Applied Mathematics, 2011, 2, 1397-1404
doi:10.4236/am.2011.211198 Published Online November 2011 (http://www.SciRP.org/journal/am)
Existence of Competitive Equilibria without Standard
Boundary Behavior
Francesco Ruscitti
Department of Political and Social Sciences, John Cabot University, Rome, Italy
Received August 30, 2011; revised October 10, 2011; accepted October 17, 2011
Abstract
We study the existence of competitive equilibria when the excess demand function fails to satisfy the stan-
dard boundary behavior. We introduce alternative boundary conditions and we examine their role in proving
the existence of strictly positive solutions to a system of non-linear equations (competitive equilibium prices).
In addition, we slightly generalize a well-known theorem on the existence of maximal elements, and we un-
veil the link between the hypothesis of our theorem and one of the boundary conditions introduced in this
work.
Keywords: Excess Demand Function, Competitive Equilibrium, Set-Valued Functions, Selections, Fixed
Points, Maximal Elements
1. Introduction
The purpose of this paper is twofold. In general, we
provide a set of sufficient conditions in order for a sys-
tem of N non-linear equations in N unknown to possess a
strictly positive solution. In particular, since we deal with
excess demand functions (vector fields) defined on suit-
able price-domains, from the standpoint of Economic
Theory the natural interpretation of our results is the
existence of a price-vector that clears every markets that
are assumed perfectly competitive. In other words, the
existence of a vector of strictly positive prices such that
demand equals supply on every market. Such state of the
economy is called competitive equilibrium.
Note that the literature about the existence of compe-
titive equilibria is vast, and a survey of the numerous and
remarkable contributions would hardly do justice to them.
So, why yet another paper on the existence of compe-
titive equilibria? To answer this question, first it is worth
recalling briefly the established literature.
Let :
N
N
Z  be a function satisfying the fol-
lowing properties:
1) Z is continuous.
2) ()( )


Z
pZp for all
N
p  and all 0


(homogeneity of degree zero).
3) for all
() 0pZp
N
p  (Walras law)
4) There exists a s > 0 such that ()
i
Z
p
5) If , where p ≠ 0 and pi = 0 for some i, then
n
pp




1
Max, ,
nNn
ZpZ p.
Recall that any finite-dimensional economy with con-
tinuous, strictly convex and strictly monotonic prefe-
rences, and with production sets that are closed, strictly
convex, bounded above, and such that a strictly-positive
aggregate consumption bundle is producible from the
aggregate endowment, gives rise to an aggregate excess
demand function enjoying the above properties (see, e.g.,
Aliprantis et al. [1], Arrow and Hahn [2], Mas-Colell et
al. [3]). Notice that property is the standard bound-
ary behavior. Clearly, a competitive equilibrium price
vector is a
5)
N
p


 such that .

=0Zp

Obviously, with constant returns-to-scale production,
the production set is neither strictly convex nor bounded
above. We borrow the formulation of the economy and
the definition of competitive equilibrium from Geana-
koplos [4].
Specifically, assume that preferences are continuous,
strictly convex and strictly monotonic. Then, a constant
returns-to scale economy can be formalized by an agg-
regate net demand function-technology pair


,
Z
Y,
where :
N
N
Z  and is a closed and
convex cone that allows for free disposal. Clearly, we
must restrict attention to the set of
N
Y
N
p  such that
0pY

, i.e., 0py

for all
y
Y. One can assume,
without much loss in generality, that the set of
N
p



s for all
N
p  and all . 1, 2,,iN

F. RUSCITTI 1398
such that is nonempty.1 Under these assump-
tions,
0pY 
Z
still satisfies properties 1) through 5) above. A
competitive equilibrium for a constant returns-to-scale
production economy can now be defined as a price
N
p such that

