Convex preferences

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 diminishing marginal utility without requiring utility functions.

Notation

Comparable to the greater-than-or-equal-to ordering relation ${\displaystyle \geq }$  for real numbers, the notation ${\displaystyle \succeq }$  below can be translated as: 'is at least as good as' (in preference satisfaction).

Similarly, ${\displaystyle \succ }$  can be translated as 'is strictly better than' (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 ${\displaystyle \succeq }$  on the consumption set X is called convex if for any

${\displaystyle x,y\in X}$  where ${\displaystyle y\equiv x}$ 

and for every ${\displaystyle \theta \in [0,1]}$ :

${\displaystyle \theta y+(1-\theta )x\succ x}$ .

${\displaystyle \theta y+(1-\theta )x\succ y}$ .

That is, the preference ordering ${\displaystyle \succeq }$  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.^[1]

= Alternative definition

Use x and y to denote two consumption bundles. A preference relation ${\displaystyle \succeq }$  is called convex if for any

${\displaystyle x,y,z\in X}$  where ${\displaystyle y\succeq x}$  and ${\displaystyle z\succeq x}$ ,

and for every ${\displaystyle \theta \in [0,1]}$ :

${\displaystyle \theta y+(1-\theta )z\succeq x}$ .

Strict convexity

A preference relation ${\displaystyle \succeq }$  is strictly convex if for any

${\displaystyle x,y,z\in X}$  where ${\displaystyle y\succeq x}$ , ${\displaystyle z\succeq x}$ , and ${\displaystyle y\neq z}$ ,

and for every ${\displaystyle \theta \in [0,1]}$ :

${\displaystyle \theta y+(1-\theta )z\succ x}$ 

Thus the preference ordering ${\displaystyle \succeq }$  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 ${\displaystyle y\geq x}$  and ${\displaystyle z\geq x}$ , then every weighted average of y and z is also ${\displaystyle \geq x}$ .

2. Consider an economy with two commodity types, 1 and 2. Consider a preference relation represented by the following ordinal utility function:

${\displaystyle u(x_{1},x_{2})=x_{1}\cdot x_{2}}$ 

Relation to indifference curves and utility functions

A set of convex-shaped indifference curves 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-concave utility functions, although these are not necessary for the analysis of preferences.

References

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

See also

• Convex function
• Level set
• Quasi-convex function
• Semi-continuous function
• Shapley–Folkman lemma
• Wold, Herman O.

1. Nicholas J. Sanders, "Preference and Utility - Basic Review and Examples".