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". ItThe concept roughly corresponds to [[marginal utility#The “law” of diminishing 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 Formally''x'', if''y'', and <math>\succeq</math>''z'' to isdenote three consumption bundles (combinations of various quantities of various goods). Formally, a preference relation ''P'' 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>.

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.

<math>\succeq</math> is ''strictly'' convex if for any

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

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

then it is also true that

:<math>\thetax, y, +z (1-\theta)in zX</math> where <math>y \succsucceq x </math> for any, <math>z \thetasucceq \inx (0</math>,1) and <math> y \neq z</math>.,

then it is also true that

It can be translated as: 'is better than relation' (in preference satisfaction).

:<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.

A convex shaped indifference curve displaying convex preferences thus means that the agent prefers, in terms of consumption bundles, averages over extremes (agents express a basic inclination for diversification).

A set of [[Convex function|convex]]-shaped [[indifference curve]]s displays convex preferences: Given a convex indifference curve containing the set of all bundles (of two or more goods) that are all viewed as equally desired, the set of all goods bundles that are viewed as being at least as desired as those on the indifference curve is a [[convex set]].

Convex preferences with their associated convex indifference mapping arise from [[Quasi-convex function|quasi-concave]] utility functions, although these are not necessary for the analysis of preferences.

==References==