Convex preferences: Difference between revisions

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{{Short description|Concept in economics}}

In [[economics]], '''convex preferences''' are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "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.

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In [[economics]], '''convex preferences''' are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". This implies that the consumer prefers a variety of goods to having more of a single good. The concept roughly corresponds to the concept of [[marginal utility#Diminishing marginal utility|diminishing marginal utility]] without requiring [[utility function]]s.

== Notation ==

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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>

== Alternative definition ==

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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> yx \neq zy</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>.

That is, for any two bundles that are viewed as being equivalent, a weighted average of the two bundles is better than each of these bundles.<ref name=Sanders>[{{cite web |url=http://njsanders.people.wm.edu/100A/Prefs_and_Utility_Examples.pdf |archivedate=March 20, 2013 |archiveurl=https://web.archive.org/web/20130320003154/http://njsanders.people.wm.edu/100A/Prefs_and_Utility_Examples.pdf |first=Nicholas J. |last=Sanders, "|title=Preference and Utility - Basic Review and Examples"]. |work=College of William & Mary }}</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>, then every weighted average of ''y'' and ''ס'' 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 [[Leontief utility function]]:

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

This preference relation is convex. PROOF{{em|Proof}}: suppose ''x'' and ''y'' are two equivalent bundles, i.e. <math>\min(x_1,x_2) = \min(y_1,y_2)</math>. If the minimum-quantity commodity in both bundles is the same (e.g. commodity 1), then this implesimplies <math>x_1=y_1 \leq x_2,y_2</math>. Then, any weighted average also has the same amount of commodity 1, so any weighted average is equivalent to <math>x</math> and <math>y</math>. If the minimum commodity in each bundle is different (e.g. <math>x_1\leq x_2</math> but <math>y_1\geq y_2</math>), then this implies <math>x_1=y_2 \leq x_2,y_1</math>. Then <math>\theta x_1 + (1-\theta) y_1 \geq x_1</math> and <math>\theta x_2 + (1-\theta) y_2 \geq y_2</math>, so <math>\theta x + (1-\theta) y \succeq x,y</math>. This preference relation is convex, but not strictly-convex.

3. A preference relation represented by [[linear utility]] functions is convex, but not strictly convex. Whenever <math>x\sim y</math>, every convex combination of <math>x,y</math> is equivalent to any of them.

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4. Consider a preference relation represented by:

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

This preference relation is not convex. PROOF{{em|Proof}}: let <math>x=(3,5)</math> and <math>y=(5,3)</math>. Then <math>x\sim y</math> since both have utility 5. However, the convex combination <math>0.5 x + 0.5 y = (4,4)</math> is worse than both of them since its utility is 4.

== 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]].

Convex preferences with their associated convex indifference mapping arise from [[Quasi-convex function|quasi-concave]] utility functions, although these are not necessary for the analysis of preferences. For example, Constant Elasticity of Substitution (CES) utility functions describe convex, homothetic preferences. CES preferences are self-dual and both primal and dual CES preferences yield systems of indifference curves that may exhibit any degree of convexity.<ref>{{Cite journal |last=Baltas |first=George |date=2001 |title=Utility-consistent Brand Demand Systems with Endogenous Category Consumption: Principles and Marketing Applications |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1540-5915.2001.tb00965.x |journal=Decision Sciences |language=en |volume=32 |issue=3 |pages=399–422 |doi=10.1111/j.1540-5915.2001.tb00965.x|url-access=subscription }}</ref>

== See also ==

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* [[Semi-continuous function]]

* [[Shapley–Folkman lemma]]

* [[Herman Wold|Wold, Herman O.]]

==References==

{{reflist}}