Convex preferences: Difference between revisions

Revision as of 19:52, 24 September 2008

Revision as of 19:53, 24 September 2008

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Comparable to the greater-than-or-equal-to [[Order theory#Partially ordered sets|ordering]] relation <math>\geq</math> for real numbers, the notation <math>\succeq</math> below can be translated as: 'is at least as good as' (in [[preference]] satisfaction). Formally, if <math>\succeq</math> is a preference relation on the [[consumption set]] X, then <math>\succeq</math> is [[Convex set|convex]] if for any <math>x, y, z \in X</math> where <math>y \succeq x </math> and <math>z \succeq x </math>, then it is the case that <math>\theta y + (1-\theta) z \succeq x </math> for any <math> \theta \in [0,1] </math>.

<math>\succeq</math> is ''strictly'' convex if for any <math>x, y, z \in X</math> where <math>xy \succeq yx </math> and <math>z \succeq x </math>, and <math> y \neq z</math> then it is also true that <math>\theta y + (1-\theta) z \succ x </math> for any <math> \theta \in (0,1) </math>. It can be translated as: 'is better than relation' (in preference satisfaction).

An indifference curve displaying convex preferences thus means that the agent prefers, in terms of consumption bundles, averages over extremes.