201Lecture32006

201Lecture32006 - Economics 201BSecond Half Lecture 3 The...

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Unformatted text preview: Economics 201BSecond 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 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 = 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 i + T i , so p = p ( x 1 +...
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201Lecture32006 - Economics 201BSecond Half Lecture 3 The...

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