Title:Core equivalence theorem: countably many types of agents and commodities in
L (µ)
Author: Anna Martellotti, Dipartimento di Matematica ed Informatica, Università degli
Studi di Perugia. 1
Abstract: We prove a core-walras equivalence result for a finitely additive confederate
economy with commodity space L1 (µ) and a measurable bounded map of extremely desirable commodities: when the map of extremely desirable commodities is simply bounded
the properness of preferences is no more equivalent to the existence of just one extremely
desirable commodity as assumed in the countably additive model by Rustichini and Yannelis.
Proposed running head: Core equivalence theorem
AMS subject classification: primary: 91B50, 91B54; secondary 28B20, 46A22, 28A25
JEL classification: D51; C02
1
1
Anna Martellotti, Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1,
06123, Perugia (ITALY), e-mail: [email protected]
1
CORE EQUIVALENCE THEOREM:
COUNTABLY MANY TYPES OF AGENTS
AND COMMODITIES IN L1 (µ)
Anna Martellotti
2
AMS subject classification: primary: 91B50, 91B54; secondary 28B20, 46A22, 28A25
JEL classification: D51; C02
Abstract
We prove a core-walras equivalence result for a finitely additive confederate economy with commodity
space L1 (µ) and a measurable bounded map of extremely desirable commodities: when the map of extremely
desirable commodities is simply bounded the properness of preferences is no more equivalent to the existence
of just one extremely desirable commodity as assumed in the countably additive model by Rustichini and
Yannelis.
1
Introduction
Several extensions of the classical core-walras equivalence have appeared in the literature. In
particular we shall focus on generalizations in two directions: infinite-dimensional commodity
space X and finitely additive space of agents (A, Σ, m).
Finitely additive economies have been investigated by several authors for both finite and
infinite dimensional commodities spaces; we quote for instance Basile (1993), Basile-Graziano
(2001), Basile- De Simone-Graziano (2005), Cheng (1991), Vind (1964). In this framework
authors commonly search coalitional versions of the equivalence, which usually does not
completely coincide with the individualistic approach.
The individualisic approach to the core-walras equivalence for a finitely additive economy
in Armstrong and Richter (1985) and Weiss (1981); a finitely additive individualistic corewalras equivalence has been proved by Angeloni-Martellotti (2007) when the commodity
space is a reflexive and separable Banach lattice.
When the commodity space has infinite dimensions, and the positive cone has an empty
interior, it is well known that the core-walras equivalence may fail (see e.g. RustichiniYannelis (1991)), unless suitable assumptions on the preferences are adopted. Rustichini
2
Anna Martellotti, Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1,
06123, Perugia (ITALY), e-mail: [email protected]
2
and Yannelis proposed the assumption of the existence of a so called extremely desirable
commodity. Again Angeloni-Martellotti (2007) have assumed the same condition, but in an
equivalent form that apparently allows a greater flexibility in the choice of the extremely
desirable commodity: more precisely we had shown that if there exists a measurable map
v : A → X + such that v(a) is an extremely desirable commodity only for agent a, provided
the range v(A) has suitably small measure of non compactness, then the economy admits in
fact one extremely desirable commodity.
As an intermediate step in the same paper we had considered the confederate model, namely
economies where the grand coalition A can be decomposed into countably many pairwise
disjoint subcoalitions forming an m-exhaustion π, so that in each subcoalition of π agents
have the same preferences; in other words the system of preferences can be represented as a
formal series.
The value of these confederate models is essentially the fact that in them allocations can
be uniformly approximated by means of formal series that are pointwise preferred to the
allocation (approximation “from above”): the strong argument upon which this crucial
approximation technique is based is the weak compactness of bounded subsets of X.
In this paper we continue the investigation started in the quoted paper, and we achieve an
individualistic core-walras equivalence result for a confederate finitely additive economy with
commodity space L1 (µ). The paper is not a mere generalization of the previous (AngeloniMartellotti (2007)) for two main reasons.
The lack of weak compactness of bounded sets is surrogated by the use of the so called
Optimization without compactness (see Bukhvalov (1995)), but the resulting approximation is
done by formal series that the approximated allocation pointwise prefers to them, that is with
the converse inequality. The technique however has no symmetry, and so this approximation
“from below”, instead than the one “from above” represents a serious complication in the
argument of the proof.
The second novelty in the paper is the most important generalization in it; in fact we
considerably extend the flexibility in the choice of the extremely desirable commodity, by
assuming that there exists a Σ-measurable simply bounded map v : A → X + such that
v(a) is extremely desirable for agent a. Since we do not know how large the bounded set
v(A) can be, this assumption genuinely extends the properness condition of Rustichini and
Yannelis. What is shown in the paper is that we can replace v by means of a formal series,
namely that the model coincides with a deeper confederate model where agents within each
of the subcoalitions forming the m-exhaustive decomposition of the grand coalition adopt
the same utilities and the same extremely desirable commodity. Although given in the
framework of L1 (µ), the same extension could be analogously proven in the confederate
3
case (Angeloni-Martellotti (2007)). The proof is largely inspired to the original proof of
Rustichini and Yannelis, but with substantial adaptements to the more general assumption
on preferences.
