Journal of Mathematical Economics 1 (1974) 237-246. © North-Holland Publishing Company
AN EQUILIBRIUM EXISTENCE THEOREM WITHOUT
COMPLETE OR TRANSmVE PREFERENCES*
Andreu MAS-COLELL
University of California, Berkeley, Calif. 94720, U.S.A.
Received May 1974, revised version received August 1974
The Walrasian equilibrium existence theorem is reproved without the assumptions of complete
or transitive preferences.
1. Introduction
The aim of this paper is described in its title - to demonstrate that for the
general equilibrium Walrasian model to be well defined and consistent (i.e., for
it to have a solution), the hypotheses of completeness and transitivity of consumers' preferences are not needed.
Schmeidler (1969) proved the existence of equilibria in a model with a continuum of traders and incomplete preferences; he asked if completeness could be
similarly dropped in the case of a finite number of traders; the result here
answers this question in the affirmative. We refer to Aumann (1964, also 1962),
for forceful arguments in favor of relaxing completeness as~umptions on decisionmakers' preferences.
For the case where preferences are complete, Sonnenschein (1971) showed how
it was possible to obtain continuous demand functions without making any use
of transitivity.
Let Q be the consumption set. Our hypotheses on preferences are: >- is an
open subset of Q x Q (continuity), which is irreflexive (i.e., x >- x never holds)
and such that, for every x e Q, {ye Q: y >- x} is non-empty and convex. We do
not assume that >- is asymmetric or transitive and the stringent convexity
hypothesis '{ye Q: (x, y) rt >-} is convex' is not made; this last condition lacks
*The content of this paper has been presented at seminars in Berkeley and Stanford; I am
indebted to its participants for many helpful comments. I want to thank R. Aumann, G.
ebreu, D. Gale, R. Mantel, and B. Peleg for their suggestions; in particular, D. Gale pointed
out an error in an earlier version, and R. Aumann and B. Peleg suggested a simplification of
the proof. Needless to say, they are innocent of any remaining shortcomings. Research has
been supported by NSF Grants GS-40786X and GS-35890X which is gratefully acknowledged.
238
A. Mas-Cole//, An equilibrium existence theorem
intuitive appeal in a context where preferences may not be complete. In a few
words, the only things of substance we are postulating are non-saturation and the
convexity of'preferred than' sets.
The problem at hand seems to require an existence proof of a novel type. Even
assuming transitivity and monotonicity of preferences, their incompleteness may
severely destroy the convex-valuedness of the demand correspondence [which is
an irrelevant consideration in the continuum of traders context; this is the fact
exploited by Schmeidler (1969)]; actually, we shall argue in the appendix, by an
example, that an attempt to a proof through demand correspondence is completely barren. Of course, the demonstration we give is a fixed-point one, but
the mapping constructed does not appear to have been used before. Perhaps the
closest relative to the approach taken here, is Smale's {1974, appendix) existence
proof; the specifics are very different, but there is some analogy in the nature of
the problems being solved. This will become clear in the text.
For the sake of clarity and conciseness the analysis is limited to pure exchange
economies. There is no difficulty in extending the results to, for example, the
private ownership economies of Debreu's Theory of value (l 959).
2. The model and statement of theorems
There are
Q
=
t commodities, indexed by h, anq N consumers, indexed by i;
R~.1
In section 2.1, a model where consumers are described by preference relations
is given and a theorem is stated. In section 2.2, an alternative model, differing
only in the specification of consumers, is described and another theorem stated;
it is shown then that the last implies the former.
2.1
Every consumer i is specified by (X 1 , >-;, w 1), where X;
and ro; e R 1• We denote (z, v) e >-;by z >- 1 v.
c
R 1, >-; c X 1 x X 1,
(C.1)
For every i, X; (the consumption set) is a non-empty, bounded below,
closed, convex set.
