6Pareto Optimality

6Pareto Optimality - 1 ,...,I } . Adding (2) over all i ,...

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Pareto Optimality Let A be a set (of alternatives) and i } I i =1 a family of preference relations on A . ˆ a A is Pareto Optimal if there is no a 0 A with a 0 ± i ˆ a for i = 1 ,...,I and a 0 ² j ˆ a for some j ∈ { 1 ,...,I } . Suppose that, for each i = 1 ,...,I , ± i is represented by a utility function u i on A . Lemma 1 Any solution to max a A I X i =1 u i ( a ) (1) is Pareto Optimal. Proof : Suppose that ˆ a is not Pareto Optimal. Then there is an a 0 A with u i ( a 0 ) u i a ) (2) for i = 1 ,...,I and u j ( a 0 ) > u j a ) (3) for some j ∈ {
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Unformatted text preview: 1 ,...,I } . Adding (2) over all i , and using (3), we get I X i =1 u i ( a ) > I X i =1 u i (ˆ a ) so that ˆ a does not solve (1). ± More generally, let W : R I → R be a strictly increasing function. Any maximizer of W ( u 1 ( a ) ,...,u I ( a )) on A is Pareto optimal . The lemma is the special case in which W = ∑ u i . 1...
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This note was uploaded on 11/19/2010 for the course ECON 202 taught by Professor Schlee during the Spring '10 term at ASU.

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