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 [[Convexcalled set|''convex]]'' if for any

:<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) x \succ x </math>.

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

That is, the preference ordering <math>\succeq</math> is convex if for any two goods bundles that are each viewed as being at least as good as a third bundleequivalent, a weighted average of the two bundles is alsobetter viewedthan aseach beingof atthese leastbundles.<ref asname=Sanders>[http://njsanders.people.wm.edu/100A/Prefs_and_Utility_Examples.pdf goodNicholas asJ. theSanders, "Preference and Utility - Basic Review thirdand bundleExamples"].</ref>

=== 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) z \succeq x </math>.

That is, the preference ordering <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 ==

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Thus the preference ordering <math>\succeq</math> 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.

== 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>

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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

== See also ==