EQUILIBRIA IN PRODUCTION ECONOMIES
CHARALAMBOS D. ALIPRANTIS1 , MONIQUE FLORENZANO2 , AND RABEE TOURKY3
1
Department of Economics, Krannert School of Management, Purdue University, 403 West State Street,
W. Lafayette, IN 47907-2056, USA;aliprantis@mgmt.purdue.edu
2
CNRS–CERMSEM, MSE Université Paris 1,106-112 boulevard de l’Hôpital, 75647 Paris Cedex 13, FRANCE;
monique.florenzano@univ-paris1.fr
3
Department of Economics and Commerce, University of Melbourne, Parkville 3052, Melbourne, AUSTRALIA;
rtourky@unimelb.edu.au
Abstract. This paper studies production economies having a locally convex topological vector
commodity space ordered by a closed and generating convex cone such that the order intervals
are topologically bounded. The generally assumed lattice properties on the commodity–price
duality are replaced by an assumption of uniform properness of the Riesz–Kantorovich functional
associated with a list of continuous linear functionals. On such an economy, our assumptions are
quite general. In particular, consumers preferences are non-ordered, not necessarily monotone,
and we do not assume free-disposal. The equilibrium existence theorem established in this paper
is the most general for production economies in the literature.
JEL classification: C61; C62; D20; D46; D51
Keywords: Production economies; Equilibrium; Edgeworth equilibrium; Properness; Riesz–
Kantorovich formula; Sup-convolution
1. Introduction
This paper is the last in a series of papers [8, 9, 10] studying Walrasian equilibria without vector
lattice assumptions on the commodity–price duality of an economy. More precisely, this paper
extends to production economies results obtained in [10] for exchange economies.
In [9] and [10], we were addressing the equilibrium existence problem for exchange economies
having ordered topological vector commodity spaces. The main motivation for going from lattice
ordered topological vector spaces, the case usually considered in the literature, to this setting was
coming from Finance. In security models, one is given with a pair of function spaces, L and X,
where L is the portfolio space and X is the space of contingent claims, together with a linear
Date: January 29, 2005.
A first version of this paper has been presented at the 11th Conference on Real Analysis and Measure Theory
in Ischia (Italy, 2004). The paper has benefitted of the comments of the audience. We thank particularly V. Filippe
da Rocha and Yiannis Polyrakis for useful suggestions about the properness assumptions used in this paper. The
research of C. D.Aliprantis is supported by the NSF Grants EIA-0075506 and SES-0128039. The research of R.
Tourky is funded by the Australian Research Council Grant A00103450.
2
operator R : L → X, one-to-one and positive, which pulls back the order intervals of R(L) to
closed and bounded subsets of L. The portfolio dominance ordering, a notion introduced in [4, 5],
is the only ordering which is relevant for purposes of economic analysis, especially for studying the
arbitrage freeness of security prices. But, when the portfolio space is reordered by the portfolio
dominance ordering, it is seldom a vector lattice and the existence of equilibrium cannot be deduced
from theorems which require lattice properties on the commodity space. Minimal conditions of
compatibility between order and topology of the commodity (portfolio) space were given in [9] and
[10] for completing a general equilibrium analysis of exchange models.
In this paper, we study the existence of general equilibrium for a production economy having an
ordered topological vector commodity space. To go from exchange models to production economies
is never easy. It suffices to recall the competitive challenge which lead after the seminal paper of
Mas-Colell [27] to the publication in 1987 of Zame’s paper[39] and Aliprantis–Brown–Burkinshaw’s
paper[3] and in 1989 of Richard’s paper [34]. Our assumptions on the economy are quite general.
In particular, preferences are non-ordered and we do not assume any monotonicity of preferences
or free disposal in production. Under (most often) stronger assumptions on the characteristics of
the economy, the equilibrium existence problem has been solved in two consecutive settings. In
the first one, introduced by Mas-Colell [27], the commodity space is is a topological vector lattice.
In the second one, introduced by Mas-Colell–Richard [29], the commodity space is an ordered
locally convex topological vector space such that both commodity space and price space are lattice
ordered. In these two settings, the equilibrium existence is obtained under properness assumptions
on the characteristics of the economy. Since its first introduction to economics by Mas-Colell in
[27] and [28], the formulation of properness has become more and more general [33, 23, 37, 38],
but the lattice properties of the commodity–price duality are always used in a nontrivial way in
the few papers [3, 39, 34, 38, 23] which address the equilibrium existence problem for production
economies.
In this paper, we replace these lattice properties by a condition of compatibility between order
and topology of the commodity space, stated in terms of properness of the Riesz–Kantorovich
functional associated with a finite list of continuous linear functionals, which is obviously satisfied
under these properties. The condition stated in this paper is stronger than the one used in our
previous papers for the exchange case. Also, since the commodity space is not assumed to be a
vector lattice, at least on the production side, properness has to be reformulated in a way which
does not depend on lattice properties of the underlying space.
Our paper is organized as follows. In section 2, we recall the notion of the Riesz–Kantorovich
functional associated with a finite list of (linear) functionals, a notion that we put in relation with
the concept of sup-convolution, well-known inoptimization. In section 3, we define the model and
posit assumptions as well on the commodity space as on the characteristics of the economy. The
commodity space is assumed to be a locally convex topological vector space ordered by a closed
and generating convex pointed cone such that the order intervals are topologically bounded. Mild,
but classical, assumptions are made on the economy. Let us emphasize here that we do not assume
any kind of monotonicity or transitivity on consumers’ preferences or free disposal in production.
Properness is defined as well for preferences as for production. Under a classical compactness
assumption, Edgeworth equilibria exist. Section 4 is devoted to their decentralization with linear
prices. For going from non-linear to linear decentralization, we introduce a condition of compatibility between order and topological structure of the commodity space stated in terms of uniform
properness of the Riesz–Kantorovich functional. A by-product of the decentralization result is a
quasiequilibium existence result. Existence of equilibrium is obtained under standard conditions
3
of nontriviality and of irreducibility of the economy. In view of the generality of our assumptions,
this result extends any previous equilibrium existence result obtained under properness in the
Mas–Colell or in the Mas–Colell–Richard settings. Section 5 is devoted to examples which show
that our result allows to obtain existence of equilibrium in settings of economic interest which are
covered by no previous equilibrium existence result.
2. Mathematical preliminaries
For details regarding Riesz spaces that are not explained below we refer the reader to [6] and
[7]. This paper will utilize the notion of the Riesz–Kantorovich formula that was introduced in
[13] and used extensively in [14, 9, 10]. We shall briefly introduce this formula here and refer the
reader to [12] for a complete discussion regarding the Riesz–Kantorovich formula.
We start with the following classical result from the theory of partially ordered vector spaces
due to F. Riesz and L. V. Kantorovich.
Theorem 2.1 (Riesz–Kantorovich). If L is an ordered vector space with a generating cone and
the Riesz Decomposition Property,1 then the order dual L∼ is a Riesz space and for each f, g ∈ L∼
and x ∈ L+ its lattice operations are given by:
©
ª
1. [f ∨ g](x) = sup© f (y) + g(z) : y, z ∈ L+ and y + z = xª .
2. [f ∧ g](x) = inf f (y) + g(z) : y, z ∈ L+ and y + z = x .
