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. You may wish to arrive at the answers mathematically, or by using a graph (the graph is not required to be presented), either way, please provide a brief description of how you arrived at your results.
To find the profit-maximizing quantity and price for the monopoly firm, we need to determine the point where marginal revenue (MR) is equal to marginal cost (MC).
To start, let's calculate the marginal revenue. For a monopoly, the marginal revenue is not equal to the price like in a perfectly competitive market. It is determined by the change in total revenue resulting from selling an additional unit. Since the demand curve equation is given as P = $500 − 10Q, we know that total revenue (TR) is equal to price (P) multiplied by quantity (Q):
TR = P * Q
TR = ($500 - 10Q) * Q
TR = $500Q - 10Q²
To find the marginal revenue, we calculate the derivative of the total revenue equation with respect to quantity:
MR = d(TR) / dQ
MR = d($500Q - 10Q²) / dQ
MR = $500 - 20Q
Now, let's set MR equal to MC to find the profit-maximizing quantity:
MR = MC
$500 - 20Q = $100
Simplifying the equation, we have:
-20Q = -$400
Q = 20
So, the profit-maximizing quantity for the monopoly firm is Q = 20 units per day.
To find the corresponding price, we substitute the value of Q back into the demand curve equation:
P = $500 - 10Q
P = $500 - 10(20)
P = $300
Therefore, the profit-maximizing price for the monopoly firm is P = $300 per unit.
The monopoly firm will maximize its profit by producing and selling 20 units per day at a price of $300 per unit.