2
The economic model and the main result
Throughout this paper X will denote the space L1 (µ) = L1 (Ω, O, µ), where (Ω, O, µ) is a
measure space such that L1 (µ) is separable, with cone X + and unit ball X1 . The topological
dual of X will be denoted by X ∗ .
We define a (purely exchange) economy a quadruple
E = {(A, Σ, m); X + ; e; {a }a∈A },
where:
(E1) (A, Σ, m) is the space of agents, with Σ a σ-algebra of subsets of A, m : Σ −→ [0, 1] a
finitely additive measure;
(E2) X + is the consumption set of each agent;
(E3) e : A −→ X + is the initial endowment, with e ∈ L1X (m), where L1X (m) denotes the
space of all the Bochner integrable functions g : A → X (for the theory of integration
of vector valued functions when m is only finitely additive, we refer to the classical
book of Dunford and Schwartz);
(E4) {a }a is the preference relation associated to agent a ∈ A, namely it is an irreflexive
pre-order.
As usual one defines the classical economic entities. In particular we shall define the following
items
- We say that the coalition S ∈ Σ can improve the allocation f if there exists an allocation
g such that
(1) g(a) a f (a) m − a.e. in S;
Z
Z
(2) m(S) > 0 and
g dm =
e dm.
S
S
- The core of E, C(E), is the set of all the feasible allocations that cannot be improved by
any coalition.
4
- The budget set of an agent a ∈ A for the price p is
Bp (a) = {x ∈ X + : p(x) ≤ p(e(a))}.
- A Walras equilibrium of E is a pair (f, p) ∈ L1X (A, A, m) × ((X ∗ )+ \ {0}) such that f is
a feasible allocation and f (a) is a maximal element of a in the budget set Bp (a), for
m-almost all a ∈ A.
- A walrasian allocation is a feasible allocation f which admits a price p so that the pair
(f, p) is a Walras equilibrium. W (E) is the set of all the walrasian allocations of E.
We shall assume the following hypotheses:
(A.1) (Perfect competition) m is semiconvex, that is, every B ∈ Σ contains B 1 ∈ Σ such
2
1
that m(B 1 ) = m(B).
2
2
For the initial endowement e ∈ L1X + (m) we assume
Z
e(a) dm is strictly positive,
(A.2) (Resource availability) the aggregate initial endowment
and we write
A
Z
e dm 0,
A
where we say that x ∈ X is strictly positive if x∗ (x) > 0 whenever x∗ is a non-zero
element of (X ∗ )+ , where (X ∗ )+ denotes the positive cone of the dual space X ∗ .
Analogously to Angeloni-Martellotti (2007) we shall assume the following hypothesis on the
finitely additive measure m:
(A.3) The ideal of m-null sets is stable under countable unions.
According to the results of Candeloro-Martellotti (1992) for every a ∈]0, +∞) there exists
a semiconvex finitely additive measure on A = [0, 1] × [0, a] fulfilling (A.3) without being
countably additive; we refer to Angeloni-Martellotti (2007) for a more detailed discussion on
this condition.
We now introduce the preferences’ structure. Let us consider a sequence of functions
un : X + → R that are concave, uniformly continuous on bounded subsets of X + and strictly
monotonic, where strictly monotonic means that
(a) un (y) > 0, for every y ∈ X + \ {0} and un (0) = 0;
(b) for every x, y ∈ X + , z ∈ X + , un (x) > un (y) ⇒ un (x + z) > un (y + z).
5
Then the family {un , n ∈ N} defines a sequence of transitive and reflexive orderings on X + ,
{n , n ∈ IN } by means of x n y iff un (x) ≥ un (y). As usual we shall also consider the strict
preference associated to each un by x n y iff un (x) > un (y).
Let π = {An , n ∈ IN } be an m-exhaustion of A, that is a!countable collection of measurable
∞
[
pairwise disjoint subsets of A such that lim m
An = 0.
k→∞
n=k
Then, we shall set
(A.4) x a y if and only if un (x) > un (y) whenever a ∈ An (briefly we shall say that E is a
confederate model).
Note that in this way a satisfies the usual monotonicity, continuity and measurability
assumptions on preferences and that the strict monotonicity that we have assumed implies
the usual monotonicity assumption of preferences as, for instance, in Angeloni-Martellotti
(2004).
We shall need two extra assumptions on preferences.
Definition 2.1 We remind that Mas-Colell (1986) defines a preference to be proper at
v ∈ X + with respect to the neighbourhood of 0, W if for every x ∈ X + , z ∈ X, α > 0, the
following implication holds:
x + z − αv x ⇒ z 6∈ αW.
We shall briefly say that is (v, W )- proper.
We note the following obvious consequences of the definition:
- if is (v, W )-proper, and if α ∈]0, 1[ then is (v, αW )-proper;
- if is (v, W )-proper and monotone, and if w ≥ v, then is (w, W )-proper.
We shall consider the following assumption
(A.5) there exist a bounded measurable function v : A −→ X + and r > 0 such that for
every a ∈ A, a is (v(a), rX1o )-proper for every a ∈ A.