Z
pY and 0pY.
To summarize: if one is dealing with general pro-
duction economies, then
Z
defined above is the (pro-
duction-inclusive) aggregate excess demand function. If,
rather, one is dealing with constant returns-to-scale
economies, then
Z
is the aggregate net demand func-
tion that stems solely from consumers’ preference maxi-
mization. In either cases, when preferences are con-
tinuous, strictly convex and strictly monotonic,
Z
sati-
sfies properties 1) through 5) above.
Now, let us turn to our contribution in this paper: we
weaken the continuity of the excess demand function, we
drop the standard boundary behavior (replacing it with
alternative boundary conditions), and we prove a new
mathematical theorem which is then utilized to study the
existence of competitive equilibria. More precisely, fol-
lowing in Tian’s footsteps [5], we do not assume that the
aggregate excess demand function is lower semicon-
tinuous, whereas in the literature the excess demand fun-
ction is typically continuous (see above).
Moreover, we address hypothetical economies in
which the standard boundary behavior of the aggregate
excess demand function (property 5) above) is not ne-
cessarily satisfied, and we prove two theorems on the
existence of competitive equilibria. Indeed, to motivate
our work, in Section 3 we exhibit two model-economies:
in the former, the standard boundary behavior fails. In
the latter, the sufficient conditions for the standard boun-
dary behavior are violated, and therefore the standard
boundary behavior may or may not be satisfied. On the
other hand, it is well-known that, whenever the excess
demand function is defined on a relatively-open price-
domain2 (as it is the case in this paper), some sort of
boundary conditions are needed to demonstrate the exi-
stence of a zero for such a function. In fact, loosely put,
proper boundary conditions remedy the lack of compact-
ness of the price-domain, and thus enable the application
of specific fixed-point theorems. For these reasons, we
introduce two alternative boundary conditions and we
study how they are related to one another. Our alterna-
tive assumptions on the boundary behavior formalize
inward-pointing conditions of the aggregate excess de-
mand function. The former condition is formalized by
means of the projection mapping (see Section 2), and the
latter by means of convex combinations. This comes in
handy because this method of modelling the boundary
conditions enables us to retain the central idea of the first
existence theorem in the proof of the second one, which
thus becomes a variant of the first theorem. Hence, this
approach offers a somewhat unified framework for two
seemingly different problems
Finally, in the context of Hilbert spaces, we prove a
slight generalization of a theorem on the existence of
maximal elements due to Yannelis and Prabhakar [6]
(Theorem 5.1). Interestingly, one of the assumptions in
our theorem lends itself to be interpreted in terms of
boundary behavior 1 defined in Section 2. This strong
analogy enables us to prove again the existence of a stri-
ctly positive equilibrium price vector as a short corollary
of our new theorem.
Clearly, in this work we treat the excess demand fun-
ction as the primitive of the economy at hand. This may
be regarded by economists as an unconventional route to
proving the existence of competitive equilibria. Never-
theless, the approach we follow, based directly upon the
excess demand function, is well-suited to highlight the
mathematical aspects of our contribution. Indeed, in
Section 4 we shall develop a unifying treatment that can
handle both general production economies and constant
returns-to-scale economies. We shall detail the proof of
the existence of competitive equilibria only for the for-
mer, since virtually the same method may be used to
analyze constant returns-to-scale production economies
as well.
The lay-out of the paper is as follows: In Section 2, we
set our notation and we develop some background. Also,
we introduce two alternative boundary conditions on the
excess demand function. We also explain how our con-
ditions relate to the literature we know of, and finally we
state the main mathematical theorem that will be used in
this paper. It is a celebrated selection theorem due to
Michael [7]. In Section 3, we construct two model eco-
nomies for which the standard boundary behavior of the
excess demand function is not necessarily satisfied. This
justifies our interest in proving existence of competitive
equilibria under alternative boundary conditions. In Sec-
tion 4, we prove two theorems on the existence of com-
petitive equilibria or, rather, on the existence of a strictly
positive solution to a finite system of non-linear equa-
tions. In the process, we also compare our approach to
the relevant literature. In subsection 4.1, we prove a
theorem on the existence of maximal elements for corre-
spondences3 whose domain is different from the range,
and domain and range are both subset of a Hilbert space.
It is a natural generalization of Yannelis and Prabhakar
[6], and thus it may be interesting in its own right. In
subsection 4.2 we employ our new theorem to study the
existence of competitive equilibria. In Section 5 we con-
clude and we outline a few directions for future research.
1See [4].
2For instance, when the excess demand function stems from consumers
with strictly-monotonic preferences.
3Throughout this paper, by correspondence we mean a set-valued func-
tion.

