I'm self studying intermediate macroeconomics by reading a textbook and I came across a relationship that I'm not quite sure how it is derived.

Let $F(K,L)$ be a production function where $K$ is capital stock and $L$ is the labor force. Assuming constant returns to scale, I'm not sure how to derive $$F(K,L) = F_K(K,L)K + F_L(K,L)L,$$ where $F_x$ denotes the partial derivative of $F$ with respect to $x$.


2 Answers 2


The result follows from Euler's Theorem on homogenous equations. This theorem states that if a function $f(x,y)$ is homogeneous of degree $\lambda$ then the following holds:

$\lambda f(x,y)= x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}$

A (production) function with constant returns to scale is homogeneous of degree 1 by definition. The definition of constant returns to scale is basically the same as the definition of homogeneity of degree 1. That means $\lambda=1$, which proves the result you need.


Constant returns to scale simply means that if K and L are increases by 10%, Y will also be increases by 10%. Y will never be increases by more than F(K, L), so profit for additional labor is not possible in that simplified model. I’m assuming this textbook could be Macroeconomics by Mankiw because this concept looks familiar? So Fk and FL would increase by the same % and your output (Y) increases by the same percentage in a constant returns to scale model.

  • $\begingroup$ This answer does not address the question and is written poorly. $\endgroup$ Apr 15, 2018 at 0:21
  • $\begingroup$ How so? I think that’s an interesting response, that should be backed by specifics. It’s a vague comment. $\endgroup$ Apr 15, 2018 at 14:49
  • $\begingroup$ "Y will also be increases", "Y will never be increases", "would increase by the same %", and other sentence fragments, just to name a few issues. It's essentially illegible. An answer that addresses the question would talk about homogeneity, which is what you were trying to explain but avoided saying (perhaps because you didn't know). $\endgroup$ Apr 15, 2018 at 15:33
  • $\begingroup$ If you don't understand that if (k, L) increase, Y will also increase, that may reflect that you don't understand the question or answer more than anything. Will all due respect, if it doesn't make sense to you, you would in theory, be confused by the answer. Constant returns to scale mean that for every % increase in (K, L) Y (output) increases by the same %. I read this verbatim to double check. I think that you're fundamentally confused, with all due respect. $\endgroup$ Apr 15, 2018 at 15:39
  • $\begingroup$ I think you are the one confused, with all due respect. Obviously I know what constant returns is. Your answer missed tying that to Euler's theorem, which is what the question was about. The question asked for a derivation, not an intuitive explanation. Please improve your reading comprehension. $\endgroup$ Apr 15, 2018 at 15:48

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