There's a common misperception that if a function (usually a production function or a consumer valuation function) scales proportionally (meaning that when all inputs double, for example, the output doubles), then that function must be a linear function.
Sadly, some of the people falling for this error make important decisions in market research and in capacity planning, two areas where this kind of behavior (scaling proportionally) happens a lot and where the error in considering only linear models may have serious consequences for the bottom line.
(And we should always strive to make fewer math errors, of course.)
Let's start with the simple part, using a two-variable function:
\[ f(x,y) = a \, x + b \, y \]
If we scale $x$ and $y$ by a constant $c$ we get
\[ f(cx,cy) = a \, cx + b \, cy = c (a \, x + b \, y) = c \, f(x,y).\]
Clearly, linear functions scale proportionately. What about the other part, the ``must be a linear function'' part?
That's wrong. And we need no more than an example to show it. Voilá:
\[ f(x,y) = x^\alpha \, y^{(1-\alpha)}\]
for an $\alpha \in (0,1)$.
That function is not linear at all; here's a plot for $\alpha = 0.5$ and $x,y$ each in $[0,10]$ (that makes the $z = f(x,y)$ variable also in $[0,10]$, obviously):
And yet,
\[ f(cx,cy) = (cx)^\alpha \, (cy)^{(1-\alpha)} = c^{\alpha + (1-\alpha)} \, x^\alpha \, y^{(1-\alpha)} = c f(x,y). \]
So, now we know not to limit ourselves to linear functions when describing systems that exhibit proportional scaling.
Nerd note: a function that scales proportionally is called ``homothetic'' or ``homogeneous of degree one.''
