Convex preferences

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In economics, convex preferences are a property of utility functions commonly represented in an indifference curve as a bulge toward the origin for normal goods (for unwanted goods, the curve bulges away from the origin). It roughly corresponds to the "law" of diminishing marginal utility but uses modern theory to represent the concept.

Comparable to the greater-than-or-equal-to ordering relation $\geq$ for real numbers, the notation $\succeq$ below can be translated as: 'is as at least as good as' (in preference satisfaction). Formally, if $\succeq$ is a preference relation on the consumption set X, then $\succeq$ is convex if for any $x, y, z \in X$ where $y \succeq x$ and $z \succeq x$ , then it is the case that $\theta y + (1-\theta) z \succeq x$ for any $\theta \in [0,1]$ .

$\succeq$ is strictly convex if for any $x, y, z \in X$ where $y \succeq x$ and $z \succeq x$ , and $y \neq z$ then it is also true that $\theta y + (1-\theta) z \succ x$ for any $\theta \in (0,1)$ . It can be translated as: 'is better than relation' (in preference satisfaction).