Cobb-Douglas functions | Wiskunde op Tilburg University

Definition: A function of the form

$z(x,y) = kx^{\alpha}y^{1-\alpha}, (x>0, y>0)$
where

$k>0$ ,

$0 \leq \alpha \leq 1$ , is called a Cobb-Douglas function.

Remark 1: Sometimes, also functions of the form

$z(x,y)=kx^{\alpha}y^{\beta}$ , with

$k>0$ ,

$\alpha>0$ and

$\beta >0$ are called Cobb-Douglas functions. Given this definition also

$z(x,y) = 4x^2y^4$ is a Cobb-Douglas function for example.

Remark 2: Cobb-Douglas functions often appear in microeconomics dues to desirable properties.

Example: An example of a Cobb-Douglas function is

$U(x,y) = 2x^{\tfrac{1}{3}}y^{\tfrac{2}{3}}$ .