A new lemma on the existence of maximal elements of binary relations is proved and applied to a revealed preference relation on price vectors. The resulting maximal elements are equilibrium prices. This technique allows one to generalize results of Aliprantis and Brown [1982], Neuefeind [1980], and Geistdoerfer-Florenzano [1982]

Border, Kim C.

Caltech Authors - Main

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA. CALIFORNIA 91125
ON EQUILIBRIA OF EXCESS DEMAND CORRESPONDENCES 
Kim C. Border 
,�s'1\lUTE Op 
�'!!{- }'�<":
� � 
.§ � 
c:c 
0 
'-' C'I . -< -I 
. 
� tt;
� 
�
l-1: "� � '" SJtALL t.\�"'� 
SOCIAL SCIENCE WORKING PAPER 460 
September 1982
Revised January 1983
ABSTRACT 
A new lemma on the existence of maximal elements of binary 
relations is proved and applied to a revealed preference relation on 
price vectors. The resulting maximal elements are equilibrium prices. 
Th.is technique allows one to generalize results of Aliprantis and 
Brown [1982], Neuefeind [1980], and Geistdoerfer-Florenzano [1982]. 
m EQUILm.R.IA OF EXCESS DEMAND CORRESPONDENCES. 
Kim C. Border 
Introduction. 
In this note we present a new technique for generalizing some 
well known theorems on tho existence of equilibrium prices for excess 
demand correspondences. Tho basic result in this area is duo to 
Debreu [19561. Galo [1955) . Kuhn [1956) . and Nikaido [1956) . Recent 
variations are duo to Grandmont [1977) and Nouef eind [1980) who 
introduce boundary conditions on the behavior of excess demand. and by 
Geistdoorfer-Floronzano [1982) who allows a weakening of Walras' law 
and the continuity condition. Aliprantis and Brown [1982) extend the 
equilibri1111 theorem to certain infinite-dimensional spaces for the 
case of excess demand functions (singleton-valued correspondences). 
This paper extends tho results of Geistdoerfer-Florenzano to 
correspondences satisfying a boundary condition. The method of proof 
is based on the fact noted by Arrow and Burwicz [1958) that if the 
excess demand correspondence is singleton�valued and satisfies a gross 
substitutability condition. then the equilibri11111 price vector 
minimizes a revealed preference relation. Following Aliprantis and 
Brown we make use of a generalization of a lemma of Sonnenschoin 
[1971) to prove the existence of equilibri11111 prices by means of 
maximizing a revealed preference type of relation for price vectors. 
• I have benefited greatly from the comments of Don Brown and Ed 
Green. 
2 
We then use a separating hyperplane arg1111ent to extend their result to 
correspondences. We motivate tho technique with a discussion of tho 
trivial case of the one person pure exchange economy. 
Tho Qno Person Pure Ex£hango B£opomy. 
Consider the case of an individual with a well behaved utility 
function and a fixed endowment w of commodities. All prices are taken 
to lie in the unit simplex. At prices p tho individual receives 
income p ·w and demands the commodity vector h(p). Define the excess 
demand function E by E(p) = h(p) - w. An equilibri11111 price vector p 
satisfies E(p) � o. i.e •• h(p) � w. It is well-known from the theory 
of compensated demand curves that for any price vector p with relative 
prices different from p the commodity vector h(p) will be preferred to 
h(p) = w. By tho nature of the budget constraint. w is always 
affordable. so it follows from tho weak axiom of revealed preference 
that h(p) cannot be afforded at prices p. that is. we must have that 
p • h(p) > p w. In terms of excess demands this becomes 
p • E(p) > O. If we define the binary relation U between price 
vectors by q U p  if q • E(p) > 0 then tho equilibrium.price vector is 
the largest element of U .  Conversely, let q be a maximal element of 
U ,  i.e., for all other prices p. not p U q. Then p • E(q) i 0 for 
all p. It then follows from Walras' law that E(q) = O. Thus if the 
weak axiom of revealed preference holds the equilibrium price is 
unique. This approach reduces the search for equilibrium prices to 
the search for maximal elements of a revealed preference relation. 
