91Ó°ÊÓ

Average cost is the total cost divided by the quantity of goods produced, while marginal cost is the additional cost of producing one more unit of the good. In mathematical terms, average cost (AC) can be written as: \[ AC = \frac{Total\ Cost}{Quantity} \] On the other hand, marginal cost (MC) is the derivative of the total cost with respect to the quantity produced, represented as: \[ MC = \frac{d(Total\ Cost)}{d(Quantity)} \]

Here, we need to determine if the Average Cost (AC) is decreasing when Marginal Cost (MC) is decreasing. Let's analyze it in terms of their slopes. If the slope of MC is negative and the slope of AC is also negative, it would mean that both are decreasing. The slope of AC with respect to quantity can be represented as: \[ \frac{d(AC)}{d(Quantity)} = \frac{d(\frac{Total\ Cost}{Quantity})}{d(Quantity)} \] And similarly, the slope of MC can be represented as: \[ \frac{d(MC)}{d(Quantity)} \]

To prove that the Average Cost is necessarily decreasing when the Marginal Cost is decreasing, we must show that the slope of AC is negative when the slope of MC is negative. However, that's not always true. For instance, consider a case when the Marginal Cost is greater than the Average Cost i.e. \( MC > AC \), and Marginal Cost is decreasing. In such a scenario, the average cost is increasing despite the fact that the marginal cost is decreasing. On the other hand, if the Marginal Cost is less than the Average Cost i.e. \( MC < AC \) and Marginal Cost is decreasing, then the average cost will also be decreasing. Hence, we can't conclusively say that the average cost is necessarily decreasing when the marginal cost is decreasing as it depends on the specific relationship between the average cost and the marginal cost for the given scenario.