201Lecture62006 - Economics 201BSecond Half Lecture 6 The...

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Economics 201B–Second Half Lecture 6 The Second Welfare Theorem in the Arrow Debreu Economy Theorem 1 (Second Welfare Theorem) (Pure Exchange Case) If x is Pareto Optimal in a pure exchange economy, with strongly monotone, continuous, convex preferences, there exists a price vector p and an income transfer T such that ( p ,x ,T ) is a Walrasian Equilibrium with Transfers. Outline of Proof : Let B i = { x 0 i x i : x 0 i ± i x i } B = I X i =1 B i = { b 1 + ··· + b I : b i B i } Then 0 6∈ B (if it were, we’d have a Pareto improvement). By Minkowski’s Theorem, Fnd p 6 =0suchthat inf p · B 0 Show ± R L + \{ 0 } ² B i and hence p 0. Show inf p · B i =0foreach i . DeFne T to make x i a±ordable at p : T i = p · x i p · ω i Show I i =1 T i =0and x i Q i ( p ,T ) 1
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Use strong monotonicity to show that p ± 0. Show p ± 0 Q i ( p ,T )= D i ( p ,T ) Now, for the details: Let B i = { x 0 i x i : x 0 i ² i x i } B = I X i =1 B i = { b 1 + ··· + b I : b i B i } Claim: 0 6∈ B If 0 B , there exists b i B i such that I X i =1 b i =0 Let x 0 i = x i + b i Since x 0 i x i = b i B i ,wehave x 0 i ² i x i I X i =1 x 0 i = I X i =1 ( x i + b i ) = I X i =1 x i + I X i =1 b i = I X i =1 x i ω Therefore, x 0 is a feasible allocation, x 0 Pareto improves x ,so x is not Pareto Optimal, contradiction. Therefore, 0 6∈ B .
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201Lecture62006 - Economics 201BSecond Half Lecture 6 The...

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