Rubinstein2005-page64

# Rubinstein2005-page6 - Claim Any homothetic continuous and monotonic preference relation on the commodity bundle space can be represented by a

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October 21, 2005 12:18 master Sheet number 62 Page number 46 46 Lecture Four Figure 4.3 Homothetic preferences. Homothetic Preferences: A preference ± is homothetic if x ± y implies α x ± α y for all α 0. (See ﬁg. 4.3.) The preferences represented by ± k = 1, ... , K x β k k , where β k is positive, are homothetic. In fact, any preference relation represented by a utility function u that is homogeneous of any degree λ is homo- thetic. ( x ± y iff u ( x ) u ( y ) iff α λ u ( x ) α λ u ( y ) iff u x ) u y ) iff α x ± α y ). Note that lexicographic preferences are also homothetic.
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Unformatted text preview: Claim: Any homothetic, continuous, and monotonic preference relation on the commodity bundle space can be represented by a utility func-tion that is homogeneous of degree one. Proof: We have already proven that any bundle x has a unique bundle ( t ( x ) , . . . , t ( x )) on the main diagonal so that x ∼ ( t ( x ) , . . . , t ( x )) , and that the function u ( x ) = t ( x ) represents ± . By the assumption that...
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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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