Unit 7 The firm and its customers

7.4 Production and costs: The cost function for Beautiful Cars

When we analyse a firm’s decision on how much output to produce and what price to set, we need to know how the firm’s production costs vary with its output level: that is, the firm’s cost function. In the example of Apple Cinnamon Cheerios in Section 7.2 we make the simplest possible assumption: the unit cost of a pound of Cheerios is the same, irrespective of the scale of production. In other words, the firm has constant returns to scale.

But as Section 7.3 explains, costs per unit of output may vary with the level of production. How does this affect the firm’s price and quantity decision?

Imagine a firm that manufactures cars. Compared with Volkswagen, which produces more than 9 million vehicles a year, this firm produces specialty cars and is rather small. We will call it Beautiful Cars.

Think about the costs of producing and selling cars. The firm needs premises (a factory) equipped with machines for casting, machining, pressing, assembling, and welding car body parts. It may rent them from another firm, or raise financial capital to invest in its own premises and equipment. Then it must purchase the raw materials and components, and pay production workers to operate the equipment. Other workers will be needed to manage the production process, and to market and sell the finished cars.

cost function

The relationship between a firm’s total costs and its quantity of output. The cost function C(Q) tells you the total cost of producing Q units of output (including the opportunity cost of capital).

opportunity cost

What you lose when you choose one action rather than the next best alternative. Example: ‘I decided to go on vacation rather than take a summer job. The job was boring and badly paid, so the opportunity cost of going on vacation was low.’

The firm’s owners—the shareholders—would usually not be willing to invest in the firm if they could make better use of their money by investing and earning profits elsewhere. What they could receive if they invested elsewhere, per dollar of investment, is an example of opportunity cost, in this case called the opportunity cost of capital. Part of the cost of producing cars is the amount that has to be paid out to shareholders to cover the opportunity cost of capital—that is, to induce them to continue to invest in the assets that the firm needs to produce cars.

opportunity cost of capital

The opportunity cost of capital is the amount of income an investor could have received, per unit of investment spending, by investing elsewhere.

variable costs

Costs of production that vary with the number of units produced.

average cost

The total cost of producing the firm’s output divided by the total number of units of output produced.

marginal cost

The increase in total cost when one additional unit of output is produced. It corresponds to the slope of the total cost function at each point.

The different costs of production facing the firm can be classified as either fixed costs or variable costs. Costs are fixed if the firm has to pay them irrespective of the number of cars it produces and sells. For Beautiful Cars, we will assume that the size of the factory is fixed, so the associated costs are fixed too—either rental payments to another firm under a long-term contract, or the opportunity cost of capital invested in the factory. These will be the same whether it produces many cars, or none at all. Likewise, R&D to develop future models also carries fixed costs. We assume that other costs, such as wages, raw materials, and equipment costs are variable and increase with output: if the firm decides to increase the number of cars produced per day, it will need to increase all of these variable inputs—raising total variable costs (including the wage bill, and the opportunity cost of investing in equipment).

Suppose that Beautiful Cars has fixed costs, F, and that its variable costs are directly proportional to the quantity of cars it produces. So its cost function, giving the total cost of producing Q cars, is:

where c is the cost per car.

The upper panel of Figure 7.7 represents the total cost function C(Q) of Beautiful Cars graphically. It shows how total costs depend on the quantity of cars, Q, produced per day, when F = $80,000 per day, and c = $14,400 per car. From the total costs, we have worked out the average cost per car, and how it changes with Q; the average cost (AC) function is plotted in the lower panel.

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Figure 7.7 Beautiful Cars: total cost and average cost.

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The cost function

The top panel shows the cost function, C(Q). It shows the total cost for each level of output, Q.

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Fixed costs

Beautiful Cars has fixed costs of $80,000 per day. The firm incurs these costs irrespective of output. When Q = 0, the only costs are the fixed costs: C(0) = 80,000.