F. RUSCITTI1399
2. Background, Notation, and Boundary
Conditions
Let :
N
N
Z  be the aggregate excess demand
function of a general production economy. The question
we are after in this paper is:
Problem: Does there exist some
N
p

 such that
? ()=0Zp

To set the stage for the subsequent analysis, let
, where 1 is the dimensional

=:
N
pp

 1
(1,1, ,1)

=1
=1
N

N
 vector , and let .

=:Int pp

 1
Also, given any >0

, let

=:pp



1.
Finally, given any >0

, we let
.

= for 

=:some  pp = 1, 2,i N
 i
Normalization: Clearly, since we are searching for a
N
p

 such that , by virtue of property 2)
above we can restrict the domain of
()=0Zp

Z
to
I
nt

. We
choose this normalization over other admissible ones (for
example, one might have
Z
defined on the intersection
of the unit sphere with
N

) because convexity is very
handy in our proofs.
Recall that the standard boundary behavior (property
5)) plays a crucial role in proving the existence of a
strictly positive vector, , such that (see,
e.g., Aliprantis et al. [1], and Mas-Colell et al. [3]).
When production exhibits constant returns-to-scale, the
standard boundary behavior of the aggregate net demand
still comes into play to prove the existence of competi-
tive equilibria. For details see, for example, Geanakoplos
[4].
p()=0Zp

For our purposes it will be convenient to use a for-
mulation of the standard boundary behavior which does
not involve asymptotic conditions. To this end, the fol-
lowing result is a straightforward consequence of the
standard boundary behavior of the excess demand func-
tion (and of property 4). It is not difficult to prove:
Proposition 2.1: Let :
N
ZInt be a map satis-
fying properties 4) and 5) listed above. Assume
that is such that , with

=1
nn
pIn
tn
pp
=0
i
p for some i. Then, for any π
I
nt there exists
a such that . nπ()>0
n
Zp
As the examples in Section 3 demonstrate, one can
conceive of an economy for which the standard boundary
behavior may fail.
Therefore, we still wish to provide an affirmative
answer to Problem above, but we have to drop the
standard boundary behavior hypothesis. To this end, we
shall now introduce two alternative boundary conditions
for the aggregate excess demand function, but first we
need to provide some mathematical background.
For any >0

, define the restriction of the (metric)
projection mapping to , that is :P.

 It is
known that is well-defined and continuous, and that P


π=πP for all π


 (see, e.g., Aliprantis and Bor-
der [8], pp. 247-249). It’s easy to see that


πP



for each π\


. We are now ready to introduce:
Boundary behavior 1: There exists a >0

such
that


()Pp 0pZ

 for all . p
Remark 2.1: To the best of our knowledge, the
projection mapping was introduced in Economics by
Todd [9] in a general equilibrium model of production
with activity analysis. It was used also by Kehoe [10]. In
this paper we utilize the projection function in a different
manner. Basically, boundary behavior 1 formalizes the
assumption that the excess demand function is “inward-
pointing” on


. A different “inward-pointing” condi-
tion on the excess demand function was introduced by
Neuefeind [11], and Husseinov [12]. We stress, however,
that Neuefeind works with continuous excess demand
functions, whereas in the next section we are able to
address Problem above without assuming continuity of
the excess demand function.
Boundary behavior 2: There exists a >0

such
that if π


 and , with

π>0pZp

, then
there is a 0< <1

such that

1πp

  
.
It is logical to investigate the relationship between
boundary behavior 1 and boundary behavior 2. The next
theorem, due to Iryna Topolyan4, serves this purpose:
Theorem 2.1 (Topolyan): If :
N
ZInt satisfies
properties 3) above, and boundary behavior 2, then it
also satisfies boundary behavior 1.
Proof: Assume that
Z
satisfies boundary behavior 2.
Let >0

be such that if π

 and


π>0pZ,
with p

, then there is a 0< <1

such that


π1p



. Begin by noticing that, by 3), for
each p



))
we have that . We claim
that
((pZP))p=0
(( Pp 0pZ

 for all . Indeed, suppose,
by contradiction, that there exists a such that
p
p

((pZPp))>0


. Then, must lie in p
\

. Put
π=()Pp



. By assumption, there exists 0<<1

such that


π1p




. Hence, by definition of
projection mapping,
 
π1ppp  
 π
 , but
the latter inequality implies that 0

, which is im-
possible. The proof is complete.
In the proofs presented in Section 4 we shall invoke
the following selection theorem due to Michael [7],
(Theorem 3.1’’’):5
Theorem 2.2 (Michael): Let X be a perfectly normal
1pological space, and let Y be a separable Banach
space. Let
T to


DY be the collection of all nonempty and
convex subsets of Y which are either finite-dimensional,
4Personal communication.
5For our purposes it will suffice to state the theorem as done in Tian
[5].

