Interpretation of $x c '(x)$ Consider a cost function that is continuous, differentiable and (possibly) convex: $c(x):\mathbb{R}^+\to \mathbb{R}$. I was wondering if there is a &quot;common&quot; way to interpret the expression:
$$xc'(x)$$
in terms of economic intuitions. Any suggestions or references?
If your cost function is also homogeneous of degree $k$ (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem,
$$ x c'(x) = k c(x).$$
That is, $x c'(x)$ is your cost itself, up to some scaling factor $k$ (for example, if $c(x) = ax$ so that $c(x)$ is homogeneous of degree 1, then $x c'(x) = ax = c(x)$).
Without homogeneity, I am not sure there is any insightful interpretation you can give to $x c'(x)$, but others may have better ideas. The reason I am pessimistic is the &quot;most natural&quot; way for $x c'(x)$ to appear is in
$$[xc(x)]' = xc'(x) + c(x),$$
which gives
$$\frac{[xc(x)]'}{c(x)} = xc'(x).$$
However, it's not clear to me how to interpret $\frac{[xc(x)]'}{c(x)}$ either. Why should anyone care about $xc(x)$ in the first place? Provided $c(x)$ is the total cost (and not the per unit cost), why would you want multiply your total cost by the number of units you are interested in (what does &quot;the total cost of producing 5 units times 5&quot; means?).
If you let $c(x)$ be the per unit cost, maybe you can make some interpretative sense of  $\frac{[xc(x)]'}{c(x)}$ as $xc(x)$ then becomes the total cost?