Rubinstein2005-page38

Rubinstein2005-page3 - x and superior to y Since Y is dense there exists z 1 ∈ Y such that x  z 1  y Similarly there exists z 2 ∈ Y such

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October 21, 2005 12:18 master Sheet number 36 Page number 20 20 Lecture Two Proposition: Assume that X is a convex subset of < n that has a countable dense subset Y .I f % is a continuous preference relation, then % has a (continuous) utility representation. Proof: By a previous claim we know that there exists a function v : Y [− 1, 1 ] , which is a utility representation of the preference relation % restricted to Y . For every x X , de±ne U ( x ) = sup { v ( z ) | z Y and x  z } . De±ne U ( x ) =− 1 if there is no z Y such that x  z , which means that x is the minimal element in X . (Note that for z Y it could be that U ( z )< v ( z ) .) If x y , then x  z iff y  z . Thus, the sets on which the supre- mum is taken are the same and U ( x ) = U ( y ) . If x  y , then by the lemma there exists z in X such that x  z  y . By the continuity of the preferences % there is a ball around z such
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Unformatted text preview: x and superior to y . Since Y is dense, there exists z 1 ∈ Y such that x  z 1  y . Similarly, there exists z 2 ∈ Y such that z 1  z 2  y . Finally, U ( x ) ≥ v ( z 1 ) (by the de±nition of U and x  z 1 ), v ( z 1 ) > v ( z 2 ) (since v represents % on Y and z 1  z 2 ), and v ( z 2 ) ≥ U ( y ) (by the de±nition of U and z 2  y ). Bibliographic Notes Recommended readings : Kreps 1990, 30–32; Mas-Colell et al. 1995, chapter 3, C. Fishburn (1970) covers the material in this lecture very well. The example of lexicographic preferences originated in Debreu (1959) (see also Debreu 1960, in particular Chapter 2, which is available online at http://cowles.econ.yale.edu/P/cp/p00b/p0097.pdf.)...
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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