Externalities, Nonconvexities, and Fixed Points

We say that m(z) ∈ C w ∗ f(X) is minimally essential in C w ∗ f(X) if for any otherm 0 (z) ∈ C w ∗ f(X) essential for Γ(z), m 0 (z) ⊆ m(z) implies that m 0 (z) =m(z).We will denote by E Γ(z) (C w ∗ f(X)) the collection of sets in C w ∗ f(X) essentialfor Γ(z), and we will denote by M Γ(z) (C w ∗ f(X)) the collection of sets in C w ∗ f(X)minimally essential for Γ(z).Minimal essentiality relative to P w ∗ f(X) is defined similarly. In the followingLemma we establish that for any minimal USCO, ϕ(·) ∈ [Γ] ρZ -w ∗, Γ ∈ U ρ Z -w ∗,andfor any z, ϕ(z) is minimally essential in P w ∗ f(X) for the set, Γ(z). Thus, we willshow thatϕ(z) ∈ M Γ(z) (P w ∗ f(X)) for all z ∈ Z.The following result relating minimal USCOs and minimal essential sets, for minimalUSCOs defined on the parameter space is similar to our prior result, relatingminimal USCOs and minimal essential sets, for minimal USCOs definedonthehyperspaceof Ky Fan sets.Lemma 14 (ϕ(z) is minimally essential in P w ∗ f(X) for Γ(z))Suppose assumptions [A-1] hold and let Γ ∈ U ρZ -w ∗ be an USCO. Then for anyminimal USCO, ϕ ∈ [Γ] ρZ -w ∗,andforanyz ∈ Z, the subset given by ϕ(z) ∈ P w ∗ f(X)is minimally essential for Γ(z) in P w ∗ f(X).Proof. Because ϕ(·) is an USCO, for each z ∈ Z, ϕ(z) is essential for Γ(z). Infact,ϕ(z) is essential for ϕ(z) - i.e., for all ε > 0 there exists a δ > 0, such thatϕ(z 0 ) ∩ B w ∗(ε, ϕ(z)) 6= ∅ for all z 0 ∈ B δ (z).To see that ϕ(z) is minimal, suppose that for some z 0 ∈ Z there is a nonempty,closed, and proper subset m(z 0 ) of ϕ(z 0 ) such that for any ε > 0 there exists a δ > 0such that for all z contained in B δ (z 0 )ϕ(z) ∩ B w ∗(ε,m(z 0 )) 6= ∅. (51)Because m(z 0 ) is a proper subset of ϕ(z 0 ),wecanchooseε 0 > 0 so that ϕ(z 0 )∩B w ∗(ε 0 ,m(z 0 )) is a proper subset of ϕ(z 0 ), and by (51) we can choose δ 0 > 0 so thatdefine the mapping q(·) as follows:ϕ(z) ∩ B w ∗(ε 0 ,m(z 0 )) 6= ∅ for all z ∈ B δ 0(z 0 ),q(z) :=½ ϕ(z) ∩ Bw ∗(ε 0 ,m(z 0 )) if z ∈ B δ 0(z 0 )ϕ(z)otherwise.By Lemma 2(ii) in Anguelov and Kalenda (2009), q(·) is an USCO, and moreover,Grq(·) is a proper subset of Grϕ(·), a contradiction of the fact that ϕ(·) ∈ [Γ] ρZ -w ∗.Thus, for each z ∈ Z, ϕ(z) is minimally essential in P w ∗ f(X) for ϕ(z), and becauseϕ(z) ⊆ Γ(z), ϕ(z) is minimally essential in P w ∗ f(X) for Γ(z).29