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Total costs are increasing

As Q increases, the firm needs to employ more production workers and purchase more raw materials. Total costs rise by $14,400 per car produced, so the cost function is a straight line. At point A, 10 cars are produced at a cost of $224,000.

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https://www.core-econ.org/microeconomics/07-firm-and-customers-04-beautiful-cars.html#figure-7-7d

Average cost

The average cost of a car is the total cost divided by the number of cars. If the firm produces 10 cars per day, the average cost is AC = $224,000/10 = $22,400. We have plotted the average cost at point A on the lower panel.

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Falling average cost

As output rises above A, the average cost falls. At point B, the total cost is $512,000, and average cost is $17,067. At point D, average cost is lower still: $16,000.

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The average cost function

We can calculate the average cost at every value of Q to draw the average cost (AC) function in the lower panel.

The slope of the cost function tells us how much total cost increases for each additional car produced. The increase in costs when output increases by one unit is called the marginal cost (MC). For Beautiful Cars the slope, and therefore the marginal cost, is a constant, c. So whatever the number of cars it decides to make, the marginal cost of a car (the cost of producing one more) is c = $14,400.

Whenever a firm has a cost function with fixed costs and constant marginal cost, the average cost of a unit of output falls as output rises. Figure 7.7 illustrates this for Beautiful Cars; we can also deduce this by writing:

So the average cost of a car is its marginal cost plus a share of the fixed costs. The average cost is always greater than the marginal cost, but as output increases the fixed costs are shared between more and more cars, and average cost decreases. Figure 7.8 shows both the average cost and marginal cost functions—also called the AC and MC curves—for Beautiful Cars. In the figure, average cost slopes downward, getting closer and closer to the constant marginal cost, $14,400.

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Figure 7.8 Beautiful Cars: average and marginal cost.

Costs in the short run and the long run

short run

The term does not refer to a specific length of time, but instead to what happens while some things (such as prices, wages, capital stock, technology, or institutions) are assumed to be held constant (they are assumed to be fixed, or exogenous). For example, the firm’s stock of capital goods may be fixed in the short run, but in the longer run the firm could vary it (by selling some, or buying more).

Firms’ marginal costs are not always constant—particularly if some inputs are difficult to change. Remember that the marginal cost is the cost of making one more unit of output. A car manufacturer might reach the point where the only way to raise output with its current stock of equipment is to introduce overtime shifts on the assembly line. If overtime wage rates are higher, the marginal cost of a car will be higher. We say that its marginal cost increases with output in the short run—that is, while its stock of equipment is fixed—and then it may be higher than the average cost.

exogenous

Exogenous means ‘generated outside the model’. In an economic model, a variable is exogenous if its value is set by the modeller, rather than being determined by the workings of the model itself. See also: endogenous.

endogenous

Endogenous means ‘generated by the model’. In an economic model, a variable is endogenous if its value is determined by the workings of the model (rather than being set by the modeller). See also: exogenous.

long run

The term does not refer to a specific length of time, but instead to what is held constant and what can vary within a model. The short run refers to what happens while some variables (such as prices, wages, or capital stock) are held constant (taken to be exogenous). The long run refers to what happens when these variables are allowed to vary and be determined by the model (they become endogenous). A long-run cost curve, for example, refers to costs when the firm can fully adjust all of the inputs including its capital goods.

economies of scope

Cost savings that occur when two or more products are produced jointly by a single firm, rather being produced in separate firms.

The cost function we have described for Beautiful Cars is a long-run cost function, in the sense that we assume it can increase the amount of equipment as well as the size of the workforce when it wants to increase output, so that its marginal cost remains constant.

But we have also assumed that it has substantial fixed costs, including the costs of the factory. We could analyse the firm’s decisions in what we might call the very long run, in which it could vary the size of the factory, too. For manufacturing firms like Beautiful Cars, a high proportion of total costs will be variable in the very long run. But there are other types of firms that do have high long-run fixed costs: we will cover some examples in Section 7.11.