In Angeloni-Martellotti (2007) the properness of preferences has been assumed with respect
to a suitable neighbourhood of 0, namely for an open, convex, circled and solid W such that
W c ∩ X + is convex; note that, due to the structure of L1 (µ) here we can directly assume
that W is on open ball.
6
Moreover in the same paper we had assumed that the range of v is small in the sense of
r
the Hausdorff measure of non compactness χ, namely that χ[v(A)] ≤ . This proved to be
2
equivalent to Rustichini and Yannelis’ original assumption on preferences.
For the sake of simplicity in the notation we shall denote by W the open ball rX1o .
We come now to our last assumption. We shall assume that
(A.6) for every n ∈ N and for every ε > 0 there exists yn,ε ∈ X1 ∩ X + , yn,ε 6= 0 and αn,ε > 0
such that
un (x + tyn,ε ) − un (x) ≥ tαn,ε
(1)
for every x ∈ X + ∩ εX1 .
Condition (A.6) says that within each coalition An ∈ π for every ε > 0 there exists a
privileged bundle yn,ε so that any additional increment in the direction of yn,ε of commodities
of “small” value (i.e. not exceeding ε) implies a superlinear increasing of the preferences.
If (A.6) holds we shall find that for every x ∈ X + ∩ εX1
(un )0+ (x)(yn,ε ) = lim+
t→0
un (x + tyn,ε ) − un (x)
≥ αn,ε > 0,
t
and hence that
inf
x∈(X + ∩rX1 )
(un )0+ (x)(yn ) ≥ αn,ε > 0.
(2)
Condition (2) is similar to assumption (A.6) in Angeloni-Martellotti (2007).
An interesting example of preferences satisfying both (A.5) and (A.6) are those stemming
from radial utilities, defined as follows: Let f : [0, +∞) → [0, +∞) be a strictly increasing
concave function; define u : X + → [0, +∞) as u(x) = f (kxk). Then u is also concave, since
for every x1 , x2 ∈ X + and every t ∈ [0, 1] we have ktx1 + (1 − t)x2 k = tkx1 k + (1 − t)kx2 k
(this is a property of the L1 (µ) norm on the positive cone X + ) and f is concave. Also
strict monotonicity and uniform continuity on bounded sets are easily inherited from the
properties of f .
To prove (A.5) choose any v ∈ X + , v 6= 0 and choose r < kvk. Then u is (v, W )-proper;
in fact, for every x ∈ X + , α > 0 we have that f (kx + αvk + t) > f (kxk) for every
t ∈] − αkvk, +∞).
Now, if z ∈ W then, setting t = kx + αv − zk − kx + αvk we have |t| ≤ kzk < αr < αkvk
and so kx + αv − zk = kx + αvk + t with t ∈ [−αkvk, +∞), whence
u(x + αv − z) = f (kx + αv − zk) = f (kx + αvk + t > f (kxk) = u(x).
7
To prove (A.6) let ε > 0 be fixed and choose any yε ∈ X + with kyε k = 1. Consider α = f−0 (ε).
By concavity, for every x ∈ X + ∩ εX1 , x 6= 0 and every t > 0 it is
u(x + tyε ) − u(x) ≥ u(ξ + tyε ) − u(ξ)
εx
; thus we can prove (A.6) only for bundles x with kxk = ε. Then again, since
kxk
for t > 0 |x + tyε ||kxk + tkyε k = ε + t, we have that
where ξ =
u(x + tyε ) − u(x) = f (t + ε) − f (ε) ≥ αt.
Our main result will be the following:.
Theorem 2.1 (Main theorem) Under assumptions (A.1)-(A.6), C(E) = W(E).
3
Proof of the main theorem
Let f ∈ L1X + (m) be fixed, and consider the multifunctions Γ, F defined as
Γ(a) = {x ∈ X + |x n f (a)},
a ∈ An ;
F (a) = (Γ(a) − e(a)) ∪ {0},
a ∈ An .
Given a functional x∗ ∈ X ∗ and a constant α ∈ R, we shall use the following notations for
the half-spaces:
G+ (x∗ , α) = {x ∈ X : x∗ (x) > α},
G− (x∗ , α) = {x ∈ X : x∗ (x) < α},
F + (x∗ , α) = {x ∈ X : x∗ (x) ≥ α},
F − (x∗ , α) = {x ∈ X : x∗ (x) ≤ α}.
Angeloni-Martellotti (2007) obtained for reflexive and separable commodity spaces an approximating procedure of Γ by means of formal series, from inside. The technique was heavily
based upon the weak compactness of bounded sets.
In this framework we will also reach an approximation result, but we shall use the µclosedness of some bounded sets as a surrogate of the weak compactness; therefore in this
paper the approximation will be from outside. More precisely we have the following result.
Lemma 3.1 Under assumption (A.4) for every ε > 0 there exists a formal series
X
ϕε =
xn,k 1En,k
n,k∈N
such that
8
i) ϕε (a) n f (a) for every a ∈ An ;
ii) for every n ∈ N the family {En,k , k ∈ N} is an m-exhaustion of An ;
iii) kϕε − f k1 ≤ ε uniformly in A.