F. RUSCITTI 1400
or closed, or have an interior point. Suppose that

:

F
XDY
dence. Then, th
is a lower hemicontinuous correspon-
ere exists a continuous function
:
f
XY such that
 
f
xFx for all
x
X

.
. Illustrative Examples
e now present two examples of economies whose
ple 1: Consider a competitive economy with one
re
3
W
excess demand function does not necessarily satisfy pro-
perty 5). The former example is very simple, admittedly
artificial, but its virtue is to convey the main ideas. As
for the latter, we refer the reader to Impicciatore et al.
[13]. The key ingredient in both models is that not all of
the goods and services traded affect consumers prefe-
rences, while agents are endowed with strictly-mono-
tonic preferences over a subset of commodities and ser-
vices.
Exam
presentative consumer and one representative firm.
There are two commodities. A consumption good, de-
noted by c, which is produced by the firm with linear
technology and consumed by the consumer. We let p be
the price of the consumption good. The second com-
modity, denoted by
x
, is a production input, owned by
the consumer, which is available in fixed and limited
quantity, say
x
. The production input is not produced
and is supplied by the consumer to the firm. We let w
be the price of the input production services. T
consumer is endowed with
he
x
units of the production
input, but she is not endowed ith the consumption good.
Consumer’s preferences are represented by the utility
function :u which is a function only of the
consumptnd is assumed to be strictly in-
creasing. The production technology is such that
w
ion good a
x
units of the production input yield ax units of th
consumption good, with >0a.
Thus, the firm’s profit-ization p
e
maximroblem is
axi-
0
ax xpax wx
, and the consumer’s preference m
can be described as

m
mization problem
(, )0
max cx uc
..
=
0
st
pc wx
xx
It is very easy to see that, if a competitive equilibrium
pr ice vector (, )pw

exists, then we must have
(, )0pw

. So, in equilibrium prices must be strictly
is why in Section 4 we will be concerned
with the existence of strictly positive equilibrium price
vectors. It is routine matter to check that the consumer’s
net demand function is:

positive. This
,=,
w
Z
pwx x

.
p


for this economy the standard
e net demand function does not
ho
Now we show that
boundary behavior of th
ld. This is why in Section 4 we shall put forward a
method for proving existence of competitive equilibria
under alternative boundary conditions of the aggregate
net demand functions. To see that the standard boundary
behavior fails, note that in view of Proposition 2.1 above
it will suffice to exhibit a price vector ˆˆ (, )0pw  such
that, for each >0

with ˆ>p

and ˆ>w

, there
exists a vector




,=1,pw

  satisfying

ˆˆ ,,0x x


1
pw






Indeed, simple calculations reveal that, if we take


.

11
ˆˆ ,=,
22

pw 

, then 11 ,,<0xx




for any
22 1 

  
1
0< <2
 .
One might ask what goes wrong in this model, with
regard to the standard boundary behavior. Basically, in
this example what causes the standard boundary behavior
of the net demand function to fail is the presence of a
non-reproducible input available in fixed quantity. Also,
consumer’s preferences over both goods are convex, but
not strictly-convex, and monotonic, but not strictly-
monotonic (see Section 1).
Example 2: Another example one might think of,
deals with a mainstream reformulation of the original
Walras’ theory of savings and capital accumulation. We
refer the reader to Impicciatore et al. [13] for the details.
Here we just sketch informally a few elements of the
model.
Time-horizon is finite with two periods, =0,1t. In
each period there are C consumption goods, and
J
labor/leisure services; there are M capital goods, as
as a consistent number of capital goods’ production
services.
There exists a complete array of Arrow-Debreu for-
ward markets open at =0t. Consumers purchase capi-
tal
well
ods i
ec
goods produced at =0t in order to sell their pro-
duction services at =1t. We assume that consumers
have to store capital gon order to supply their ser-
vices to the production stor in the next period.
There is a finite number
H
of consumers, indexed
by =1, ,hH. We assume that capital goods are not
consumed, nor do they affect agents’ preferences. Hence,
consumers’ preferences are defined on the consumption
set ) =J
h
X . Preferences on h
2( C
X
are continuous,
strictly increasing and strictly quasi-concave.
At =0t each consumer is endowed with labor/lei-
sure services and capital goods. Similarly, at =1t each
consumer is endowed with labor/leisure services. At
=0 ces from owned capital goods are inelasti- t servi

