<p>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. </p>

CC BY-SA 3.0
2018-04-15T00:21:19.043Z
This answer does not address the question and is written poorly.
Kenneth Rios
CC BY-SA 3.0
2018-04-15T14:49:05.937Z
How so? I think that’s an interesting response, that should be backed by specifics. It’s a vague comment.
Simeon Ikudabo
CC BY-SA 3.0
2018-04-15T15:33:31.513Z
"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).
Kenneth Rios
CC BY-SA 3.0
2018-04-15T15:39:48.693Z
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.
Simeon Ikudabo
CC BY-SA 3.0
2018-04-15T15:48:32.760Z
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.
Kenneth Rios
CC BY-SA 3.0
2018-04-15T15:53:37.397Z
Imho you don’t understand the question and lacked a point with any substance in your comment. Saying “I do understand it” without explaining how you understand, and making a vague argument is not intuitive by any definition. Please improve your understanding of words like “intutitive”, and please refrain from vague comments that make no sense, with all due respect. I think that’s reasonable.
Simeon Ikudabo
CC BY-SA 3.0
2018-04-15T15:58:51.743Z
Failing to address my clear point about you misreading the question signals to me that you have no valid response. Straw-mans claiming that I do not understand words such as "intuitive" are both baseless and ironic coming from an individual that struggles with basic English. The question asked for a derivation, you offered an explanation of what constant returns is. With all due respect, I won't be reading further replies.
Kenneth Rios
CC BY-SA 3.0
2018-04-15T16:29:43.813Z
You couldn’t respond to anything that I actually stated, so you’re wasting my time. Your last comment is riddled with false premises such as, “an individual that struggles with basic English”, although I’m clearly speaking to you in English. It’s comments such as these that make your reply truly remarkable, and baseless. I was under the false notion that you had something valuable to add, but you are wasting my time. With all due respect, I will not be responding to further replies @KennethRios
Simeon Ikudabo
CC BY-SA 3.0
2018-04-14T20:05:44.787Z
2018-04-14T20:05:44.787Z
Simeon Ikudabo
<p>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:</p>

<p>$\lambda f(x,y)= x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}$</p>

<p>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.</p>

CC BY-SA 3.0
2018-04-14T20:20:20.397Z
2018-04-14T20:25:32.647Z
2018-04-14T20:25:32.647Z
BB King
<p>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. </p>

<p>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 $x$. </p>

<macroeconomics>
Why Constant Returns to Scale Suggests Certain Relation
lithium123
# Why Constant Returns to Scale Suggests Certain Relation

(Asked by: username_0 on 2018-04-14)

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 $x$.

**Tags:** macroeconomics

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## 2 answers

### Answer by username_3 on 2018-04-14. Score: 5 | Accepted answer

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.


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### Answer by username_1 on 2018-04-14. Score: -2

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.

> **Comments:**
> **username_2**: This answer does not address the question and is written poorly.  
> **username_1**: How so? I think that’s an interesting response, that should be backed by specifics. It’s a vague comment.  
> **username_2**: "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).  
> **username_1**: 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.  
> **username_2**: 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.  
> **username_1**: Imho you don’t understand the question and lacked a point with any substance in your comment. Saying “I do understand it” without explaining how you understand, and making a vague argument is not intuitive by any definition. Please improve your understanding of words like “intutitive”, and please refrain from vague comments that make no sense, with all due respect. I think that’s reasonable.  
> **username_2**: Failing to address my clear point about you misreading the question signals to me that you have no valid response. Straw-mans claiming that I do not understand words such as "intuitive" are both baseless and ironic coming from an individual that struggles with basic English. The question asked for a derivation, you offered an explanation of what constant returns is. With all due respect, I won't be reading further replies.  
> **username_1**: You couldn’t respond to anything that I actually stated, so you’re wasting my time. Your last comment is riddled with false premises such as, “an individual that struggles with basic English”, although I’m clearly speaking to you in English. It’s comments such as these that make your reply truly remarkable, and baseless. I was under the false notion that you had something valuable to add, but you are wasting my time. With all due respect, I will not be responding to further replies @KennethRios  

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