GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
C. D. ALIPRANTIS1 , M. FLORENZANO2 , V. F. MARTINS DA ROCHA3 AND R. TOURKY4
1
2
3
4
Department of Economics, Krannert School of Management, Purdue University, W. Lafayette IN 47907-1310,
USA; [email protected]
CNRS–CERMSEM, Université Paris 1, 106-112 boulevard de l’Hôpital, 75647 Paris Cedex 13, FRANCE;
[email protected]
CERMSEM, Université Paris 1, 106-112 boulevard de l’Hôpital, 75647 Paris Cedex 13, FRANCE;
[email protected]
Department of Economics and Commerce, University of Melbourne, Parkville 3052, Melbourne, AUSTRALIA;
[email protected]
Abstract. An F -cone of a real vector space is a pointed and generating convex cone, the union
of a countable family of finite dimensional polyhedral convex cones such that each one is an
extremal subset of the following one. In this paper, we study security markets with countably
many securities and arbitrary finite portfolio holdings. If, under the portfolio dominance order,
the positive cone of the portfolio space is an F -cone, then Edgeworth allocations and nontrivial
quasi-equilibria do exist. This result extends the case where, as in Aliprantis–Brown–Polyrakis–
Werner (1998), the positive cone is a Yudin cone.
Keywords: Security markets; Edgeworth equilibrium; nontrivial quasi-equilibrium; inductive
limit topology; F -cone; Riesz–Kantorovich functional.
1. Introduction
Let us consider a finance model with m investors trading securities and having identical expectations on the security payoffs. Let E be a portfolio (vector) space. Given an ordered vector payoff
space X, typically some Lp (Ω, Σ, P ) for 1 ≤ p ≤ ∞ and an underlying probability space (Ω, Σ, P ),
let a linear operator R : E → X define for each portfolio x the random payoff R(x) an investor
expects to receive when holding the portfolio. If we suppose that R is one-to-one, it is natural to
order the portfolio space E by the portfolio dominance ordering:
z ≥R z 0 whenever R(z) ≥ R(z 0 ).
Let us now assume that investors are constrained to have non-negative end-of-period wealth or,
equivalently, that their portfolio set is equal to the cone K = {x ∈ E : x ≥R 0} of positive
payoff portfolios. If each investor has a strict preference over K described by the correspondence
Pi : K → K and an initial endowment of securities ωi ∈ K, then a financial economy is given as:
E = (E, K), (K, Pi , ωi )m
i=1 .
An equilibrium definition for this economy requires the definition of a security price space E 0 , in
duality with E. Relative to this duality, an equilibrium is classically a pair (p, x) of a nonzero price
m
p ∈ E 0 and a market clearing allocation x = (xi )m
of investors’optimal portfolios given
i=1 ∈ K
Date: September 2, 2002.
1
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
2
prices of the securities. Our paper is a contribution to the equilibrium existence problem in this
economy.
The existence of equilibrium in such a model was first studied by Brown–Werner [6], then by
Aliprantis–Brown–Polyrakis–Werner [2], who introduced in the same paper the notion of portfolio
dominance ordering. As in Brown–Werner [6], Aliprantis–Brown–Polyrakis–Werner assume that
there are countably many securities, but in [2], each investor is restricted to portfolios with nonzero holdings of only finitely many securities. The portfolio space of the model is thus the vector
space Φ of all eventually zero sequences. Assuming that the cone K of positive payoff portfolios
is a lattice cone, Aliprantis–Brown–Polyrakis–Werner prove that the portfolio space Φ, equipped
with the inductive limit topology1 relative to the family of all finite dimensional subspaces of Φ, is
a topological vector lattice. As the portfolio sets of investors coincide with the positive cone K of
the portfolio space, in the line of Mas-Colell [10], the existence of a quasiequilibrium with a price
in RN , the topological dual of Φ, is mainly based on assumptions of uniform properness on K of
the preferences of the investors. Our purpose in this paper is to show that, in the same setting
and under comparable properness assumptions, the key assumption that the cone K of positive
payoffs portfolios is a lattice cone can be seriously weakened.
As in [2], we assume that the portfolio space is Φ, ordered by the portfolio dominance ordering
and equipped with the inductive limit topology. As in [2], we assume that the agents have the
positive cone as portfolio sets and make properness assumptions on their preferences, but we replace
the lattice ordering hypothesis by the strictly weaker assumption that the positive cone is an F cone. By this assumption, on which we will comment later, we mean that, under the portfolio
dominance ordering,
the positive cone of the portfolio space is a convex, pointed and generating
S∞
cone K = n=1 K n , the union of a countable family of finite dimensional polyhedral convex
cones K n such that each K n is an extremal subset (or a face) of K n+1 . Under this assumption,
equilibrium exists. Moreover, the countably many extremal vectors of the positive cone have the
same economic interpretation as the vectors of the Yudin basis assumed in [2]. The corresponding
portfolios can be thought of as mutual funds that investors trade at equilibrium.