3 
lhon there is more than one consumer. the excess demand function need 
not obey the weak axiom of revealed preference. Remarkably enough,tho 
'
aame approach works even in this case, though the equilibrium need not 
be unique. Thus we next present a lemma on the existence of maximal 
elements of binary relations. 
Kaxiptal Elements of Binary Relations. 
A binary relation on a set E: can be described in terms of a 
correspondence U : E: °'°' E:, where U(p) is interpreted to be the set 
elements of E: which are larger than p (the upper contour sot). A.!!'= 
maximal element of E: is a p a E: for which U{p) = 4. With each binary 
relation U we associate the correspondence u-l : E: °'°' E: by 
q a U-l(p) if and only if p a U(q). The following lemma is a 
strengthening of results of Fan (1961,Lemma 4) and Sonnenschein 
(1971,Theorem 4). 
!.2l!l!!ll.a. Let E: be a compact convex nonempty subset of a Hausdorff 
topological vector space and let U : E: °'°' E: be a binary relation 
satisfying for all p 1 E: 
(i} x J co U(p) (where co denotes convex hull). 
(ii) if q a u-1(p), then there is a p' (possibly = p) 
such that q a int u-1(p').
Then the set of U-lil&ximal elements of E: is compact and nonempty. 
� By definition, the aet of U-maximal elements is just 
n (E:\U-l(p)). By hypothesis (ii). 
pal: 
n (E:\U-l{p)) • n (E:\int U-l(p')). 
pal: p'aE: 
4 
This latter intersection is clearly compact, being the intersection of 
compact sets. 
We now show that it is nonempty. For each p, put 
F(p) = E:\(int u-1(p)). As noted above each F(p) is compact. We first 
D . 
claim that if q a co {pi i = 1 • • • •  ,n}, then q a U F(p1): Suppose 
i=l 
D i -1 i i that q I UF(p ). Then q a U (p ) for all i, so p a U(q) for all i. 
i•l 
But then q a co {pi} C co U(q), which violates (i). It then follows 
from the E:naster-E:uratowsld-Kazurkitnricz lemma as extended by Fan 
(1961,Lemma 1) that n F(p) � 4. 
pal: 
EJ;cess Demand Correspondences. 
Q.E.D. 
Let Jl. n be the commodity space. Denote by Jl. : the positive 
cone of ltn, i.e., • : = {z a JRD :zi 2 0 for i • 1, ••• n}, and N 
denote the negative cone, -RE:. Let 
A = {p I JR n: t p ... 1. p. 2 0 
1=1 1 1 
i • 1, • • •  ,n} and. 
S = {p a JRn:p » 0} where p » q means p. > q., 
1 1 i = 1,, • • •  ,,n. The 
domain of the excess demand correspondence will be denoted by D, and 
it will be either equal either to S or to A. 
A correspondence E : D °'°' JR n satisfies the 
yeak weak form of Walras' law if for each p, 
(IWWL) p • z .i 0 for some z c E(p). 
A correspondence E : D '' lR n satisfies the 
strong form of !alras' lay if for each p. 
(SWL) p • z = 0 for all z a E(p). 
In a private ownership economy. Walras' law will be satisfied if all 
consumers spend exactly all of their income. 
A correspondence E : S �' lR n is said to satisfy the 
boundary condition (B) if the following holds. 
s 
(B) if for every q c A. there is a p a A such that for every q 
sequence {�}CS with �' q c A\S. there is a subsequence {!!pl
such that for all p. pq 
• z > 0 for all z c E(q�). 
Boundary condition (B) is weaker than the condition introduced by 
Neuefeind [1980,Lemma 2) . which requires p to be independent of q.q 
It is however, stronger than the condition used by Aliprantis and 
Brown [1982) or Grandmont [1977) . Their condition allows p to depend q 
on the sequence as well as its limit. 
