Convex preferences: Difference between revisions

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{{Short description|Concept in economics}}

In [[economics]], '''convex preferences''' is a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "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''' is a property ofare an individual's ordering of various outcomes, whichtypically roughlywith correspondsregard to the ideaamounts 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 ==

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 ==

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 \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 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>

=== 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>,

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

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

Thus the preference ordering <math>\succeq</math> is strictly convex ifi.e., 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 strictlyat betterleast thanas good as the third bundle.

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

== Strict convexity ==

A preference relation <math>\succeq</math> is ''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>,

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 ==

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.

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 \equivsucceq 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 whenever

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

then 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 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> and <math>z \geq x </math>, then every weighted average of ''y'' and ''zס'' 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 [[ordinalLeontief utility]] function]]:

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

This preference relation is convex. {{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 implies <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.

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. {{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>

==References==

* [[Hal R. Varian]]; ''Intermediate Microeconomics A Modern Approach''. New York: W. W. Norton & Company. ISBN 0-393-92702-4

*

== See also ==

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* [[Semi-continuous function]]

* [[Shapley–Folkman lemma]]

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

==References==

[[Category:Consumer theory]]

{{reflist}}

[[Category:Economics curves]]

[[Category:Microeconomics]]

[[Category:Utility function types]]