(C.2)
For every i, >- 1 (the preference relation) is a relatively open set such that,
1 Commodities will be denoted by superscripts while subscripts will be reserved for
(consumption, production) vectors; x ~ y means x" > y1' for all h, x > y means xh > yh for all
h and x ¥- y, x ;;;., y means x > y or x = y; co D, Int D, 'iJD stand for the convex hull, the
interior, and the boundary of D c R", respectively. The Euclidean norm is JI II; for x,y ER",
xy denotes the inner product. If B,D c R", B+D = {z 1 +z 2 : z 1 EB, z 2 ED}, BD = {zy:
zEB, yE D}. When there is no ambiguity, we write b instead of {b}. If B c R" and sE R",
B ~ s means b ~ s for every b EB; analogously for B ;;;., s; [ ], [ ), ... denote segments in
the usual way.
A. Mas-Co/ell, An equilibrium existence theorem
239
for every x 1 e X 1, { z e X 1 : z >- 1 x 1} is non-empty, convex (non-saturation
and convexity) and does not contain x 1 (irreflexivity).
(C.3)
For every i, w 1 ~ x 1 for some x 1 e X 1•
An economy Sis identified with{(X1,
N
Definition. (x, p) e [] X 1 x .:1 is an equilibrium for G = {(x 1,
i=l
(E.l)
(E.2)
(E.3)
N
I
i=
l
>- 1, w1)}f= 1 • Let .:1 = {p e Q: Ip'' = 1.}
h=l
>-;. w1)}f= 1 if:
N
L
i=
w,;
for every i, px 1 = pw 1;
1
x, ~
1
>- 1 x 1, thenpz > px1•
If S = {(x 1, >- 1, w1)}f= 1 satisfies (C.J), (C.2), (C.3), (C.4), then
for every i, ifz
Theorem I.
there is an equilibrium for G.
!!xii
Let S = {x e R 1 :
= I} and G be an economy satisfying the conditions of
the theorem. For every i define g 1 : X 1 -+ S by g 1(x 1) = {p e S: if z >- 1 x 1, then
pz ~ px 1}; then equilibrium condition (E.3), in the presence of (C.3), amounts
to requiring (I/llPll)P eg1(x 1). This heuristic comment motivates the more general
model of the next section.
2.2
A set H c R" is contractible if the identity map on H is homotopic to a
constant map. For the present purposes it will suffice to know that convex sets
and intersections of S with convex cones which are not linear subspaces, are
contractible. A correspondence is said contractible-valued if its values are
contractible sets. The product of two contractible sets is contractible.
For every i let g 1 : X 1 -+ S be a correspondence and, in the definitions of
section 2.1, substitute >- 1 throughout by g 1•
Replace (C.2) by
(C.2') for every i, g 1 : X 1 -+ S is an u.h.c., contractible-valued correspondence
such that.for every x 1 e X 1, the (possibly empty) set {p e S: pz ~ pxifor
all z e Xi} is a subset ofg 1(x 1);
and (E.3) by
(E.3') for every i,
(I/llPl!)P e g1(x,).
We can state:'
Theorem 2. If S = {(X1, g 1, w1)}f= 1 satisfies (C.I), (C.2'), (C.3), (C.4), then
there is an equilibrium for <ff.
240
A. Mas-Co/ell, An equilibrium existence theorem
Theorem 2 implies Theorem 1. Let an economy ti= {(X,, >-,, roJ}r= 1
satisfying (C.l), (C.2), (C.3), (C.4) be given. For every i and x 1 e X 1 define
g1(x,) = {p e S: if z >- 1 x,, then pz ~ px1}; since >- 1 is irreflexive and {z e X 1 :
z >- 1 x 1} is a non-empty, open, convex subset of the convex set X 1, the set
{p e R 1: if z >- 1 x 1, thenpz ~ px 1} is a non-empty, convex cone which cannot be
a linear subspace; therefore, g 1(x,) is non-empty and contractible. Obviously,
{p ES: pz ~ px1 for all Z E X1} C g,(x,).
The correspondence g 1 : X 1 -+ S so defined is u.h.c.
Proof. Theset{(z, x 1,p) e X 1 x X 1 x S: z >- 1 x,,pz < px.} is open. The graph
of g 1 is the complement of the projection of this set on the last two coordinates;
hence it is closed. Q.E.D.
Therefore ti' = {(X,, g,, ro 1)}r= 1 satisfies the hypothesis of Theorem 2, and so
there is an equilibrium for ti'.
An equilibrium for ti', is an equilibrium for 8.
