Why Constant Returns to Scale Suggests Certain Relation - Economics Stack Exchange most recent 30 from economics.stackexchange.com 2019-12-08T03:36:55Z https://economics.stackexchange.com/feeds/question/21476 https://creativecommons.org/licenses/by-sa/4.0/rdf https://economics.stackexchange.com/q/21476 3 Why Constant Returns to Scale Suggests Certain Relation lithium123 https://economics.stackexchange.com/users/18006 2018-04-14T19:37:56Z 2018-04-15T00:26:05Z <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> https://economics.stackexchange.com/questions/21476/-/21477#21477 -2 Answer by Simeon Ikudabo for Why Constant Returns to Scale Suggests Certain Relation Simeon Ikudabo https://economics.stackexchange.com/users/16329 2018-04-14T20:05:44Z 2018-04-14T20:05:44Z <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> https://economics.stackexchange.com/questions/21476/-/21479#21479 5 Answer by BB King for Why Constant Returns to Scale Suggests Certain Relation BB King https://economics.stackexchange.com/users/5557 2018-04-14T20:20:20Z 2018-04-14T20:25:32Z <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>