We say that the correspondence E : D '°' lR n is upper demi-
continuous if for each open half-space B in Jln, 
E+[B] E {p : E(p) C B} is open in D. (This terminology is from 
Browder [1967) . For closed-valued correspondences this definition is 
equivalent to what Geistdoerfer-Florenzano calls upper hemi-
continuity. For finite-dimensional spaces every neighborhood of a 
6 
point contains a neighborhood which is a finite intersection of open 
half-spaces. in which case demi-continuity reduces to hemi-continuity 
in the sense of Hildenbrand [1974) .) 
A free disposal equilibrium price of the correspondence 
E : D '' JR n is a price p such that E(p) n N # rl. An eauilibrium 
�p is a price p 1 S such that 0 s E(p). 
Existence of Equilibrium Prices. 
We will treat tlfo cases. In the first case the excess demand 
correspondence is assumed to be defined for all nonnegative prices and 
to satisfy the weak weak !alras' law and to take on only compact 
convex values. The existence of free disposal equilibriua prices is 
established. In the second case, the excess demand correspondence is 
defined only for strictly positive prices and is assumed to exhibit 
nice boundary behavior. In particular, we assume boundary condition 
(B). The strong fo:i::m of Walras' law is assumed, though the' values of 
the correspondence are only assumed to be closed and convex. 
Existence of equilibriua price� rather than just free disposal 
equilibriua prices is established. In both cases the excess demand 
correspondence is assumed be upper demi-continuous and to be 
homogeneous of degree zero, so that only noDDalized prices are 
considered. 
The outline of the proof in each case is as follows. We 
define a binary relation between prices based on the revealed 
prof eronco araument for the one person case and show that the maximal 
olomonta must bo equilibrium prices. Thia part of the proof uses 
7 
Walraa' law and the fact that the excess demand correspondence assumes 
closed convex values. and is baaed on a separating hyperplane 
aramaent. We then show that the relation satisfies the hypotheses of 
the leana. and so it has a nonempty compact set of maximal elements. 
Thia part of the proof uses Walras' law. demi-continuty and the 
boundary condition. One can view this approach as an attempt to reduce 
tho general case to the one person pure exchange economy. 
Theorem 1. Let E : A �� lit n be upper demi-continuous with nonempty 
compact convex values and satisfy the weak weak fora of Walraa' law. 
Then the set of free disposal equilibri1111 prices for E is compact and 
nonempty. 
EI22!... For each p a A set 
U(p} • {q : q • z > 0 for all z a E(p}}. 
Then U satisfies the hypotheses of the lemma: For U(p) is 
' -1 clearly convex for each p and by Walraa' law pr U(p}. Also U (p} is 
open for each p: 
-1 For if q a U (p) we have that p • z > 0 for all z a E(q). 
Then since E is upper demi-continuous E+[{x : p • x > 0)) is a 
neighborhood of q in u-1(p). 
Now p is U-maximal if and only if 
for each q a A. there is a z a E(p} with q • z i O. (•} 
Wo now show that E(p} RN # '  if and only if (•}. Suppose 
z a E(p} R N. Then q 0 z i 0 for all q a A, so (*) holds. If on tho
other hand, E(p} R N s '· then we can strictly separate E(p} which is
compact and convex from N which is closed and convex (Schaefer (1971. 
9.2)). That is there exists some q a lR.n and some c a  JR satisfying 
x a N and z a E(p} imply q • z > c > q • x 
Since N is the neaative cone. c > 0 and q a lit:. Thus q • z > 0 for 
all z a E(p). and (*} does not hold. 
It thus follows from the lemma that the set of free disposal 
equilibrium prices is compact and nonempty. 
8 
Q.E.D. 
