Is it constant returns to scale if the output of a production function is purely a function of one variable?

Question

For $Y=F(K,L)= 2L$

If I multiply them by an constant $z$:

$Y= F(zK,zL0)$, i'll get $2(zL) = z(2L)$. Inputs increase proportionally therefore constant returns to scale.

This doesnt seem right because the outputs are determined by one variable.

Answer 1

Returns to scale is a concept that we use to think about how output changes as we continually add more inputs. It does not matter if your production function takes one input or $N$ inputs. All that matters is how inputs behave within the function itself. Another way to think about this is to consider that a production function with a single input can exhibit decreasing, constant, or increasing returns to scale.

For example:

1. $f(L) = L^2$ exhibits increasing returns to scale.
2. $f(L) = L$ exhibits constant returns to scale
3. $f(L)= L^{\frac{1}{2}}$ exhibits decreasing returns to scale

Your production function is linear in its one input [case number 2 above] and so it exhibits constant returns to scale. That is, the answer to your question is yes.

Answer 2

Return to scale is a long run concept and as underlying production function is dependent upon a single variable i.e. labour so there must be discussion upon whether it is a increasing returns to a factor or constant return to a factor. Because it is only short run in which we have one input and other inputs are fixed.