Convex preferences: Difference between revisions

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== Definition ==

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 called '''convex''' if for any

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

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

:<math>\theta y + (1-\theta) xz \succsucceq x </math>.

i.e, for any two bundles that are each viewed as being at least as good as a third bundle, a weighted average of the two bundles is viewed as being at least as good than the third bundle.

:<math>\theta y + (1-\theta) x \succ y </math>.

A preference relation <math>\succeq</math> is called ''strictly''strictly convex''' if for any

That is, the preference ordering <math>\succeq</math> is convex if for any two goods bundles that are each viewed as being equivalent, a weighted average of the two bundles is better than each of these bundles.<ref name=Sanders>[http://njsanders.people.wm.edu/100A/Prefs_and_Utility_Examples.pdf Nicholas J. Sanders, "Preference and Utility - Basic Review and Examples"].</ref>

:<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>:

=== Alternative definition ==

:<math>\theta y + (1-\theta) z \succ x </math>

Use ''x'' and ''y'' to denote two consumption bundles. A preference relation <math>\succeq</math> is called ''convex'' if for any

i.e, for any two distinct 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.

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

=== Alternative definition ==

Use ''x'' and ''y'' to denote two consumption bundles. A preference relation <math>\succeq</math> is called '''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>:

:<math>\theta y + (1-\theta) zx \succeq x </math>.

That is, if a bundle ''y'' is preferred over a bundle ''x'', then any mix of ''y'' with ''x'' is still preferred over ''x''. <ref name=Board>[http://www.econ.ucla.edu/sboard/teaching/econ11_09/econ11_09_lecture2.pdf Simon Board. "Preferences and Utility]</ref>

== Strict convexity ==

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

A preference relation is called '''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>,

:<math>x, y \in X</math> where <math>y \equiv x </math>

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

:<math>\theta y + (1-\theta) zx \succ x </math>.

:<math>\theta y + (1-\theta) x \succ y </math>.

Thus the preference ordering <math>\succeq</math>That is strictly convex if, for any two distinct goods bundles that are each viewed as being at least as good as a third bundleequivalent, a weighted average of the two bundles (includingis abetter positivethan amounteach of eachthese bundle)bundles.<ref isname=Sanders>[http://njsanders.people.wm.edu/100A/Prefs_and_Utility_Examples.pdf viewedNicholas asJ. beingSanders, strictly"Preference betterand thanUtility the- thirdBasic bundleReview and Examples"].</ref>

== Examples ==

1. If there is only a single commodity type, then any weakly-monotonically-increasing preference relation is convex. This is because, if <math>y \geq x </math> and <math>z \geq x </math>, then every weighted average of ''y'' and ''z'' is also <math>\geq x </math>.

2. Consider an economy with two commodity types, 1 and 2. Consider a preference relation represented by the following [[ordinal utility]] function:

:<math>u(x_1,x_2) = x_1 \cdot x_2</math>