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Quoting from the question: $F(aK,aL)$ : simultaneous and proportional inputs increase $aF(K,L)$ : output increase But $F(aK,aL)$ is an output increase, that is why it starts with $F$. It is the output increase due to the simultaneous and proportional increase of inputs. At the same time, $aF(K,L)$ is a hypothetical output, the output were this if ...


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Returns to scale is directly related to homogeneous functions: For a homogeneous function $F(x,y)$ given $\theta>0$, for simplicity, $$F(\theta x,\theta y) = \theta^r F(x,y)$$ where would we refer to $F$ being a homogeneous function of degree $r$. Here it is much clearer to see that if $r>1$, we have increasing returns to scale because for a given $\...


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