# A monopoly firm faces a demand curve given by the following equation: P = \$500 − 10Q, where Q equals quantity sold per day. Its marginal cost curve is MC = \$100 per day. Assume that the firm faces no fixed cost. What is the firm's maximum profit?

The maximum profit earned by the company is \$6150 when 25 items are produced and sold at \$250

The demand curve is a function of price (P) and the number of items that buyers will be at that price, Q. The Cost curve is a function C(Q) that gives the total cost of producing a quantity Q. This includes the fixed cost incurred by the seller which remains the same irrespective of the quantity produced as well as the marginal cost incurred for each unit produced.

The demand curve for the monopoly firm is given by the equation P = \$500 − 10Q, where Q equals quantity sold per day. The marginal cost curve is MC = \$100 per day.

If a quantity is produced, the profit of the company is

`Pr = P*Q - MC*Q`

`=(500 - 10Q)*Q - 100`

`= 500Q - 10Q^2 - 100`

The profit is maximized at `(dPr)/(dQ) = 0` and value of `(D^2Pr)/(dQ)` at that point is negative.

`(dPr)/(dQ) = 500 - 20Q`

`500 - 20Q =0`

=> `Q = 500/20 = 25`

At Q = 125, `(D^2Pr)/(dQ)` = -4.

To maximize profit, the company should manufacture a quantity 25. The selling price of these would be \$250. The maximum profit earned by the company is \$6150.