Proof. Let (x, y, p) be an equilibrium for ti'. It has to be shown that, for
every i, if z >- 1 x 1, then pz > px 1• Let pz ~ px 1, z >- 1 x,, for some i. Pick z ~
ro 1, z e X 1; then (since pz < pro1 ~ px,) for every z' e [z, z] we have "z' e X 1 and
pz' < pxt and, for z' e [z, z] sufficiently close to z, z' >- 1 x 1; hencep 'fi g 1(x 1). A
contradiction. Q.E.D.
It is worth emphasizing that the model with consumers specified by the g/s
correspondences admits of more interpretations than the one of (global)
preference maximization. For example, it encompasses Smale's notion of an
'extended equilibria' for non-convex economies [in which case g 1 would be a
normalized gradient vector field; see Smale (1974)] or the various concepts of
'local preference satisfaction' and 'preference fields' to be found in the nonintegrability literature [see Georgescu-Roegen (1936); Katzner (1970, ch. 6)].
See also Debreu (1972) from where the notation g 1 is taken.
3. Proof of the theorems
It suffices to prove Theorem 2.
Let 8 = {(X1, g 1, ro 1)}r= 1 satisfy (C.l), (C.2'), (C.3), and (C.4). Denote e=
(I, ... , 1) e R 1• For every i, define g1 : U -+ S by
if x 1 e X 1;
if x 1 'fi X 1, x 1 ~ se;
ifmin x~ < s.
"
The correspondence g1 satisfies (C.2') with respect to U. Moreover, if x 1 e
then g1(x 1) = {p e S: px 1 = O}.
au,
241
A. Mas-Co/ell, An equilibrium existence theorem
/f(x,p) e o,N x '1 is an equilibrium for t!' = {(Q, gi,
equilibrium fort!.
wi)}r= 1 ,
then it is an
Proof For every i, pxi ~ pwi > Inf pX1• Let xi¢ Xi; since p e giCx 1), it
follows by the construction of gi that if z e Xi, thenpz ~ pxi, i.e.,pxi ~ Inf pX1;
a contradiction. Therefore xi e X 1,p e gi(x 1), for every i. Q.E.D.
In view of this we assume for the rest of the proof that t! satisfies:
(C.5)
For every i, Xi
= Q and.for all X1 E oQ, glx1)xi
~
0.
The fixed-point theorem to be used (an immediate corollary of the EilenbergMontgomery theorem) is contained in the:
Lemma. If K c Rn is a non-empty, convex, compact set and F: K -+ Rn is an
u.h.c. contractible-valued correspondence, then there exist x e Kandy e F(x) such
that(y-x)(z-x) ~ Oforall z e K.
Proof Let F(K) c H, where H is taken convex and compact. Since K is
closed and convex, we can define a continuous function aK: H-+ K by letting
aK(x) e K be the II II-nearest element toxin K. For every x e H, z e K, one has
(x-aK(x))(z-aK(x)) ~ 0. The correspondence F o aK: H-+ H is u.h.c. and
contractible-valued. By the Eilenberg-Montgomery fixed-point theorem
[Eilenberg (1946); see also, Debreu (1952)], there is x e H such that x e F(aK(x)).
Call y
Q.E.D. 2
= aK(x). Then x e F(y), ye Kand (x-y)(z-y)
~
0 for every z e K.
Pick an arbitrary e > 0 and let re > Ir= 1 wi + ee, k = rt(t N + 2) and <p : '1 -+
R 1 be a continuous function such that: (i) p<p(p) ~ 0 for all p e ,1; (ii) <p(J) ~ ee;
(iii) if ph = 0 then <ph(p) ~ - k, for all p e '1. It is clear that such a function exists.
By an obvious limiting argument the theorem will be proved if we show the
existence of(x,p)eQNxJ such that L,r= 1 x 1 ~L:r=iwi+ee andpegi(x 1),
IPxi- P w1I < eforall i.
Denote rxiCp) = pwi+(I/N)p<p(p) and define the u.h.c., contractible-valued
correspondences <Pi: [O, r] 1 x A -+ R 1, 1 ~ i ~ N, <P.tJ: [O, r] 1N x A -+ R 1 by
N
<P.tJ(x,p) = p+
2
I,
i= 1
(xi-wi)-<p(p).
