Suppose the demand curve for a monopolist is Qd = 500 – P, and the marginal revenue function is MR = 500 -2Q. The monopolist has a constant marginal and average total cost of $50 per unit.
To find the monopolist's profit-maximizing level of output and price, we need to determine the quantity at which marginal revenue equals marginal cost.
1. Start by calculating the monopolist's marginal cost (MC). Since the average total cost is constant at $50 and the marginal cost is equal to the average total cost, the marginal cost is also $50.
2. Set the marginal revenue (MR) equal to the marginal cost (MC) to find the profit-maximizing quantity. In this case, MR = MC implies 500 - 2Q = 50.
Solve the equation:
500 - 2Q = 50
2Q = 500 - 50
2Q = 450
Q = 225
The monopolist's profit-maximizing quantity is 225 units.
3. Substitute the quantity back into the demand equation to find the price at the profit-maximizing quantity. The demand equation is Qd = 500 - P.
Rewrite the equation to solve for P:
P = 500 - Q
P = 500 - 225
P = 275
The monopolist's profit-maximizing price is $275 per unit.
Therefore, the monopolist will produce and sell 225 units of output at a price of $275 per unit to maximize their profit.
To find the monopolist's profit-maximizing level of output and price, we need to equate marginal revenue (MR) to marginal cost (MC). In this case, we know that MC is equal to the constant marginal and average total cost of $50 per unit.
Step 1: Calculate the monopolist's marginal cost (MC):
Since the marginal and average total cost is constant at $50 per unit, the MC function is MC = $50.
Step 2: Equate marginal revenue (MR) to marginal cost (MC):
We know that MR = 500 - 2Q and MC = $50. Let's set them equal to each other:
500 - 2Q = 50
Step 3: Solve for Q (the monopolist's profit-maximizing level of output):
Subtract 500 from both sides of the equation:
-2Q = 50 - 500
-2Q = -450
Divide both sides of the equation by -2:
Q = (-450) / (-2)
Q = 225
The monopolist's profit-maximizing level of output is 225 units.
Step 4: Calculate the monopolist's profit-maximizing price:
To determine the price, substitute the value of Q into the demand curve equation, Qd = 500 - P:
Q = 500 - P
225 = 500 - P
Subtract 500 from both sides of the equation:
-275 = -P
Multiply both sides of the equation by -1:
275 = P
The monopolist's profit-maximizing price is $275 per unit.
Therefore, the monopolist should produce and sell 225 units at a price of $275 per unit to maximize profits.