In particular, note that if L has the Riesz Decomposition Property, then for any finite collection
of linear functionals f1 , f2 , . . . , fm ∈ L∼ their supremum in L∼ at each x ∈ L+ is given by
m
m
m
h_
i
nX
o
X
fi (x) = sup
fi (xi ) : xi ∈ L+ for each i and
xi = x .
(?)
i=1
i=1
i=1
For any positive integer m and x ∈ L+ define
m
n
o
X
m
Am
=
(x
,
.
.
.
,
x
)
∈
L
:
xi = x .
1
m
x
+
i=1
The formula (?) that gives the supremum of the order bounded linear functionals f1 , . . . , fm is
called the Riesz–Kantorovich formula of these functionals. The useful observation here is that if
each fi is an arbitrary function from L+ to (−∞, ∞], then the right-hand side of (?) still defines
an extended real number for each x ∈ L+ . That is, the formula appearing in (?) defines a function
from L+ to (−∞, ∞] called the Riesz–Kantorovich functional of the m-tuple of functions
f = (f1 , . . . , fm ) and denoted Rf . In other words, Rf : L+ → (−∞, ∞] is defined by
m
nX
o
Rf (x) := sup
fi (xi ) : (x1 , . . . , xm ) ∈ Am
x
i=1
for each x ∈ L+ .
If each fi is a function that carries order intervals to bounded from above subsets of R, then the
Riesz–Kantorovich functional is real-valued. Moreover, if each fi is super-additive and positively
1 The Decomposition Property states that if x, y , y ∈ L
+ satisfy 0 ≤ x ≤ y1 + y2 , then there exist elements
1
2
x1 and x2 such that 0 ≤ x1 ≤ y1 , 0 ≤ x2 ≤ y2 and x = x1 + x2 . The order dual L∼ is the vector space consisting
of all linear functionals on L which map order intervals of L to order bounded subsets of R, ordered by the relation
f ≥ g whenever f (x) ≥ g(x) for all x ∈ L+ .
4
homogeneous, then the Riesz–Kantorovich functional Rf is also super-additive and positively homogeneous; in particular, it is a concave function. Using as prices the Riesz–Kantorovich functional
of a list of personalized prices, a new theory of value was presented in [14].
Let us now associate with each fi : L+ → (−∞, ∞) the function fbi : L → [−∞, ∞) defined by
½
fi (x)
if x ∈ L+
fbi (x) :=
−∞
otherwise .
If each fi is finite-valued, it is easy to recognize in Rf (x) for x ∈ L+ the value at x of the
sup-convolution2 of functions fbi defined by
m
m
nX
o
X
£ m ¤
∇ fbi (x) := sup
fbi (xi ) :
xi = x .
i=1
i=1
i=1
Definition 2.2.PWe will say that Rf is exact at x with respect to a vector (x1 , . . . , xm ) ∈ L
Ãm
+
m
satisfying x = i=1 xi , if the sup-convolution is exact at x with respect to (x1 , . . . , xm ) ∈ Lm
+.
That is, if
m
m
X
X
£
¤
bi (xi ) =
bi (x) =
Rf (x) = ∇m
fi (xi ) .
f
f
i=1
i=1
i=1
X
Let hX, X 0 i be a dual system and let f ∈ R . Recall that a vector y 0 ∈ X 0 is called a
supergradient of f at x if f (x) is finite and f (y) − f (x) ≤ hy − x, y 0 i for all y ∈ X. The (possibly
empty) set of all supergradients of f at x is called the superdifferential at x of the function f
and denoted ∂f (x).
The following result (see for example [26, Proposition 6.6.4]) will be used in our work.
Theorem 2.3 (Moreau). Assume that hX, X 0 i is an arbitrary dual system. For each i = 1, . . . , m
let gi : X → [−∞, ∞] be a non identically equal to −∞ function. If the
sup-convolution ∇m
i=1 gi is
Pm
m
exact at x with respect to some (x1 , . . . , xm ) ∈ X that satisfies x = i=1 xi , then
m
\
£
¤
∂ ∇m
g
(x)
=
∂gi (xi ).
i
i=1
i=1
3. The economic model
In what follows, if M is an ordered linear space, then for the sake of notational convenience,
M + or M+ will denote the positive cone of M .
The commodity space of our model is an ordered linear vector space L equipped with a
Hausdorff locally convex topology τ such that:
A1: The positive cone L+ of L is generating (i.e., L = L+ − L+ ) and τ -closed.
A2: The order intervals of L are τ -bounded.
L
L
Let f ∈ R and g ∈ R be extended
real-valued functions. Using theªconvention +∞ + (−∞) = −∞ + (+∞) =
©
−∞, the formula [f ∇g](x) = sup f (y) + g(z) : y, z ∈ L and y + z = x defines an extended real-valued function
f ∇g called the sup-convolution of f and g. The expression f ∇g is also called by Rockafellar and Wets [35] the hypoaddition of functions f and g, because if hypo f denotes the hypograph of f , one has hypo(f ∇g) = hypo f + hypo g,
as long as the supremum defining [f ∇g](x) is attained when finite.
2
5
The topological dual of (L, τ ) (i.e., the vector space of all τ -continuous linear functionals on L)
will be denoted L0 . The algebraic dual of L (i.e., the vector space of all linear functionals on L) is
denoted L∗ . The order dual of L (i.e., the vector space of all order-bounded linear functionals on
L) is denoted L∼ . Since every order interval of L is τ -bounded, it follows that L0 ⊆ L∼ ⊆ L∗ .
Assumptions A1 and A2 are satisfied by the following ones used by Mas-Colell–Richard [29]
and many others:
B1: L is a vector lattice whose positive cone L+ is τ -closed; and
B2: L0 is a vector sublattice of L∼ .
Both sets of assumptions are implied by Mas-Colell’s assumption in [27] and [28] that L is a
topological vector lattice. To see this, note that if L is a topological vector lattice, then B1 simply
follows from the continuity of lattice operations, and B2 follows from the Nakano–Roberts theorem
(see [7, Theorem 2.22]). On the other hand, B2 implies that the cone L0+ is generating, which
is equivalent the fact that the cone L+ is a normal cone for the weak topology σ(L, L0 ), and the
latter implies that the order intervals are σ(L, L0 )-bounded, thus τ -bounded (see [32, Chapter 2,
Corollary 1.23 and Proposition 1.4, and 2.1 for the definition of normal cones]). The assumption
that L is a vector lattice and B1 obviously imply A1.
On L as commodity space, let us consider a private ownership production economy
³¡
´
¢
E = Xi , Pi , ωi i∈I , (Yj )j∈J , (θij )i∈I,j∈J
where I = {1, . . . , m} is a finite set of m (≥ 2) consumers, J = {1, . . . , n} is a finite set of n (≥ 1)
producers. Each consumer i is characterized by a non-empty consumption set Xi ⊆ L, an
initial endowment ωi ∈ Xi and an irreflexive preference correspondence Pi : Xi →
→ Xi , i.e.,
xi ∈
/ Pi (xi ) for each xi ∈ Xi . Each producer j is characterized by a non-empty production set
Yj ⊆ L. For every producer and each consumer, the firm shares 0P≤ θij ≤ 1 classically represent a
contractual claim of consumer i to the profit of producer j and i∈I θij = 1 for each j ∈ J. In a
core and Edgeworth equilibrium approach, the relative shares θij reflect consumers’ stockholdings
that represent proprietorships of production possibilities and θij Yj is the portion of the j producer’s
technology setPat i’s disposal.
