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 anywhenever

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

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

:<math>\theta y + (1-\theta) z \succeq 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 as the third bundle.

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

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

andthen for every <math>\theta\in(0,1)</math>:

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

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.<ref name=Varian>[[Hal R. Varian]]; ''Intermediate Microeconomics A Modern Approach''. New York: W. W. Norton & Company. {{ISBN|0-393-92702-4}}</ref><ref name=Mas>[[Andreu Mas-Colell|Mas-Colell, Andreu]]; [[Michael Whinston|Whinston, Michael]]; & [[Jerry Green (economist)|Green, Jerry]] (1995). ''Microeconomic Theory''. Oxford: Oxford University Press. {{ISBN|978-0-19-507340-9}}</ref>

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

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

:<math>\theta y + (1-\theta) x \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>{{cite web |url=http://www.econ.ucla.edu/sboard/teaching/econ11_09/econ11_09_lecture2.pdf |first=Simon |last=Board |title=Preferences and Utility |date=October 6, 2009 |work=Econ 11. Microeconomic Theory. Autumn 2009 |publisher=University of California, Los Angeles }}</ref>

A preference relation is called '''strictly convex''' if for anywhenever

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

andthen 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>.