Externalities, Nonconvexities, and Fixed Points

Because E n → E 0 , we have by Proposition 3.2.2 in Klien and Thompson (1984)hw ∗ ×w ∗that(X × X)\E 0 = Li w ∗ ×w ∗{(X × X)\En }. (82)Given (82), we have, therefore, that£D(E 0 ) × X ¤ ∩ £ (X × X)\E 0¤ = £ D(E 0 ) × X ¤ ∩ Li w ∗ ×w ∗{(X × X)\En }. (83)But now we have a contradiction. We haveThus,(y 0n ,x 0 ) −→w ∗ ×w ∗ (y0 ,x 0 ) ∈ £ D(E 0 ) × X ¤ ∩ Li w ∗ ×w ∗{(X × X)\En }.(y 0 ,x 0 ) ∈ £ D(E 0 ) × X ¤ ∩ £ (X × X)\E 0¤ .But by assumption,(y 0 ,x 0 ) ∈ £ D(E 0 ) × X ¤ ∩ £ E 0¤ ,a contradiction.(2) It is easy to see that D(K(z)) = R(K(z)) = Φ(z). Thus (Z1) holds. Letz ∈ Z be any parameter vector. We must show that K(z) ∈ S. RecallthatK(z) :={(y, x) ∈ Φ(z) × Φ(z) :ϕ(z,(y, x)) ≤ 0},and note that for all y ∈ Φ(z), ϕ(z, (y, y))=0. Thus, (Z2) holds.Toseethat(Z3)holdsobservethatbecauseϕ(z, (·,x)) is quasiconcave, y ∈ Φ(z)such that(y, x) /∈ K(z)is given by the set,{y ∈ Φ(z) :ϕ(z,(y, x)) > 0},and this set is convex (or empty).11 Appendix 3: The Proof of Lemma 11Because S is an h w ∗ ×w∗-compact metric space and hence a Baire space and becausen(·) is a minimal USCO, there is an h w ∗ ×w∗-dense subset, D, ofS such n(E) is asingleton for all E ∈ D (see Anguelov and Kalenda, 2009, Lemma 7). Moreover, forany E e ∈ S, D eE := D ∩ S eE is dense in S eE .Suppose n(·) ∈ [N(·)] but that for some E e ∈ S, n eE (·) is not a minimal USCO ofNE e (·) on S eE . Then, there is a minimal USCO m(·) of n eE (·) (i.e., m(·) ∈ [n (·)]) sucheEthat for some E 0 ∈ SE e , m(E0 ) is a proper subset of nE e (E0 ). By Lemma 11, becausem(·) ∈ [n eE (·)], wehaveforanyE 0 ∈ S eE , m(E 0 ) ∈ M n eE (E 0 )(S eE ),implyingthatforall ε 0 > 0 there exists δ 0 > 0 such that for allE ∈ B hw ∗ ×w ∗ (δ 0 ,E 0 ) ∩ SE e , ⎫⎬(84)n eE (E) ∩ B w ∗(ε 0 ,m(E 0 ⎭)) 6= ∅.51