Suppose we have a production function $f(z)=2$.
I am asked to determine whether the function exhibits increasing, decreasing, constant or no returns to scale.
For $t>0$, $f(tz)=2$.
I'm not sure about the answer: should I say the function exhibits no returns to scale whatsoever or take different values for $t$ ($0<t<1 \implies$decreasing returns to scale, $t=1 \implies$ constant returns to scale, $t>1 \implies$ increasing returns to scale)?
You want to find a relation between $tF(z)$ and $F(tz)$ for all $t>1$ (or $0$ for CRS).
So since $2t=tF(z)>F(tz)=2$ for all $t>1$, we see decreasing returns to scale.
I would say it exhibits no returns to scale. What follows is a counter argument to the idea that it can exhibit returns to scale. A 'return to scale' means that production will change in response to a change in the input. A 'constant return to scale' is a straight-line function (or a portion thereof) including the origin (0, 0), with the input on the horizontal axis and the output on the vertical axis. That is, a directly proportional relationship. A 'decreasing return to scale' means that, as more and more of the input is used, production changes with a decreasing effect. This means that the graph will have a gradient that gets more and more flat as the input increases. An 'increasing return to scale' means that production changes with increasing effect as the input increases - a function who's gradient increases.
In conclusion, if the input has no effect on production, there is no 'return' from the input.
