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By Ewald C.-O.

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Example text

E. find x ∈ bi (p, Mi ) such that Ui (x) ∩ bi (p, Mi ) = ∅. 51 Such an x is called a demand vector and is a solution to the consumers problem given prices p and income Mi . Since we interpret p and Mi as parameters this gives us another correspondence, the so called demand correspondence for consumer ”i” di : Rm + × R+ →→ Xi (p, Mi ) → { demand vectors for consumer ”i” given prices p and income Mi }. A supply vector1 y ∈ Yj for supplier ”j” specifies the quantities of each commodity supplied ( positive entry ) and the amount of each commodity used as an input ( negative entry ).

Then there exists a continuous selection of γ. 6 Fixed Point Theorems for Correspondences One can interpret Brouwer’s fixed point theorem as a special case of a fixed point theorem for correspondences where the correspondence is in fact given by a map. That this is not the only case where fixed points of correspondences are guaranteed is shown in this section. The main fixed point theorem for correspondences is the Kakutani fixed point theorem. It will follow from the following theorem. 1. Let K ⊂ Rm be compact, nonempty and convex and µ : K →→ K a correspondence.

E. y ∈ γ(x). t. U is itself the interior of a polytope. , yn ))◦ . Since γ has open sections the sets Vi := γ − [{yi }] are open for all i. Clearly for all z ∈ Vi we have yi ∈ γ(z) and furthermore x ∈ Vi for all i. The set V := ni=1 Vi is nonempty and open and furthermore contains x. W := V × U is open in X × Y . Let (x′ , y ′ ) ∈ W . Then yi ∈ γ(x′ )∀i. , yn )◦ ⊂ γ(x′ ) ⇒ (x′ , y ′ ) ∈ Gr(γ). Therefore W ⊂ Gr(γ) and W is an open neighborhood of (x, y) ∈ Gr(γ). 8. Let γ : X →→ Y be a correspondence.

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