Question

Suppose the long-run total cost function for an industry is given by the cubic equation \(\mathrm{TC}=\mathrm{a}+\mathrm{bq}+\mathrm{cq}^{2}+\mathrm{d} q^{3}\) 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 \(d\).

Short answer

The total cost function \(TC=a+bq+cq^{2}+dq^{3}\) will yield a U-shaped average total cost curve for values of \(a, b, c,\) and \(d\) such that the second derivative of the ATC curve is positive.

Step-by-step solution

Calculate the Marginal Cost (MC)

The marginal cost is the derivative of the total cost function with respect to quantity. It can be computed using the following formula: \(MC=\frac{dTC}{dq}\). Applying this to the given total cost function we get: \(MC = \frac{d}{dq}(a+bq+cq^2+dq^3) = b+2cq+3dq^2\).

Calculate the Average Total Cost (ATC)

The average total cost is defined as the total cost divided by the quantity. Compute this by: \(ATC = \frac{TC}{q} = \frac{a+bq+cq^2+dq^3}{q} = \frac{a}{q}+b+cq+dq^2\).

Determine if ATC has a minimum point

A U-shaped average cost curve means that the average total cost has a minimum point. To find this, we first find the derivative of the average total cost with respect to the quantity denoted by \(ATC' = \frac{dATC}{dq}\). This yields: \(ATC' = -\frac{a}{q^{2}} + c + 2dq\). Then, we set the derivative equal to zero and solve for \(q\) to find the turning point of the curve. If the second derivative of the ATC with respect to \(q\) is positive at this turning point, then the curve has a minimum point.

Calculate the second derivative of ATC

The second derivative of the average total cost is given by: \(ATC'' = \frac{d^2ATC}{dq^2} = \frac{2a}{q^{3}} + 2d\). If this value is positive for the \(q\)-values obtained in the previous step, then the total cost function is consistent with a U-shaped average cost curve.

Key concepts

Long-run Total Cost Function

Understanding the long-run total cost (LRTC) function is essential for businesses and economists as it helps determine the least possible cost of production for different production levels. When we discuss the long-run total cost, we refer to a period where all inputs are variable, and firms can adjust all factors of production. The LRTC function typically represents the total expense incurred by a firm in the production of a certain quantity of output in the long run.

In calculus, the LRTC function can often be depicted as a cubic function, like in our exercise, expressed as \( \mathrm{TC} = \mathrm{a} + \mathrm{bq} + \mathrm{cq}^{2} + \mathrm{d}q^{3} \), where \(q\) denotes the quantity produced, and \(a\), \(b\), \(c\), and \(d\) are constants. This function helps in analyzing how total costs change with changes in output level over time. Furthermore, the shape of the LRTC curve provides insights into economies and diseconomies of scale.

Marginal Cost

The concept of marginal cost is pivotal in economics and decision-making. Marginal cost (MC) is the additional cost incurred when producing one more unit of a good or service. It represents the change in total cost that arises when the quantity produced is incremented by one unit. That is, it's the cost of producing one extra item of a product.

Using calculus, the marginal cost is found by taking the derivative of the total cost function with respect to the quantity, which is expressed mathematically as \( MC = \frac{dTC}{dq} \). For the given cubic total cost function, we calculate the MC to be \( b+2cq+3dq^2 \). The relationship between MC and the total cost function is crucial since it indicates the direction in which costs are moving as production changes, which in turn can dictate pricing, expansion, and production strategies.

Average Total Cost

Average total cost (ATC) is a concept that relates directly to the overall efficiency of production. It's defined as the total cost per unit of output, or in other words, it's the cost on average for each unit produced. Mathematically, ATC is computed as the total cost divided by the quantity of goods produced \( \left( ATC = \frac{TC}{q} \right) \).

When expressed in terms of our function, ATC becomes \( \frac{a}{q}+b+cq+dq^2 \). The U-shaped average cost curve depicts how ATC decreases with an increase in output up to a certain point (achieving economies of scale) and then increases as output continues to grow (diseconomies of scale). This U-shape is due to the spread of fixed costs over more units and the eventual increase in variable costs due to factors such as overutilization of resources or the need for more expensive inputs.

Calculus in Economics

Calculus plays a fundamental role in economics. It is used to analyze and model the behavior of functions, which represent economic measures such as cost, revenue, and profit. Calculus helps economists understand how these measures change with varying levels of production or consumption, which is vital for optimization problems.

For instance, to locate the minimum or maximum points of cost functions and to understand the responsiveness of consumers and producers to changes in price (elasticity), we use derivatives. In our exercise, the use of derivatives helped us calculate both the marginal cost and the first and second derivatives of the average total cost. By setting these derivatives to zero, we can find turning points, which, in the case of the ATC, revealed that the total cost function can indeed be consistent with a U-shaped average cost curve under certain conditions. These mathematical tools are indispensable for forming strategies and making informed economic decisions.