Curso de competencias para el ámbito audiovisual

Let K = G(γ ) ⊆ G denote the trading set. Consider an individual trader with endowment vectore∈RK₊and so the effective trading setJ= {g∈ K |eg>0}. Let

P_KJ :=

p ∈PKpJ >0J

(83) denote the subset of the price simplex inRK where at least one goodg ∈ J has a positive price. For the effective trading setJ, aJ -regularnet demand correspondence was defined in Sect.6.2as a multifunctionP_K0 p→→Z(p;Z)⊂RK satisfying:

(i) [budget exhaustion] p z=0 for all p∈ P_K0 and allz∈ Z(p;Z).

(ii) [feasibility]:z−(e¯J,0K\J)for all p∈ P_K0 and for allz∈ Z(p;Z).

(iii) [continuity]: The correspondenceZhas a graph

ΓZ:= {(p,z)∈ P_K0 ×RK |z∈ Z(p;Z)} (84) which is a relatively closed subset of P_K0 ×RK.

(iv) [boundary condition]: If the infinite sequence(pn,zn)n∈NinΓZis such that pn

converges to a point p¯on the boundary of P_KJ, then the sum_g∈Kzn_g→ +∞.

Lemma 1 Let e ∈ RK _{be any endowment vector satisfying e e, let J¯ := {g ∈}

K | eg > 0} denote the associated endowment set, and let be the restriction

toRK × {0G\K}of any preference ordering onR₊G that is continuous and strictly monotone. Define the extended Walrasian budget correspondence

P_KJ p→→BJ(p;e):= {z∈RK |z−e and p z≤0} (85)

along with the extended Walrasian net demand correspondence

P_KJ p→→ZJ(p;e):= {z∈ BJ(p;e)z˜z⇒ p˜z>0} (86) with graph ΓJ _{:= {(} p,z)∈ P_KJ×RK |z∈ ZJ(p;e)} (87) Then: 1. ZJ(p;e)= ∅if p∈bdPK =PK\P_K0;

2. ΓJ is a relatively closed subset of P_KJ ×RK;

3. the restrictionZof p→→ZJ(p;e)to the subdomain P_K0is a J -regular net demand correspondence.

Proof (i) Ifz∈ BJ(p;e)withp z<0, then there existsz˜zsatisfying pz˜<0. Monotonicity implies that˜zz, and soz∈/ ZJ(p;e). The contrapositive implies

the budget exhaustion condition requiring thatp z=0 for everyz∈ ZJ(p;e).

(ii) The feasibility condition is an obvious implication of (86).

(iii) Suppose that(pn,zn)n∈N is a sequence of points inΓJ which converges to a

limit(p¯,z¯)with p¯ ∈ P_KJ. Then the definitions (86) and (85) evidently imply thatz¯ ∈ BJ(p¯;e). Also, because of the budget exhaustion property (i) that we

have already demonstrated, one has pnzn =0 for alln ∈N. Taking the limits

asn → ∞, it follows that p¯z¯=0.

Consider any net trade vectorz˜−esatisfyingz˜ ¯z. For any scalarγ ∈(0,1), define the convex combination

zγ :=γz˜+(1−γ )(−e) (88) Continuity of preferences now implies that: (a) there exists a scalarγ ∈(0,1) such thatzγ ¯z; (b)zγ znfor all largen ∈ N. Because(pn,zn)∈ ΓJ for

alln ∈ N, combining the definition (86) with budget exhaustion implies that

pnzγ > 0 = pnznfor all largen ∈ N. Taking the limit asn → ∞implies

that p z¯ γ ≥ 0= ¯pz¯. By hypothesis, one haseJ 0J andp¯ ∈ P_KJ, implying that p¯J >0J and so p¯JeJ >0. Because in addition p¯z¯ =0, it follows that

¯

p(¯z+e) >0 and also, because of the definition (88), that

γ p¯(˜z− ¯z)= ¯p[zγ −(1−γ )(−e)−γz¯] ≥(1−γ )p¯(¯z+e) >0 Hence p¯z˜>0.

Finally, since this argument holds for everyz˜ ¯z, definition (86) implies that

¯

z∈ ZJ(p¯;e). This proves thatΓJis a relatively closed subset ofP_KJ×RK, and also thatΓ_Zdefined by (84) is a relatively closed subset ofP_K0 ×RK.

(iv) Suppose that(pn,zn)n∈N is any sequence of points in ΓJ with the property

that (pn)n∈N converges to a limit p¯ on the boundary of PKJ. Then p¯g = 0

for at least one good g ∈ K. Hence, strict monotonicity of preferences implies that ZJ(p¯;e) = ∅. Because the graph ΓJ of the correspondence

P_KJ p→→ZJ(p;e) is a relatively closed subset of P_KJ ×RK, it follows that the sequence(zn)n∈Ncannot have any limit point. In particular, it can have

no convergent subsequence, and so_g∈K|zn,g| → ∞as n → ∞. Because

zn−efor alln∈N, it follows that

g∈Kzn,g→ +∞asn→ ∞.

The space of regular net demand correspondences

This section of the appendix uses the abbreviation AB to indicate the frequent page references to the book byAliprantis and Border(1999).

Let(RK,T)denote the finite-dimensional Euclidean spaceRK with its Euclid- ean metric topologyT. We give an alternative characterization of regular net demand correspondences using the standard mathematical concept of theone-point compactifi- cationof(RK_{,T)— see AB, p. 56. This is defined as the topological space(R}K

∞,T∞) where:

1. the setR_∞K :=RK∪ {∞}is defined by appending to the Euclidean spaceRK one extra “point at infinity” denoted by∞;

2. the topologyT_∞is made up of the topologyT ofRK, together with the comple- mentR_∞K\V of each compact subsetV ⊂RK.

Note that each compact subset of(RK,T)remains a compact subset of(R_∞K,T_∞). Note too that(RK,T)is locally compact, in the sense that every point ofRK has a compact neighbourhood. It is also separable—i.e. the closure of a countable subset, such as the setQK of all points with rational coordinates. It follows (AB, Corollary 3.33, p. 88) that(R_∞K,T_∞)is metrizable. Letd_∞denote any metric. Then(R_∞K,d_∞)

is a compact metric space.

We now give the price simplex P ⊂RK the Euclidean metricdP ofRK, and we

give the Cartesian productP×R_∞K the sup metric defined by

d((p,z), (p˜,z˜)):=max{dP(p,p˜),d_∞(z,z˜)}

This makes(P×RK_∞,d)a compact separable metric space.

Following (36), letZK denote the set of all net demand correspondencesP_K0 p→→Z(p)⊂RK that areJ-regular for some J ⊆K. We identify each correspon- denceZ ∈ ZK with the closureΓ¯_Zin the metric space(PK ×R_∞K,d)of its graph

ΓZ⊂P_K0×RK_{. Because of the boundary condition (iv), this closure is easily seen to}

be

¯

ΓZ=ΓZ∪(bdPK× {∞})

This identification establishes an injectionZ→Γ_Zfrom the setZK into the space K(PK×RK_∞)of compact subsets of the metric space(PK×RK_∞,d). Giving the latter

space its Hausdorff metricdHmakes(K(P×R_∞K,dH)a Polish space (AB, p. 111). Then we can define the metricdZonZKso that the distancedZ(Z,Z)between any

pair of regular demand correspondencesZ,Z ∈ZK equals the Hausdorff distance

dH(Γ¯_Z,Γ¯_Z)between the closures of their respective graphs. This construction makes (ZK_,d

Z)a Polish space.