Homogeneous function

A mathematical function is called homogeneous of degree n if the function value changes by a factor with a proportional change of all variables to the proportionality factor.

Functions of this type are important, for example, in economics and in the natural sciences.

Definition

A function on the real vector space k-dimensional

Is called homogeneous of degree n if and only if for all

Applies.

Examples from economics

Individual demand functions provide a link between prices p, income E and the quantities demanded for the goods x dar. If, for instance, following a currency changeover (from DM to Euro ) to a monetary halving of all prices and income, and this is of individuals fully taken into account (freedom from money illusion ), then the quantities demanded will not change. That is, it is

Demand functions are homogeneous of degree zero thus in the price and income variables (zero homogeneity ).

Production functions establish a relationship between inputs and the corresponding output y. If, then possibly in chemical manufacturing in each case proportional change (eg duplication ) of all inputs to a proportional change (doubling) of the output then:

Such a production function would then homogeneously with the homogeneity degree 1 (linear homogeneous).

Positive homogeneity

To weaken the definition from such that the condition

Calls only for positive, so you call the function positively homogeneous ( of degree n ).

For such functions are the Euler's theorem (or Euler 's theorem ) positively homogeneous functions of an equivalent characterization of:

A positively homogeneous function can thus be represented in a simple manner by means of the partial derivatives, and coordinates.

This fact is very often used in physics, especially in thermodynamics, since the occurring there intensive and extensive state variables are homogeneous functions of zeroth and first order. This is actually used in, for example, the derivation of Euler 's equation for the internal energy.

In economics, it follows from the Euler theorem for production functions of homogeneity degree 1 in factor prices and the goods price p

With linear homogeneous production functions, the value of the product is equal to the factor cost ( Ausschöpfungstheorem ).

Derivation of Euler 's theorem

Be a positively homogeneous function of degree n, so that with.

Construct now.

This results in:

Since,

Follows by the chain rule:

This expression can be used when the output equation is derived by partial.

As follows:

It follows the Euler theorem: