By Ewald C.-O.

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Each consumer has an income at some rate which we denote with Mi . e. Mi ≥ 0. We assume that he cannot purchase more then his income, that is he cannot take any credit. This determines the budget set for player i. {x ∈ Xi : p(x) ≤ Mi } These budget sets depend on the two parameters price p and income Mi and hence can be interpreted as correspondences 50 bi : Rm + × R+ →→ Xi (p, Mi ) → {x ∈ Xi |p(x) ≤ Mi } We do now come to a very important point in game theory and mathematical economics, so called utility.

Exercise ! 2. (Sequential Characterization of Hemi Continuity) Let X ⊂ Rm , Y ⊂ Rk and γ : X →→ Y be a correspondence. 1. Assume ∀x ∈ X that γ(x) is compact. Then γ uhc ⇔ for every sequence xn → x and yn ∈ γ(xn ) there exists a convergent subsequence ynk → y and y ∈ γ(x). 2. γ is lhc ⇔ xn → x and y ∈ γ(x) implies there exists a sequence yn ∈ γ(xn ) with yn → y. Proof. 1. “⇒” : Assume xn → x and yn ∈ γ(xn ). Since γ(x) is compact it has a bounded neighborhood V . t. γ(U ) ⊂ V . t. ∀n ≥ n0 we have xn ∈ U .

3 Then W1 ∩ W2 is a neighborhood of (x, g) disjoint from Gr(E), since for all (˜ x, g˜) ∈ W1 ∩ W2 we have Ui (˜ x, g˜) ∩ Si (˜ x, g˜) = ∅ for the simple reason that Ui (˜ x, g˜) contains V . Since in both cases we can 3 V is an open neighborhood of zi , then definition lhc 58 find neighborhoods of (g, x) which are still contained in Gr(E)c we have that Gr(E)c is open and hence Gr(E) is closed. 1 ( second part ) that E is uhc. That it has compact values is clear. 3. Let K ⊂ Rm be compact, G ⊂ Rk and let γ : K × G →→ K be closed correspondence.

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