F. RUSCITTI1401
pplie
the producor.
feasible qu
tored atvai-
lable to the pr
or ea
Each consumes a
bundle of consumtion gor/leisure and capital
goods that maximes his uty. The consumect
to
e of equilibria. It h underscorin
th
cally supplied. Capital goods purchased at are
stored for one period; at =1t their services are su-
d to the production sector.
Since we are concerned with the behavior the
aggregate net demand functie shall omit the forma-
lization of and the assumptions o
=0t
of
tion sect
vices a
e
er is subj
ng that i
on, w
n
ods, labo
tili
is wort
We assume that consumers are endowed with indi-
vidual storage technologies, formalized as follows: for
each

1, 2,,hH and each capital good

1, 2,, M, the storage function :
h
m



maps anyantity of the capital good purchased
and sthe quantity of ser
m
=0t
odu
into
ction sector at =1t.
A capacity constraint on the storage technology is in
place. That is, fch

1, 2,,hH and each
m

1, 2,, M, there exists a ˆ>0
h
m
k such that
ˆ
:0,
hh
mm
k 




.
r takes prices as given, and choos
p
iz
the storage capacity constraint and the budget con-
straint.
The authors then define the notion of virtual aggregate
net demand function, which is instrumental in proving
existenc
is model the virtual aggregate net demand function
may fail to satisfy the standard boundary behavior. To
see this, note that we may well think of each consumer as
being equipped with monotonic, and convex preferences
defined over every goods and services traded in the
economy. On the other hand, we know from Section 1
that sufficient conditions for the standard boundary
behavior are strictly convex and strictly monotonic pre-
ferences. In other words, the sufficient conditions for the
standard boundary behavior are violated. Furthermore,
suppose we are given an arbitrary sequence


πn of
strictly positive prices that converges to π0, where
π belongs to the boundary of
N

. By the capacity
constraint on storage, the demand for capital goos is
bounded above, and one can prove that ast one
nsumers’ income is finite and positive. In a nutshell,
these are the reasons why the virtual aggregate net
demand function does not necessarily satisfy the standard
boundary behavior. Therefore, as we pointed out above,
we seek a method to prove existence of equilibria that
does not hinge on the standard boundary behavior.
4. Main Existence Theorems
d
leat
co
Su pose we are given a funpction :
N
ZInt
ence
. In t
of
his
some
such that
ing assumption
section we are concerned with the exist
pInt
 ()=0. Zp 
We begin by making the follow:
Assumption 4.1: :
N
ZInt satisfies the Wal-
hat is ()pZ ras law, t=0p

for all pInt. Also, Z
sa h that
th
tisfies boundary behavior 1. Moreover, Z is suc
e correspondence
  
:defined by
π:=: π>0



 
U
UppZ

(4.1)
is lower hemicontinuous.6
Remark 4.1: If :
N
ZInt
o see that U
is low
tinuous, then it’s easy t is lower
assumption is weaker than
asontinuity of Z.
4.1 bel
e, the first part of
ou
nal version of Browder’s selection
th
er semicon-
:


