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 diminishing marginal utility without requiring utility functions.
Notation
Comparable to the greater-than-or-equal-to ordering relation for real numbers, the notation below can be translated as: 'is at least as good as'. Similarly, can be translated as 'is strictly better than', and Similarly, can be translated as 'is equivalent to'.
Definition
Use x, y, and z to denote three consumption bundles. Formally, a preference relation on the consumption set X is called convex if for any 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 as the third bundle. A preference relation is called strictly convex if for any 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 is viewed as being strictly better than the third bundle.
Alternative definition
Use x and y to denote two consumption bundles. A preference relation is called convex if for any 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, then every weighted average of y and ס is also. 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. : supposex and y are two equivalent bundles, i.e.. If the minimum-quantity commodity in both bundles is the same, then this imples. Then, any weighted average also has the same amount of commodity 1, so any weighted average is equivalent to and. If the minimum commodity in each bundle is different, then this implies. Then and, so. 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, every convex combination of is equivalent to any of them. 4. Consider a preference relation represented by: This preference relation is not convex. : let and. Then since both have utility 5. However, the convex combination is worse than both of them since its utility is 4.
A set of convex-shaped indifference curves displays convex preferences: Given a convex indifference curve containing the set of all bundles 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.