Let ω =
i∈I ωi be the total endowment, and let Aω be the set of all feasible (or
attainable) allocations of E, that is,
n
Y
X
X o
¡
¢ Y
Aω = (x, y) = (xi )i∈I , (yj )j∈J ∈
Xi ×
Yj :
xi = ω +
yj .
Q
i∈I
j∈J
i∈I
j∈J
3
The set Xω , the projection of Aω on i∈I Xi , is the set of all feasible consumption allocations.
We recall the following standard notions of equilibria for an economy E.
Definition 3.1. A 3-tuple (x, y, p) consisting of a feasible allocation (x, y) and a non-zero linear
functional p is said to be:
1. a quasi-equilibrium, if
P
(a) for every i ∈ I we have p(xi ) = p(ωi ) + j∈J θij p(y j ) and xi ∈ Pi (xi ) implies p(xi ) ≥
p(xi ), and
(b) for every j ∈ J and every yj ∈ Yj we have p(yj ) ≤ p(y j );
2. an equilibrium, if it is a quasi-equilibrium and if xi ∈ Pi (xi ) implies p(xi ) > p(xi ).
3
©
Q
P
P
Q
That is, Xω = x = (xi )i∈I ∈ i∈I Xi :
i∈I xi = ω +
j∈J yj for some (yj )j∈J ∈
j∈J Yj }.
6
Definition 3.2. A quasi-equilibrium (x, y, p) is said to be nontrivial, if for some i ∈ I we have
inf{p(zi ) : zi ∈ Xi } < p(xi ).
In this paper, we will be interested only in nontrivial quasi-equilibria.¡ If (x, y,
¢ p) is some trivial quasi-equilibrium, then for every feasible allocation (x, y), the pair (x, y), p is also a quasiequilibrium. On the other hand, if the quasi-equilibrium (x, y, p) is nontrivial, then it is well-known
that, under some continuity condition on preferences or concavity for utility functions, and some
irreducibility assumption on the economy, the quasi-equilibrium (x, y, p) is actually an equilibrium.
The following notions of optimality are also standard.
Definition 3.3. A (feasible) consumption allocation x ∈ Xω is said to be:
1. weakly Pareto optimal, if there is no feasible consumption allocation x ∈ Xω satisfying
xi ∈ Pi (xi ) for each i ∈ I,
2. a core allocation, if it cannot
Q be blocked by any coalition in the sense that there is no
coalition
S
⊆
I
and
some
x
∈
i∈S Xi such that:
P
P
P
P
(a)
x
∈
ω
+
i∈S i
i∈S i
i∈S
j∈J θij Yj , and
(b) xi ∈ Pi (xi ) for all i ∈ S,
3. an Edgeworth equilibrium, if it belongs to the core of every r-fold replica of Q
E,4
4. a fuzzy core allocation, if there exist no τ = (τi )i∈I ∈ [0, 1]I \ {0} and x ∈ i∈I Xi such
that:P
P
P
P
(a)
j∈J θij Yj , and
i∈I τi
i∈I τi ωi +
i∈I τi xi ∈
(b) xi ∈ Pi (xi ) for all i ∈ I with τi > 0.
From now on, we impose on E the following conditions.
ω-properness: That is,
P
1. For each consumer i, the consumption set Xi is convex, and ω = i∈I ωi > 0.
2. For each i and every weakly Pareto optimal consumption allocation x = (xi )i∈I , xi ∈
cl Pi (xi ), Pi (xi ) is open in Xi for some linear topology on L (or is induced by a concave
utility function) and is ω-proper at xi in the following sense adapted from Tourky [37]:
there exist in L a convex set Pbi (xi ) and a convex set Zi (xi ) such that
(a) the vector xi + ω is a τ -interior point of Pbi (xi ),
(b) Pbi (xi ) ∩ Zi (xi ) = Pi (xi ), and
(c) xi , 0, ωi ∈ Zi (xi ), Zi (xi ) + L+ ⊆ Zi (xi ),
(d) for every u > 0 and every couple {zi , zi0 } of elements of Zi (xi ), if −u ≤ zi , −u ≤ zi0
then there exists z ∈ Zi (xi ) such that −u ≤ z ≤ zi and −u ≤ z ≤ zi0 .
3. For each producer j, the production set Yj is convex and 0 ∈ Yj .
4. For each j and every yj ∈ Yj associated with a weakly Pareto optimal consumption
allocation, Yj is ω-proper at yj in the following sense adapted from Tourky [38]:
there exist in L a convex set Ybj (yj ) and a convex set Zj (yj ) such that
(a) the vector yj − ω is a τ -interior point of Ybj (yj ),
(b) Ybj (yj ) ∩ Zj (yj ) = Yj , and
(c) 0 ∈ Zj (yj ) and Zj (yj ) − L+ ⊆ Zj (yj ),
4 The ideas in this definition go back to Debreu–Scarf [19]. An important reference is also [17]. Edgeworth
equilibria were first introduced and studied in [2].
7
(d) for every u > 0 and every couple {zj , zj0 } of elements of Zj (yj ), if zj ≤ u, zj0 ≤ u
then there exists z ∈ Zj (yj ) such that zj ≤ z ≤ u and zj0 ≤ z ≤ u.
Compactness: For some Hausdorff linear topology σ on L, the set Xω of all feasible consumption allocations is σ m -compact and preferences have σ-open (in Xi ) lower sections
Pi−1 (xi ) = {x0i ∈ Xi : xi ∈ Pi (x0i )}.
At this stage, some comments on our properness assumptions are in order.
In our hypothesis, ω-properness of preferences at a component of a (feasible) weakly Pareto
optimal consumption allocation is stated in a general form for the sake of generality and symmetry
with our definition of ω-properness of production sets. In the classical case, the consumption sets
Xi and the sets Zi (xi ) all coincide with the positive cone L+ of the commodity space L which
obviously satisfies the assumptions required for the sets Zi (xi ). The interest of our definition is to
allow for more general consumption sets. It is worth noticing that if the commodity space L is a
Riesz space, then our assumptions on the sets Zi (xi ) are equivalent to Tourky’s assumption in [37]
that each Zi (xi ) is a convex lattice containing 0, ωi , xi and satisfying Zi (xi ) + L+ ⊆ Zi (xi ). Let
us stress that our assumptions do not imply that L+ is a lattice cone.
The sets Zj (yj ) which are involved in the definition of ω-properness of production sets at a
production component of a weakly Pareto optimal allocation correspond to the pretechnology sets
defined by Mas-Colell [28] and used by Richard [34]. In Mas-Colell [28] and Richard [34], the
pretechnology sets depend on j but not on a particular yj . As above, if the commodity space L
is a Riesz space, then our assumptions on the sets Zj (yj ) are equivalent to Tourky’s assumption
in [38] that each Zj (yj ) is a convex lattice containing 0, yj and satisfying Zj (yj ) − L+ ⊆ Zj (yj ).
When the commodity space is an ordered vector space but not a Riesz space, then pretechnology
sets equal to the negative cone −L+ satisfy obviously our assumptions but are not the only one
such examples.
To conclude this section, let us remark that it easily follows from the previous definitions that
every equilibrium consumption allocation is an element of the fuzzy core and, consequently, an
Edgeworth equilibrium, a core allocation and a weakly Pareto optimal allocation. The following
lemma follows from the previous assumptions and Florenzano [21].
Lemma 3.4. In our economy, Edgeworth equilibria exist and belong to the fuzzy core.