hemicontinuous. Thus, our
suming lower-semic
Before we state and prove Theoremow, let us
comment on the strategy of the proof, and on the relation
to the established literature. By and larg
r proof is inspired by Tian [5] and Ewald’s approach
to proving the basic Ky-Fan theorem (see Ewald [14],
Lemma 3.6.1, and Theorem 3.6.5). Our proof, though,
departs from Ewald’s in two significant ways. First of all,
the correspondences defined in [14] are assumed to have
open lower sections. In contrast, we posit the assumption
of lower hemicontinuity (see Assumption 4.1 above)
because, in general, it is weaker. Moreover, we assume
lower hemicontinuity to facilitate a comparison with the
approach followed by Tian [5], and because we believe it
is a more natural assumptions when dealing with Eco-
nomic models.
Secondly, since Ewald deals with correspondences
with open lower sections, he finds it natural to employ
the finite-dimensio
eorem. In contrast, we work directly with a lower
hemicontinuous correspondence, U defined in (4.1), and
therefore we shall employ Michael’s selection theorem
(Theorem 2.2 above). Incidentally, Theorem 2.2 above is
a generalization of the finite-dimensional version of
Browder’s selection theorem used by Ewald.
Theorem 4.1: If Assumption 4.1 holds, then there
exists a ˆ π


 such that

ˆ
π=0Z.
Proof: Clearly, :U


 is convex-valued. By
Walras law we have that


ππUfor all π


 (4.2)
Put




=π:πWU


. If, then it’s
easy to see that we are done. So, assume,
generality, that
=W
without loss in

W. Pick anitrary y arbπW

. By
finiti  

Now, take de on of, there exists a W

πU.
any open neighborhood of

 in , say





.
Clearly,




πU 




. Since U is lowecon-
tinuous, there ets an open neighbf π
 in

r hemi
xisorhood o

,
6ε appearing in the definition of the above correspondence is the ε in-
volved in the definition of boundary behavior 1.

F. RUSCITTI 1402
say

πV, such that

r everyππUV

 
fo  π

. (4.3)
It followm (4.3) that

πVW. Hence, W is
n in
s fro
ope



, that is
nuous selection, t
.
7 Nn
Th above. Th, admits a conti-
hat is therexists a cnuous function
ext, consider the restrictioof U to
W
|:
W
UW
It should be clear that |W
U meets the conditions of
eorem 2.2
.
erefore
e
|W U
onti
 
:such thatππforall π.
f
WfUW
Now define a new correspondence :


 as
follows:
 
πif π
π
if π
f
W


W




Clearly,

is convex and compact valued. We wish
to prove that

is upper hemicontinu. To this end,
d graph theorem it will suffice to show that
ous
by the close

has clos graph. To see this, let

ed


π,
nn

be a se-
quence satisfying

π
nn
 , for all n, ππ
n, and
n

 . We must show that

π  er

. So
=
, we consi
. Sin
d
o cases: 1) If πW, then

π  ce tw

is
closed, it followsthat at once




=π  . 2)
because ππ
n andopen in πW: W is


(see
above), there is a h that or all nN.
Thus, for all nN we have that
N sucπnW f



π
nn n
.
by continu
π=f  
Hence,ity of
f
, we g

π=
lim n
n
ff 

Thus,
  
π= .f   Now,
et

=π.

π take the composi-
tion of

with P (P is the projection function de-
finen 2), and define the new correspondence d in Sectio
:P


Since P is continuous and

is upper hemiconti-
nuous, P

 is upper hemicontinuous. By construction,
P


va
is no
th


. Th
=f
n-empty valued, co-valued and closed-
en, ion, e,
nve
ta
x
lued. Iten follows from Kakuni’s fixed point theo-
rem that there exists a π such that

(π)P 

. Now, set

ˆ π=πP. We claim that
ˆ
πW . To see this, assume, by contradiction, that
ˆ
πW 
π
by constructˆ =πf. Henc
 

ˆ
π 
 
ˆˆ
ππU , which implies that π


>0.

ˆ
π= 
s,
ˆ
ππZ 
e preceding inequality contradicts boundary

But
havi
entails

ˆ
π

th
or
pZ
be-
1. Therefore, ˆ πW  an, whichd so U
0 for all p. Thu


ˆ
π0
Z

,
and since ˆ π0, Walras law immediately implies that


ˆ
π=0Z. The proof ihed.
Rema Theorem 4ould be coo
Theorem 4.1 and Theorem 6.1 in Tian [5]. Tian relaxes the
of lower semicontinuity
s finis
4.2: Our.1 shmpared t
of Z too. However, he
rk
assumption
assumes that the excess demand function is defined on
the whole

