201Lecture32006

# 201Lecture32006 - Economics 201B–Second Half Lecture 3...

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Unformatted text preview: Economics 201B–Second Half Lecture 3 The Welfare Theorems in the Edgeworth Box Walrasian Equilibrium with Transfers : An income trans- fer is a vector T ∈ R 2 such that T 1 + T 2 = 0 (budget balance). B i ( p, T ) = { x ∈ R 2 + : p · x ≤ p · ω i + T i } D i ( p, T ) = { x ∈ B i ( p, T ) : x i y for all y ∈ B i ( p, T ) } A Walrasian Equilibrium with Transfers is a triple ( p, x, T ) where • x is an exact allocation; • T is an income transfer; • x i ∈ D i ( p, T ) ( i = 1 , 2). Observe that ( p, x, 0) is a Walrasian Equilibrium with Transfers if and only if ( p, x ) is a Walrasian Equilibrium. 1 Theorem 1 (Weak First Welfare Theorem, Edgeworth Bo In the Edgeworth Box, every Walrasian Equilibrium with Trans- fers is weakly Pareto Optimal. Proof: Let ( p, x, T ) be a Walrasian Equilibirum with Transfers. Suppose x is not weakly Pareto Optimal. Then we can find x ∈ ( R 2 + ) 2 such that x 1 + x 2 = ¯ ω x i i x i ( i = 1 , 2) x i i x i implies x i 6 i x i ; since x i ∈ D i ( p, T ), p · x i > p · ω i + T i , so p · ¯ ω = p · ( x 1 +...
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201Lecture32006 - Economics 201B–Second Half Lecture 3...

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