“Euler’s theorem”的概念、定义、翻译、参考文献-科学参考

Euler’s theorem

Economics

单词	Euler’s theorem
释义	Euler’s theorem Economics If f (x₁, …, x_n) is homogeneous of degree λ‎ then $x_{1} \frac{\partial f}{\partial x_{1}} + … + x_{n} \frac{\partial f}{\partial x_{n}} = λ f (x_{1}, …, x_{n}) .$ One use of the theorem is to demonstrate that with constant returns to scale and competitive factor markets the total payment to factors equals the revenue of the firm. Assume there are two factors, capital, K, and labour, L. Constant returns to scale implies the production function is homogeneous of degree one so Euler’s theorem gives $K \frac{\partial f}{\partial K} + L \frac{\partial f}{\partial L} = f (K, L) .$ If the price of output is p, it follows that $K p \frac{\partial f}{\partial K} + L p \frac{\partial f}{\partial K} = p f (K, L) .$ Competition on the factor market ensures that $r = p \frac{\partial f}{\partial K}$ and $w = p \frac{\partial f}{\partial L} .$ Hence, $r K + w L = p f (K, L)$ so total payment to factors equals revenue. This argument has been used as the basis for a theory of distribution.
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