Convex preferences: Difference between revisions

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In [[economics]], '''convex preferences''' is 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#Diminishing marginal utility|diminishing marginal utility]] without requiring [[utility function]]s.

== Notation ==

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

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

Similarly, <math>\succ</math> can be translated as 'is strictly better than' (in preference satisfaction).

it is the case that

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

== Definition ==

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.

Use ''x'', ''y'', and ''z'' to denote three consumption bundles (combinations of various quantities of various goods). Formally, a preference relation <math>\succeq</math> on the [[consumption set]] ''X'' 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>,

and for every <math>\theta\in[0,1]</math>:

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

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

That is, the preference ordering ''P''<math>\succeq</math> 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.

== Strict convexity ==

Moreover,A preference relation <math>P\succeq</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>,

and for every <math>\theta\in[0,1]</math>:

it is also true that

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

hereThus the preference ordering <math>\succsucceq</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 strictly better than the third bundle.

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.

== Relation to indifference curves and utility functions ==

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]].