Rubinstein2005-page63

# Rubinstein2005-page63 - x | f x ≥ f y is convex Obviously...

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October 21, 2005 12:18 master Sheet number 61 Page number 45 Consumer Preferences 45 As usual, the above property also has a stronger version: Strict Convexity: The preference relation % satisFes strict convexity if for every a % y , b % y , a 6= b and λ ( 0, 1 ) imply that λ a + ( 1 λ) b Â y . Example: The preferences represented by x 1 + x 2 satisfy strict convexity. The preferences represented by min { x 1 , x 2 } and x 1 + x 2 satisfy con- vexity but not strict convexity. The lexicographic preferences satisfy strict convexity. The preferences represented by x 2 1 + x 2 2 do not sat- isfy convexity. We now look at the properties of the utility representations of convex preferences. Quasi-Concavity: A function u is quasi-concave if for all y the set { x | u ( x ) u ( y ) } is convex. The term’s name derives from the fact that for any concave func- tion f and for any y the set
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Unformatted text preview: { x | f ( x ) ≥ f ( y ) } is convex. Obviously, if a preference relation is represented by a utility func-tion, then it is convex iff the utility function is quasi-concave. How-ever, the convexity of % does not imply that a utility function representing % is concave. (Recall that u is concave if for all x , y , and λ ∈ [ 0, 1 ] , we have u (λ x + ( 1 − λ) y ) ≥ λ u ( x ) + ( 1 − λ) u ( y ) .) Special Classes of Preferences Often in economics, we limit our discussion of consumer prefer-ences to a class of preferences possessing some additional special properties. ±ollowing are some examples of “popular” classes of preference relations discussed in the literature....
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