<p>If your cost function is also homogeneous of degree <span class="math-container">$k$</span> (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem,</p>
<p><span class="math-container">$$ x c'(x) = k c(x).$$</span></p>
<p>That is, <span class="math-container">$x c'(x)$</span> is your cost itself, up to some scaling factor <span class="math-container">$k$</span> (for example, if <span class="math-container">$c(x) = ax$</span> so that <span class="math-container">$c(x)$</span> is homogeneous of degree 1, then <span class="math-container">$x c'(x) = ax = c(x)$</span>).</p>
<p>Without homogeneity, I am not sure there is any insightful interpretation you can give to <span class="math-container">$x c'(x)$</span>, but others may have better ideas. The reason I am pessimistic is the &quot;most natural&quot; way for <span class="math-container">$x c'(x)$</span> to appear is in</p>
<p><span class="math-container">$$[xc(x)]' = xc'(x) + c(x),$$</span></p>
<p>which gives</p>
<p><span class="math-container">$$\frac{[xc(x)]'}{c(x)} = xc'(x).$$</span></p>
<p>However, it's not clear to me how to interpret <span class="math-container">$\frac{[xc(x)]'}{c(x)}$</span> either. Why should anyone care about <span class="math-container">$xc(x)$</span> in the first place? Provided <span class="math-container">$c(x)$</span> is the <em>total</em> cost (and not the <em>per unit</em> 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?).</p>
<p>If you let <span class="math-container">$c(x)$</span> be the <em>per unit</em> cost, maybe you can make some interpretative sense of  <span class="math-container">$\frac{[xc(x)]'}{c(x)}$</span> as <span class="math-container">$xc(x)$</span> then becomes the total cost?</p>

CC BY-SA 4.0
2020-06-23T19:08:05.020Z
2020-06-24T12:21:48.053Z
2020-06-24T12:21:48.053Z
FZS
<p>Consider a cost function that is continuous, differentiable and (possibly) convex: <span class="math-container">$c(x):\mathbb{R}^+\to \mathbb{R}$</span>. I was wondering if there is a &quot;common&quot; way to interpret the expression:</p>
<p><span class="math-container">$$xc'(x)$$</span></p>
<p>in terms of economic intuitions. Any suggestions or references?</p>

<microeconomics><mathematical-economics><production-function><cost-functions>
Interpretation of $x c '(x)$
Weierstraß Ramirez
# Interpretation of $x c '(x)$

(Asked by: username_0 on 2020-06-23)

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 "common" way to interpret the expression:

$$xc'(x)$$

in terms of economic intuitions. Any suggestions or references?

**Tags:** microeconomics, mathematical-economics, production-function, cost-functions

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## 1 answers

### Answer by username_1 on 2020-06-23. Score: 2

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 "most natural" 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 "the total cost of producing 5 units times 5" 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?


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