The main tool of this extension is a result of Aliprantis–Florenzano–Tourky [3, Theorem 5.1]
giving conditions for decentralizing Edgeworth allocations of a proper exchange economy of which
the commodity space is not a vector lattice. The existence of Edgeworth allocations is obtained
under classical continuity assumptions on the preferences of the agents. Their decentralization
as nontrivial quasi-equilibria requires properness assumptions that we shall precise later. The
double assumption on the topology of the portfolio space and its order structure guarantees that
we can apply the result in [3], after verifying the sufficient condition for this decentralization as
stated in [3]. This condition, expressed in terms of properness of the Riesz–Kantorovich functional
associated to a list of continuous linear functionals on Φ, should be understood as a condition (here
verified) of compatibility between the topology and the order structure of the portfolio space.
The paper is organized as follows: In Section 2, we recall the definition and properties of
the inductive limit topology ξ on Φ generated by the family of all its finite dimensional vector
subspaces. We also define F -cones and establish some properties of ξ when Φ is ordered by an
F -cone. Edgeworth and nontrivial quasi-equilibria of our economy are studied in Section 3. The
properness of the Riesz–Kantorovich functional is proved in the last section. With the appendix,
this section is the only technical part of the paper.
1The definition and properties of this topology will be stated later in Section 2.
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
3
2. Inductive limit topology and F -cones
2.1. The inductive limit topology. Let E be a real vector space and let F be the family
(directed by inclusion) of all finite dimensional vector subspaces F of E. The inductive limit
topology ξ on E generated by F is the finest locally convex topology on E for which, for each
F ∈ F, the natural embedding iF : F ,→ E is τF -ξ continuous, where τF is the unique Hausdorff
linear topology on F .
The fact that, as a finite dimensional space, each F ∈ F admits a unique Hausdorff linear
topology τF has several remarkable consequences easy to derive:2
• If (Fγ )γ∈Γ is a family of finite dimensional vector subspaces such that each F ∈ F is
contained in some Fγ , then ξ is also the finest locally convex topology on E for which, for
each γ ∈ Γ, the natural embedding iFγ : Fγ ,→ E is τFγ -ξ continuous.
• The topology ξ is also the finest locally convex topology on E. Consequently, ξ is Hausdorff
and the topological and algebraic duals, E 0 and E ∗ coincide.
• If (ei )i∈I is a Hamel basis of E, E is the direct sum of the one dimensional vector spaces
Rei and ξ is the finest locally convex topology on E for which each natural embedding
Rei ,→ E is τRei -ξ continuous.
• For each ε = (εi )i∈I ∈]0, +∞[I , let
Vε := {x ∈ E : ∀i ∈ I
|xi | < εi }.
The family of all subsets Vε is a base of neighborhoods of 0 for the topology ξ.
The following result is also classical (see [5, Chap. II, § 4, exerc. 8]). We give its proof for the sake
of completeness of the paper.
Proposition 2.1. Assume that E is a real vector space with a countable Hamel basis. A subset A
of E is ξ-closed if and only if for each finite dimensional vector subspace F of E, the set A ∩ F is
closed for the unique Hausdorff linear topology on F .
In particular, if F is the family of all finite dimensional subspaces of E, then
ξ = {V ⊂ E : ∀F ∈ F
V ∩ F is open in F }.
Proof. Let A be a ξ-closed subset of E. For each finite dimensional subspace F of E, i−1
F (A) = A∩F
is closed in F .
Conversely, let A be a subset of E such that for each finite dimensional subspace F of E, A ∩ F
is closed. Let (en )n∈N be a countable basis of E and let x ∈ E \ A. We propose to prove that there
exists ε = (εi )i∈N ∈]0, +∞[N such that x + Vε ⊂ E \ A. For each n ∈ N, we note E n the vector
space spanned by the vectors {e1 , . . . , en }, and for each finite family (B1 , . . . , Bn ) of subsets of R,
we let
(
)
n
n
Y
X
n
Bi := x ∈ E : x =
xi ei with xi ∈ Bi ∀i = 1, . . . , n .
i=1
i=1
Without any loss of generality we can suppose that x ∈ E 1 , then for each n ≥ 1
x ∈ E n \ (A ∩ E n ).
As A ∩ E n is closed in E n , it follows that there exists (ε1 , . . . , εn ) ∈]0, +∞[n such that
n
Y
(2.1)
x+
] − εi , +εi [⊂ E n \ (A ∩ E n ).
i=1
2For details about inductive limit topologies, see the books [5], [9], [12].
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
4
Let α > 1.