Exactly as in [15], one can prove that if preference correspondences are derived from quasiconcave
utility functions defined on general consumption sets, our compactness condition can be replaced
by a weaker compactness assumption made in the utility space on the “utility set”.5
The next section is devoted to look at sufficient conditions on the order structure of the commodity space that will guarantee for our economy the validity of the core equivalence theorem
(first stated by Debreu–Scarf [19] for a finite dimensional economy).
4. Decentralizing Edgeworth equilibria
For the discussion of this section, we fix an Edgeworth equilibrium consumption allocation
(xi )i∈I , thus a fuzzy core consumption allocation of E, and (y j )j∈J , the associated production
5
That is, the set of feasible and individually rational utility vectors.
8
¢
¡
allocation. Such an allocation (xi )i∈I , (y j )j∈J exists by Lemma 3.4. The proof of its possible
decentralization by prices in L0 will adapt ideas of [22, Chapter 5.3.4] where is proved the decentralization of Edgeworth equilibria of an ω-proper production economy defined on a vector lattice
commodity space satisfying B1 and B2. For technical details not covered in this paper, we refer
to this chapter.
Since L+ is by A1 generating, we can choose u > 0 in L such that the order interval [−u, u]
contains
xi , ωi , y j for all i ∈ I and j ∈ J. Now consider the ordered vector subspace Lu =
S
λ>0 λ[−u, u] equipped with the order topology. This space is Archimedean (a property inherited
from L of which the positive cone is τ -closed) and has u as an order unit.6 Its order topology (i.e.,
the finest locally convex topology on Lu for which every order interval is bounded) is normable
(see Schaefer [36, Chapter V 6.2]). More precisely, the gauge k · ku of [−u, u], defined for each
z ∈ Lu by
©
ª
kzku = inf λ > 0 : − λu ≤ z ≤ λu ,
is a norm on Lu that generates the order topology whose closed unit ball is precisely the order
interval [−u, u] and u is an interior point of L+
u = Lu ∩ L+ . Moreover, it follows from A2 (namely
from the fact that [−u, u] is τ -bounded) that on Lu , the order topology is finer than the topology
7
induced by τ . In addition, it is not difficult to see that the cone L+
u is k · ku -closed in Lu .
Let Eu be the economy E restricted to Lu in an obvious way. For each consumer i, Xiu = Xi ∩Lu ,
and for each xi ∈ Xiu , Piu (xi ) = Pi (xi ) ∩ Lu ; for each producer j, Yju = Yj ∩ Lu . It is easily seen
that the consumption allocation (xi )i∈I is an Edgeworth equilibrium and belongs to the fuzzy core
of Eu . It follows from the ω-properness of E that for each i and any 0 < α ≤ 1, the vector xi + αω
belongs to the norm-interior of Piu (xi ). It thus follows from the standard decentralization result in
presence of an interiority property that there exists a nonzero pu ∈ (Lu , k · ku )0 such that (x, y, pu )
is a quasi-equilibrium of the restricted economy Eu and pu (ω) > 0.
Before going further, let us recall an extension lemma due to Podczeck [33] whose a proof can
be found in [22, Lemma 5.3.1, p. 134]. In [33], this lemma is used for deducing the existence of
equilibria for a proper exchange economy defined on a vector lattice commodity space satisfying
B1 and B2 from the equilibrium existence in the economy restricted to the order ideal generated
by the total endowment.
Lemma 4.1 (Podczeck). Let (L, τ ) be an ordered topological vector space, let K be a vector subspace of L (endowed with the induced order), let A be a convex subset of K such that A + K+ ⊆ A,
let V be a convex τ -open subset of L such that V ∩ A 6= 6°, and let a ∈ A ∩ cl V . If p is a linear
functional on K satisfying
p · a ≤ p · x, for all x ∈ V ∩ A,
then there exists some π ∈ L0 such that π|K ≤ p, and
p · (a − x) = π · (a − x) for each x ∈ A with x ≤ a .
The following obvious consequence of the preceding lemma was stated in [23].
6 That
is, the order interval [−u, u] is radial at the origin.
1
1
1
Indeed, if {xn } ⊆ L+ satisfies − n
u ≤ x − xn ≤ n
u for all n, then from x = (x − xn ) + xn ≥ x − xn ≥ − n
u
and the Archimedean property it follows that x ≥ 0.
7
9
Corollary 4.2. Let (L, τ ) be an ordered topological vector space , let K be a vector subspace of L
(endowed with the induced order), let A be a convex subset of K such that A − K+ ⊆ A, let V be
a convex τ -open subset of L such that V ∩ A 6= 6°, and let a ∈ A ∩ cl V . If p is a linear functional
on K satisfying
p · a ≥ p · x, for all y ∈ V ∩ A ,
then there exists some π ∈ L0 such that π|K ≤ p, and
p · (a − y) = π · (a − y) for each y ≥ a, y ∈ A .
Our next step associates
with the¢ quasi-equilibrium price pu a finite list of τ -continuous linear
¡
functionals π u = (πiu )i∈I , (πju )j∈J defined on the whole commodity space. These continuous
linear functionals can be thought of as personalized supporting prices for each consumer and each
producer at the corresponding component of the allocation (x, y).
Proposition 4.1. In our economy, there exist (πiu )i∈I , (πju )j∈J in L0 such that:
1. For
(a)
(b)
(c)
each consumer i we have
πiu ≤ pu on Lu ,
πiu · Pbi (xi ) ≥ πiu · xi , and
if zi ∈ Zi (xi ) ∩ Lu satisfies zi ≤ xi , then
pu · (xi − zi ) = πiu · (xi − zi ) = Rπu (xi − zi ) .
2. For
(a)
(b)
(c)
each producer j we have
πju ≤ pu on Lu ,
πju · Ybj (y j ) ≤ πju · y j , and
if zj ∈ Zj (y j ) ∩ Lu satisfies zj ≥ y j , then
pu · (zj − y j ) = πju · (zj − y j ) = Rπu (zj − y j ) .
3. Moreover,
(a) Rπu ≤ pu on Lu , and
hP ¡
¢ P ¡
¢i
P
P
(b) if v ≤ ω satisfies v = i∈I vi − j∈J wj ∈
j∈J Zj (y j ) ∩ Lu ,
i∈I Zi (xi ) ∩ Lu −
then
pu (ω − v) = Rπu (ω − v) .
Proof. Using Lemma 4.1 and its corollary with K = Lu and for each i, a = xi , A = Zi (xi ) ∩ Lu ,
V = int Pbi (xi ), for each j, a = y j , A = Zj (y j )∩Lu , V = int Ybj (y j ), the existence of (πiu )i∈I , (πju )j∈J
in L0 satisfying (1), (2) and (3)(a) is a straightforward consequence of the fact that (x, y, pu ) is
a quasi-equilibrium of the restricted economy Eu and of the ω-properness assumptions at each
component of the allocation (x, y).
P
P
To prove (3)(b), assume that v ≤ ω, v = i∈I vi − j∈J wj with each vi ∈ Zi (xi ) ∩ Lu and
each wj ∈ Zj (y j ) ∩ Lu . Let zi ∈ Zi (xi ) ∩ Lu be such that zi ≤ {vi , xi } and zj ∈ Zj (yj ) ∩ Lu be
such that zj ≥ {wj , y j }. Such zi and zj exist in view of the definition of Lu and our assumptions
on sets Zi (xi ) and Zj (y j ). On one hand, using (1)(c), (2)(c) and the superadditivity of Rπu , it
10
follows from (1)(a) and (2)(a) that
(pu − Rπu )(ω − v)
(pu − Rπu )
=
(pu − Rπu )
≤
³X
xi −
i∈I
³X
i∈I
X
j∈J
yj −
(xi − zi ) +
X
vi +
i∈I
X
j∈J
X
wj
j∈J
´
´
(zj − y j ) = 0 .
