Euler's theorem

The Euler's theorem (sometimes called Euler's identity or set of Euler over homogeneous functions ) is a set of the analysis , the correlation of a (total) differentiable , and (positive) homogeneous function with their partial derivatives describes. The theorem is widely used in economics , especially in microeconomics . There it is also known under the name Wicksteed-Euler theorem or exhaustion theorem.

history

The set is named after Leonhard Euler (1707–1783). The Euler theorem was integrated into economics by the economist Philip Wicksteed . He used Euler's theorem in his book The Co-ordination of the Laws of Distribution , published in 1894 .

statement

Let the function be (totally) differentiable and (positive) homogeneous in degree . The latter means that for everyone and is. Then applies to all : ${\displaystyle f\colon \mathbb {R} ^{k}\to \mathbb {C} }$ ${\displaystyle \lambda \in \mathbb {R} }$ ${\displaystyle f(tx)=t^{\lambda }f(x)}$ ${\displaystyle t\in \mathbb {R} _{>0}}$ ${\displaystyle x\in \mathbb {R} ^{k}}$ ${\displaystyle x\in \mathbb {R} ^{k}}$

${\displaystyle \lambda \cdot f(x)=\sum _{i=1}^{k}{\frac {\partial f}{\partial x_{i}}}(x)\cdot x_{i}={\frac {\partial f}{\partial x_{1}}}(x)\cdot x_{1}+\ldots +{\frac {\partial f}{\partial x_{k}}}(x)\cdot x_{k}}$.

Derivation

Consider the function . From the multidimensional chain rule follows: ${\displaystyle \mathbb {R} _{>0}\to \mathbb {C} ,\;t\mapsto f(tx)}$

${\displaystyle \sum _{i=1}^{k}{\frac {\partial f}{\partial x_{i}}}(x)\cdot x_{i}={\frac {\mathrm {d} f(tx)}{\mathrm {d} t}}{\bigg \vert }_{t=1}={\frac {\mathrm {d} t^{\lambda }f(x)}{\mathrm {d} t}}{\bigg \vert }_{t=1}=\lambda t^{\lambda -1}f(x){\bigg \vert }_{t=1}=\lambda f(x)}$,

where the second equality is true because of the assumed homogeneity of . ${\displaystyle f}$

Application in economics

Let be the (totally) differentiable production function with constant returns to scale of a company. Mathematically this means that (positive) is homogeneous of degree one. Then it follows from Euler's theorem: ${\displaystyle f\colon \mathbb {R} _{\geq 0}^{k}\to \mathbb {R} }$ ${\displaystyle f}$

${\displaystyle f(x)=\sum _{i=1}^{k}{\frac {\partial f}{\partial x_{i}}}(x)\cdot x_{i}={\frac {\partial f}{\partial x_{1}}}(x)\cdot x_{1}+\ldots +{\frac {\partial f}{\partial x_{k}}}(x)\cdot x_{k}}$

Assuming perfect competition in all factor markets , each factor of production is rewarded in market equilibrium according to its marginal yield . For everyone this means that the factor wages correspond to the -th production factor . This implies that the company under consideration can not make a profit in the market equilibrium , since the entire production is used to pay for the factors of production ,,. ${\displaystyle x_{1},\ldots ,x_{k}}$ ${\displaystyle x^{*}\in \mathbb {R} _{\geq 0}^{k}}$ ${\displaystyle i=1,\ldots ,k}$ ${\displaystyle i}$ ${\displaystyle {\frac {\partial f}{\partial x_{i}^{*}}}(x^{*})}$ ${\displaystyle x^{*}}$ ${\displaystyle f(x^{*})}$ ${\displaystyle \sum _{i=1}^{k}{\frac {\partial f}{\partial x_{i}}}(x^{*})\cdot x_{i}^{*}}$

A concrete example: The Cobb-Douglas production function is given , where and here the factors represent capital and labor. is obviously differentiable and homogeneous of degree one, since applies to all . According to Euler's theorem: ${\displaystyle f\colon \mathbb {R_{\geq 0}^{2}} \to \mathbb {R} ,\;(K,L)\mapsto {\sqrt {KL}}}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle f}$ ${\displaystyle f(\alpha K,\alpha L)=\alpha f(K,L)}$ ${\displaystyle \alpha \in \mathbb {R} _{>0}}$

${\displaystyle K{\frac {\partial f}{\partial K}}(K,L)+L{\frac {\partial f}{\partial L}}(K,L)=K\cdot {\frac {1}{2}}{\frac {\sqrt {L}}{\sqrt {K}}}+L\cdot {\frac {1}{2}}{\frac {\sqrt {K}}{\sqrt {L}}}={\sqrt {KL}}=f(K,L)}$

See also

• Homogeneous function

Web links

• Eric W. Weisstein : Euler's Homogeneous Function Theorem . In: MathWorld (English).
• Euler's theorem in Gabler's economic dictionary

Individual evidence

1. ↑ ^{a b} Konrad Königsberger: Analysis 2 . 5th edition. Springer, Berlin 2004, ISBN 3-540-20389-3 , Chapter 2.8 Exercise 4 .
2. ↑ Andreu Mas-Collel, Michael D. Whinston, Jerry R. Green: Microeconomic Theory . Oxford University Press, New York 1995, ISBN 978-0-19-510268-0 , pp. 929 .