Proof. Let ε > 0 be fixed. For p ∈ N let δp = δp (ε) < ε be the parameter of the uniform
continuity of up . As in the proof of Lemma 3.2 of Angeloni-Martellotti
(2007), thanks to the
X
1
separability of L (µ) we can choose a formal series fε,p =
xp,k 1Ep,k such that
k∈N
α) {Ep,k , k ∈ N} is an m-exhaustion of Ap ;
1
β) kfε,p − f k1 ≤ δp (ε) uniformly in Ap .
2
As in the mentioned proof, we shall now change the values of fε,p keeping the decomposition
{Ep,k , k ∈ N}.
In order to do this we shall state the following
Claim: for every n ∈ N there exists a positive functional x∗n ∈ X ∗ such that for every
x ∈ X +,
F − (x∗n , x∗n (x)) ⊂ {y ∈ X + : un (y) ≤ un (x)}.
Let n ∈ N, xo ∈ X + be fixed.
From the continuity and the concavity of un , the set {y ∈ X + |un (y) ≥ un (xo )} is closed and
convex, and xo is a boundary point; therefore there exists a supporting functional x∗n ∈ X ∗
such that {y ∈ X + |un (y) ≥ un (xo )} ⊂ F + (x∗n , x∗n (xo )). This immediately implies that x∗n is
a positive functional: in fact for every z ∈ X + from the strict monotonicity of un we have
that un (xo + z) ≥ un (xo ) and hence x∗n (xo + z) ≥ x∗n (xo ) namely x∗n (z) ≥ 0.
Then G− (x∗n , x∗n (xo )) ⊂ {y ∈ X + |un (y) < un (xo )}, and taking the strong closure on both
sides we reach F − (x∗n , x∗n (xo )) ⊂ {y ∈ X + |un (y) ≤ un (xo )}, that is the required inclusion at
xo .
We shall now prove that the same inclusion holds true with the same x∗n at each other bundle
in X + .
Fix then z ∈ X + and choose any w ∈ G− (x∗n , x∗n (z)), that is x∗n (w) < x∗n (z).
Write w = z − xo + y; then in y = w − z + xo we have x∗n (y) < x∗n (xo ) and therefore
y ∈ G− (x∗n , x∗n (xo )). Hence, from the previous inclusion, un (y) < un (xo ).
Assume by contradiction that un (w) > un (z), and apply the strict monotonicity; then
un (w + xo ) > un (z + xo ) or else un (z + y) > un (z + xo ), while we have proved that un (y) <
un (xo ) whence again the strict monotonicity yields that un (y + z) < un (xo + z).
9
In conclusion, G− (x∗n , x∗n (z)) ⊂ {y ∈ X + |un (y) ≤ un (z)} and taking the strong closure we
reach the required inclusion.
We now turn again to the construction of the new series. Let p ∈ N be fixed. Since fε,p is
1
constant in Ep,k , from (β) we have that f (Ep,k ) ⊂ xp,k + δp (ε)X1 . Let us consider for every
2
a ∈ Ep,k the set
1
La = xp,k + δp (ε)X1 ∩ X + ∩ F − (x∗p , x∗p (f (a))).
2
Clearly La 6=Ø, for f (a) ∈ La for each a ∈ Ep,k . Moreover La is convex and τµ −closed.
Indeed, if xn → x µ−a.e. then,Z from the previous Claim, there exists hp ∈ L∞ (µ), h ≥ 0
µ−a.e. and such that x∗p (x) =
hp x dµ, x ∈ X. Then xn · hp → x · hp µ−a.e. and, since
Ω
everything is in the cone X + , by Fatou’s Lemma
Z
Z
∗
hp xn dµ = lim inf x∗p (xn ) ≤ x∗p (f (a)),
xp (x) =
hp x dµ ≤ lim inf
n
Ω
n
Ω
that is, x ∈ F − (x∗p , x∗p (f (a))),. Analogously, since |xn − xp,k | → |x − xp,k | µ−a.e., we have
Z
Z
1
|x − xp,k | dµ ≤ lim inf
|xn − xp,k | dµ = lim inf kxn − xp,k k1 ≤ δp (ε),
kx − xp,k k1 =
n
n
2
Ω
Ω
1
namely x ∈ xp,k + δp (ε)X1 .
2
It is also evident that (La )a∈Ep,k has the finite intersection property.
In fact, if a1 , . . . , ar ∈ Ep,k , then x∗p (f (aj0 )) = min{x∗p (f (a1 )), . . . , x∗p (f (ar ))} for some suitr
\
∗
− ∗
∗
− ∗
La i ,
able j0 . Then F (xp , xp (f (aj0 ))) ⊂ F (xp , xp (f (ai ))), i = 1, . . . , r, whence Laj0 =
i=1
for the dependence upon a ∈ A in La is precisely given by the half space F − (x∗p , x∗p (f (a))),
and we have already noticed that L(a) 6=Ø, for \
every a ∈ A.