On the other hand, it follows also from (1)(a) and (2)(a) that (pu − Rπu )(ω − v) ≥ 0, which implies
(pu − Rπu )(ω − v) = 0
The following corollary can be seen as an analogue for the production economy Eu of Proposition
5.1 in [9] which extends at several instances statement (1) of Theorem 7.5 in [14] .
Corollary 4.3. We have in addition:
1. xi ∈ Piu (xi ) =⇒ Rπu (xi − zi ) ≥ Rπu (xi − zi ) for every zi ∈ Zi (xi ) ∩ Lu , zi ≤ {xi , xi }.
2. yj ∈ Yju =⇒ Rπu (zj − yj ) ≥ Rπu (zj − y j ) for every zj ∈ Zj (y j ) ∩ Lu , zj ≥ {yj , y j }.
3. If for each i, zi0 ∈ Zi (xi ) ∩ Lu , zi0 ≤ {ωi , xi }, if for each j, zj0 ∈ Zj (y j ) ∩ Lu , zj0 ≥ {0, y j },
then
P
P
P
P
(a) Rπu (ω − i∈I zi0 + j∈J zj0 ) = i∈I πiu (xi − zi0 ) + j∈J πju (zj0 − y j ), and
P
P
(b) for each i, πiu · (xi − zi0 ) + j∈J θij πju · (zj0 − y j ) ≥ Rπu (ωi − zi0 + j∈J θij zj0 ).
Proof. To prove (1), let us assume xi ∈ Piu (xi ) and zi ∈ Zi (xi ) ∩ Lu ,zi ≤ {xi , xi }. Recall that
Pi (xi ) = Pbi (xi ) ∩ Zi (xi ), thus that Piu (xi ) = Pbi (xi ) ∩ Zi (xi ) ∩ Lu . We then easily deduce from
Proposition 4.1 that:
Rπu (xi − zi ) ≥ πiu · (xi − zi ) ≥ πiu · (xi − zi ) = Rπu (xi − zi ).
The proof of (2) is done symmetrically. The proof of (3) goes as follows.
Using the superadditivity of Rπu , the last assertion of Proposition 4.1, and assuming that for
each i, zi0 ∈ Zi (xi ) ∩ Lu , zi0 ≤ {ωi , xi }, and for each j, zj0 ∈ Zj (y j ) ∩ Lu , zj0 ≥ {0, y j },
we get:
X
X
X
X
X
X
pu (ω −
zi0 +
Rπu (zj0 − y j )
zj0 ) = Rπu (ω −
zi0 +
zj0 ) ≥
Rπu (xi − zi0 ) +
i∈I
j∈J
=
X
i∈I
i∈I
πiu (xi − zi0 ) +
X
j∈J
j∈J
j∈J
i∈I
πju (zj0 − y j ) = pu (ω −
X
zi0 +
i∈I
X
zj0 ),
j∈J
which proves the first assertion of (3).
Finally, recall that (x, y, pu ) is a quasi-equilibrium of Eu . We thus have for every i,
X
(4.1)
pu · x i = p u · ω i +
θij pu · y j .
j∈J
From (4.1), using (1)(c) and (2)(c) of Proposition 4.1, we deduce easily from that for each i,
X
X
θij Rπu (zj0 − y j ) = pu · (xi − zi0 ) +
θij pu · (zj0 − y j )
Rπu (xi − zi0 ) +
j∈J
= pu · (ωi − zi0 ) +
j∈J
X
j∈J
X
¡
¢
θij pu · zj0 ≥ Rπu ωi − zi0 +
θij zj0 ,
j∈J
11
which completes the proof.
Let us now recall some notions of properness for functions on L+ .
Definition 4.4. Let v ∈ L+ be such that v > 0. We say that a function f : L+ → R is:
1. v-proper at some x ∈ L+ , if there exists a convex set F such that:
(a) x + v is an interior point of F , and
(b) F ∩ L+ = {y ∈ L+ : f (y) > f (x)}.
2. v-pointwise proper at x ∈ L+ , if there exists an open pointed convex cone Γx such that:
(a) −v ∈ Γx , and
©
ª
6 .
(b) (x − Γx ) ∩ y ∈ L+ : f (y) > f (x) = °
Considering the set {y ∈ L+ : f (y) > f (x)} as a preferred set for a preference correspondence
defined on L+ by the utility function f , the reader recognizes in these definitions the usual notions
of v-pointwise properness as defined by Mas-Colell [27] and of v- properness as defined by Tourky
[37]. Mas-Colell [27] defines uniform properness on a subset X of L+ as properness at every x ∈ X
with a properness vector and a properness cone independent of x.8 The definition of uniform
v-properness given here is quite similar.
Definition 4.5. The function f : L+ → R is uniformly v-proper on X ⊂ L+ if for every x ∈ X
there exist a convex set Fx and a τ -neighborhood V of 0 (independent of x) such that
a. x + v + V ⊆ Fx , and
b. Fx ∩ L+ = {y ∈ L+ : f (y) > f (x)}.
We now introduce the following additional assumption of compatibility between the order structure and the topology τ of the commodity space of our economy.
A3: For any finite list f of continuous linear functionals (fk )K
k=1 such that fk (ω) > 0 for each
k, the Riesz–Kantorovich functional Rf is uniformly ω-proper at any point ω 0 ≥ ω, provided
that it is exact at this point.
Remark 4.6. This hypothesis is automatically satisfied under the assumptions B1 and B2, and
a fortiori if the commodity space is a locally solid Riesz space. Indeed, it follows from Theorem
2.1 and Assumptions B1, B2 that the Riesz–Kantorovich functionals are linear and τ -continuous.
Assume now that f is a finite list of continuous linear functionals (fk )K
k=1 such that fk (ω) > 0 for
each k (which implies Rf (ω) > 0). For each ω 0 ∈ L+ , define Fω0 = {y ∈ L : g(y) > g(ω 0 )}, where
g is the continuous linear functional which coincides with Rf on L+ . If V is a τ -neighborhood of
0 such that g(ω + V ) > 0, then g(ω 0 + ω + V ) > g(ω 0 ), thus ω 0 + ω + V ⊆ Fω0 , while Fω0 ∩ L+ =
{y ∈ L : Rf (y) > Rf (ω 0 )}.
Proposition 4.2. Under A3, there exists a price system π u ∈ L0 such that π u ·ω > 0 and (x, y, π u )
is a quasi-equilibrium of Eu .
P
P
Proof. Let us consider the set Ω of all ω 0 = ω− i∈I zi0 + j∈J zj0 , where for each i, zi0 ∈ Zi (xi )∩Lu ,
zi0 ≤ {0, ωi , xi }, and for each j, zj0 ∈ Zj (y j ) ∩ Lu , zj0 ≥ {0, y j }. This set is nonempty in view of the
definition of Lu and our assumptions on Zi (xi ) and Zj (y j ). It is directed by the relation
ω 00 ≥ ω 0 if and only if for each i and j, zi00 ≤ zi0 and zj00 ≥ zj0 .