An example: The cost function of a university

Economists Rajindar and Manjulika Koshal studied the cost functions of universities in the US.² They estimated the marginal and average costs of educating graduate and undergraduate students in 171 public universities in the academic year 1990–91. (Work through Exercise 7.2 to explore how average costs and marginal costs vary with the number of graduate and undergraduate students.) They found that the universities benefited from what are termed economies of scope: there were cost savings from producing several products together—graduate education, undergraduate education, and research.³

Extension 7.4 Cost functions for the case when marginal costs are increasing

In the main part of this section, we assume that Beautiful Cars has linear variable costs: they increase in direct proportion to output \(Q\). In this extension, we describe cost functions for more general cases, using calculus (differentiation) to analyse how costs change as output rises.

In the main part of the section, Beautiful Cars has total cost function:

\[C(Q)=F+cQ\]

where the parameter, \(F\), represents its fixed costs and \(c\) is its constant marginal cost. But as we have explained, marginal cost may not always be constant, particularly in the short run. For example, if increasing the number of units requires more intensive production, machines may break down and workers may get tired, and the firm may need to pay overtime rates. These effects would increase the marginal cost. In other circumstances, a larger scale of production could enable the firm to use its inputs more efficiently, in which case marginal cost might fall as \(Q\) increases.

As explained in Extension 3.3, we use derivatives to measure marginal changes in all the mathematical extensions.

In the main part of this section, we defined marginal cost as the addition to total cost of producing one more unit of output. To describe cost functions more generally, we will treat \(Q\) as a continuous variable so that we can differentiate the cost function \(C(Q)\). Then the marginal cost corresponds to the derivative of the cost function: the rate of change in costs in response to an infinitesimal increase in \(Q\). In general, total costs must increase with the quantity of output produced, so the marginal cost is positive:

\[\text{MC}=C'(Q) > 0\]

The upper panel of Figure E7.1 shows an alternative, non-linear cost function for Beautiful Cars. This panel shows the following:

• The firm has fixed costs, represented by the point where the cost function crosses the vertical axis: if no cars are produced, the cost is \(C(0) = F > 0\).
• For \(Q > 0\), \(C\) is increasing and also convex: the slope of the cost curve increases as \(Q\) increases.

The convexity of the cost curve implies that marginal cost is an increasing function of output:

\[\text{MC} =C'(Q) \Rightarrow \frac{d\text{MC}}{dQ}=C''(Q)>0\]

The lower panel shows the upward-sloping marginal cost function corresponding to the total cost function in the upper panel. The firm’s marginal costs increase rapidly with the quantity of cars produced.

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Figure E7.1 An alternative cost function for Beautiful Cars.

The average cost (AC) is defined as the total cost divided by the number of cars produced. Therefore, if \(Q\) cars are produced:

\[\text{AC} = \frac{C(Q)}{Q}\]

In the upper panel, the average cost of producing \(Q\) cars is the slope of the line from the origin to the point, \((Q, C(Q))\). The diagram shows that this slope varies with \(Q\): AC is itself a function of \(Q\). In the lower panel, we have plotted the average cost that corresponds to each point on the cost function: in other words, the graph of the function, \(AC(Q)\).

The particular cost function shown in Figure E7.1 is: \(C=118,750 + 1,500Q + 190Q^2\). If you wish you can check the properties described above for this particular case. Since it is quadratic, it has one property that does not hold more generally. The marginal cost function is:

\[\text{MC} =C'(Q) =1,500 + 380Q\]

which is not only upward-sloping, but also a straight line.

Figure E7.1 represents a case where marginal costs rise as output increases. But we can use the same approach to draw the cost curves for a firm with decreasing marginal costs: as done in Exercise 7.2.

The shape of the average cost function, and the relationship between MC and AC

Remember that in general, at any point, \(Q\):

• the marginal cost corresponds to the slope of the cost function \(C(Q)\), and
• the average cost corresponds to the slope of the ray from the origin to \(C(Q)\).

The upper panel of Figure E7.1 shows that average cost is high when \(Q\) is low; it then decreases gradually until point B, where \(Q=25\), before increasing again. This is reflected in the lower panel by the U-shaped AC curve, with a minimum value at \(Q=25\).

