201Lecture82006

201Lecture82006 - Economics 201B–Second Half Lecture 8...

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Unformatted text preview: Economics 201B–Second Half Lecture 8 Existence of Walrasian Equilibrium (Continued) Proposition 1 (17.C.1) Debreu-Gale-Kuhn-Nikaido Lemma Sup- pose z : Δ → R L is a function satisfying 1. continuity 2. Walras’ Law ∀ p ∈ Δ p · z ( p ) = 0 3. bounded below: ∃ x ∈ R L ∀ p ∈ Δ z ( p ) ≥ x 4. Boundary Condition: If p n → p where p ∈ Δ \ Δ , then | z ( p n ) | → ∞ Then there exists p ∗ ∈ Δ such that z ( p ∗ ) = 0 1 Outline of proof: • Define a correspondence f : Δ → Δ (so f ( p ) ∈ 2 Δ ) by f ( p ) = { q ∈ Δ : q · z ( p ) ≥ q · z ( p ) for all q ∈ Δ } f identifies the goods in highest excess demand. • Extend the domain of f to Δ to make it have closed graph. • Verify that if p ∗ ∈ f ( p ∗ ), then p ∗ ∈ Δ and z ( p ∗ ) = 0. • Check that f satisfies the hypotheses of Kakutani’s Theorem. • By Kakutani’s Theorem, there exists p ∗ ∈ Δ such that p ∗ ∈ f ( p ∗ ), so p ∗ ∈ Δ and z ( p ∗ ) = 0. Details of proof : • Define a correspondence f : Δ → Δ (so f ( p ) ∈ 2 Δ ) by f ( p ) = { q ∈ Δ : q · z ( p ) ≥ q · z ( p ) for all q ∈ Δ } f identifies the goods in highest excess demand. ∀ ` = ` z ( p ) ` > z ( p ) ` ⇒ f ( p ) = { (0 , . . . , , 1 , , . . . , 0) } ↑ ` z ( p ) ` = z ( p ) ` 1 > z ( p ) ` for all ` ∈ { ` , ` 1 } ⇒ f ( p ) = { (0 , . . . , , α...
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201Lecture82006 - Economics 201B–Second Half Lecture 8...

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