Now, from Theorem 1.3 of Bukhvalov (1995),
La 6= ∅.
a∈Ep,k
Let us choose zp,k ∈
\
L(a), and set ϕp,ε =
a∈Ep,k
X
zp,k 1Ep,k . Thus for a ∈ Ap
k∈N
1
kϕp,ε (a) − f (a)k1 ≤ kϕp,ε (a) − fp,ε (a)k1 + kfp,ε (a) − f (a)k1 ≤ kxp,k − zp,k k1 + δp (ε)
2
1
1
≤
δp (ε) + δp (ε) = δp (ε) ≤ ε
2
2
X
whence iii) holds for ϕε =
ϕp,ε .
p∈N
Finally note that from the Claim it is also immediate that up (ϕp,ε (a)) ≤ up (f (a)), for a ∈ Ap ,
and hence i) also holds.
2
10
As already noticed, if in (A.5) we replace v by another function w which is m-a.e. greater
than v, we shall find that a is m-a.e. (w(a), W )-proper.
We shall in fact replace v by a suitable formal series.
Lemma 3.2 UnderX
assumptions (A.4), (A.5) and (A.6), for any δ > 0 there exists a bounded
formal series w =
wn,k1En,k such that
n,k∈N
i. a is (w(a), W )-proper m-a.e.;
ii. for every n ∈ N the family {En,k , k ∈ N} is an m-exhaustion of An ;
iii. d(w(a), W ) > δ for m-a.e. a ∈ A.
Proof. Fix ε > 0. Since v(A) is bounded, there exists R > 0 such that v(A) + εX1 ⊂ RX1 .
Applying (A.6) we can choose yk ∈ X + ∩ X1 , µk > 0 such that
uk (x + εyk ) − uk (x) ≥ εµk
(3)
for each x ∈ X + ∩ RX1 . By assumption each uk is uniformly continuous in v(A) + εX1 ; so
we can choose positive δk = δk (εµk ) < ε such that for x, y ∈ v(A) + εX1 with kx − yk < δk ,
|uk (x) − uk (y)| < µk ε.
Thus we can
apply the technique of Lemma 3.1 to v in each Ak , and determine a formal
X
xn,k 1En,k such that
series ϕk =
n∈N
i. ϕk (a) a v(a) for every a ∈ Ak ;
ii. kϕk (a) − v(a)k < δk for a ∈ Ak ;
iii. for every k ∈ N the family {En,k , n ∈ N} is an m-exhaustion of Ak .
From (3) we have that
uk (xn,k + εyk ) − uk (xn,k ) ≥ εµk ;
Indeed xn,k ∈ v(A) + δk X1 ⊂ v(A) + εX1 by ii.
Moreover we have from ii. that |uk (xn,k ) − uk (v(a))| < εµk for every a ∈ Ak .
Finally, for a ∈ Ak
uk (xn,k + εyk ) ≥ uk (xn,k ) + εµk > uk (v(a)).
Hence the formal series z =
∞ X
∞
X
(xn,k + εyk )1En,k satisfies z(a) a v(a) pointwise in A.
k=1 n=1
11
From (A.5) and what already noticed for the pair of properness, a is (z(a), W )-proper for
each a ∈ A.
Note that for every n and k, xn,k + εyk 6∈ W . In fact, since xn,k + εyk ∈ X + we have that
for every x ∈ X +
x + xn,k + εyk − α(xn,k + εyk ) a x
for every a ∈ An and every α ∈]0, 1[. By (A.5) xn,k + εyk 6∈ αW for α ∈]0, 1[. As W is open
this implies that xn,k + εyk 6∈ W .
Hence replacing each xn,k + εyk with wn,k = xn,k + εyk + δy for some δ > 0 and y ∈ X + we
obtain another formal series
X
w=
wn,k 1En,k
n,k∈N
such that a is (w(a), W )-proper for every a ∈ A, and such that d(w(a), W ) > δ.
Finally note that
kw − vk < 2ε + δ
by virtue of ii. and the definition of w; therefore w is bounded.
We shall write
w(a) =
∞
X
wk 1Fk
2
(4)
k=1
with a suitable rearrangement of the indexes so that {Fk , k ∈ N} is an m-exhaustion of A
and in each coalition Fk agents adopt the same utility uk .
Consider now, for each k ∈ N the open convex cone
[
α(wk + W )
Ck =
α>0
and define H to be the set of absolutely converging series
P
ak with ak ∈ Ck , k ∈ N.
Lemma 3.3 The set H is convex, has non-empty interior and for every x ∈ H and λ ∈]0, 1[
λx ∈ H.
Proof. The convexity of H immediately follows from
Xthat of each Ck .
To prove that H has non-empty interior, note that
wk m(Fk ) + W ⊂ H.
k
X
In fact, as
m(Fk ) = m(A) = 1 (for {Fk , k ∈ N} is an m-exhaustion of A) we have that
k
u
∈ W. Then we can write
k m(Fk )
X
X
u
wk m(Fk ) + u =
m(Fk ) wk + P
k m(Fk )
k
k
for every u ∈ W , P
12
(5)
and note that
m(Fk ) wk + P u
≤ m(Fk ) kwk k + P u
≤ m(Fk )(R + r)
m(F
)
m(F
)
k
k
k
k
X
and since
m(Fk ) converges, we have proven that the series on the right-hand side of (5)
k
X
converges, and hence that
wk m(Fk ) + u ∈ H.
k
The final property of H is inherited by the analogous property of every Ck .