Theorem 2. Let E : S �� lit n be upper demi-continuous with nonempty. 
closed. convex values and satisfy the strong from of Walras' law and 
the boundary condition (B} (relative to S and A}: 
(SWL) 
(B) 
p • z .. 0 for all z a E(p). 
if for every q a A. there is a p a A such that for every q 
net {q�} CS with �� q a A\S. there is a subnet {� } such that 
for all p. pq • z > 0 for all z a E(� ). 
Then the set of equilibrium prices for E is compact and nonempty. 
Proof. Define the binary relation U on A by 
9 
p z > 0 for all z I E(q) and p,q e S 
p I U(q) if or 
p I S, q I A\S. 
We first sho. that the U-•axiaal elements are precisely the 
equilibria. prioes. First suppose that p is U-•azi•al. i.e •• 
U(p) • d. Since U(p) = S for all p a A\S. we have that p a S. Since 
p a S and U(p) = 4 we have 
for each q a s. there is a z a E(p) with q • z i O. (1) 
le now sho. that (1) implies 0 a E(p). Suppose by way of 
oontradiction that 0 J E(p). Then we can strictly separate {O) which 
is compact and convex from E(p) which is closed and convex with 
- D - - - -p s It 
• 
Th'IUI p • z > 0 for all z a E(p). Put pl E lp + (1 - l)p. 
Then for z a E(p). pl • z s l; • z + (1 - l)p • z = l; 
• z > 0 for
l > o. (ll.ecall that p • z = 0 for z a E(p) by Walras' law.) For 
l > 0 small enouah. pl >> 0 so that the normaliz ed price vector 
P1 = <[: <P1>i>-
1P1 a S and pl • z > 0 for all z a E(p). which violates 
(1). 
Conversely, if p is an equilibri'llll prioe. then 0 a E(p) and 
since p • 0 s 0 for all p. U(p) • d. 
We now verify that U satisfies the hypotheses of the le11111&: 
(ia) p J U(p): For p a S this follo.s from Walras' law. For 
pa A\S, p JS= U(p). 
(ib) U(p) is convex: For p a s. let q1• q2 a U(p). i.e. 
q1 • z > O. q2 • z > 0 for z a E(p). Then [lq1 + (1 - l)q2] • z > 0 
10 
as well. For p a A\S, U(p) • S which is convex. 
(ii) If q I u-1(p). then there is a p' with q I int u-1(p'): 
There are two cases: (a) q e S and (b) q a A\S. 
(a) q 8 s n u-1(p). Then p • z > 0 for all z I E(q). Let 
H be the open half space {x : p • x > 0). Then by upper demi­
oontinuity, E+(H] is a neiahborhood of q contained in u-1(p). 
(b) q a (A\S) n U-l(p). By boundary condition (B) 
-1 1 -1 q a int U (pq): Suppose q, int U (pq). Then there is a net
q
4 
"' q with q
4 
J u-1(pq) for all a. This implies that q4 
a S
and so there is some z
4 
a E(11g) with pq • z4 
i O. 
(B). 
This violates 
Q.E.D. 
11 
REFERENCES 
AlipraRtis. C.D •• aRCI D.J. Brown (1982). Equilibria iR aarketa with a 
Riesz apace of commodities. Caltech Social ScieRce WorkiR& 
Paper #427. 
Arrow. �.J. and L. Hurwicz (1958). On the stability of the competitive 
equilibri11a. I. Econometriqa 26. 522-552. 
Browder. F.E. (1967). A new geReralization of the Schauder fixed point 
theorem, Hath . Ann· 174. 285-290. 
Debreu. G. (1956). Market equilibri1111, Proq. Natl. Acad. Sci. 38. 
386-393, 
Fan, �. (1961). A generalization of Tychonoff's fixed poiRt theorea. 
Hath. Apa, 142� 305-310. 
Gale. D. (1955), The law of supply IJl.d dem1J1.d. Hath. Sca114inayica 3, 
155-169. 