, and demonstrates the existence of a
q

 such that ()0Zq


. Neuefeind [11] formalizes
a condition that the excess demand function is
“inward-poting” close to the boundary of the p
. His assumoes not require the excess
demand function to satisfy property 5) above. However,
Neuefeind assumes that the excess demand function is
continuous. We dispense with the standard boundary
condition on the excess demand function too, but unlike
Neuefeind we relax the assumption of continuity of the
excess demand function.
Now, we assume that :
inrice-
ption dsimplex
N
ZInt satisfies boun-
dary behavior 2. Specifically:
Assumption 4.2: :
N
ZIn satisfies the Wal- t
ras law, that is


=0 for pZ ppInt . Also, all
Z
satisfies boundary behavior 2. Moreover,
Z
is such
e that the correspondenc




π

:definedby π:= :>0

 UUppZ
is lower hemicontinuous.
Rema
4.1, the fo
sh
rk 4.3: In light of Theorem 2.1, and Theorem
llowing theorem is obvious, and requires n
ive a direct proof in order to
o
proof. However, we will g
ow that our way of proving existence is quite general,
and can handle different types of boundary conditions.
Incidentally, in the proof of Theorem 4.2 we will make
use of a fixed point theorem due to Halpern and Berg-
man; it is of some interest to observe that, to the best of
our knowledge, such a theorem has never been used
before to prove existence of competitive equilibria.
Theorem 4.2: If Assumption 4.2 holds, then there
exists a ˆ
π


 such that


ˆ
π=0 Z.
Proof: The first part of the proof up to the construc-
tion of the correspondence :


 is identical to
the prooheorem 4.1tha is f of T. Recall t :




defined as follows:



π
π
if π
if π
f
W
W







Clearly,

is convex and compact valued. Further-
more,

Theorem 4.
is u
1).
pper hemicontinuous (see the proof of
Now, we will prove that

is inward
pointing.8 Tthis end, pick any πo

 . If πW

, then


π= 

, and therefore

πππ

  for any
>0

. On the other hand, if πW, then






π=πf  , and therefore

o, we

ππ>0fZ. S
7So far we have demonstrated that, if U is lower hemicontinuous, then
8For the definition of inward pointing correspondence see Aliprantis
its domain is open. This is a well-known fact, but we have worked out
the proof for the sake of completeness. and Borde
r
[8], Definition 17.53.

F. RUSCITTI1403
consider two cases. 1): If π\


 , then clearly
 
ππfπ

  for so>0me

small enough.
2): If π

 then, since

fboundary

ππZ
n
>0,
behavior 2 immediately implies the existece of a >0

such tπ. hat

πf

π

  , Hence

is inward
pointingfore we have established that . There

meets
the sufficient conditions of Halpern-Bergman fixed point
theorem (see Alipra [8], Theorem 17.54).
Thus,
ntis and Border

has a fixed point. That is, there exists a
ˆ π
9

 such that

ˆˆ ππ  . Now, if ˆ πW, then
 
ˆˆ ˆ ˆ
ππ=ππfU  , which contradicts (4.2) above.
Thus, we must have that ˆ πW. Hence, by definition of
W and U, we obtain

ˆ
π0Zp

 for all p

,

ˆ
πther hand, since ˆ π0,
Walras law immediately implies that

ˆ
π=0Z, as was
to be proven.
4.1. Some Instrumental Results
We begin this
and so 0Z. On the o
section by reminding the reader an open
which will be used
6] (Proposition 2).
graph theorem for correspondences
later on. It was first proven by Zhou [1
Theorem 4.1.1: Let :
N
UX  be a correspon-
dence, where
X
is a topological space. Assume that
U has convex and open upper sections. Then, U is
lower hand only if U hemicontinuous if as open graph.
In the context of Hilbert spaces, we can now prove a
theorem which is a natural generalization of Theorem 5.1
in Yannelis and Prabhakar [6]. It is a generalization in
that we do not require the domain and range of the
correspondence at hand to be the same. Moreover, in
place of the first condition in Theorem 5.1 of Yannelis
and Prabhakar [6], we posit a more general assumption
(see assumption 1) below). It is more general in the sense
that it collapses to Yannelis and Prabhakar’s condition
whenever the domain and range of the correspondence
coincide. Interestingly, our assumption 1) below bears a
natural economic interpretation in terms of boundary
behavior 1 defined in Section 2. Consequently, we shall
show, in Section 4.2, that our Theorem 4.1.2 can be
employed to provide another short proof of Theorem 4.1.
Let  be a Hilbert space, and let
X
and Y be
non-empty, convex and compact subsets of , with
X
Y. Let :PY X be the (metric) projection
mapping defined in Section 2. For any subset
A
of , 
we denote by conA the convex-hull of
A
.
Theorem 4.1.2: Let :UXY be a correspondence
such that: 1) For each
y
Y,

