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Unformatted text preview: Economics 201BSecond 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 > 0, there exists x i X i such that | x i x i | < 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 i + J X j =1 ij p y j + T i W i is the income available to person i . Observe that I X i =1 W i = I X i =1 p i + I X i =1 J X j =1 ij p y j + I X i =1 T i = p I X i =1 i + J X j =1 I X i =1 ij p y j + 0 = p + J X j =1 p y j 1 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...
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This note was uploaded on 08/01/2008 for the course ECON 201B taught by Professor Anderson during the Spring '06 term at University of California, Berkeley.
- Spring '06