2
We shall now consider the Aumann integral
Z
F dm.
I=
A
Lemma 3.4 Let H be an open set. Under the assumptions (A.4) and (A.6)
Z we have that if
I ∩H 6= Ø, then there exist a simple selection ψ of Γ and E ∈ Σ such that (ψ − e)dm ∈ H.
Furthermore there exists no ∈ N such that E ⊂
no
[
E
Fk , where {Fk , k ∈ N} is the m-exhaustion
k=1
in (4).
Proof. Since I ∩ H 6=Ø, there exist Eo ∈ Σ and so ∈ SΓ1 (i.e. the set of integrable selections
of Γ) such that
Z
zo =
(so − e)dm ∈ H.
Eo
Since H is open, there exists ro > 0 such that zo + ro X1 ⊂ H.!
no
[
Fk there holds
Choose now no ∈ N in such a way that on E1 = Eo ∩
k=1
Z
zo −
< ro .
(s
−
e)dm
o
E1
Z
This can obviously be done since (Fn )n is an m-exhaustion, and
(so − e)dm m.
Then again
Z
(so − e)dm ∈ H;
z1 =
E1
let r1 > 0 be such that z1 + r1 X1 ⊂ H.
In choosing E1 we have restricted our attention to finitely many utilities.
Since so ∈ L1X (m) we have that (see Brooks-Martellotti (1991)) lim m({kso k > t}) = 0.
t→+∞
13
Hence τo > 0 can be chosen in such a way that, setting
E2 = {a ∈ E1 : kso (a)k ≤ τo },
there holds
Z
r1
(so − e)dm
≤ 2.
E1 \E2
In this way again
Z
(so − e)dm ∈ H,
z2 :=
E2
r1
X1 ⊂ H.
2
According to (2) we have that
and necessarily z2 +
µ := min inf{(un )0+ (x)(yn )|x ∈ X + ∩ (τo + ro )X1 } > 0.
1≤n≤no
Without loss of generality we can always assume that µ = 1, since the no preferences
1
1
u1 , . . . , uno can always be replaced by the multiples u1 , . . . , uno without any change in
µ
µ
the perferences structure.
r1 By our assumptions, u1 , . . . , uno are uniformly continuous on V := X + ∩ τo +
X1 . Hence
2
r 1
we can choose δ
such that for each pair x, y ∈ V with kx − yk < δ,
8
|uk (x) − uk (y)| <
r1
8
(6)
for k = 1, . . . , no .
r r
1
1
< .
Moreover we can choose δ
8
4
X
Applying Lemma 3.1 there exists a formal series ϕ =
xn,k 1En,k such that
n,k∈N
i) ϕ(a) n so (a) for every a ∈ An ;
ii) for every n ∈ N the family {En,k , k ∈ N} is an m-exhaustion of An ;
r 1
iii) kϕ − so k1 ≤ δ
uniformly in A.
8
In this way
Z
(ϕ − e)dm ∈ H
z3 :=
E2
Z
(for kz2 − z3 k ≤
kϕ − so kdm <
E2
r1
r1
r1
m(E2 ) < ) and z3 + X1 ⊂ H.
4
2
4
14
Z
By means of ii) and the absolute continuity
(ϕ − e)dm m choose now no integers,
k(1), . . . , k(no ) ∈ N such that setting
E3 = E2 ∩
no k(n)
[
[
En,k
n=1 k=1
we have
Z
(ϕ − e)dm − z3 < r1
4
E3
so that
Z
(ϕ − e)dm ∈ H
z4 :=
E3
and z4 +
r1
X1 ⊂ H.
8
Note that in this way ϕ reduces to a simple function on E3 : ϕ|E3 =
k(n)
no X
X
xn,k 1En,k .
n=1 k=1
Observe now that, from the choice of τo , and since E3 ⊂ E2 , so (a) ∈ V whenever a ∈ E3 ;
then by iii), also ϕ(a) ∈ V for a ∈ E3 . Moreover, thanks to (6) and iii) again, we deduce
|uk (so (a)) − uk (ϕ(a))| <
r1
8
(7)
for each a ∈ E3 and for k = 1, . . . , no .
We shall now replace ϕ by another simple function ψ such that
j) ψ a so for a ∈ E3 ;
Z
r1
≤ .
jj) (ψ
−
e)dm
−
z
4
8
E3
According to how ϕ has been written on E3 , and applying (A.6) to each An , n = 1, . . . , no ,
k(n) no X
X
r1 r1
consider ψ =
xn,k + yn 1En,k so that kψ − ϕk ≤ . Hence
8
8
n=1 k=1
Z
Z
r1
(ψ − e)dm −
≤ m(E3 ) ≤ r1 ,
(ϕ
−
e)dm
8
8
E3
E3
and this proves jj).
Moreover, from the definition of µ = 1, we have that, for n = 1, . . . , no
r1
r1
r1 un xn,k + yn − un (xn,k ) ≥ (un )0+ (xn,k )(yn ) ≥
8
8
8
or else, for a ∈ E3 ,
un (ψ(a)) ≥ un (ϕ(a)) +
15
r1
.
