A monopolist has the cost function of C(q)=675+50q+1.5q2C(q)=675+50q+1.5q2 and its inverse demand function is given by p=150−qp=150−q. Up until what market quantity is it a natural monopoly?

Let us determine the marginal cost (MC). It is a derivative of the cost function.

C(q)=675+50q+1.5q2MC=50+(2)1.5qMC=50+3qC(q)=675+50q+1.5q2MC=50+(2)1.5qMC=50+3q

Let us make determine the total revenue.

TR=p×qTR=(150−q)×qTR=150q−q2TR=p×qTR=(150−q)×qTR=150q−q2

Let us determine the marginal revenue (MR). It is a derivative of the total revenue function.

TR=150q−q2MR=150−2qTR=150q−q2MR=150−2q

A natural monopoly will occur where the marginal revenue is equal to the marginal cost. This is the level where a firm maximizes its profit. Let us compute the quantity (q).

150−2q=50+3q3q+2q=150−505q=100q=100÷5q=20150−2q=50+3q3q+2q=150−505q=100q=100÷5q=20

The natural monopoly occurs up until the market quantity is 20 units.