8 If v is the properness vector, recall that Mas-Colell’s uniform properness on L implies v-properness (see Tourky
+
[37]).
12
Moreover, if ω 0 ∈ Ω then zi0 ≤ 0 for each i and zj0 ≥ 0 for each j imply ω 0 ≥ ω > 0. From (3)
in Corollary 4.3, it follows that the Riesz–Kantorovich functional Rπu is exact at each ω 0 ∈ Ω.
From Proposition 4.1, it is easily deduced that for each i, either πiu = 0 or πiu · ω > 0, that for
each j, either πju = 0 or πju · ω > 0, that Rπu (ω) = pu (ω) > 0, thus that at least one of the πiu ,
πju is nonzero. Set P (ω 0 ) = {z ∈ L+ : Rπu (z) > Rπu (ω 0 )}. Applying Assumption A3, there exist
a 0-neighborhood V and for each ω 0 ∈ Ω, a convex set Pb(ω 0 ) such that ω 0 + ω + V ⊆ Pb(ω 0 ) and
P (ω 0 ) = L+ ∩ Pb(ω 0 ).
Moreover,
Rπu (ω 0 + αω) ≥ Rπu (ω 0 ) + αRπu (ω) > Rπu (ω 0 ) every 0 < α ≤ 1,
0
0
so that ω 0 belongs to the closure of Pb(ω 0 ). Now, if z ∈ Pb(ω 0 )∩L+
u , then Rπu (z) > Rπu (ω ) = pu (ω ).
0
p
·
z
≤
p
·
ω
)},
From Rπu ≤ pu on Lu , it follows that pu · z > pu · ω 0 . If we define X = {z ∈ L+
:
u
u
u
the last observation can be rephrased as X ∩ Pb(ω 0 ) = 6°, so that to each ω 0 ∈ Ω, we can associate
a nonzero π ω0 ∈ L0 which separates Pb(ω 0 ) and X, that is,
(4.2)
π ω0 · X ≤ π ω0 · ω 0 ≤ π ω0 · Pb(ω 0 ).
Since ω 0 + ω is an interior point of Pb(ω 0 ), we have π ω0 · ω > 0, and we can normalize prices letting
pu · ω = Rπu (ω) = π ω0 · ω = 1.
S
Let Lω0 be the ordered vector subspace Lω0 = λ>0 λ[−ω 0 , ω 0 ]. Clearly, Lω0 ⊂ Lu . We first claim
that π ω0 = pu on Lω0 . Indeed, for every z ∈ L+
u , we know that
(4.3)
pu · z ≤ pu · ω 0 =⇒ π ω0 · z ≤ π ω0 · ω 0 .
So, using the existence of Lagrange multipliers for a convex programming problem,9 there exist two
real numbers λ1 ≥ 0 and λ2 ≥ 0 not all equal to zero such that λ1 [π ω0 ·z −π ω0 ·ω 0 ] ≤ λ2 [pu ·z −pu ·ω 0 ]
for every z ∈ L+
u . From π ω 0 ·ω = pu ·ω > 0, we easily deduce that (λ1 , λ2 ) À 0. Letting successively
z = 0 and z = 2ω 0 , we also see that for some λ > 0, πω 0 · ω 0 = λpu · ω 0 and π ω0 · z ≤ pu · z for every
+
0
z ∈ L+
u . The previous inequality holds in particular for every z ∈ Lω 0 . Since ω is an interior point
+
(for the order topology) of Lω0 , it follows that π ω0 = λpu on Lω0 . Recalling that π ω0 · ω = pu · ω,
it follows that λ = 1.
P
Recalling that for each i, we have pu · xi = pu · ωi + j∈J θij pu · y j , that is,
X
X
pu · (xi − zi0 ) +
θij pu · (zj0 − y j ) = pu · (ωi − zi0 ) +
θij pu · zj0 ,
j∈J
j∈J
we deduce from our first claim:
X
X
π ω0 · (xi − zi0 ) +
θij π ω0 · (zj0 − y j ) = π ω0 · (ωi − zi0 ) +
θij π ω0 · zj0 ,
j∈J
j∈J
0
and, thus, for each i and for every ω ∈ Ω,
(4.4)
π ω 0 · xi = π ω 0 · ωi +
X
j∈J
θij π ω0 · y j .
We next claim that for some µ > 0 (depending on ω 0 ), µπ ω0 is a supergradient of Rπu at
ω . Indeed, from Rπu (z) ≥ Rπu (ω 0 ) =⇒ π ω0 (z) ≥ π ω0 (ω 0 ) for every z ∈ L+ , one deduces, as
0
9 A simple proof of the existence of Lagrange multipliers for a convex programming problem can be found in
Barbu and Precupanu [18, Chapter 3, Theorem 1.1]. See also [20] and [1, Chapter 5, Theorem 5.77].
13
previously, the existence of two real numbers µ1 ≥ 0 and µ2 ≥ 0 not all equal to zero such that
µ1 [π ω0 ·z−π ω0 ·ω 0 ] ≥ µ2 [Rπu (z)−Rπu (ω 0 )] for every z ∈ L+. As previously, from Rπu (ω) = π ω0 ·ω >
0, one deduces (µ1 , µ2 ) À 0, so that for some µ > 0, µπ ω0 · ω 0 = Rπu (ω 0 ) and µπ ω0 · z ≥ Rπu (z)
for every z ∈ L+ , which proves the claim.
P
P
Q
Q
0
+ j∈J zj0 0 where for each i,
Fix now (x, y) ∈ i∈I Piu (xi ) × j∈J Yju . Let ω00 = ω − i∈I zi0
0
0
≤ {0, xi , ωi , xi }, and for each j, zj0 0 ∈ Zj (y j ) ∩ Lu , zj0 0 ≥ {yj , 0, y j }. Once
∈ Zi (xi ) ∩ Lu , zi0
zi0
again, this is possible in view of the definition of Lu and our assumptions on Zi (xi ) and Zj (y j ).
For every ω 0 ≥ ω 0 in Ω, applying Theorem 2.3 and using (1) and (2) of Proposition 4.1, we get:
(4.5)
xi ∈ Piu (xi ) =⇒ π ω0 · (xi − zi0 ) ≥ π ω0 · (xi − zi0 ) =⇒ π ω0 · xi ≥ π ω0 · xi
and for each i, πiu (xi − zi0 ) = π ω0 · (xi − zi0 ),
(4.6)
yj ∈ Yju =⇒ π ω0 · (zj0 − yj ) ≥ π ω0 · (zj0 − y j ) =⇒ π ω0 · yj ≤ π ω0 · y j
and for each j, πju (zj0 − y j ) = π ω0 · (zj0 − y j ).
Let V be the τ -neighborhood of 0 referred to in Assumption A3. We can assume that V is
convex and circled. From (4.2), we deduce that π ω0 · V ≤ π ω0 · ω = 1, thus that each π ω0 belongs
to V 0 , the polar set of V in L0 . Since L is locally convex, it follows from Alaoglu–Bourbaki’s
theorem that V 0 is τ -equicontinuous, thus σ(L0 , L)-compact. Passing to a subnet if necessary, we
σ(L0 ,L)
−−−→ π u ∈ L0 such that π u ·ω = 1. Passing to limit in the relations
can assume that π ω0 −
P (4.5), (4.6)
and (4.4), we get π u · xi ≥ π u xi , π u · yj ≤ π u y j , and for each i, π u · xi = π u · ωi + j∈J θij π u · y j ,
which completes the proof that (x, y, π u ) is a quasi-equilibrium of Eu .
