Rubinstein2005-page60 - monotonicity and con-tinuity can be...

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October 21, 2005 12:18 master Sheet number 58 Page number 42 42 Lecture Four Figure 4.1 % satisfy continuity if for all a , b X ,if a  b , then there is an ε> 0 such that x  y for any x and y such that d ( x , a )<ε and d ( y , b )<ε . Existence of a Utility Representation Debreu’s theorem guarantees that any continuous preference rela- tion is represented by some (continuous) utility function. If we assume monotonicity as well, we then have a simple and elegant proof: Claim: Any consumer preference relation satisfying
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Unformatted text preview: monotonicity and con-tinuity can be represented by a utility function. Proof: Let us Frst show that for every bundle x , there is a bundle on the main diagonal (having equal quantities of all commodi-ties), such that the consumer is indifferent between that bun-dle and the bundle x . (See Fg. 4.1.) The bundle x is at least as good as the bundle 0 = ( 0, . . . , 0 ) . On the other hand, the bundle...
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