Suppose the long-run total cost function for an industry is given by the cubic equation \(\mathrm{TC}=\mathrm{a}+\mathrm{b} q+\mathrm{c} q^{2}+\mathrm{d} q^{3}\). Show (using calculus) that this total cost function is consistent with a U-shaped average cost curve for at least some values of \(a, b, c\), and \(d\).

Understanding the average cost curve is essential in economics, as it represents the average cost per unit of output produced. It is derived from the total cost (TC) function, which in our exercise was given in the form of a cubic equation: \( TC = a + bq + cq^2 + dq^3 \).

When we speak about average cost (AC), we're essentially calculating the cost on a per-unit basis. The AC is found by dividing TC by the quantity (q) of goods produced, leading us to \( AC = a/q + b + cq + dq^2 \). This curve typically takes on a 'U' shape, which indicates that initially, as production begins, the average cost of production decreases. But, after reaching a certain level of production, the average cost starts to increase with each additional unit produced.

This U-shape reflects the presence of economies of scale at lower production levels—where increasing production can lower costs—and diseconomies of scale at higher production levels—where costs increase with production. Managers use this curve to determine the optimal production level where costs are minimized.

In calculus, the first derivative is a powerful tool that represents the rate of change of a function. To analyze the behavior of the average cost curve, we resort to the first derivative in calculus to determine whether the curve is increasing or decreasing at certain points.

The steps provided in our exercise involve finding the first derivative of the AC function. This is done by differentiating \( AC = a/q + b + cq + dq^2 \) with respect to q, which yields \( AC' = -a/q^2 + c + 2dq \). This derivative can tell us a lot about the shape of the average cost curve. For example, when \( AC' < 0 \), the average cost is decreasing, and when \( AC' > 0 \), the average cost is increasing. By finding where the first derivative is negative, positive, and zero, we can sketch the behavior of the AC curve and identify points such as where the average cost is at its minimum—a crucial factor in cost management.

The minimum average cost is a point of significant interest in the study of production and cost efficiency. It represents the point at which a company is producing goods at the lowest possible average cost. In the context of our exercise, this point occurs where the average cost curve changes from decreasing to increasing—think of it as the bottom of the 'U' shape on the graph.

To find this point mathematically, we examine the second derivative of AC, denoted as \( AC'' \). Since the first derivative \( AC' \), given by \( -a/q^2 + c + 2dq \), tells us the slope of the AC curve, the second derivative will indicate the concavity or convexity at a particular point. In this case, \( AC'' = 2a/q^3 + 2d \). The condition for the minimum average cost is where this second derivative switches from negative to positive, meaning \( AC'' \) equals 0. At this point, q tells us the quantity of production at which average cost is at its minimum—enabling businesses to optimize their production levels and achieve cost effectiveness in their operations.