Homogeneous function

A mathematical function is called homogeneous in degree if the function value changes by the factor when all variables change proportionally by the proportionality factor . ${\displaystyle \lambda }$ ${\displaystyle t}$ ${\displaystyle t^{\lambda }}$

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

definition

A function on the -dimensional real coordinate space ${\displaystyle n}$

${\displaystyle \Phi :\mathbb {R} ^{n}\to \mathbb {R} }$

is called homogeneous in degree if for all and ${\displaystyle \lambda \in \mathbb {R} ^{+}}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle t\in \mathbb {R} }$

${\displaystyle \Phi (t\cdot x)=t^{\lambda }\cdot \Phi (x)}$

applies. Is , the function is called super-linearly homogeneous, with linear homogeneous and otherwise ( ) sub-linear homogeneous. ${\displaystyle \lambda >1}$ ${\displaystyle \lambda =1}$ ${\displaystyle \lambda <1}$

Examples from microeconomics

In microeconomics homogeneous play production functions an important role. They establish a connection between production factors  and the associated production . In the case of a linearly homogeneous production function, an increased / decreased use of all production factors leads to an increased / decreased production in the same proportion, because it follows ${\displaystyle y=f(x_{1},\dotsc ,x_{n})}$ ${\displaystyle x_{i}}$ ${\displaystyle y}$ ${\displaystyle \lambda =1}$

${\displaystyle t\cdot f(x_{1},\dotsc ,x_{n})=f(tx_{1},\dotsc ,tx_{n})}$.

Such a production function is homogeneous with the degree of homogeneity  1 (linear homogeneous). An example of a homogeneous production function of degree 1 is the Cobb-Douglas production function. In the case of homogeneous production functions, the degree of homogeneity corresponds to the elasticity of the scale (only in one direction). Over-linearly homogeneous production functions show increasing, linearly homogeneous constant and under-linearly homogeneous decreasing economies of scale . The reverse conclusion, inferring the degree of homogeneity from returns to scale, is not possible, however, because the ratio of factors used can also be changed for returns to scale, but not to determine the homogeneity property. ${\displaystyle F(K(t),L(t))=A\cdot K(t)^{a}L(t)^{1-a}\;,a\in (0,1)}$

Another example are individual demand functions . They represent a connection between prices  , income  and the demanded quantities  . If, for example, in the course of a currency conversion (from DM to Euro) all prices and incomes are halved and this is fully taken into account by the individuals (freedom from illusion of monetary value ) the demanded quantities will not change. That means that: ${\displaystyle x=x(p,E)}$ ${\displaystyle p}$ ${\displaystyle E}$ ${\displaystyle x}$

${\displaystyle x(tp,tE)=t^{0}\cdot x(p,E)=x(p,E)}$

Demand functions are thus homogeneous of degree 0 in prices and income ( zero homogeneity ).

Homothety

In the case of ordinal utility functions , the assumption of homogeneity does not make sense, because a strictly monotonically growing transformation of  a utility function represents the  same preferences as the function  itself. A homothetic utility function is a strictly monotonically growing transformation of a homogeneous utility function. For utility functions with this property, the Engel curves are linear. ${\displaystyle T(u)}$ ${\displaystyle u}$ ${\displaystyle u}$

Example: Be and . Obviously the utility function is linearly homogeneous. Their transformation is inhomogeneous, but homothetic; it represents the same order of preference. ${\displaystyle u(x,y)={\sqrt {xy}}}$ ${\displaystyle T(u)=\ln {u}}$

Positive homogeneity

A function  is called positive homogeneous of degree  , if ${\displaystyle \Phi :\mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} }$ ${\displaystyle \lambda \in \mathbb {R} }$

${\displaystyle \Phi (tx_{1},\dotsc ,tx_{n})=t^{\lambda }\cdot \Phi (x_{1},\dotsc ,x_{n})}$

applies to everyone and everyone . ${\displaystyle t>0}$ ${\displaystyle x\in \mathbb {R} ^{n}\setminus \{0\}}$

In contrast to homogeneous functions, positive homogeneous functions only need to be defined on and the degree of homogeneity  can be any real number. ${\displaystyle \mathbb {R} ^{n}\setminus \{0\}}$ ${\displaystyle \lambda }$

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

A differentiable function  is positively homogeneous of degree  if and only if applies ${\displaystyle \Phi :\mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} }$ ${\displaystyle \lambda >0}$

${\displaystyle \lambda \cdot \Phi (x)=\sum _{i=1}^{n}{\frac {\partial \Phi }{\partial x_{i}}}(x)\cdot x_{i}=\langle {\text{grad }}\Phi (x),x\rangle =D_{x}{\Phi (x)}\quad \mathrm {(Eulersche\ Homogenit{\ddot {a}}tsrelation)} }$

for everyone . Here, the partial derivatives denote from to the -th component of , the directional derivative at the point  in the direction of the vector  and the gradient from . ${\displaystyle x\in \mathbb {R} ^{n}\setminus \{0\}}$ ${\displaystyle {\tfrac {\partial \Phi }{\partial x_{i}}}}$ ${\displaystyle \Phi }$ ${\displaystyle i}$ ${\displaystyle x}$ ${\displaystyle D_{x}{\Phi (x)}}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle {\text{grad }}\Phi (x)}$ ${\displaystyle \Phi (x)}$

A positively homogeneous function can therefore be represented in a simple way by the partial derivatives and coordinates.

