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indifference curves can be funky shaped, this is specific to ones representing convex preferences/quasi-concave utility |
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In [[economics]], '''convex preferences''' refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes". It roughly corresponds to [[marginal utility#The “law” of diminishing marginal utility|the "law" of diminishing marginal utility]] but uses modern theory to represent the concept.
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]] satisfaction). Formally, if <math>\succeq</math> is a preference relation on the [[consumption set]] X, then <math>\succeq</math> is [[Convex set|convex]] if for any
:<math>x, y, z \in X</math> where <math>y \succeq x </math> and <math>z \succeq x </math>, then it is the case that
:<math>\theta y + (1-\theta) z \succeq x </math> for any <math> \theta \in [0,1] </math>. <math>\succeq</math>
is ''strictly'' convex if for any
:<math>x, y, z \in X</math> where <math>y \succeq x </math> and <math>z \succeq x </math>, and <math> y \neq z</math>
then it is also true that
:<math>\theta y + (1-\theta) z \succ x </math> for any <math> \theta \in (0,1) </math>.
It can be translated as: 'is better than relation' (in preference satisfaction).
A convex shaped indifference curve displaying convex preferences thus means that the agent prefers, in terms of consumption bundles, averages over extremes (agents express a basic inclination for diversification).
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