To go further, we now consider the family U of all u > 0 in L such that the order interval [−u, u]
contains xi , ωi , y j for all i ∈ I and j ∈ J and notice that U is a directed set. We will apply the
previous result to the economies Eu defined as above and will pass to limit.
The next proposition is the main result of this paper.
Proposition 4.3. Under A3, there exists a price system π ∈ L0 such that π · ω > 0 and (x, y, π)
is a quasi-equilibrium
of E. This quasi-equilibrium is nontrivial if for some λ > 0, λω ∈ ω +
P
P
Y
−
X
.
j
i
j∈J
i∈I
Proof. For each u ∈ U , in view of the previous proposition and of its proof, let π u ∈ L0 such that
π u · ω 0 ≤ π u · Pb(ω 0 ) and (x, y, π u ) is a quasi-equilibrium of Eu . Let V be the convex and circled
τ -neighborhood of 0 such that ω 0 + ω + V ⊆ Pb(ω 0 ). As previously, π u · V ≤ π u · ω = 1, and passing
σ(L0 ,L)
−−−→ π ∈ L0 such that π · ω = 1. Q
to a subnet if necessary, we can assume that π u −
Q We now claim that π supports the allocation (x, y). To see that, fix now (x, y) ∈ i∈I Pi (xi ) ×
j∈J Yj . By construction of U , all xi , yj belong to some Lu0 for u0 ∈ U and consequently to any
Lu ⊃ Lu0 . Passing to limit in the relations π u · xi ≥ π u xi and π u · yj ≤ π u y j , we get π · xi ≥ πxi
and π · yj ≤ πy j .
P
Passing to limit in the relations π u · xi = π u · ωi + j∈J θij π u · y j , we get that for each i,
P
π · xi = π · ωi + j∈J θij π · y j , proving that (x, y, π) is a quasi-equilibrium of E.
P
P
Finally, assume that for some λ > 0, λω ∈ ω + j∈J Yj − i∈I Xi . From π · ω = 1, we deduce
P
Q
P
Q
that there exist x ∈ i∈I Xi , y ∈ j∈J Yj such that π · (ω + j∈J yj − i∈I xi ) > 0. We then
14
have for some i0 ,
π · x i0 < π · ω i0 +
X
j∈J
θ i0 j π · y j ≤ π · ω i0 +
X
j∈J
θ i 0 j π · y j = π · x iO ,
which proves that the quasi-equilibrium is nontrivial.
Remark 4.7. The condition for nontriviality is in particular satisfied if inaction is possible as well
for consumers (0 ∈ Xi ) as for producers (0 ∈ Yj ).
Recalling that preferences are non-ordered and that we do not assume any monotonicity property
of preferences or free-disposal in production, a consequence of the last proposition is the following
theorem which extends any previous equilibrium existence result obtained in the Mas-Colell or
Mas-Colell–Richard settings:
Theorem 4.8. Assume A1, A2, A3 on the commodity space. Then an ω-proper and compact
production economy in which inaction is possible as well for consumers as for producers has a
nontrivial quasi-equilibrium.
Remark 4.9. As usual, an irreducibility condition on the economy guarantees that the nontrivial
equilibrium is actually an equilibrium. A very simple condition, inspired by Arrow–Hahn [16], is
the following:
IR: For any non-trivial partition {I1 , I2 }Q
of the set I of consumers and for any feasible consumption allocation x, there exist x0 ∈ i∈I Xi such that
0
• x
∈P
I1 ;
P
Pi ∈ Pi0(xi ) ∀i
0
0
x
∈
ω
+
•
i∈I2 ωi .
j∈J Yj with, for some λ > 0, (ω − ω) ≤ λ
i∈I i
In the next section, we show with examples that this paper is more than a unified proof for old
results and allows to obtain the existence of equilibrium in settings of economic interest which are
covered by no previous equilibrium existence result.
5. Examples
There is a class of examples of commodity spaces which satisfy Assumptions A1 and A2 and
are not vector lattices. We concentrate here on a few examples of commodity spaces that satisfy
also Assumption A3.
Example 5.1. Consider the vector space
ª
©
L = f ∈ C[0, 2] : f (1) = 21 [f (0) + f (2)] .
Clearly, L is a closed vector subspace of the Banach lattice C[0, 2], where C[0, 2] is equipped with
the sup norm kf k∞ = supx∈[0,2] |f (x)|. The ordered vector space has the following properties.
1.
2.
3.
4.
5.
L has order units; for instance constant function 1 ∈ L is an order unit.
The positive cone L+ is closed, generating, with a non-empty interior.
L is not a vector lattice.
L satisfies the Riesz Decomposition Property.10
The order intervals are norm bounded.
10 This
was shown by I. Namioka [31, p. 45]; see also [32, Example 1.7, p. 14].
15
In particular, it should be clear from the above properties that assumptions A1 and A2 are
satisfied.
Moreover, in this case the Decomposition Property guarantees that the Riesz–Kantorovich functionals are additive and their linear extensions are continuous (since the closed unit ball of L is
the order interval [−1, 1]). As in Remark 4.6, it is easily deduced that for any ω > 0 and every
m-tuple (f1 , . . . , fm ) of continuous linear functionals satisfying fi (ω) > 0 its Riesz–Kantorovich
functional is uniformly ω-proper on L+ .
Example 5.2. Let L = C k [0, 1], the vector space of all real-valued functions on [0, 1] which are k
times continuously differentiable. With the pointwise ordering and the sup norm, L is an ordered
topological vector space such that:
1. L is not a vector lattice.
2. L has order units; for instance 1 ∈ L (and hence the positive cone L+ is generating).
3. The positive cone is norm closed.
4. The order intervals are norm bounded.
5. L satisfies the Riesz Decomposition Property. 11
6. L0 is a vector lattice and its lattice operations are given by the Riesz–Kantorovich formulas.
We indicate here how to prove (6) without using (5). One thus has to prove separately the two
claims contained in Assertion (6). Actually, both can be deduced from the fact that every positive
linear functional on L has a unique extension to a positive linear functional on all of C[0, 1]. It
follows from this that the norm dual of L is order isomorphic to the norm dual of C[0, 1] and thus,
since L∼ = ca[0, 1] is a Riesz space, L0 is likewise a Riesz space. Using the same extension property
and the norm density of L in C[0, 1], it is easily proved that the lattice operations of L0 satisfy the
Riesz–Kantorovich formula.
Clearly, Assumptions A1 and A2 are satisfied. Once more, since the Riesz–Kantorovich functionals are additive, A3 follows from similar arguments to the ones used in Remark 4.6.
Remark 5.3. In the two examples above, the ω-uniform properness of the Riesz Kantorovich
functional is implied by the facts that L0 is a vector lattice and its lattice operations are given by
the Riesz–Kantorovich formulas. One could wonder whether these conditions are equivalent to the
ω-uniform properness condition. The following example shows that the answer is not.
Example 5.4. Let L = R` be ordered by a closed convex pointed and generating cone K. It
follows that K has a non-empty interior and that the order intervals of L are compact. In this
case, it is well-known that K has the decomposition property if and only if K has exactly ` extremal
rays, i.e., if and only if K is a lattice cone.