Geistdoerfer-FlorenzaRo. H. (1982). The G�le-Ni.kaido-Debreu leama and 
the existence of transitive equilibrium with or without the 
free disposal assuaption. J, Hath. Ecop. 9, 113-134. 
Grandmont, J.M. (1977). Temporary general equilibri1111 theory. 
Eqop911ptrica 4S, S35-S12. 
Hildenbrand, W. (1974). •core and equilibria of a large economy.• 
Princeton University Presa, Princeton. 
�uhR. H.W. (1956). A Rote OR 'The law of auppl7 aRd deaaRd'. Il
Sc11l4iRayica 3. 155-169. 
12 
NeuefeiRd, W. (1980). Notea OR exiateRce of equilibri1111 proofs aRCI the 
bo11Rdary behavior of supply. Econometrica 48, 1831-1837. 
Nikaido. H. (1956), On the classical aultilateral exchange problem, 
Hetroeco9911ica 8. 135-145. 
Schaefer. H.H. (1971), •Topoloaioal vector spaces.• Sprinaer-Verlag. 
New York. 
Sonnenschein, H, (1971). Demand theory without transitive preferences. 
with applications to the theory of competitve equilibri1111. 
in •Preferenqps. utility. and demand.• ed. J.S. Chipman et. 
al., Harcourt. Brace, Jovanovich. New York. 


On Equilibria of Excess Demand Correspondences

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA. CALIFORNIA 91125
ON EQUILIBRIA OF EXCESS DEMAND CORRESPONDENCES 
Kim C. Border 
,�s'1\lUTE Op 
�'!!{- }'�<":
� � 
.§ � c:c 0 '-' C'I . -< -I . 
� tt;
� 
�
l-1: "� � '" SJtALL t.\�"'� 
SOCIAL SCIENCE WORKING PAPER 460 
September 1982
Revised January 1983
ABSTRACT 
A new lemma on the existence of maximal elements of binary 
relations is proved and applied to a revealed preference relation on 
price vectors. The resulting maximal elements are equilibrium prices. 
Th.is technique allows one to generalize results of Aliprantis and 
Brown [1982], Neuefeind [1980], and Geistdoerfer-Florenzano [1982]. 
m EQUILm.R.IA OF EXCESS DEMAND CORRESPONDENCES. 
Kim C. Border 
Introduction. 
In this note we present a new technique for generalizing some 
well known theorems on tho existence of equilibrium prices for excess 
demand correspondences. Tho basic result in this area is duo to 
Debreu [19561. Galo [1955) . Kuhn [1956) . and Nikaido [1956) . Recent 
variations are duo to Grandmont [1977) and Nouef eind [1980) who 
introduce boundary conditions on the behavior of excess demand. and by 
Geistdoorfer-Floronzano [1982) who allows a weakening of Walras' law 
and the continuity condition. Aliprantis and Brown [1982) extend the 
equilibri1111 theorem to certain infinite-dimensional spaces for the 
case of excess demand functions (singleton-valued correspondences). 
This paper extends tho results of Geistdoerfer-Florenzano to 
correspondences satisfying a boundary condition. The method of proof 
is based on the fact noted by Arrow and Burwicz [1958) that if the 
excess demand correspondence is singleton�valued and satisfies a gross 
substitutability condition. then the equilibri11111 price vector 
minimizes a revealed preference relation. Following Aliprantis and 
Brown we make use of a generalization of a lemma of Sonnenschoin 
[1971) to prove the existence of equilibri11111 prices by means of 
maximizing a revealed preference type of relation for price vectors. 
• I have benefited greatly from the comments of Don Brown and Ed 
Green. 
2 
We then use a separating hyperplane arg1111ent to extend their result to 
correspondences. We motivate tho technique with a discussion of tho 
trivial case of the one person pure exchange economy. 
Tho Qno Person Pure Ex£hango B£opomy. 
