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

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.

• So? As long as the definition is satisfied it is satisfied. Feb 18 '18 at 15:37

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.
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