A function is called by degrees homogeneous the n, if during proportinaler change of all variables by the porportionality factor \ alpha itself the function value around the factor \. Formally:

A function on that k-dimensional real vector space

\ Phi: \ mathbb {R} ^k \ rightarrow \ mathbb {R}

is called homogeneous from degrees the n exactly if to all \ alpha applies, x_i \ in \ mathbb {R}

\ Phi (\ alpha x_1, \ ldots, \ alpha x_k) = \ alpha^n \ cdot \ Phi (x_1, \ ldots, x_k)

Functions of this type are importantly e.g. in the economic science and in the natural sciences.

Examples from the economic science:

Individual demand curves x=x (p, E) represent a connection between prices p, incomes E and the inquired quantities of x after the goods. If it comes e.g. in the course of a currency reform (from DM to euro) to a halving according to amount of all prices and the incomes and if this by the individuals is completely considered (liberty of money illusion), then the inquired quantities will not change. That is called it applies

\ alpha^0 x=x (\ alpha p, \ alpha E)

Demand curves are thus homogeneous of degrees of zero in the variables of prices and incomes.

Production functions y=f (x_1, \ dots, x_n) manufacture a connection between inputs x_i and that associated output y. It comes then in chemistry production when proportional change (e.g. duplication) all inputs to an appropriate proportional change (duplication) of the output possibly applies in such a way in each case:

\ alpha^1 y=f (\ alpha x_1, \ dots, \ alpha x_n)

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

Euler theorem

The Euler sentence over homogeneous functions represents an equivalent characterisation:

n \ cdot \ Phi (x_1, \ ldots, x_k) = \ sum_ {i=1} ^k \ frac {\ partial \ Phi} {\ partial x_i} \ cdot x_i \; \; \ Leftrightarrow \; \; n \ cdot \ Phi (\ vec {x}) = \ vec {x} \ cdot \ nabla \ Phi (\ vec {x})

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

This fact is very frequently used in physics, particularly in thermodynamics, since the intensive and extensive variables of state arising there are homogeneous functions of zeroth and/or first degree.

In the economic science follows the goods price p from the Euler theorem for production functions of the homogeneity degree of 1 with the factor prices q_i and

y= f (x_1, \ ldots, x_k) = \ sum_ {i=1} ^k \ frac {\ partial f} {\ partial x_i} \ cdot x_i = \ sum_ {i=1} ^k \ frac {q_i} {p} \ cdot x_i \; \; \ Rightarrow \; \; p \ cdot y = \ sum_ {i=1} ^k q_i \ cdot x_i

With linear homogeneous production functions the value of the product is equal to the factor costs (exhaustion theorem).

Articles in category "Homogeneous function"

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