Consider the case of an individual with a well behaved utility 
function and a fixed endowment w of commodities. All prices are taken 
to lie in the unit simplex. At prices p tho individual receives 
income p ·w and demands the commodity vector h(p). Define the excess 
demand function E by E(p) = h(p) - w. An equilibri11111 price vector p 
satisfies E(p) � o. i.e •• h(p) � w. It is well-known from the theory 
of compensated demand curves that for any price vector p with relative 
prices different from p the commodity vector h(p) will be preferred to 
h(p) = w. By tho nature of the budget constraint. w is always 
affordable. so it follows from tho weak axiom of revealed preference 
that h(p) cannot be afforded at prices p. that is. we must have that 
p • h(p) > p w. In terms of excess demands this becomes 
p • E(p) > O. If we define the binary relation U between price 
vectors by q U p  if q • E(p) > 0 then tho equilibrium.price vector is 
the largest element of U .  Conversely, let q be a maximal element of 
U ,  i.e., for all other prices p. not p U q. Then p • E(q) i 0 for 
all p. It then follows from Walras' law that E(q) = O. Thus if the 
weak axiom of revealed preference holds the equilibrium price is 
unique. This approach reduces the search for equilibrium prices to 
the search for maximal elements of a revealed preference relation. 
3 
lhon there is more than one consumer. the excess demand function need 
not obey the weak axiom of revealed preference. Remarkably enough,tho 
'
aame approach works even in this case, though the equilibrium need not 
be unique. Thus we next present a lemma on the existence of maximal 
elements of binary relations. 
Kaxiptal Elements of Binary Relations. 
A binary relation on a set E: can be described in terms of a 
correspondence U : E: °'°' E:, where U(p) is interpreted to be the set 
elements of E: which are larger than p (the upper contour sot). A.!!'= 
maximal element of E: is a p a E: for which U{p) = 4. With each binary 
relation U we associate the correspondence u-l : E: °'°' E: by 
q a U-l(p) if and only if p a U(q). The following lemma is a 
strengthening of results of Fan (1961,Lemma 4) and Sonnenschein 
(1971,Theorem 4). 
!.2l!l!!ll.a. Let E: be a compact convex nonempty subset of a Hausdorff 
topological vector space and let U : E: °'°' E: be a binary relation 
satisfying for all p 1 E: 
(i} x J co U(p) (where co denotes convex hull). 
(ii) if q a u-1(p), then there is a p' (possibly = p) 
such that q a int u-1(p').
Then the set of U-lil&ximal elements of E: is compact and nonempty. 
� By definition, the aet of U-maximal elements is just 
n (E:\U-l(p)). By hypothesis (ii). 
pal: 
n (E:\U-l{p)) • n (E:\int U-l(p')). 
pal: p'aE: 
4 
This latter intersection is clearly compact, being the intersection of 
compact sets. 
We now show that it is nonempty. For each p, put 
F(p) = E:\(int u-1(p)). As noted above each F(p) is compact. We first 
D . 
claim that if q a co {pi i = 1 • • • •  ,n}, then q a U F(p1): Suppose 
i=l 
D i -1 i i that q I UF(p ). Then q a U (p ) for all i, so p a U(q) for all i. 
i•l 
But then q a co {pi} C co U(q), which violates (i). It then follows 
from the E:naster-E:uratowsld-Kazurkitnricz lemma as extended by Fan 
(1961,Lemma 1) that n F(p) � 4. 
pal: 
EJ;cess Demand Correspondences. 
Q.E.D. 
Let Jl. n be the commodity space. Denote by Jl. : the positive 
cone of ltn, i.e., • : = {z a JRD :zi 2 0 for i • 1, ••• n}, and N 
denote the negative cone, -RE:. Let 
A = {p I JR n: t p ... 1. p. 2 0 
1=1 1 1 
i • 1, • • •  ,n} and. 
S = {p a JRn:p » 0} where p » q means p. > q., 
1 1 i = 1,, • • •  ,,n. The 
domain of the excess demand correspondence will be denoted by D, and 
it will be either equal either to S or to A. 
