Sembrado, haza, sementero

A.1 Mathematical preliminaries

A correspondence ϕ : X → 2^Y is a function from X to the family of all subsets of Y . A correspondence ϕ : X → 2^Y is compact-valued (nonempty-valued, convex-valued) if ϕ(x) is a compact (nonempty, convex) subset of Y for every x ∈ X. Let X and Y be sets. The graph G_Φ of a correspondence Φ : X → 2^Y is the set Gr_Φ = {(x, y) ∈ X × Y : y ∈ Φ(x)}.

If X and Y are topological spaces, φ : X → 2^Y is said to be lower semi continuous (l.s.c.) if the set {x ∈ X : φ(x) ∩ V 6= ∅} is open in X for every open subset V of Y . The correspondence φ : X → 2^Y is said to be upper semi continuous (u.s.c.) if the set {x ∈ X : φ(x) ⊂ V } is open in X for every open subset V of Y . It is known that a closed-valued correspondence with compact Hausdorff range space has a closed graph if and only if it is upper hemi continuous.

If (X, α) and (Y, β) are measurable spaces, Φ is said to have a measurable graph if GrΦbelongs to the product σ-algebra α ⊗ β. We are often interested in the situation where (X, α) is a measurable space, Y is a topological space and β = β(Y ) is the Borel σ-algebra of Y . A correspondence φ : X → 2^Y from a measurable space (X, α) into a topological space Y is said to be lower measurable if {x ∈ X : φ(x) ∩ V 6= ∅} ∈ α for every V open in Y . It is known that if the correspondence φ is lower measurable and closed valued, then it has a measurable graph. Furthermore, if the measure space (X, α, µ) is complete and the correspondence φ from (X, α, µ) to 2^Y has a measurable graph, then it is lower measurable ( see for example Castaing-Valadier (1977)).

Let (Ω, F , µ) be a measurable space and Y be a separable metric space. From Kuratowski and Ryll-Nardzewski Measurable Selection Theorem it follows that if φ : Ω → 2^Y is a lower measurable, closed and nonempty-valued correspon-dence, then there exists a measurable function f : Ω → Y such that f (ω) ∈ φ(ω) for all ω ∈ Ω.

A.2 Proof of the auxiliary theorem

For all i ∈ I, let B_i : Ω × 4 → 2^IR`+ be the random budget set of agent i, defined by

B_i(ω, q) = {x_i(ω) ∈ X_i(ω) : q · x_i(ω) ≤ q · e_i(ω)}.

Let D_i : Ω × 4 → 2^IR`+ be the maximin random demand set of agent i, defined by

Di(ω, q) = {xi(ω) ∈ Bi(ω, q) : ui(ω, xi(ω)) = max

yi∈B_i(ω,q)ui(ω, yi)}.

Lemma A.1 For each i, D_i(·, ·) has a jointly measurable graph, that is GrDi(·,·) ∈ F ⊗ B(4) ⊗ B(IR^`₊).

Proof: Since for all i ∈ I and ω ∈ Ω, e_i(ω) is an interior point of X_i(ω), it follows that B_i(ω, ·) is lower hemi continuous. Moreover, it is easy to verify that B_i(ω, ·) has closed graph, with a compact range space, therefore it is upper hemi continuous. Consequently, for all i ∈ I and ω ∈ Ω, Bi(ω, ·) is continuous. From Berge’s maximum theorem, it follows that for each ω ∈ Ω, Di(ω, ·) is upper semi continuous with non empty and compact values. Consequently, D_i(ω, ·) has closed graph and since Ω is finite, it follows that it has a jointly measurable graph. That is, for all i ∈ I,

Gr_Di(·,·) = (ω, q, x_i) ∈ Ω × 4 × IR^`₊ : ω ∈ Ω and (q, x_i) ∈ Gr_Di(ω,·)

= [

ω∈Ω

{ω} × Gr_Di(ω,·) .

Since {ω} and Gr_Di(ω,·) are closed, it follow that Gr_Di(·,·) ∈ F ⊗ B(4) ⊗ B(IR^`₊).  Define the aggregate maximin random excess demand for the economy E by

Z(ω, q) =

n

X

i=1

D_i(ω, q) −

n

X

i=1

e_i(ω).