The lemma can be proved by appealing only to the Brouwer fixed-point theorem. Suppose
Fis a function. Then there is x e K such that x = <i1<.(F(x)), i.e., (F(x)-x)(z-x) ~ 0 for all
z e K; the result follows then by a continuity argument and the fact [proved, for example, in
Mas-Colell (1974b)] that, with the hypothesis made, there is for any e > 0 a continuous
function/: K
-+
R" which graph is contained in the e-neighborhood of the graph of F.
242
A. Mas-Co/ell, An equilibrium existence theorem
Let then the u.h.c., contractible-valued correspondence
R 1<N+ 1 >be given by
(J :
[O, r] 1N x ..1 -+
Applying the lemma to (J we obtain p e ..1, x e QN, and, for every i, Pi e g(x 1)
such that, denoting x = ~J = 1 x 1, Q) = ~J = 1 ro,,
(a)
for every i, if z e [O, r] 1, then (z-x 1) ( a. 1(p)p't-px 1
(b)
if q e ..1, then (q-p)(x-U}-cp(p))
~
&i) ~
O;
O.
We show that (x, p) is as desired; let y = Q)+cp(p).
If p e 0..1, then
which contradicts (b). Therefore p ;;;i: 0 and, by (b), x- y = A.e for some A. e R.
If x, E ao, then by taking z = 0 (resp. z = re) if x, #: 0 (resp. x, = 0), we
contradict (a). Therefore, for every i, x 1 ~ 0 and so, by (a), a.l,p)p't ;;;i: (pxJllPii)p.
Hence a.,(p) ;;;i: px 1 for every i, which implies A. ~ 0. From this we get, for every
i, x 1 ~ x ~ y < re, and so, again by (a), a. 1(p)p't ~ (pxJllPll>P· Therefore,
for every i, a. 1(p)p't = (pxJllPll>P which yields a. 1(p) = px,,p =Pi· Since, then,
px = py, we also have A.= 0. Hence ~J = 1 x 1 ~ L,f = 1 ro 1+ee and p e g1(x 1),
pro 1 ~ px 1 ~ pro 1+e for all i. This concludes the proof.
4. Remarks
4.1
Let >- be a preference relation on D (to make things specific) satisfying the
hypotheses of Theorem 1, i.e., (C.2). For pe..1,p ~ 0, we [O, oo) define
h> (p, w) = {x e D: if y >- x, thenpy > px}; this set is non-empty [this follows
from Sonnenschein' s proof in ( 1971); although his result is phrased in terms of a
complete preorder ;:::- , the proof of the non-emptyness of h'>- (p, w) uses only
the convexity of the induced >-]. Hence a demand correspondence h> : Int ..1 x
[O, oo)-+ D is well defined; it is also u.h.c. Given initial endowments roe 0,
h>."': Int ..1 -+ D stands for the excess demand correspondence generated by
If", i.e., li>"·"'(p) = h(p, pro).
A. Mas-Co/ell, An equilibrium existence theorem
243
In the appendix we give an example of a quite simple >- which is transitive,
monotone and satisfies (C.2), but for some c.o e !2, no u.h.c. subcorrespondence
of h>·"' is connected-valued (say, that a correspondence is connected-valued if
its values are connected sets; F: A -+Bis a subcorrespondence of G: A -+ B if
F(t) c G(t) for all t e A); also, >- possesses a continuous utility function [i.e., a
function u: a-+ R such that if x >- y, then u(x) > u(y); this is Aumann's term
(1964)], but not a quasi-concave one. Thus, the example shows that, even if
transitivity is assumed, Theorem 1 cannot be obtained as a corollary of available
existence results and, also, that a proof by the way of demand correspondences
is not possible.
4.2
Schmeidler (1969) proved that, in the continuum of agents case, equilibria
exist for economies which (in addition to other hypotheses) have consumers with
transitive, incomplete, not necessarily convex, preferences. Trivial examples
[every consumer has preferences on R! given by {(x,y):'x 1 >y 1-l and
x 2 > y 2 ' or 'x 1 > y 1 and x 2 > y 2 -1'}1 show that, unless convexity of preferences is assumed, this result cannot be improved upon by dropping transitivity. This'is, we believe, a very good reason to keep transitivity (of>-) among
the standard assumptions of equilibrium analysis.