Claim 2.1. There exists εn+1 > 0 such that
x+
n+1
Yi
i=1
−
εi h
εi
,+
⊂ E n+1 \ (A ∩ E n+1 ).
α
α
Proof. Let α > 1; suppose that Claim 2.1 is not true. Then we can construct a sequence (v k )k∈N
of E n+1 such that for each k ∈ N,
n i
Y
εi
εi h
1
1
k
v ∈
− ,+
× − ,+
and x + v k ∈ A ∩ E n+1 .
α
α
k
k
i=1
The sequence (v k )k∈N lies in a compact subset of E n+1 . Passing to a subsequence if necessary, we
can suppose that (v k )k∈N converges to some v ∈ E n+1 . Note that
n h
Y
εi
εi i
− ,+
v∈
× {0}.
α
α
i=1
It follows that
x+v ∈A∩
n h
Y
εi
εi i
− ,+
x+
α
α
i=1
!
⊂A∩
x+
n
Y
!
] − εi , +εi [ .
i=1
This contradicts (2.1).
In order to apply the previous claim, we consider a sequence (αn )n∈N ∈]1, +∞[N such that3
lim
N →∞
N
Y
αn =
n=1
∞
Y
αn < ∞.
n=1
Applying inductively Claim 2.1, we can construct a sequence (εn )n∈N ∈]0, +∞[N such that for each
n ∈ N,
n Y
εi
εi
(2.2)
x+
− Qn
, + Qn
⊂ E n \ (A ∩ E n ).
α
α
i
i
i=1
i=1
i=1
Q∞
Now consider β ∈ R such that β > n=1 αn and let V be the following ξ-neighborhood of 0,
(
)
X
εi
V := x ∈ E : x =
xi ei , |xi | <
∀i ∈ N .
β
i∈N
We assert that x + V ⊂ E \ A. Indeed, suppose that (x + V ) ∩ A 6= 6. Then there exists n ∈ N
such that (x + V ) ∩ A ∩ E n 6= 6. But
n n Y
Y
εi
εi
εi
εi
− Qn
, + Qn
.
(x + V ) ∩ E n ⊂ x +
− ,+
⊂x+
β
β
i=1 αi
i=1 αi
i=1
i=1
This contradicts (2.2).
As a consequence,
3For instance, let α = exp(1/(n2 )).
n
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
5
Corollary 2.2. Let E be a real vector space with a countable Hamel basis. A subset A of E is
ξ-closed if and only if there exists a family (Fγ )γ∈Γ of finite dimensional subspaces of E such that:
(1) each finite dimensional subspace F of E is contained in some Fγ ,
(2) for every γ ∈ Γ, A ∩ Fγ is closed in Fγ .
Proof. Assume first that A is ξ-closed. It follows from Proposition 2.1 that the family F of all
finite dimensional vector subspace of E satisfies the conditions 1 and 2.
Assume conversely that a family (Fγ )γ∈Γ satisfies the conditions 1 and 2 of the present corollary.
In order to apply Proposition 2.1, we propose to prove that for each finite dimensional vector
subspace F of E, A ∩ F is closed in F . Let F ∈ F; then there exists γ ∈ Γ such that F ⊂ Fγ .
Note that
A ∩ F = F ∩ (A ∩ Fγ ).
Since A ∩ Fγ is closed in Fγ , the subset A ∩ F is closed in F .
2.2. F -cones.
Definition 2.3. Let E be a real vector space. A (convex) cone K of E is called
S∞ an F -cone if K
is pointed (i.e. K ∩ (−K) = {0}), generating (i.e. K − K = E) and if K = n=1 K n is the union
of a countable family of finite dimensional polyhedral convex cones K n such that each K n is an
extremal subset4 of K n+1 .
Remark 2.4. Note that if K is an F -cone of E, then E has a countable Hamel basis. Moreover,
observe that an F -cone K is generated by a countable family of vectors which is the union of
the extremal directions of each K n . Countably generated Yudin cones5 are a particular case of
F -cones, but an F -cone is not necessarily a Yudin cone. For example, let E be a vector space with
a countable Hamel basis (en )n∈N . Let K be the cone generated by the family {e3 + e1 , e3 + e2 , e3 −
e1 , e3 − e2 } ∪ {en }n≥4 . Then K is not a Yudin cone, but it is an F-cone.
Proposition 2.2. Let E be a real vector space and let K be an F -cone of E. Then K is ξ-closed.
Proof. Since K is generating, for each finite dimensional subspace F of E, F ⊂ (K n −K n ) for some
0
n. Moreover, it is easy to see that each K n is an extremal subset of K n for every n0 > n. Now
we propose to prove that for each n ∈ N, K ∩ (K n − K n ) = K n . Indeed, let x ∈ K ∩ (K n − K n ),
0
then there exists y, z ∈ K n and n0 > n such that x ∈ K n and x = y − z. It follows that y = x + z.
