### 10.2 Homogeneous functions

Definion:
A function $f:{ℝ}^{n}↦ℝ,\stackrel{⃗}{x}↦f\left(\stackrel{⃗}{x}\right)$ is called homogeneous of degree $h\in ℝ$, if for all $\stackrel{⃗}{x}\in {ℝ}^{n}$ applies:

If the input variables $\stackrel{⃗}{x}{\sum }_{i}^{b}$ are multiplied by a positive number $k\ge 0$, the function value is multiplied by the factor ${k}^{h}$.

Homogeneous functions have special properties, which are briefly listed below and illustrated on the following pages for functions of two variables. For homogeneous functions, the following two theorems apply in particular:

Euler’s theorem
Homogeneity of the derivatives

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de