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Rubinstein2005-page59 - • The preFerence represented by x...

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October 21, 2005 12:18 master Sheet number 57 Page number 41 Consumer Preferences 41 Monotonicity: The relation satisfies monotonicity if for all x , y X , if x k y k for all k , then x y , and if x k > y k for all k , then x y . In some cases, we will further assume that the consumer is strictly happier with any additional quantity of any commodity. Strong Monotonicity: The relation satisfies strong monotonicity if for all x , y X if x k y k for all k and x = y , then x y . Of course, in the case that preferences are represented by a util- ity function, preferences satisfying monotonicity (or strong mono- tonicity) are represented by monotonic increasing (or strong monotonic increasing) utility functions. Examples: The preference represented by min { x 1 , x 2 } satisfies monotonicity but not strong monotonicity
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Unformatted text preview: . • The preFerence represented by x 1 + x 2 satisfes strong mono-tonicity . • The preFerence relation | x − x ∗ | satisfes nonsatiation , a related property that is sometimes used in the literature: For every x ∈ X and For any ε > 0 there is some y ∈ X that is less than ε away From x so that y  x . Every monotonic preFerence re-lation satisfes nonsatiation, but the reverse is, oF course, not true. Continuity We will use the topological structure oF < K + (induced From the stan-dard distance Function d ( x , y ) = p ∑ ( x k − y k ) 2 ) to apply the defni-tion oF continuity discussed in Lecture 2. We say that the preFerences...
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