()
y
conUP y; 2) U
has open lower sections in
X
. Then, there exists a
ˆ
x
X such that Ux
Proof: Assume, by wining a contradiction,
that for every


ˆ=.
ay of obta
x
X

, . Then, th

Ux e corre-
spondence :
X
Y

, ded by

efin


=
x
conU
ery
x
x
in
X
for ev, is co
va
nvevaluex-d and nonempty-
lued. By assumption 2), it’s easy to see that

has
open lower sectioX.10 Browder selection
theorem (se [17]) there exists
function
ns in
rowder
Hence, by
e B
:
a continuous
f
XY such that
 
f
xx for any
x
X

. Now, consider the composition of maings
:
pp
f
PY Y. By Brouwer-Schauder-Tychonoff theo-
rem, :
f
PY Y has a fixed point. That is, there exi-
sts a
y
Y


such that










=PyPyconUPy
 
 , which con-
tradic1). The proof is finished.
4.2. Bound
=y f

ts assumption
ary Behavior and Existence of
Maximal Elements
onsider, again, the correspondence defined in (4.1)
whose
exisundary behavior 1. Define
e auxiliary correspondence
CU
above. Recall that ε therein is the positive number
tence is guaranteed by bo
th



:byπ:=: π>0




NN
UUppZ
and let :
N
S

 be defined by


π=S for every
π


. Clearly,
=UUS
.
Assum : :
N
ption 4.2.1ZInts the Wal-
ras law and bounbehavior 1. Mo re
satisfie
dary over,
Z
is such
: that
N
U


Next we show
  is tinuous.
that Tbe established as
a rollary of Theo
sponds to ass
lower hemicon
heorem 4.can 1
simple and short corem 4.1.2. In this
regard, it is interesting to notice that boundary behavior 1
correumption 1) in Theorem 4.1.2.
Corollary 4.2.1: If Assumption 4.2.1 holds, then there
exists a ˆ π


 such that ˆ (π)=0Z.
Proof: By Theorem 4.1.1, :
N
U



 has open
graph. Obviously, :
N
S

 has open lower sec-
tions. Thus, =UUS

 has open lower sections as well.
Now, noat boundaryr 1 imptice th be
exists a ˆ π
havio
orem 4.1.2.
lies that
sa
U
tisfies assumption 1) of TheHence, by
Theorem 4.1.2, there

 such that
ˆ (π)=U

. Tˆ (π)0Z hat is, p

 for all p. As in
the proof of Theorem 4.1, Walras law readily impl
that ˆ (π)=0Z.
From dp
ies
5. Concluding Remarks
the stanoint of applied mathematics, we believe
at this work is self-contained. From the perspective of
be improved and ex-
th
economic theory, this paper can
9We remark that the existence of a fixed point for γ may be proven also
by invoking Theorem 4 in Tian [15]. 10See the proof of Lemma 5.1 in Yannelis and Prabhakar [6].

F. RUSCITTI 1404
nded. Let us outline what it would be worth under-
D. Aliprantis, D. J. Brown and O. Burkinshaw, “Exis-
tence and Optimality of Competitive Equilibria,” Spri
990. doi:10.1007/978-3-642-61521-4
te
taking for future research. First of all, one should in-
vestigate the relationship between the standard boundary
behavior and our boundary behavior 1. Secondly, if it
turns out that neither of them implies the other, or that
the standard boundary behavior implies our boundary be-
havior 1, then it would be interesting to construct rele-
vant economic models for which the standard boundary
behavior does not hold, but our boundary conditions are
satisfied by the excess demand function of the model
itself.
6. References
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] K. Arrow and F. Hahn, “General Competitive Analysis,”
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: An Intertemporal Equilib-
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