This fact is very often used in physics, especially in thermodynamics , since the intensive and extensive state variables occurring there are homogeneous functions of zeroth or first degree. Specifically, this is used for B. in deriving the Euler equation for the internal energy .

In economics, it follows from Euler's theorem for production functions of degree of homogeneity 1 for factor prices  and the price of goods  ${\displaystyle q_{i}}$ ${\displaystyle p}$

${\displaystyle y=f(x_{1},\dotsc ,x_{k})=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\cdot x_{i}=\sum _{i=1}^{n}{\frac {q_{i}}{p}}\cdot x_{i}\;\;\Rightarrow \;\;p\cdot y=\sum _{i=1}^{n}q_{i}\cdot x_{i}}$.

In the case of linearly homogeneous production functions, the value of the product is equal to the factor costs (see also: exhaustion theorem ).

Derivation of the Euler theorem

→ Main article : Euler's theorem

First, a positively homogeneous differentiable function is given  . So it applies . Differentiation of the left side yields with the chain rule ${\displaystyle \Phi :\mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} }$ ${\displaystyle \Phi (t\cdot x)=t^{\lambda }\Phi (x)}$ ${\displaystyle t}$

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}\Phi (t\cdot x)=\sum _{j=1}^{n}{\frac {\partial \Phi }{\partial x_{j}}}(t\cdot x)\cdot x_{j}}$.

Differentiation of the right side yields, however ${\displaystyle t}$

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}t^{\lambda }\Phi (x)=\lambda \cdot t^{\lambda -1}\cdot \Phi (x)}$.

By inserting Euler's homogeneity relation follows. ${\displaystyle t=1}$

Conversely, let us now give a differentiable function  that fulfills Euler's homogeneity relation. For the given we consider the real function  . Because of the homogeneity relation, the ordinary first-order differential equation is satisfied ${\displaystyle \Phi :\mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} }$ ${\displaystyle x\in \mathbb {R} ^{n}\setminus \{0\}}$ ${\displaystyle f(t):=\Phi (t\cdot x),t>0}$ ${\displaystyle f}$

${\displaystyle f'(t)={\frac {\text{d}}{{\text{d}}t}}\Phi (t\cdot x)=t^{-1}\sum _{j=1}^{k}{\frac {\partial \Phi }{\partial x_{j}}}(t\cdot x)\cdot x_{j}t{\overset {\text{Euler Relation}}{=}}{\frac {\lambda }{t}}\Phi (t\cdot x)={\frac {\lambda }{t}}f(t)}$

with the initial condition

${\displaystyle f(1)=\Phi (x)}$.

A solution to this initial value problem is and according to a uniqueness theorem for ordinary differential equations the solution in the domain is unique. But that means . ${\displaystyle f(t)=t^{\lambda }\cdot \Phi (x)}$ ${\displaystyle t>0}$ ${\displaystyle \Phi (t\cdot x)=t^{\lambda }\cdot \Phi (x)}$

See also

• Homogeneous polynomial
• Euler's theorem (economics)

literature

• LD Kudryavtsev: Homogeneous function . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ↑ ^{a b} Homogeneous function: I / III . In: W. Gelert, H. Kästner, S. Neuber (Eds.): Fachlexikon ABC Mathematik . Harri Deutsch, Thun / Frankfurt-Main 1978, ISBN 3-87144-336-0 .
2. ↑ This is a linear homogeneous function because of ${\displaystyle F(\alpha K(t),\alpha L(t))=A\cdot (\alpha K(t))^{a}(\alpha L(t))^{1-a}=A\cdot \alpha ^{a}K(t)^{a}\alpha ^{1-a}L(t)^{1-a}=\alpha A\cdot K(t)^{a}L(t)^{1-a}=\alpha F(K(t),L(t)).}$
3. ^ Hal R. Varian: Microeconomic Analysis. 1992, p. 146 f.
4. ^ A. Ostrowski : Lectures on differential and integral calculus . 2nd Edition. tape  2 differential calculus in the field of several variables . Birkhäuser Verlag, Basel 1961, Chapter IV Supplement to Differential Calculus, § 12 Partial Derivatives of Higher Order, Section 63 Euler's Theorem on Homogeneous Functions , p. 169 ff .