A correspondence E : D °'°' JR n satisfies the 
yeak weak form of Walras' law if for each p, 
(IWWL) p • z .i 0 for some z c E(p). 
A correspondence E : D '' lR n satisfies the 
strong form of !alras' lay if for each p. 
(SWL) p • z = 0 for all z a E(p). 
In a private ownership economy. Walras' law will be satisfied if all 
consumers spend exactly all of their income. 
A correspondence E : S �' lR n is said to satisfy the 
boundary condition (B) if the following holds. 
s 
(B) if for every q c A. there is a p a A such that for every q 
sequence {�}CS with �' q c A\S. there is a subsequence {!!pl
such that for all p. pq • z > 0 for all z c E(q�). 
Boundary condition (B) is weaker than the condition introduced by 
Neuefeind [1980,Lemma 2) . which requires p to be independent of q.q 
It is however, stronger than the condition used by Aliprantis and 
Brown [1982) or Grandmont [1977) . Their condition allows p to depend q 
on the sequence as well as its limit. 
We say that the correspondence E : D '°' lR n is upper demi-
continuous if for each open half-space B in Jln, 
E+[B] E {p : E(p) C B} is open in D. (This terminology is from 
Browder [1967) . For closed-valued correspondences this definition is 
equivalent to what Geistdoerfer-Florenzano calls upper hemi-
continuity. For finite-dimensional spaces every neighborhood of a 
6 
point contains a neighborhood which is a finite intersection of open 
half-spaces. in which case demi-continuity reduces to hemi-continuity 
in the sense of Hildenbrand [1974) .) 
A free disposal equilibrium price of the correspondence 
E : D '' JR n is a price p such that E(p) n N # rl. An eauilibrium 
�p is a price p 1 S such that 0 s E(p). 
Existence of Equilibrium Prices. 
We will treat tlfo cases. In the first case the excess demand 
correspondence is assumed to be defined for all nonnegative prices and 
to satisfy the weak weak !alras' law and to take on only compact 
convex values. The existence of free disposal equilibriua prices is 
established. In the second case, the excess demand correspondence is 
defined only for strictly positive prices and is assumed to exhibit 
nice boundary behavior. In particular, we assume boundary condition 
(B). The strong fo:i::m of Walras' law is assumed, though the' values of 
the correspondence are only assumed to be closed and convex. 
Existence of equilibriua price� rather than just free disposal 
equilibriua prices is established. In both cases the excess demand 
correspondence is assumed be upper demi-continuous and to be 
homogeneous of degree zero, so that only noDDalized prices are 
considered. 
The outline of the proof in each case is as follows. We 
define a binary relation between prices based on the revealed 
prof eronco araument for the one person case and show that the maximal 
olomonta must bo equilibrium prices. Thia part of the proof uses 
7 
Walraa' law and the fact that the excess demand correspondence assumes 
closed convex values. and is baaed on a separating hyperplane 
aramaent. We then show that the relation satisfies the hypotheses of 
the leana. and so it has a nonempty compact set of maximal elements. 
Thia part of the proof uses Walras' law. demi-continuty and the 
boundary condition. One can view this approach as an attempt to reduce 
tho general case to the one person pure exchange economy. 
Theorem 1. Let E : A �� lit n be upper demi-continuous with nonempty 
compact convex values and satisfy the weak weak fora of Walraa' law. 
Then the set of free disposal equilibri1111 prices for E is compact and 
nonempty. 
EI22!... For each p a A set 
U(p} • {q : q • z > 0 for all z a E(p}}. 
Then U satisfies the hypotheses of the lemma: For U(p) is 
' -1 clearly convex for each p and by Walraa' law pr U(p}. Also U (p} is 
open for each p: 
-1 For if q a U (p) we have that p • z > 0 for all z a E(q). 
Then since E is upper demi-continuous E
+
[{x : p • x > 0)) is a 
neighborhood of q in u-1(p). 