Finally let W : Ω → 2⁴be the (free-disposal) equilibrium correspondence defined by¹⁶

W(ω) = {q ∈ 4 : Z(ω, q) ∩ IR^`₋6= ∅}.

16IR^{`</sup><sub>−</sub> denotes the negative cone of IR<sup>`}.

Since each D_i(·, ·) has jointly measurable graph and Ω is finite¹⁷, from Castaing-Valadier (1977, p.80), it follows that D_i(·, ·) is jointly lower measurable, and so is Z(·, ·). Therefore the set B = {(ω, q) ∈ Ω × 4 : Z(q, ω) ∩ IR^`₋ 6= ∅} belongs to F ⊗ B(4). Notice that the set B coincides with the graph of the correspondence W, i.e.,

B = GW = {(ω, q) ∈ Ω × 4 : q ∈ W(ω)}.

Hence, GW belongs to F ⊗ B(4), i.e., W(·) has a measurable graph, and since Ω is complete, it follows from Castaing-Valadier (1977, p.80) that W(·) is also lower measurable.

Since for each fixed ω ∈ Ω, D_i(ω, ·) is upper semi continuous, convex, nonempty and compact valued, so is Z(ω, ·). Furthermore, for any fixed ω ∈ Ω, Z(ω, ·) satisfies Walras law, i.e., for all q ∈ 4 and for every z ∈ Z(ω, q), q · z ≤ 0.

It follows from the Gale-Nikaido-Debreu lemma that, for each fixed ω ∈ Ω, W(ω) is nonempty, i.e., W(·) is a nonempty valued correspondence.

Since for each fixed ω ∈ Ω, Z(ω, ·) is upper semi continuous, it follows that the set {q ∈ 4 : Z(ω, q) ∩ V 6= ∅} is closed for all V closed subset of IR^{`</sup>. Hence for all ω ∈ Ω the set {q ∈ 4 : Z(ω, q) ∩ IR<sub>−</sub><sup>`} 6= ∅} = W(ω) is closed.

Consequently, W(·) satisfies the conditions of the Kuratowski and Ryll-Nardzewski Measurable Selection Theorem and hence it admits a measurable selection. This means that there exists a F -measurable function p^∗ : Ω → 4 such that p^∗(ω) ∈ W(ω) for all ω ∈ Ω, i.e.,

Z(ω, p^∗(ω)) ∩ IR^`₋ 6= ∅ for all ω ∈ Ω.

Notice that this means that:

(1⁰) for all i and ω, x^∗_i(ω) maximizes ui(ω, ·) subject to p^∗(ω) · x^∗_i(ω) ≤ p^∗(ω) · ei(ω) (2⁰) X

i∈I

x^∗_i(ω) ≤X

i∈I

e_i(ω) for all ω ∈ Ω.

17Since Ω is finite, it follows that (Ω, F , π_i) is complete for each i ∈ I.

Consider for all i ∈ I, G_i = F_i∨ σ(p^∗). To complete the proof we must show that (p^∗, x^∗) is a free disposal maximin REE. Clearly, condition (2⁰) above holds and for each i ∈ I and ω ∈ Ω, x^∗_i(ω) ∈ B_i(ω, p^∗(ω)). Therefore, all it remains to be shown is that condition (ii) in Definition 3.3 is satisfied. Assume to the contrary that condition (ii) in Definition 3.3 does not hold, then there exist an agent i ∈ I, a state ¯ω ∈ Ω and an allocation y ∈ L_Xi such that y ∈ B_i^∗(¯ω, p^∗) and

u^REE_i (¯ω, y) > u^REE_i (¯ω, x^∗_i).

Since Ω is finite, there exists a state ω ∈ E^Gi(¯ω) such that ui(ω, x^∗_i(ω)) = u^REE_i (¯ω, x^∗_i). This implies that

u_i(ω, y(ω)) ≥ u^REE_i (¯ω, y) > u^REE_i (¯ω, x^∗_i) = u_i(ω, x^∗_i(ω)),

which contradicts condition (1⁰). Hence, (p^∗, x^∗) is a free disposal maximin REE.



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