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201Lecture52006

# 201Lecture52006 - Economics 201BSecond Half Lecture 5 The...

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Economics 201B–Second Half Lecture 5 The Welfare Theorems in the Arrow Debreu Economy Local nonsatiation : The preference relation i on the con- sumption set X i is locally nonsatiated if, for every x i X i and every ε > 0, there exists x i X i such that | x i x i | < ε and x i i x i . Note that this is a substantial weakening of monotonicity Important especially with production, since we want to al- low for input goods which provide no direct consumption utility Theorem 1 (First Welfare Theorem) If preferences are lo- cally nonsatiated and ( x , y , p , T ) is a Walrasian Equilibrium with Transfers, then ( x , y ) is Pareto Optimal. Proof: Let W i = p · ω i + J j =1 θ ij p · y j + T i W i is the income available to person i . Observe that I i =1 W i = I i =1 p · ω i + I i =1 J j =1 θ ij p · y j + I i =1 T i = p · I i =1 ω i + J j =1 I i =1 θ ij p · y j + 0 = p · ¯ ω + J j =1 p · y j 1

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By the definition of Walrasian Equilibrium with Transfers, ( x , y ) is a feasible allocation. Suppose ( x , y ) is not Pareto Optimal. Then there is a feasible allocation ( x , y ) such that x i i x i ( i = 1 , . . . , I ) x i i x i for some i, WLOG i = 1 x 1 1 x 1 x 1 B 1 ( p , y , T ) p · x 1 > W 1 We claim that p · x i
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201Lecture52006 - Economics 201BSecond Half Lecture 5 The...

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