Now fix ω > 0 (i.e., ω 6= 0 and ω ∈ K) and pick m linear functionals (f1 , . . . , fm ) such that
fi (ω) > 0 for each i. If ω is in the interior of L+ = K, then it is easily seen that Rf is ω-uniformly
proper on L+ .12 However, generically on the finite list of linear functionals, their Riesz–Kantorovich
is not additive.
This case is not of real interest for equilibrium existence since when ω is in the interior of the
positive cone then the equilibrium of an ω-proper economy can be proved using classical theorems.
11 This is asserted by several authors (see for instance [24, p. 9] and [25, pp. 18–20]). For the case k = 1, one
can find a proof in unpublished lecture notes of C.D. Aliprantis.
12 Indeed, Let V be a τ -neighborhood of 0 such that ω+V ⊆ K and f (ω+V ) > 0 for each i. Then R (ω+V ) > 0,
i
f
thus Rf (ω 0 + ω + V ) ≥ Rf (ω 0 ) + Rf (ω + V ) > Rf (ω 0 ).
16
When ω is on the boundary of the positive cone, equilibrium may not exist (see an example
in [11]) and the Riesz–Kantorovich functionals may not be ω-proper. An interesting case is the
case where L+ = K is a polyhedral convex cone (with more than ` extremal rays) and ω does
not belong to the interior of K. It should be noted that for any finite list of continuous linear
functionals (fk )K
k=1 such that fk (ω) > 0 for each k, Rf is ω-proper at ω. However, we do not know
at this time if these functionals are uniformly proper at any point ω 0 ≥ ω.
References
[1] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, 2nd Edition, Springer–
Verlag, Heidelberg and New York, 1999.
[2] C. D. Aliprantis, D. J. Brown, and Burkinshaw, Edgeworth equilibria, Econometrica 55 (1987), 1109–1137.
[3] C. D. Aliprantis, D. J. Brown, and Burkinshaw, Edgeworth equilibria in production economies J. Econom.
Theory 43 (1987), 252–291.
[4] C. D. Aliprantis, D. J. Brown, I. A. Polyrakis, and J. Werner, Portfolio dominance and optimality in infinite
security markets, J. Math. Econom. 30 (1998), 347–366.
[5] C. D. Aliprantis, D. J. Brown, and J. Werner, Minimum-cost portfolio insurance, J. Econom. Dynam. Control
24 (2000), 1703–1719.
[6] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York and London, 1985.
[7] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, American
Mathematical Society, Mathematical Surveys and Monographs, Volume 105, 2003.
[8] C. D. Aliprantis, M. Florenzano, F. Martins Da Rocha and R. Tourky, Equilibrium analysis in financial markets
with countably many securities, J. Math. Econom. 40 (2004), 683–699.
[9] C. D. Aliprantis, M. Florenzano, and R. Tourky, Linear and non-linear price decentralization, J. Econom.
Theory, forthcoming.
[10] C. D. Aliprantis, M. Florenzano, and R. Tourky, General equilibrium analysis in ordered topological vector
spaces, J. Math. Econom. 40 (2004), 247–269.
[11] C. D. Aliprantis, P. K. Monteiro, and R. Tourky, Non-marketed options, non-existence of equilibria, and nonlinear prices, J. Econom. Theory, 114 (2004), 345–357.
[12] C. D. Aliprantis and R. Tourky, The super order dual of an ordered vector space and the Riesz–Kantorovich
formula, Trans.. Amer. Math. Soc. 354 (2002), 2055–2077.
[13] C. D. Aliprantis, R. Tourky, and N. C. Yannelis, The Riesz–Kantorovich formula and general equilibrium
theory, J. Math. Econom. 34 (2000), 55–76.
[14] C. D. Aliprantis, R. Tourky, and N. C. Yannelis, A theory of value: equilibrium analysis beyond vector lattices,
J. Econom. Theory 100 (2001), 22–72.
[15] N. Allouch and M. Florenzano, Edgeworth and Walras equilibria of an arbitrage-free exchange economy,
Econom. Theory, 23 (2004), 353–370.
[16] K. J. Arrow and F. H. Hahn, General Competitive Analysis, Holden-Day, San Francisco, 1971.
[17] J. P. Aubin, Mathematical Methods of Game and Economic Theory, North–Holland, Amsterdam and New
York, 1979.
[18] V. Barbu and T. Precupanu, Convexity and Optimization in Banach spaces, D. Reidel Publishing Company,
Boston and Lancaster, 1986.
[19] G. Debreu and H. Scarf, A limit theorem on the core of an economy, Internat. Econom. Rev. 4 (1963), 235–246.
[20] K. Fan, I. L. Glicksberg, and A. J. Hoffman, Systems of inequalities involving convex functions, Proc. Amer.
Math. Soc. 13 (1957), 617–622.
[21] M. Florenzano, Edgeworth equilibria, fuzzy core and equilibria of a production economy without ordered
preferences, J. Math. Anal. Appl. 153 (1990), 18–36.
[22] M. Florenzano, General Equilibrium Analysis: Existence and Optimality Properties of Equilibria, Kluwer
Academic Publishers, Boston and London, 2003.
[23] M. Florenzano and V. Marakulin, Production equilibria in vector lattices, Econom. Theory 17 (2001), 577–598.
[24] L. Fuchs, Riesz groups, Ann. Scuola Norm. Sup. Pisa 19 (1965), 1–34.
17
[25] L. Fuchs Riesz Vector Spaces and Riesz Algebras, Queen’s papers in pure and applied mathematics, No 1,
Queen’s University, Kingston, Ontario, Canada, 1966.
[26] P. J. Laurent, Approximation et Optimisation, Herman, Paris, 1972.
[27] A. Mas-Colell, The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986),
1039–1055.
[28] A. Mas-Colell, Valuation equilibria and Pareto optimum revisited, in W. Hildenbrand and A. Mas-Colell (eds.),
Contributions to Mathematical Economics. In honor of Gérard Debreu, North-Holland, Amsterdam, 317–331,
1986.
[29] A. Mas-Colell and S. F. Richard, A new approach to the existence of equilibria in vector lattices, J. Econom.
Theory 53 (1991), 1–11.
[30] J. J. Moreau, Fonctionnelles convexes, Séminaire sur les équations aux dérivées partielles, Collège de France,
1966–1967.
[31] I. Namioka, Partially Ordered Linear Topological Spaces, Mem. Amer. Math. Soc., 24, Providence, RI, 1957.
[32] A. L. Peressini, Ordered Topological Vector Spaces, Harper & Row, New York and London, 1967.
[33] K. Podczeck, Equilibria in vector lattices without ordered preferences or uniform properness, J. Math. Econom.
25 (1996), 465–485.
[34] S. F. Richard, A new approach to production equilibria in vector lattices, J. Math. Econom. 18 (1989), 41–56.
[35] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer–Verlag, Heildelberg and New York, 1998.
[36] H. H. Schaefer, Topological Vector Spaces, Springer Verlag, New York and Heildelberg, 1971.
[37] R. Tourky, A new approach to the limit theorem on the core of an economy in vector lattices, J. Econom.
Theory 78 (1998), 321–328.
[38] R. Tourky, The limit theorem on the core of a production economy in vector lattices with unordered preferences,
Econom. Theory 14 (1999), 219–226.
[39] W. R. Zame, Competitive equilibria in production economies with an infinite dimensional space, Econometrica,
55 (1987), 1075–1108.