0
But K n is an extremal subset of K n ; then x ∈ K n and K ∩ (K n − K n ) ⊂ K n . The converse is
obvious. Now recalling that a finite dimensional polyhedral convex cone is closed, Corollary 2.2
ends the proof.
We now assume that E is ordered by the F -cone K.
Proposition 2.3. The order intervals of E are ξ-compact.
4A convex subset F of a convex set C is an extremal subset (or a face) of C if for each line segment [y, z] of C
satisfying ]y, z[∩F 6= 6 then y and z belong to F . If K is a convex cone, the convex subset F is an extremal subset
of K if and only if x = y + z, x ∈ F and y, z ∈ K imply that y, z ∈ F . It follows that F is itself a convex cone.
5A convex cone C generated by a family (e )
iP
i∈I of elements of a vector space is called a Yudin cone if each x of
C has a unique representation of the form x = i∈I xi ei , where xi ≥ 0 and xi = 0 for all but finitely many i ∈ I.
C is countably generated if I is countable.
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
6
Proof. Let a ∈ K, a 6= 0. We only have to prove that the order interval [0, a] = K ∩ ({a} − K) is
ξ-compact. Let n be such that a ∈ K n . For every x ∈ [0, a], let n0 be such that x and (a − x) lie in
0
K n . If n0 ≤ n then x ∈ K n . If n0 > n, from a = x + (a − x) and since K is an F -cone, we deduce
that x ∈ K n . Thus [0, a] ⊂ K n , hence is included in the finite dimensional space (K n − K n )
(ordered by the polyhedral convex cone K ∩ (K n − K n ) = K n ). It follows that [0, a] is a compact
subset of K ∩ (K n − K n ) = K n and, in view of the definition of ξ, a ξ-compact subset of E. Remark 2.5. Let E be a vector space with a countable Hamel basis (en )n∈N . Let K be the cone
generated by the family {e1 −en , en }n≥2 . Since the order interval [0, e1 ] contains the family {en }n≥1 ,
it is not ξ-compact. In view of Proposition 2.3, the cone K is not an F -cone.
3. Equilibrium in the security market
Let us now come back to the financial economy
E = (Φ, ξ, K), (K, Pi , ωi )m
i=1
of which the portfolio space is Φ, equipped with the inductive limit topology ξ, while the positive
cone K of the portfolio dominance ordering is an F -cone. A Hamel basis of Φ consists of the
vectors (0, . . . , 0, 1, 0, . . .) corresponding to each one of the countably many securities defining the
model. The algebraic and topological dual of Φ is RN of which each element can be thought of as
a list of prices for each securitiy.
We first observe that it follows from Propositions 2.2 and 2.3 that (Φ, ξ, K) is an ordered linear
vector space equipped with a Hausdorff locally convex topology satisfying the properties:
A1: The positive cone K is generating and ξ-closed; and
A2: The order intervals of Φ are ξ-bounded
required for an economic model in Aliprantis–Florenzano–Tourky [3].
Pm
Let ω =
i=1 ωi be the total initial endowment of securities,
Pm i.e. the market portfolio. A
m
portfolio allocation is any m-tuple x = (xi )m
∈
K
such
that
i=1
i=1 xi = ω.
Definition 3.1. A pair (x, p) consisting of an allocation x and a non-zero linear functional p is
said to be
(1) a quasi-equilibrium if for every i, p(xi ) = p(ωi ), and yi ∈ Pi (xi ) implies p(yi ) ≥ p(xi ),
(2) an equilibrium, if it is a quasi-equilibrium and if yi ∈ Pi (xi ) actually implies p(yi ) >
p(xi ).
Definition 3.2. A quasi-equilibrium (x, p) is said to be nontrivial if for some i we have
inf{p(zi ) : zi ∈ K} < p(ωi ).
In this paper, we will be interested only in nontrivial quasi-equilibria.6 We now introduce some
special properties of feasible allocations.
Definition 3.3. A portfolio allocation x is said to be:
(1) individually rational if for each i, ωi ∈
/ Pi (xi ),
6If (x, p) is some trivial quasi-equilibrium, then for every allocation y, the pair (y, p) is also a quasi-equilibrium.
If the quasi-equilibrium (x, p) is nontrivial, then it is well-known that (under some additional continuity condition on
preferences or concavity for utility functions, and some irreducibility assumption on the economy) (x, p) is actually
an equilibrium.