Now p is U-maximal if and only if 
for each q a A. there is a z a E(p} with q • z i O. (•} 
Wo now show that E(p} RN # '  if and only if (•}. Suppose 
z a E(p} R N. Then q 0 z i 0 for all q a A, so (*) holds. If on tho
other hand, E(p} R N s '· then we can strictly separate E(p} which is
compact and convex from N which is closed and convex (Schaefer (1971. 
9.2)). That is there exists some q a lR.n and some c a  JR satisfying 
x a N and z a E(p} imply q • z > c > q • x 
Since N is the neaative cone. c > 0 and q a lit:. Thus q • z > 0 for 
all z a E(p). and (*} does not hold. 
It thus follows from the lemma that the set of free disposal 
equilibrium prices is compact and nonempty. 
8 
Q.E.D. 
Theorem 2. Let E : S �� lit n be upper demi-continuous with nonempty. 
closed. convex values and satisfy the strong from of Walras' law and 
the boundary condition (B} (relative to S and A}: 
(SWL) 
(B) 
p • z .. 0 for all z a E(p). 
if for every q a A. there is a p a A such that for every q 
net {q�} CS with �� q a A\S. there is a subnet {� } such that 
for all p. pq • z > 0 for all z a E(� ). 
Then the set of equilibrium prices for E is compact and nonempty. 
Proof. Define the binary relation U on A by 
9 
p z > 0 for all z I E(q) and p,q e S 
p I U(q) if or 
p I S, q I A\S. 
We first sho. that the U-•axiaal elements are precisely the 
equilibria. prioes. First suppose that p is U-•azi•al. i.e •• 
U(p) • d. Since U(p) = S for all p a A\S. we have that p a S. Since 
p a S and U(p) = 4 we have 
for each q a s. there is a z a E(p) with q • z i O. (1) 
le now sho. that (1) implies 0 a E(p). Suppose by way of 
oontradiction that 0 J E(p). Then we can strictly separate {O) which 
is compact and convex from E(p) which is closed and convex with 
- D - - - -p s It 
• 
Th'IUI p • z > 0 for all z a E(p). Put pl E lp + (1 - l)p. 
Then for z a E(p). pl • z s l; • z + (1 - l)p • z = l; 
• z > 0 for
l > o. (ll.ecall that p • z = 0 for z a E(p) by Walras' law.) For 
l > 0 small enouah. pl >> 0 so that the normaliz ed price vector 
P1 = <[: <P1>i>-
1P1 a S and pl • z > 0 for all z a E(p). which violates 
(1). 
Conversely, if p is an equilibri'llll prioe. then 0 a E(p) and 
since p • 0 s 0 for all p. U(p) • d. 
We now verify that U satisfies the hypotheses of the le11111&: 
(ia) p J U(p): For p a S this follo.s from Walras' law. For 
pa A\S, p JS= U(p). 
(ib) U(p) is convex: For p a s. let q1• q2 a U(p). i.e. 
q1 • z > O. q2 • z > 0 for z a E(p). Then [lq1 + (1 - l)q2] • z > 0 
10 
as well. For p a A\S, U(p) • S which is convex. 
(ii) If q I u-1(p). then there is a p' with q I int u-1(p'): 
There are two cases: (a) q e S and (b) q a A\S. 
(a) q 8 s n u-1(p). Then p • z > 0 for all z I E(q). Let 
H be the open half space {x : p • x > 0). Then by upper demi­
oontinuity, E+(H] is a neiahborhood of q contained in u-1(p). 
(b) q a (A\S) n U-l(p). By boundary condition (B) 
-1 1 -1 q a int U (pq): Suppose q, int U (pq). Then there is a net
q4 "' q with q4 J u
-1(pq) for all a. This implies that q4 a S
and so there is some z4 a E(11g) with pq • z4 i O. 
(B). 
This violates 
Q.E.D. 
11 
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On Equilibria of Excess Demand Correspondences

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