8
Now, from (7) we deduce that
un (ψ(a)) ≥ un (so (a))
for a ∈ E3 , which is exactly the requirement in j). The proof is thus complete.
2
We shall need the following Lemma, concerning the structure of a semiconvex finitely additive
measure
Lemma 3.5 Under assumption (A.1) for every ε > 0 and every finite decomposition of any
set E ∈ Σ, say {E1 , . . . , E` } the exists a finer decomposition of E, {F1 , . . . , Fq } such that
m(F1 ) = . . . = m(Fq );
!
q
[
m E\
Fi < ε.
i=1
Proof. To fix ideas, suppose that m(E1 ) ≤ . . . ≤ m(E` ), and let ξ = m(E1 ). Then we can
write m(Ej ) = qj ξ + τj for every j = 2, . . . , ` with qj ∈ N and τj < ξ.
By (A.1) each set Ej contains exactly qj pairwise disjoint sets F1j , . . . , Fqjj of m-measure ξ,
!
qj
[
Fij = τj .
and such that m Ej \
i=1
Consider the ` − 1 numbers τ2 , . . . , τ` .
`
X
If their sum
τi < ε the decomposition D = {E1 , Fij , j = 2, . . . , `, i = 1, . . . , qj } fulfills our
i=2
requirement.
Otherwise choose n ∈ N such that the following 2(` − 1) relationships are satisfied
ξ
+ τj0 , j = 2, . . . , `
n
ε
, j = 2, . . . , `.
τj0 <
`−1
τj = qj0
We can divide each of the sets in the decomposition D into n pairwise disjoint sets, each
ξ
ξ
of m-measure , and find other qj0 pairiwise disjoint sets of m-measure
in each relative
n q
n
j
[
complement Ej \
Fij .
i=1
Then what is left in every Ej \
qj
[
Fij is a set of m-measure τj0 , and therefore the union is a
i=1
2
set of total m-measure at most ε.
Lemma 3.6 Under assumptions (A.1), (A.4), (A.5) and (A.6), if f ∈ C(E), we have that
I ∩ (−H o ) = Ø.
16
Proof. Suppose not. Then according to Lemma 3.4 and Lemmata 3.2 and 3.5 there exists a
q
q
X
[
simple selection of Γ ψ =
yi 1Gi such that, with the notation of (4), and setting G =
Gi ,
i=1
k=1
Z
(I)
(ψ − e)dm ∈ (−H o );
G
(II) m(Gi ) = ξ for i = 1, . . . , q;
(III) there exists no ∈ N such that G ⊂
no
[
Fk ;
k=1
(IV) for every i = 1, . . . , q there exists a unique k(i) ∈ {1, . . . , no } such that Gi ⊂ Fk(i) .
Then there are two sequences (αk )k ⊂]0, +∞), (uk )k ∈ W such that
Z
(ψ − e)dm =
G
∞
X
−αk (wk − uk )
(8)
k=1
and the series in the right-hand side is absolutely convergent.
Setting
R
edm
ei = Gi
, i = 1, . . . , q
ξ
and remembering that W = −W , (8) can be rewritten as
q
X
yi ξ −
i=1
or else
q
X
i=1
q
X
ei ξ =
i=1
yi +
−αk (wk − uk )
k=1
∞
X
αk
k=1
∞
X
ξ
(wk − uk ) =
q
X
ei .
(9)
i=1
From Lemma 3.2 we have that kwk − uk k > δ, and therefore
αk
αk
δ<
kwk − uk k
ξ
ξ
which says that
∞
X
αk converges, and thus
k=1
∞
X
αk wk and
k=1
∞
X
αk uk both converge: let wo and
k=1
uo be their sums.
We can rewrite (9) as
q
X
yi + w − u =
i=1
q
X
i=1
17
ei
(10)
wo
uo
and u = . In exactly the same way as in Rustichini-Yannelis paper we
ξ
ξ
can assume without loss of generality that u ∈ X + and by means of the Riesz Decompositon Property, we can find, for every q-tuple ϑ = (ϑ1 , . . . , ϑq ) in the q-simplex ∆, q values
u1 (ϑ), . . . , uq (ϑ) in X + such that
 q

 X
ui (ϑ) = u

 i=1
ui (ϑ) ≤ yi + ϑi w, i = 1, . . . , q,
where w =
with the ui ’s uniquely determined by means of the Riesz Decomposition Property.
Define now the q maps yei : ∆ → X + as yei (ϑ) = yi + ϑi w − ui (ϑ) and consequently consider
the q maps δi : ∆ → [0, +∞) defined
as δi (ϑ)
= dist (yei (ϑ), yi + Γk(i) ) where k(i) is defined
[ αk
in (IV) above, and Γk =
α w+ W .
ξ
α>0
αk
Note that since wo ≥ wk , we have that a is w, W -proper for a ∈ Ak .
