In applied optimization, the point of maximum profit occurs at the output level where the marginal cost equals which of the following (assuming differentiability and an interior maximum)?

• A. Marginal revenue
• B. Average revenue
• C. Average cost
• D. Total revenue

Answer: (A)

Verified step by step guidance

1. Understand that in applied optimization for profit maximization, the goal is to find the output level where profit is maximized. Profit is defined as total revenue minus total cost.
2. Recall that the marginal cost (MC) is the derivative of the total cost function with respect to quantity, i.e., \(MC = \frac{dC}{dq}\), and marginal revenue (MR) is the derivative of the total revenue function with respect to quantity, i.e., \(MR = \frac{dR}{dq}\).
3. Recognize that the profit function \(\pi(q)\) is given by \(\pi(q) = R(q) - C(q)\), where \(R(q)\) is total revenue and \(C(q)\) is total cost.
4. To find the maximum profit, take the derivative of the profit function with respect to quantity and set it equal to zero: \(\frac{d\pi}{dq} = \frac{dR}{dq} - \frac{dC}{dq} = MR - MC = 0\).
5. From the above, conclude that the profit is maximized when marginal revenue equals marginal cost, i.e., \(MR = MC\).