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.
Contents
Notation
Comparable to the greater-than-or-equal-to ordering relation
Similarly,
Definition
Use x, y, and z to denote three consumption bundles (combinations of various quantities of various goods). Formally, a preference relation
and for every
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 than the third bundle.
A preference relation
and for every
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.
Alternative definition
Use x and y to denote two consumption bundles. A preference relation
and for every
That is, if a bundle y is preferred over a bundle x, then any mix of y with x is still preferred over x.
A preference relation is called strictly convex if for any
and for every
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.
Examples
1. If there is only a single commodity type, then any weakly-monotonically-increasing preference relation is convex. This is because, if
2. Consider an economy with two commodity types, 1 and 2. Consider a preference relation represented by the following Leontief utility function:
This preference relation is convex. PROOF: suppose x and y are two equivalent bundles, i.e.
3. A preference relation represented by linear utility functions is convex, but not strictly convex. Whenever
4. Consider a preference relation represented by:
This preference relation is not convex. PROOF: let
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.