Convex Preference and utility function

Question

How does convex preference imply quasi concave utility function (Intuitively and Mathematically)?

Answer 1

Let $X$ be the convex set of alternatives, let $\succeq$ be a preference relation and let $u(.)$ be a utility function that reflects these preferences, which means that $u(x) \ge u(y)$ if and only if $x \succeq y$.

The preference relation $\succeq$ is convex if

• For all $y$, $x$ and $z$ in $X$, if $x \succeq y$ and $z \succeq y$ then for all $\alpha \in [0,1]$, $\alpha x + (1-\alpha) y \succeq y$.

• Equivalently, for all $y$ in $X$, the set of all bundles that are at least as good as $y$, is a convex set.

• Equivalently, for all $y$ in $X$, the set $U_y = \{x \in X| x \succeq y\}$ is convex.

The utility function $u$ is Quasi-concave if

• For all $x$, $y$ and $z$ in $X$, if $u(x) \ge u(y)$ and $u(z) \ge u(y)$, then for all $\alpha \in [0,1]$, $u(\alpha x + (1-\alpha) z)\ge u(y)$.

• Equivalently, for all $y$ in $X$, the set of bundels that give at least as much utility as $y$ is a convex set.

• Equivalently, for all $y \in X$ the set $V_y = \{x \in X| u(x) \ge u(y)\}$ is convex.

However the set $U_y$ and $V_y$ are the same. $$ V_y = \{x \in X| u(x) \ge u(y)\} = \{x \in X| x \succeq y\} = U_y. $$ As such, convexity of preferences is identical to quasi-concavity of the utility function that reflects these preferences.

Intuitively, convexity of preferences means that all upper contour sets are convex sets. This is by definition equal to convexity of all sets that give at least as much utility level than some given fixed bundle. This is identicial to the notion of quasi-concavity.