Monopoly equilibrium with a completely inelastic demand What is the impact of an inelastic demand function on a monopoly production and pricing strategy ?

Theory states, that a monopoly will strive to maximize its profits, by arriving to a point where the difference between the revenues and costs is at a maximum

The potential revenues derive from quantity of sales at a certain price, whixh by turn, are affected by demand

If the demand is inelastic, that means that quantity will remain fixed, no matter the price

Does that narrow or broaden the monopoly's options ?
What's the impact of that type of demand on the equilibrium price ?
Perfectly inelastic demand means quantity demanded is $q$ irrespective of the price. If producing quantity $q$ costs $c$ then the monopolist's problem is

$$\max_p \{pq-c\}.$$

This problem is not well-defined because the function $pq-c$ is increasing in $p$ and has no maximum. So there is no solution to the monopolist's problem. Intuitively, since the monopolist can sell $q$ units at any price, it would want to choose a price of $+\infty$.

You remark incredulously that we do not observe infinite oil prices. If the price of fuel increased to \$1000 per gallon then people would obviously drive their cars less and demand less fuel. So the demand for oil is not perfectly inelastic!



One way to make the problem more well behaved, which is quite common in the literature, is to assume a so-called "choke-price" (a price above which demand drops to zero). Demand is therefore
$$D(p)=\begin{cases}q\text{ if }p\leq\overline{p}\\[0.3em]0\text{ if }p&gt;\overline{p}\end{cases}.$$

The monopolist's problem is then 

$$\max_p \{pq-c\}\text{ s. th. }p\leq\overline{p}.$$

This is a well-defined maxmisation problem with a unique solition: $p=\overline{p}$ (assuming $\overline{p}q&lt;c$, otherwise the firm would just shit down).