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
7
(2) a weakly Pareto optimal allocation if there is no portfolio allocation y satisfying for
each i, yi ∈ P (xi ),
(3) a core allocation, if it cannot be blocked by any coalition in the sense that there is no
S
coalition
P S ⊂ {1,P2, . . . , m} and some y ∈ K such that:
(a)
i∈S yi =
i∈S ωi and
(b) yi ∈ Pi (xi ) for all i ∈ S,
(4) an Edgeworth equilibrium, if for every integer r ≥ 1 the r-fold replica of x belongs to
the core of the r-fold replica of the economy E.7
As well-known, an equilibrium allocation of E is an Edgeworth equilibrium, hence a core, weakly
Pareto optimal and individually rational allocation.
Recall that in the financial economy E, each investor i has the cone K of positive payoffs as
her portfolio set and an initial endowment of securities ωi ∈ K. We posit on E the following
assumptions:
C1: For each i, for every xi ∈ K,
(a) Pi (xi ) is convex and xi ∈
/ Pi (xi );
(b) (Pi )−1 (xi ) := {yi ∈ K : xi ∈ Pi (yi )} is ξ-open in K.
C2: The total initial endowment ω of securities is such that ω >R 0 (i.e. ω ∈ K and ω 6= 0).
In view of Proposition 2.3 and the previous assumptions, it follows from Florenzano [8, Proposition
3] that Edgeworth equilibrium exists for E.
We now introduce the additional assumption on E which will allow to decentralize with prices
in RN , the topological dual of Φ, any Edgeworth equilibrium.
C3: For each i and every weakly Pareto optimal allocation x = (xi )m
i=1 , we have xi ∈ cl Pi (xi )
and:
(a) Pi (xi ) is ξ-open in K or Pi (xi ) = {yi ∈ K : ui (yi ) > ui (xi )} for some concave utility
function ui : K → R;
(b) There is a convex set Pbi (xi ) ⊂ Φ such that the vector xi + ω is a ξ-interior point of
Pbi (xi ) and Pbi (xi ) ∩ K = Pi (xi ).
Assumption C3 (b) states that for each i, the preference correspondence Pi is ω-proper at every
component of a weakly Pareto optimal portfolio allocation. This properness assumption was first
introduced by Tourky [13] and proved by him to be strictly weaker that the ω-Mas-Colell uniform
properness assumed by Aliprantis–Brown–Polyrakis–Werner [2] for preferences defined on K by
utility functions. The local non-satiation assumed in C3 is implied by their assumption that the
market portfolio ω is desirable.
Let x = (xi )m
i=1 be an Edgeworth equilibrium of E. In view of Assumptions C1(a), C2, C3,
it follows from Aliprantis–Florenzano–Tourky [3, theorem 5.1] that there exists p ∈ RN such that
(x, p) is a nontrivial quasi-equilibrium, provided the following assertion is satisfied:
7The ideas in this definition go back to Debreu–Scarf [7]. An important reference is also [4]. Edgeworth equilibria
were first introduced and studied in [1].
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
8
B: If f = (f1 , f2 , . . . , fm ) P
is a list of ξ-continuous linear functionals such thatP
fi (ω) > 0 for
m
m
8
m
each i, and Rf (ω) = i=1 fi (xi ) for some x = (xi )m
∈
K
such
that
i=1
i=1 xi = ω,
9
then the Riesz–Kantorovich functional Rf is ω-proper at ω.
Checking B when K is an F -cone of Φ and ξ is the inductive limit topology generated by the
finite dimensional vector subspaces of Φ will be done in Section 4.
This will prove the following theorem which extends in several respects Theorem 6.1 in Aliprantis–
Brown–Polyrakis–Werner [2].
Theorem 3.4. Assume that Φ is equipped with the inductive limit topology ξ and that the cone of
positive payoff portfolios K is an F -cone. Under the assumptions C1, C2, C3 on
E = (Φ, ξ, K), (K, Pi , ωi )m
i=1 ,
there exists a nontrivial portfolio quasi-equilibrium.
Remark 3.5. Using Remark 2.5, it is easy to construct economies E = (Φ, ξ, K), (K, Pi , ωi )m
i=1
satisfying C1, C2, C3, whose positive cone is countably generated but is not an F-cone, and for
which there exists neither weakly Pareto optimal allocation nor nontrivial quasi-equilibrium.
4. Properness of the Riesz–Kantorovich functional
Let E be a real vector space and let K be an F -cone of E. Recall that for a finite list f =
(f1 , . . . , fm ) of continuous linear functionals on (E, ξ), the Riesz-Kantorovich functional Rf is
defined on K by
m
m
X
X
xi = x}.
fi (xi ) : xi ∈ K for each i and
Rf (x) = sup{
i=1
i=1
If we let, for each ω ∈ K,
P (ω) = {ω 0 ∈ K : Rf (ω 0 ) > Rf (ω)},
then the Riesz-Kantorovich functional Rf is said ω-proper at ω if there exists a convex subset
Pb(ω) of E such that 2ω is a ξ-interior point of Pb(ω) and Pb(ω) ∩ K = P (ω).
