f e L ) satisfying T Cf) < c and f( x) L c for all x e L . Since + + ax e L for all x £ L and all a L 0, it follows that c i 0 and [5] D. J. BROWN AND L. M. LEWIS,Myopic economic agents, • f e L . But then we have 0 i f(f) < c i 0, which is impossible, + Econometrica 49(1981) , 359-368. + •• Hence, L = L holds, • + [61 R. C. BUCK, Bounded continuous functions on a locally compact For more about Banach lattices the reader can consult the book space, Michigan Math J. 5( 1958) , 95-104 . of H. H. Schaefer [16]. A locally convex topology t on L is [7] G. DEBREU, Theory of Value, Wiley (Interscience), New York, compatible with (L,L') if (L,t) has topological dual L' . The Mackey 1959. topoloRv •( L,L') on L is the finest locally convex topology on L compatible with [9] E. DIERKER, Topological Methods in Walrasian Economics, Lecture (18] H. F. SONNENSCHEIN, Demand theory without transitive Notes in Economics and Mathematical Systems, #92, Springer preferences. with applications to the theory of competitive Verlag, New York, 1974. equilibrium, In "Preferences, Utility and Demand (a Minnesota Symposium),H pp. 215-223. Harcourt Brace Jovanovich, New York, (10] N. DUNFORD AND J. T. SCHWARTZ, Linear Operators I, Wiley 1971. (Interscience), New York, 1958. (19] V. S. VARADARAJAN,Measures on topological spaces, Mat. Sb. (11] K. FAN, A generalization of Tychonoff's fixed point theorem, 55(97) (1961), 35-100. English translation: Amer, Math. Soc. Math Ann. 142(1961), 305-310. Transl. (2) 48(1965), 161-228. (12] D. M. KREPS, Arbitrage and equilibrium in economies with infinitely many commodities, J. Math. Econom. 8(1981), 15-35. [13] W. A. J, LUXEMBURG AND A. C. ZAANEN, Riesz Spaces I, North Holland, Amsterdam,1971. (14] A. MAS-COLELL, A model of equilibrium with differentiated commodities, J. Math. Econom. 2(1975), 263-295. (15] H. NIKAIDO, Conyex Structures and Economic Theory, Academic Press, New York, 1968. [16] H. H. SCH.AEFER, Ranach Lattices and Positive Ooerators, Springer-Verlag, New York-Heidelberg-Berlin, 1974. [17] F. D. SENTILLES, Bounded continuous functions on a completely regular space, Trans. Amer. Math. Soc. 168(1972), 311-336.