Convex preferences: Difference between revisions

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== 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).

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

== Definition ==

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and 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 any

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and 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]]; Whinston, Michael; & Green, Jerry (1995). ''Microeconomic Theory''. Oxford: Oxford University Press. ISBN 978-0-19-507340-9</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 \in X</math> where <math>y \succeq x </math>

and 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>[http://www.econ.ucla.edu/sboard/teaching/econ11_09/econ11_09_lecture2.pdf Simon Board. "Preferences and Utility]</ref>

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

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

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