Is It necessary for a monopoly firm to be profitable at the elastic region of the demand curve?

Question

Considering a linear demand curve, we conclude that the monopolist only produces at the elastic region of the demand curve (given, he doesn't discriminate prices). But if the cost is a function of quantity produced and increases rapidly due to increase in Q, then the monopolist may incur losses even at the elastic region, wouldn't he?

Answer 1

the monopolist only produces at the elastic region ... But ... the monopolist may incur losses even at the elastic region

There is no contradiction here. If I like chocolate cake it does not mean that I like all cake; if the profit maximizing quantity is in the elastic region it does not mean that all quantities in the elastic region are profit maximizing or even profitable.

An example

Let $D(p) = 3 - p$ so $MR(q) = 3 - 2q$, and let $$ C(q) = \frac{q^{100}}{100} $$ so $MC(q) = q^{99}$.

The profit maximizing quantity is $q = 1$, at this point $MR(1) = MC(1)$. The elastic region is $[0,1.5]$, but at $q=1.1$ the monopolist is already making heavy losses, around $-135.716$ according to this C++ program.

#include <iostream>
using namespace std;
int main() 
{
  float q = 1.1;
  double result = 1;
  for (int i = 0; i<100; i++)
  {
     result *= q;
  }
  result = (3-q)*q - result/100;
  cout << result;
  return 0;
}