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.
Building block
For an introduction to opportunity cost, read Section 2.2.
- 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:
\[C(Q)=F+cQ\]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.
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.
Average and marginal cost
At each point Q on the cost function C(Q), the average cost (AC) is the total cost of producing Q units of output, divided by number of units:
\[\text{AC} = \frac{C(Q)}{Q}\]The marginal cost (MC) is the additional cost of producing one more unit of output, which corresponds to the slope of the cost function. If cost increases by ΔC when quantity is increased by ΔQ, the marginal cost can be estimated by:
\[\text{MC} = \frac{\Delta C}{\Delta Q}\](Δ is a mathematical symbol that represents ‘the change in’.)
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:
\[\text{AC}(Q)=\frac{C(Q)}{Q}=\frac{F+cQ}{Q}=c+\frac{F}{Q}\]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.
Figure 7.8 Beautiful Cars: average and marginal cost.
Question 7.5 Choose the correct answer(s)
Suppose that the cost function for a cereal producer is C(Q) = 2Q, where Q refers to pounds of cereal. Using this information, read the following statements and choose the correct option(s).
- When Q = 0, the total cost of production is 0. If there were fixed costs of production, the total cost of production would be positive at Q = 0 (the firm incurs these costs regardless of how much they produce.)
- The slope of the total cost function is the marginal cost, which in this case is 2.
- The average cost = 2 for all outputs. There are no fixed costs, so the average cost does not fall with output.
- Both the average and the marginal cost are 2 for all values of Q.
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.
In economic models, short run and long run don’t refer to specific periods of time. In a short-run equilibrium one or more variables—typically something that takes more time to adjust—is exogenous (held constant). Modelling what will happen when such a variable becomes endogenous (can be adjusted) gives us the long-run equilibrium.
- 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
For an entertaining further discussion of costs, read chapter 7 of The Theory of Price by economist George Stigler.1
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
Exercise 7.2 Cost functions for university education
The average and marginal costs per student for the year 1990–91 that Koshal and Koshal calculated from their research are shown in the table.
Students | MC ($) | AC ($) | Total cost ($) | |
---|---|---|---|---|
Undergraduates | 2,750 | 7,259 | 7,659 | 21,062,250 |
5,500 | 6,548 | 7,348 | 40,414,000 | |
8,250 | 5,838 | 7,038 | ||
11,000 | 5,125 | 6,727 | 73,997,000 | |
13,750 | 4,417 | 6,417 | 88,233,750 | |
16,500 | 3,706 | 6,106 | 100,749,000 | |
Students | MC ($) | AC ($) | Total cost ($) | |
Graduates | 550 | 6,541 | 12,140 | 6,677,000 |
1,100 | 6,821 | 9,454 | 10,339,400 | |
1,650 | 7,102 | 8,672 | ||
2,200 | 7,383 | 8,365 | 18,403,000 | |
2,750 | 7,664 | 8,249 | 22,684,750 | |
3,300 | 7,945 | 8,228 | 27,152,400 |
- Using the data for average costs, fill in the missing figures in the total cost column.
- Plot the marginal and average cost curves for undergraduate education on a graph, with costs on the vertical axis and the number of students on the horizontal axis. On a separate diagram, plot the equivalent graphs for graduates. (Hint: For help on how to plot cost curves in Excel, check steps 1–4 of this tutorial.)
- Describe the shapes of the marginal and average cost curves for undergraduates and graduates. Verify that your answer is consistent with the authors’ findings described in this unit.
- Describe some similarity and differences between the curves for undergraduates and curves for graduates. Suggest some explanations for your observations.
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George J. Stigler. 1987. The Theory of Price. New York, NY: Collier Macmillan. ↩
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Rajindar K. Koshal and Manjulika Koshal. 1999. ‘Economies of Scale and Scope in Higher Education: A Case of Comprehensive Universities’. Economics of Education Review 18 (2): pp. 269–77. ↩
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Economies of Scale and Scope. The Economist. Updated 20 October 2008. ↩