Monopoly pricing under constant elasticity of demand While reading Ch. 24-Monopoly from Intermediate Microeconomics by Hal Varian (8 th edition), on pg. 441, he writes that a monopolist will never choose to operate where the demand curve is inelastic. I understand the argument, but, if we have constant elasticity demand curve with

$|\epsilon| &lt; 1 $

then how this effect the monopolist's choice?
A constant elasticity demand function has the form $q=p^{-\epsilon}$. Let's check this indeed gives us a constant elasticity...
$$\frac{d q}{dp}=-\epsilon p^{-\epsilon-1}$$
so, as we'd hoped, the elasticity is constant:
$$\frac{d q}{dp}\frac{p}{q}=-\epsilon p^{-\epsilon-1}\frac{p}{p^{-\epsilon}}=-\epsilon p^{-\epsilon-1+1+\epsilon}=-\epsilon p^{0}=-\epsilon.$$



Now, suppose a monopolist has constant marginal cost $c\geqslant0$ (and no fixed cost). It's profit is
$$q(p-c)=p^{-\epsilon}(p-c).$$
The effect on the firm's profits of a small increase in its price is
$$\frac{d}{d p}p^{-\epsilon}(p-c)=p^{-\epsilon -1} [c \epsilon -p\epsilon+p].$$
Now we can answer the question: if $\epsilon&lt;1$ then $c \epsilon -p\epsilon+p$ is definitely positive for every $p$. That means, no matter how high the price is, the firm would still like to increase the price further—the optimal price is $+\infty$!



Intuitively, constant  $\epsilon&lt;1$ means that the firm can always more than double its price by foregoing less than half its current sales. Provided marginal cost isn't very quickly decreasing, this is always an attractive prospect for the firm.