10.2 Homogeneous functions

A function f : n,xf(x) is called homogeneous of degree h , if for all x n applies:
f(kx) = khf(x)for allk 0+

If the input variables x ib are multiplied by a positive number k 0, the function value is multiplied by the factor kh.

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
f(x)is homogeneous of the degreehx1 x1f(x)+...+xn xnf(x) = hf(x).
Homogeneity of the derivatives
f(x)is homogeneous of the degreeh xif(x)is homogeneous of the degreeh1 for all i.

(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