Now compare the marginal and average costs in the lower panel.

• \(\text{MC} < \text{AC}\) if \(Q < 25\)

• \(\text{MC} > \text{AC}\) if \(Q > 25\)

• \(\text{MC} = \text{AC}\) if \(Q = 25\)

This example illustrates a general property of cost functions: the difference, \(\text{MC} - \text{AC}\), always has the same sign as the slope of the AC curve. In the case of the linear cost function in Figure 7.8, the AC curve slopes downwards for all values of \(Q\). Correspondingly, the marginal cost, which in that case is a horizontal line, lies below the AC curve: \(\text{MC} < \text{AC}\) for all \(Q\).

We now prove this general property: for all cost functions, whatever their shape, \(\text{MC} - \text{AC}\) has the same sign as the slope of the AC curve. Using the definition of AC as total cost divided by quantity produced, and applying the rule for differentiating a quotient, the slope of the AC curve is:

\[\frac{d}{dQ} \left( \frac{C(Q)}{Q} \right) = \frac{QC'(Q)- C(Q)}{Q^2}\]

Now \(C'(Q)= \text{MC}\) and \(C(Q) = Q \times \text{AC}\). Therefore:

\[\begin{align*} \frac{d}{dQ} \left( \frac{C(Q)}{Q} \right) &= \frac{\text{MC}}{Q} - \frac{C(Q)}{Q^2} \\ &= \frac{\text{MC}}{Q} - \frac{Q(\text{AC})}{Q^2} \end{align*}\] \[\Rightarrow \frac{d}{dQ} (\text{AC} ) = \frac{\text{MC}-\text{AC}}{Q}\]

Since \(Q>0\), it follows that the slope of the AC curve at each value of \(Q\) has the same sign as \(\text{MC} - \text{AC}\).

A consequence of this result is that if a firm’s average cost curve is U-shaped, like the one in Figure E7.1, the marginal cost curve crosses the average cost curve (\(\text{MC} = \text{AC}\)) at the point where average cost is minimized.

Question E7.1 Choose the correct answer(s)

Consider a firm with fixed costs of production. Using this information, read the following statements about the firm’s average cost (AC) and marginal cost (MC) and select the correct option(s).

• When \(\text{AC} = \text{MC}\), the AC curve has a zero slope.
• When \(\text{AC} > \text{MC}\), the MC curve is downward sloping.
• When \(\text{AC} < \text{MC}\), the AC curve is downward sloping.
• The MC curve cannot be horizontal.

• When \(\text{AC} = \text{MC}\), the cost of an additional unit equals the average cost of all existing units. Therefore, the new AC will be the same and the slope is zero.
• The MC curve can be upward sloping, horizontal, or downward sloping, irrespective of the relative size of AC and MC.
• When \(\text{AC} < \text{MC}\), the cost of an additional unit is greater than the average cost of the existing outputs. So the new AC will be larger. The AC curve is upward sloping.
• If MC is constant, then the MC curve is horizontal.

Exercise E7.1 Drawing non-linear cost functions

You are given the following two cost functions:

• Cost function #1: \(C(Q) = 5Q^2 + 3Q + 3,600\)
• Cost function #2: \(C(Q) = -2Q^2 + 4Q + 19,600\)

Do the following for each total cost function:

1. Derive an expression for i) the marginal cost curve and ii) the average cost curve (both defined for \(Q \geq 0\)).
2. Use your answers to Question 1 to draw two diagrams as in Figure E7.1: one showing the total cost curve, and one showing the marginal cost and average cost curves. Remember to plot the curves for \(Q \geq 0\) (no negative values of \(Q\)).
3. Describe the relationship between the shape of the total cost curve and the shape of the marginal cost and average cost curves.

Read more: Section 7.1 (for the quotient rule) and Sections 8.1, 8.2, and 8.4 (covering curve-sketching and convexity) of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.