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| < 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 $\epsilon<1$ 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 $\epsilon<1$ 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.
In this post you can find the algebraic steps that lead to the (standard) result mentioned in Varian's book.
Now, let's assume that, in a specific market, the consumer's preferences are such that they lead to a constant elasticity demand curve, with elasticity lower than unity in absolute terms, $|\eta| < 1$, for example
$$Q^d = AP^{\eta}, -1 <\eta < 0$$ Also, let's assume that for historical or institutional reasons this market is a monopoly. From the post mentioned above we have that profit maximization by the monopolist requires that $$P^* = \frac {|\eta|}{|\eta|-1} MC \tag{1}$$
where
$$\eta = \frac {\partial Q }{ \partial P}\cdot \frac {P}{Q} \Rightarrow \frac {\partial Q }{ \partial P} = \eta \cdot \frac {Q}{P} \tag{2}$$ and $MC$ is marginal cost. Obviously, this price is negative in our case, and so meaningless. We don't need to go into sophisticated constrained maximization procedures to see what happens here: the profit function is $$\pi = P\cdot Q(P) - C(Q(P)) \tag{3}$$ and its derivative with respect to price is $$\frac {\partial \pi}{\partial P} = Q + P\frac {\partial Q }{ \partial P} - MC\cdot \frac {\partial Q }{ \partial P} \tag{4}$$
Using $(2)$ we get
$$ \frac {\partial \pi}{\partial P}=Q + P\cdot \eta \cdot \frac {Q}{P} - MC\cdot \eta \cdot \frac {Q}{P} $$
$$\implies \frac {\partial \pi}{\partial P}= Q\cdot \left [1 + \eta - \eta \cdot \frac {MC}{P}\right]$$
$$\implies \frac {\partial \pi}{\partial P}= Q\cdot \left [1 - |\eta| + |\eta| \cdot \frac {MC}{P}\right] \tag{5}$$
From $(5)$ we see that
$$|\eta| < 1 \implies \frac {\partial \pi}{\partial P} > 0, \;\; \forall P >0 \tag{6}$$
So a profit maximizing monopolist would theoretically have the tendency to increase the price to "infinity" sending the quantity supplied to zero. Note that the Revenue function here is
$$R = P\cdot Q^d = P\cdot AP^{\eta} = AP^{1-|\eta|}, \uparrow \text{in} \;P$$
while Costs are decreasing in $Q^d$. So indeed profits would tend to infinity by selling less and less for higher and higher price.
What markets could be described by such tendencies?
