Convex preferences

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

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

An 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).

In relation to indifference curves this principle refers to the concept of diminishing marginal rate of substitution.

References

• Hal R. Varian. Intermediate Microeconomics A Modern Approach. New York: W.W. Norton & Company. ISBN 0-393-92702-4
• Mas-Colell, Andreu; Whinston, Michael; & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 978-0-19-507340-9

See also

• Convex function