In the case of the Beautiful Cars firm's linear cost function, the average cost (AC) is always greater than the marginal cost (MC). This is consistent with the general economic principle that when an average cost curve slopes downwards for all quantities, as it does here, the marginal cost will always be less than the average cost.

Average vs. Marginal Cost for Beautiful Cars' Linear Cost Function

The slope of the average cost (AC) curve can be mathematically determined by taking the derivative of the average cost function with respect to the quantity of output (Q). Since the average cost is defined as the total cost divided by quantity ($AC = \frac{C(Q)}{Q}$), finding its slope requires applying the quotient rule of differentiation.

Deriving the Slope of the Average Cost Curve

In many production scenarios, the average cost (AC) curve is U-shaped, meaning it is initially high for low output, decreases to a minimum point, and then rises as output increases further. A significant property of a U-shaped AC curve is that the marginal cost (MC) curve intersects it at its lowest point. At this specific point of intersection, the marginal cost is equal to the average cost (MC = AC).

U-Shaped Average Cost Curve

A general property for all cost functions is that the sign of the slope of the average cost (AC) curve is identical to the sign of the difference between marginal cost and average cost (MC - AC). This direct relationship holds because the quantity of output (Q) is always positive. Consequently, if the AC curve is declining, MC is less than AC; if it's rising, MC is greater than AC; and at the minimum of the AC curve where the slope is zero, MC equals AC.

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The average cost curve, denoted as AC(Q), is a graph that plots the average cost of production for every possible level of output. Each point on the curve represents the average cost associated with a specific quantity produced.

The Average Cost Curve

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

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