ξ
Choose now n1 ∈ N such that n1 r > kuo k, and consider the map f : ∆ → ∆ defined as




 ϑi + αk(i) δi 

f (ϑ) = 
q


X


1 + n1
δi
i=1
,
i=1,...,q
and note that f is continuous, since for every ϑ the ui ’s are uniquely determined, and hence
the maps ui (ϑ) are continuous; then f admits a fixed point ϑ∗ ; in ϑ∗ there holds
αk(i) δi =
ϑ∗i n1
q
X
δi ,
(11)
j=i
and, summing up,
q
X
δi αk(i) = n1
i=1
q
X
(12)
δi .
i=1
q
X
Now, exactly as done by Rustichini and Yannelis, one proves that supposing
δi (ϑ∗ ) = 0
i=1
αk(i)
one reaches a contradiction, for the pair w,
W is a properness pair for the preference
ξ
in Fk(i) .
18
Therefore (12) can be rewritten as
q
X
δi
αk(i) = n1 .
X
δj
(13)
q
i=1
j=1
Multiplying for every index i formula (11) by αk(i) we get
2
αk(i)
δi
=
ϑ∗i αk(i) n1
q
X
δj
j=1
that is
2
αk(i)
δi
X
= ϑ∗i αk(i) n1 .
q
(14)
δj
j=1
Summing up over the indexes i, and remembering (13) and the convexity of t 7→ t2 we reach

2
n1
q
X
i=1
ϑ∗i αk(i)
=
q
X
i=1
 q

X δi

δi
2
 = n21 ,
αk(i) ≥ 
α
k(i)
q
q


X
X
 i=1

δj
δj
j=1
whence
q
X
j=1
ϑ∗i αk(i) ≥ n1 .
(15)
i=1
Moreover, since αk(i) 6= 0 for each i = 1, . . . , q, we immediately have that
J := {i ∈ {1, . . . , q} such that ϑ∗i = 0} = {i ∈ {1, . . . , q} such that δi (ϑ∗ ) = 0},
and with exactly the same reasoning of Rustichini-Yannelis (1991), also that for i ∈ J,
ui (ϑ∗ ) = 0.
For i 6∈ J we have δi > 0, and so yei (ϑ∗ ) 6∈ yi + Γk(i) that is ϑ∗i w − ui (ϑ∗ ) 6∈ Γk(i) and hence
αk(i)
r
ui (ϑ∗ ) 6∈ ϑ∗i
W which in turn gives kui (ϑ∗ )k ≥ ϑ∗i αk(i) . Also it is clear that the same
ξ
ξ
inequality holds for i ∈ J since both sides vanish.
Thus, by virtue of (15)
q Z
q
X
rn1
rX ∗
∗
kuk =
ui (ϑ )dµ ≥
ϑi αk(i) ≥
ξ i=1
ξ
i=1 Ω
which is the same as kuo k ≥ rn1 which contradicts the choice of n1 . Thus we have reached
the final contradiction, and the proof is complete.
2
19
Our next result will state the convexity of the strong closure of I. However we shall not be
allowed to adopt the same result of Angeloni-Martellotti (2007) (i.e. Theorem 3.3), since
our commodity space is not reflexive, and therefore we cannot use Uhl’s result. Hence in
this case we shall need an ad hoc technique.
Lemma 3.7 Under assumption (A.1) the strong closure I is convex.
Proof. Observe first that in the proof of Lemma3.4 we have implicitly proven that for every
z ∈ I and every ε > 0 there exist a set E ⊂ A and a simple selection ψ of Γ|E such that
Z
z − (ψ − e)dm < ε.
E
It is now clear that the same technique can be applied to e, namely that we can choose
E ∈ Σ and another simple function χ such that
Z
(χ − e)dm < ε.
E
In conclusion each z ∈ I can be approximated within ε by the integral of a simple function.
Then we can directly apply the same technique as in Pucci-Vitillaro (1984) (Theorem 3.1)
where the countable additivity only counts for the strong non atomicity argument, and the
reflexivity of the space is never used.
2
We shall now apply the usual separation argment.
Theorem 3.1 Under assumptions (A.1), (A.4), (A.5) and (A.6), there exists a positive
functional x∗ ∈ X ∗ \ {0} such that x∗ (z) ≥ 0, for every z ∈ I.
Proof. From Lemma 3.6 and Lemma 3.7, I and −H o are two convex disjoint sets, with
−H o open, and hence it is possible to apply the Strong Separation Theorem. Therefore
there exists λ such that x∗ (x) ≥ λ > x∗ (y) for every x ∈ I and y ∈ −H o . As X + ⊂ I the
resulting functional x∗ is necessarily positive.
From Lemma 3.3 we have that 0 ∈ H o and hence 0 = x∗ (0) ≤ λ which concludes the proof.
2
Our final result is the Strassen formula (see Angeloni-Martellotti (2007)).
Theorem 3.2 (Strassen formula ) Under assumptions (A.1), (A.3), (A.4), (A.5) and (A.6)
Z
∗
inf{x (t), t ∈ I} =
idm
A
where x∗ is given by Theorem 3.1 and i(a) = inf{x∗ (t), t ∈ F (a)}, a ∈ A.
20
Simply note that to prove the Strassen formula we used the Castaing representation for
Γ. The same argument can be used in the present setting, since L1 (µ) is assumed to be
separable.
The final argument of the proof is standard and makes use of the remaining assumption
(A.2).
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22