We propose to prove in this section that for ω > 0 (i.e. ω ∈ K and ω 6= 0) and any finite
list f = (f1 , . . . , fm ) of continuous linear functionals on (E, ξ) such that fi (ω) > 0 for each i, the
Riesz-Kantorovich functional Rf is ω-proper at ω.
4.1. The finite dimensional case. Suppose in this subsection that E is finite dimensional. Then
the inductive limit topology ξ coincide with the unique Hausdorff linear topology on E. Moreover
the F -cone K is a pointed and generating polyhedral convex cone.
Proposition 4.1. For ω > 0 and any finite list f = (f1 , . . . , fm ) of linear functionals on ω such
that fi (ω) > 0 for each i, the Riesz-Kantorovich functional Rf is ω-proper at ω.
8It follows from the definition of R and the ξ-compactness of the order intervals of Φ that the condition
f
P
Pm
m
m such that
Rf (ω) = m
i=1 fi (xi ) for some x = (xi )i=1 ∈ K
i=1 xi = ω is satisfied for any list f = (f1 , f2 , . . . , fm )
of ξ-continuous linear functionals.
9
The precise definition of Rf is given in the following section.
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
9
Proof. The functional Rf is continuous on K (for this conclusion we need Theorems 10.2 and
20.5 in [11]). The set R(ω) := {ω 0 ∈ K : Rf (ω 0 ) ≥ Rf (ω)} is a polyhedral convex set (for this
conclusion see Theorem 19.3 and Corollary 19.3.4 in [11]). Express R(ω) as the set of solutions to
a certain system
R(ω) = {x ∈ E : ∀i ∈ I ai (x) ≥ αi },
where the finite family (ai , αi )i∈I ∈ (E ∗ )I × RI is minimal. In view of the positive homogeneity of
Rf , one can assume αi ≥ 0 for each i ∈ I. Set I 0 := {i ∈ I : αi > 0}. Since Rf (ω) > 0, it follows
that 0 6∈ R(ω) and hence the set I 0 is nonempty. Let us denote by I(ω) the level set
I(ω) := {ω 0 ∈ K : Rf (ω 0 ) = Rf (ω)}
and define
J(ω) := {ω 0 ∈ R(ω) : ∃i ∈ I 0
ai (x) = αi }.
Claim 4.1. I(ω) = J(ω).
Proof. Assume first z ∈ I(ω) and z ∈
/ J(ω). So for each i ∈ I 0 , we have ai (z) > αi . This implies
that there is some 0 < λ < 1 such that ai (λz) > αi for all i ∈ I 0 . Now if i ∈
/ I 0 , then αi = 0 and
so ai (λz) ≥ 0 and therefore λz ∈ R(ω). But z ∈ I(ω) implies
Rf (ω) = Rf (z) > λRf (z) = Rf (λz) ≥ Rf (ω),
which is a contradiction.
Conversely, assume z ∈ J(ω) and z ∈
/ I(ω). It follows that Rf (z) > Rf (ω). So if λ =
Rf (ω)/Rf (z), then 0 < λ < 1 and Rf (λz) = Rf (ω). Hence λz ∈ I(ω) ⊂ R(ω) and hence ai (λz) ≥
αi for all i ∈ I. Now notice that since z ∈ J(ω), there exists some i such that ai (z) = αi > 0. In
particular for the vector λz ∈ R(ω) we have ai (λz) = λai (z) < αi , contrary to λz ∈ R(ω).
Clearly, Claim 4.1 implies that
P (ω) = R(ω) ∩ {z ∈ E : ∀i ∈ I 0
ai (z) > αi }.
We now define
Pb(ω) = {z ∈ E : ∀i ∈ I 0 ai (z) > αi }.
It follows from this definition that Pb(ω) is a nonempty convex open set and that P (ω) ⊂ Pb(ω) ∩ K.
Claim 4.2. 2ω ∈ Pb(ω).
Proof. Since ω ∈ R(ω), for each i ∈ I 0 we must have ai (ω) ≥ αi > 0 and consequently ai (2ω) =
2ai (ω) > αi .
Claim 4.3. For each i 6∈ I 0 , there exists some u ∈ R(ω) such that ai (u) = 0. Consequently, for
any such i 6∈ I 0 , we have for each z ∈ K, ai (z) ≥ 0.
Proof. Recall first that if for some i ∈ I on has ai (z) > αi for each z ∈ R(ω), then the inequality
ai (z) ≥ αi can be removed from the expression of R(ω). Since the family (ai , αi )i∈I is supposed
to be minimal, this proves the first assertion.
