Concave and convex functions

Definition

Let $S \subseteq \mathbb{R}^n$ be convex and $f : S \to \mathbb{R}$. Then $f$ is concave if for all $x, x' \in S$ and all $\lambda \in [0, 1]$,

$f\big(\lambda x + (1-\lambda)x'\big) \ge \lambda f(x) + (1-\lambda) f(x').$

$f$ is convex if the reverse inequality ($\le$) holds, and strictly concave or convex if the inequality is strict whenever $x \ne x'$ and $\lambda \in (0, 1)$.

Read geometrically: the chord joining two points on the graph lies below a concave function and above a convex one. The domain must be convex so that $\lambda x + (1-\lambda)x'$ lies in $S$ and $f$ of it is defined — exactly the segment condition from Lesson 1.

Key properties

Sign symmetry. $f$ is convex if and only if $-f$ is concave.

Upper contour sets. If $f$ is concave, then for every $c$ the set $\{x \in S : f(x) \ge c\}$ is convex (proved below). This is the bridge back to Lesson 1: concave utility produces convex preferences.

In economics

First-order conditions become sufficient. The result the course is built toward: if $f$ is concave and $S$ is convex, any interior point $x^*$ with $\nabla f(x^*) = 0$ is a global maximum — no second-order test, no comparison of candidates.

Worked examples

Example 1 — an affine function is both concave and convex. Let $f(x) = a \cdot x + b$. Then

$f\big(\lambda x + (1-\lambda)x'\big) = \lambda\,(a \cdot x) + (1-\lambda)\,(a \cdot x') + b = \lambda f(x) + (1-\lambda) f(x').$

The defining inequality holds with equality, so both the concave ($\ge$) and convex ($\le$) conditions hold. Affine functions are the only functions that are simultaneously concave and convex.

Example 2 — a concave function’s upper contour sets are convex. Let $f : S \to \mathbb{R}$ be concave on a convex set $S$, fix $c$, and set $U = \{x \in S : f(x) \ge c\}$. Take $x, x' \in U$, so $f(x) \ge c$ and $f(x') \ge c$. For $\lambda \in [0, 1]$, concavity gives

$f\big(\lambda x + (1-\lambda)x'\big) \ge \lambda f(x) + (1-\lambda) f(x') \ge \lambda c + (1-\lambda) c = c,$

so $\lambda x + (1-\lambda)x' \in U$. Hence $U$ is convex — which is precisely why concave utility gives convex preferences.

Exercises

1. Show directly from the definitions that $f$ is convex if and only if $-f$ is concave.
2. Show that if $f$ and $g$ are concave and $\alpha, \beta \ge 0$, then $\alpha f + \beta g$ is concave.
3. Show that if $f$ is convex then each lower contour set $\{x \in S : f(x) \le c\}$ is convex.
4. Show that the pointwise minimum $\min\{f, g\}$ of two concave functions is concave.