Convex preferences: Difference between revisions

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In [[economics]], '''convex preferences''' refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes". The concept roughly corresponds to the concept of [[marginal utility#The “law” of diminishingDiminishing marginal utility|the "law" of diminishing marginal utility]] but uses modern theory to represent the concept without requiring the use of [[utility function]]s.

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 (economics)|preference]] satisfaction). Use ''x'', ''y'', and ''z'' to denote three consumption bundles (combinations of various quantities of various goods). Formally, a preference relation ''P'' on the [[consumption set]] ''X'' is [[Convex set|convex]] if for any

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:<math>\theta y + (1-\theta) z \succeq x </math> for any <math> \theta \in [0,1] </math>.

That is, the preference ordering ''P'' is convex if for any two goods bundles that are each viewed as being at least as good as a third bundle, a weighted average of the two bundles is also viewed as being at least as good as the third bundle.

Moreover, <math>P</math> is ''strictly'' convex if for any

:<math>x, y, z \in X</math> where <math>y \succeq x </math>, <math>z \succeq x </math>, and <math> y \neq z</math>,

it is also true that

:<math>\theta y + (1-\theta) z \succ x </math> for any <math> \theta \in (0,1); </math>

here <math>\succ</math> can be translated as 'is better than' (in preference satisfaction). Thus the preference ordering ''P'' is strictly convex if for any two distinct goods bundles that are each viewed as being at least as good as a third bundle, a weighted average of the two bundles (including a positive amount of each bundle) is viewed as being better than the third bundle.

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==References==

* [[Hal R. Varian]]; ''Intermediate Microeconomics A Modern Approach''. New York: W. W. Norton & Company. ISBN 0-393-92702-4

* [[Andreu Mas-Colell|Mas-Colell, Andreu]]; Whinston, Michael; & Green, Jerry (1995). ''Microeconomic Theory''. Oxford: Oxford University Press. ISBN 978-0-19-507340-9