Question

A monopoly's demand curve is P = 200 – 3Q and its MC = $20. How many customers should this company serve? What is the price paid by each customer? What will be the company's gross revenue in this venture?

Answer 1

The company will serve 30 customers, they will pay 110, and the gross revenue would be 3,300

Step-by-step explanation:

Monopolist maximize at the point where their marginal revenue equals marginal cost. We the data given by the problem we can get the marginal revenue, and use it to compare to the marginal cost given: MC=20

Total Revenue is
`TR=P * Q` (because as monopolist faces the entire demand), replacing P, we have
`(200-3Q) * Q = 200Q-3Q^2`, to find the marginal revenue, we take derivatives with respect to Q:

`(d)/(Q)(200Q-3Q^2)=`

Equalizing Marginal Revenue with Marginal Cost we can find the optimal Q*

`MR = MC\\200-6Q =20\\180=6Q\\(180)/(6)=Q\\Q*=30`

with Q* we can replace in the demand curve and find the price

`P*=200-3Q=200-3 * 30= 200-90= 110`

with this find the gross revenue
`P* * Q* = 110 * 30 =3,300`