To prove by contraposition the second assertion, assume that i 6∈ I 0 , z ∈ K and ai (z) < 0. Let
u ∈ R(ω) be such that ai (u) = 0. Since Rf (u) ≥ Rf (ω) > 0, it follows from the continuity of Rf
at u, that there exists z 0 ∈ K such that ai (z 0 ) < 0 and Rf (z 0 ) > 0. Using the positive homogeneity
of Rf , we can find some λ ≥ 1 such that Rf (λz 0 ) ≥ Rf (ω). Hence ai (λz 0 ) ≥ 0, which implies that
ai (z 0 ) ≥ 0, contrary to ai (z 0 ) < 0.
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
10
Claim 4.4. Pb(ω) ∩ K ⊂ P (ω).
Proof. Let z ∈ Pb(ω) ∩ K. In view of the definition of Pb(ω), we have only to prove that z ∈ R(ω),
i.e. that z satisfies all the inequalities defining R(ω). For i ∈ I 0 , this follows from z ∈ Pb(ω); for
i 6∈ I 0 , this follows from z ∈ K and the previous claim.
We have thus proved that Pb(ω) is an open convex set with 2ω ∈ Pb(ω) and such that Pb(ω)∩K =
P (ω).
4.2. The general case. In this subsection, the space E is not supposed toSbe finite dimensional.
∞
According to Definition 2.3, the cone K is pointed, generating and K = n=1 K n is the union
of a countable family of finite dimensional polyhedral convex cones K n such that each K n is an
extremal subset (or a face) of K n+1 . Recall that, as noticed in Remark 2.4, E has a countable
Hamel basis.
Proposition 4.2. For ω > 0 and any finite list f = (f1 , . . . , fm ) of continuous linear functionals
on (E, ξ) such that fi (ω) > 0 for each i, the Riesz-Kantorovich functional Rf is ω-proper at ω.
Proof. Let ω > 0 and let f = (f1 , . . . , fm ) be a finite list of continuous linear functionals on (E, ξ)
such that fi (ω) > 0 for each i. Without any loss of generality, we can suppose that ω ∈ K 1 . Let
n
n ≥ 1, we define f n be the finite list (f1n , . . . , fm
), where fin ∈ (E n )∗ is the restriction of fi to the
n
n
n
n
subspace E = K − K . Since each K is a face of K, we first have that Rf n coincide with the
restriction to K n of Rf , that is
∀x ∈ K n
Rf n (x) = Rf (x).
In particular, if we let P n (ω) := {ω 0 ∈ K n : Rf n (ω 0 ) > Rf n (ω)}, then P n (ω) = P (ω) ∩ K n . Now
if we let Rn (ω) := {ω 0 ∈ K n : Rf n (ω 0 ) ≥ Rf n (ω)}, then as in the proof of Proposition 4.1, there
exists a finite list (ani )i∈In of linear functionals and positive scalars (αin )i∈In such that
Rn (ω) = {x ∈ E n : ani (x) ≥ αin
∀i ∈ In }.
Moreover if the family (ai , αi )i∈I is chosen minimal, then P n (ω) = Pbn (ω) ∩ K n , where
Pbn (ω) := {x ∈ E n : ani (x) > αin
∀i ∈ In0 },
and where In0 := {i ∈ In : αin > 0}. Note that Pbn (ω) is convex, open in E n and 2ω ∈ Pbn (ω).
Moreover, following Lemma A.1 in Appendix, the construction of Pbn (ω) is independent of the
choice of the minimal family (ai , αi )i∈In .10
Claim 4.5. The sequence (Pbn (ω))n≥1 is increasing, that is
∀n ≥ 1
Pbn (ω) ⊂ Pbn+1 (ω).
Proof. Let n ≥ 1, note that following the definition of K, Rn+1 (ω) ∩ E n = Rn (ω). In particular
Rn (ω) = {x ∈ E n : [an+1
]|E n (x) ≥ αin+1
i
∀i ∈ In+1 },
10Indeed, since K n is generating in E n , it follows that Rn (ω) has an interior point in E n . Hence, from Lemma
A.1 in Appendix, the faces of Rn (ω) with dimension dim K n − 1 are exactly the convex sets Hi ∩ Rn (ω), i ∈ I where
Hi := {x ∈ E n : ai (x) = αi }.
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
11
where [an+1
]|E n ∈ (E n )∗ is the restriction of an+1
to E n . Consider a subset Jn+1 of In+1 such
i
i
n+1
n+1
that the family ([aj ]|E n ), αj )j∈Jn+1 is minimal in the definition of Rn (ω). Following the
construction of Pbn (ω), we have
Pbn (ω) = {x ∈ E n : [an+1
]|E n (x) > αjn+1
j
0
}
∀j ∈ Jn+1
0
0
0
where Jn+1
= {j ∈ Jn+1 : αjn+1 > 0}. In particular Jn+1
⊂ In+1
, and Pbn (ω) ⊂ Pbn+1 (ω).