Appendix
We give here an example of a relation >- on a such that (i) >- is transitive,
monotone, and satisfies (C.2); (ii) there is a continuous utility for >-;(iii) there
is no quasi-concave utility for>-; and (iv) for a c.o e !2, there is no u.h.c. connectedvalued subcorrespondence of h>·°'.
We define >- first in R! and show then that there is no u.h.c., connectedvalued subcorrespondence of the demand correspondence h>. This suffices since
defining a >-'in R! by '(x 1, x 2 , x 3) >-' (y 1, y 2 , y 3) if and only if (x1, x 2 ) >(y1, y 2 )' and taking c.o = (0, 0, 1) what was true of the demand correspondence
of>- will be true of the excess demand correspondence of >- '.
Hence, let t = 2. Define (utility functions) u', u": a -+ R by u'(x) = min{x 1,
2(x 1 +x 2)-2}, u"(x) = min {x 2 , 2(x 1 +x 2 )-2}; see fig. I.
Define >- c !2 x !2 by (see fig. 2) >- = {(x, y) e !2 x !2: 'x ~ y' or 'u'(x) >
u'(y) and Y1 > t. y 2 <
or 'u"(x) > u"(y) and y 2 > t. y 1 <
or 'y 1 + y 2 <
1'· and x 1 +x2 > y 1+y 2 '}. In the definition of>- only strict inequalities enter,
.and so the relation is open; clearly, it is also monotone and irreflexive. It is
easily checked that:
(a) ify 1 +y 2 < iandx >-y, thenx 1+x 2 > y 1 +y 2 ;
r
(b)
if y 1 + y 2 ;;:i: 1'· y 1 < t (resp. y 2 <
(resp. x 1 > f) and x 1+x 2 > 1'·
r
f), then x >- y implies x 2 > t
244
2
A. Mas-Co/ell, An equilibrium existence theorem
u"
2
u'
Fig. 1
Fig. 2
The fact that >- is convex is a consequence of (a) and the monotonicity of u', u"'
To see that >- is transitive, let z >- y, y >- x. If x ~ (j-, j-), then obviously z >- x.
Ifx 1 +x 2 ~ tandx 1 < j-,theny>-xmeansu"(x) > u"(y)and,by(b),y 2 > j-;
therefore z >- y implies u"(z) > u"(y) and we have z >- x. If x 1 +x 2 ~ 1 and
x 2 < j-, a symmetric argument applies. If x 1 +x 2 < 1 and y 1 +y 2 < 1, then
A. Mas-Cole//, An equilibrium existence theorem
24S
z >- x by (a) and the definition. If x 1 +x 2 < 1 and y 1 +y 2 ~ 1. then, by (b),
•
z. 1 +z 2 > 3, 1.e.,
z >- x.
Take p = (t, t), w= 1 and suppose that 'I: Int .d x [O, oo)-+ a is an u.h.c.
connected-valued subcorrespondence of If'. Forte [O, I) take p(_t) = ((t+ 1)/2,
(I - t)/2), w(t) = 1- t. For every x e a if x 1 < t and x 2 < 2, then (0, 2) >(x 1 , x 2). Therefore, for every t e [0, 1), if x e a and 0 < x 1 < ;, then x f/:
~
/f'(p(t), w(t)) and for l close enough to 1, !f'(p(l), w(l)) = {(O, 2)}. Hence
'l(P(t), w(t)) = {(O, 2)} for every t e [O, 1), and so (0, 2) e ,,(j, w). By a symmetric argument (2, 0) e '1<.P, w) and therefore [(O, 2), (2, O)] c '1<.P. w). But
this is impossible since, for example, (t, i) f/: /f'(j, w). Hence no such 'I exists.
2
2
3
-
Fig. 3
The relation>- satisfies Peleg's (1970) spaciousness condition for the existence
continuous utility function; one is represented in fig. 3. It is immediate that
the horizontal and vertical segments of fig. 3 should appear in any indifference
map of a utility for >-, hence no quasi-concave utility exists [examples of open
relations having quasi-concave but no continuous utility have been given by
Schmeidler (1969) and Peleg (1970)].
of~
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c
246
A. Mas-Co/ell, An equilibrium existence theorem
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