Mathematic notation of scale returns

Question

I'm having trouble with returns to scale mathematical notation which i found counterintuitive even I'm fine with its definition. Any help will be very appreciated.

There are three possible types of returns to scale: increasing returns to scale, constant returns to scale, and diminishing (or decreasing) returns to scale.

• If output increases by the same proportional change as all inputs change then there are constant returns to scale : $F(aK,aL) = aF(K,L)$.

• If output increases by less than that proportional change in all inputs, there are decreasing returns to scale : $F(aK,aL) < aF(K,L)$.

• If output increases by more than the proportional change in all inputs, there are increasing returns to scale $F(aK,aL) > aF(K,L)$.

If we define (with $a > 0$) :

• $F(aK,aL)$ : simultaneous and proportional inputs increase
• $aF(K,L)$ : output increase

Then, to me increasing returns to scale should be written in this way : $F(aK,aL) < aF(K,L)$ since the change in the output is greater in proportion than the change in the inputs. And the other way for decreasing returns to scale. This is not clear to me.

I hope you will help me to fix this misunderstanding.

Best, Marc

Answer 1

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 it was increased by the same proportion as the inputs. So \begin{align*} F(aK,aL) & > aF(K,L) \end{align*} shows that if you increase the inputs simultaneously and proportionally, the increase in output is larger than proportional.

Answer 2

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 $\theta$ increase in both inputs, we get a more than proportional change in total output. (e.g. if $\theta=2$, inputs double, then we get $2F(x,y)$, twice as much output if we are talking about a production function).

The classic Cobb-Douglas production has constant returns to scale and we can show this. It takes the form: $$F(K,L) =K^{1/3}L^{2/3}$$ checking for homogeneity we take a proportional increase of $\theta$:

$$F(\theta K, \theta L) = (\theta K)^{1/3}(\theta L)^{2/3}$$ $$=\theta K^{1/3}L^{2/3}$$ we have a homogeneous function of degree 1, given by the exponent on $\theta$, and thus have constant returns to scale.

Thinking about it in this way as opposed to memorizing it based on an inequality is much easier.