Now let
Pb(ω) =
[
Pbn (ω).
n≥1
Pb(ω) is obviously convex. Since 2ω ∈ Pb1 (ω) it follows that 2ω ∈ Pb(ω). For each n ≥ 1, Pb(ω)∩E n =
S
n
n
n
n
bk
bk
k≥n P (ω) ∩ E , with P (ω) ∩ E open in E . Applying Corollary 2.2, to the family (E )n∈N
of finite dimensional vector subspaces of E, it follows that Pb(ω) is ξ-open. Now we assert that
S
Pb(ω) ∩ K = P (ω). Indeed, Pb(ω) ∩ K = n≥1 Pb(ω) ∩ K n , and for each n ≥ 1,
[
[
Pb(ω) ∩ K n =
(Pbk (ω) ∩ K k ) ∩ K n =
P k (ω) ∩ K n = P n (ω).
k≥n
It follows that Pb(ω) ∩ K =
S
n≥1
k≥n
P n (ω) = P (ω).
Appendix A. Note on the faces of a polyhedral convex set
Let E be a non-trivial finite dimensional vector space and A a polyhedral convex set. By
defintion, there exists a finite family of linear functionals (ai )i∈I and of scalars (αi )i∈I such that
A = {x ∈ E : ai (x) ≥ αi
∀i ∈ I}.
The family (ai , αi )i∈I is said minimal if for each i ∈ I, the set
Ai := {x ∈ E : aj (x) ≥ αj
∀j 6= i}
is strictly bigger than A.
Recall that a convex subset F is a face (or an extremal subset) of A if for each line segment
[x, y] of A satisfying ]x, y[∩F 6= 6, then x and y belong to F .
Lemma A.1. Let F be the set of faces of A with dimension dim A − 1. If the family (ai , αi ) is
minimal and if A has an interior point then
F = {Hi ∩ A : i ∈ I},
where Hi = {x ∈ E : ai (x) = αi }.
Proof. We first prove that for each i ∈ I, Fi := Hi ∩ A is a face of A with dimension dim A − 1.
Let i ∈ I and let [x, y] ⊂ A be a line segment such that ]x, y[∩Fi 6= 6, that is, there exists λ ∈]0, 1[
such that ai (λx + (1 − λ)y) = αi . Since ai (x) ≥ αi and ai (y) ≥ αi then ai (x) = αi and ai (y) = αi .
We have thus proved that Fi is a face of A.
Now we propose to prove that dim Fi = dim A − 1. Let Ai := {x ∈ E : aj (x) ≥ αj ∀j 6= i}; then
int Ai = {x ∈ E : aj (x) > αj ∀j 6= i}. Note that (int Ai ) ∩ Hi ⊂ Fi . Since the family (ak , αk )k∈I
GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
12
11
6 . In particular, dim int(Ai ) ∩ Hi = dim Hi = dim A − 1. But
is minimal then
(int Ai ) ∩ Hi 6= (int Ai ) ∩ Hi ⊂ Fi ⊂ Hi , hence dim Fi = dim A − 1.
Now we prove that F ⊂ {Fi : i ∈ I}. Let F be a face of A with dimension dim A − 1. Then
there exists a linear functional b and a scalar β such that
F = {x ∈ E : b(x) = β} ∩ A and A ⊂ {x ∈ E : b(x) ≥ β}.
In particular
{x ∈ E : ai (x) ≥ αi ∀i ∈ I} ⊂ {x ∈ E : b(x) ≥ β}.
It follows from Rockafellar [11, Theorem 22.3] that there exists a family (λi )i∈I of non negative
scalars λi ≥ 0, such that
X
X
b=
λi ai and β ≤
λi αi .
i∈I
i∈I
But if x ∈ F , for each i ∈ I, ai (x) ≥ αi and
X
X
X
λi αi ≤
λi ai (x) = b(x) = β ≤
λi αi .
i∈I
It follows that β =
i∈I
i∈I
P
i∈I αi . Hence
F = {x ∈ E : ∀i ∈ I
ai (x) ≥ αi
and
X
λi ai (x) =
i∈I
X
λi αi }.
i∈I
That is,
F =
\
Fi
where
I 0 := {i ∈ I : λi > 0}.
i∈I 0
Let i ∈ I 0 ; then F ⊂ Fi . Since F and Fi have the same dimension, then the relative interior of F
is contained in the relative interior of Fi , which implies (see [11, Corollary 18.1.2]) F = Fi .
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11If (int Ai ) ∩ H = 6, then int Ai ⊂ {x ∈ E : a (x) > α }. It follows that int Ai ⊂ A. But Ai is striclty bigger
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i
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GENERAL EQUILIBRIUM IN INFINITE SECURITY MARKETS
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