12  Cost Functions

For more information on these topics, see Allen, Doherty, Weigelt, and Mansfield Chapter 6: The Analysis of Costs.

In the last two chapters, we went into detail about production functions. We learned about marginal product, average product, and a producer’s choice about maximizing output subject to a certain cost. In this chapter and the next, we’ll dive into the analysis of costs. We’ll talk about opportunity costs, short and long term cost functions, average and marginal costs, and economies of scale and scope.

But first, a few practice questions to review Classwork 11.

12.1 Classwork 11 Review

Practice Question: The production function relates units of inputs used (\(X_1\), \(X_2\)) to the quantity of output produced \(Q\). An isoquant is:

A curve that shows the combinations of inputs that minimize cost for a given level of output.

A curve that shows the maximum output that can be produced with a fixed amount of inputs.

A curve that shows the combinations of inputs that yield increasing returns to scale.

A curve that shows all combinations of two inputs that produce the same level of output.

Practice Question: The Marginal Rate of Technical Substitution is:

-1 times the slope of the isoquant.

The rate at which one input can be substituted for another while keeping output constant.

The ratio of the marginal products of the two inputs (\(X_1\) and \(X_2\)).

All of the above.

Practice Question: Let \(Q\) be the amount of output that can be produced using \(X\) units of an input. Then average product is equal to:

\(X + Q\)

\(Q/X\)

\(X/Q\)

\(Q - X\)

12.2 Opportunity Costs

Accountants and Economists define costs a little differently. Accountants are concerned with explicit costs like wages, rent, utilities, and materials, that are easily quantifiable and necessary for tax purposes, financial reporting, and ensuring compliance with laws and regulations. Economists, however, are more concerned with decision-making and how people respond to incentives. Therefore, economists use a broader definition of costs that includes both explicit and implicit costs.

For example, consider the the Wharton MBA program at the University of Pennsylvania, which costs around $100,000 per year for tuition, room, and board. Now, suppose a student was earning $70,000 per year in a job before enrolling. If you asked an accountant how much the MBA costs, they would likely say $100,000 per year. An economist, however, would say the cost is $170,000, adding the explicit costs of $100,000 to the implicit ones of $70,000 in the form of foregone wages, which is the student’s opportunity cost.

The $70,000 represents the value of the next best alternative the student gave up by choosing to pursue the MBA. Including such implicit costs is crucial for understanding the student’s decision. If the opportunity cost were too high, the student might reconsider attending the MBA program, just as they would if tuition were too expensive.

The Opportunity Cost is the value of the next best alternative, or the value of the opportunity forgone.

In a business context, consider a manufacturing company deciding how to allocate its resources. If a particular input, such as labor or machinery, could be used in another process that is more productive or generates higher returns, the opportunity cost of using that input in its current process should reflect the potential gains from its best alternative use.

12.3 Sunk Costs

A Sunk Cost is a resource that has already been spent and cannot be recovered. The sunk cost fallacy occurs when individuals or businesses continue an endeavor solely because they have already invested resources in it, even when it would be better to abandon it.

For example, imagine a manager has already spent $6 million on an advertising campaign. These costs are sunk because they cannot be recovered. If the manager argues that they only need to spend an additional $1 million to “turn the corner” and make the campaign successful, they are falling into the sunk cost fallacy. The decision to spend the extra $1 million should be based on comparing the expected return from this investment to the expected return of $1 million invested elsewhere. What’s already been spent should not influence the decision—it’s gone and cannot be recovered.

The key principle here is that decisions should be made based on future costs and benefits, not on past expenses that are unrecoverable.

Practice Question: The cost of the next most valuable opportunity is known as a:

Sunk cost

Opportunity cost

Comparative advantage

Absolute advantage

Practice Question: The economic definition of profit differs from the accounting definition of profit in that the economic definition includes:

Fixed costs

Variable costs

Opportunity costs

None of the above

Practice Question: Suppose that you value seeing a movie at $6 and tickets only cost $5, so you buy a ticket. But on the way to the theater, you lose your ticket. Should you buy another?

Yes

No

12.4 Classwork 12: Short-Run Cost Functions

A cost function represents the relationship between the costs of input (y-axis) and the quantity of output that can be produced (x-axis). It shows how costs change as production increases or decreases.

Cost functions can be defined for either the short-run or the long-run. In this classwork, we’ll explore short-run cost functions. In Classwork 13, we’ll explore long-run cost functions.

• The short-run is a time period so brief that some inputs, like the number of factories, office space, or specialized equipment, cannot be changed. For example, in a business setting, a manufacturing firm might not be able to quickly expand the number of machines it uses to produce goods. If demand suddenly spikes, the company may need to work with its existing machinery, even if it means incurring higher costs due to overtime wages or less efficient production.

• The long-run is a period long enough that all inputs become variable. A business can adjust its scale of production by building new factories, buying new equipment, or expanding its workforce. For instance, a tech company experiencing rapid growth could plan for the long-run by constructing new data centers or investing in state-of-the-art software.

We’ll consider three short-run cost concepts: fixed, variable, and total.

1. Some costs do not vary with output: these are called fixed costs. For example, suppose a bakery pays $3,000 per month to lease its storefront. Whether the bakery sells 10 cakes or 1,000 cakes in a given month, the rent does not change; it’s a fixed cost. Draw a ggplot to plot the bakery’s fixed cost on the y-axis and its output on the x-axis.

2. Some costs do vary with output: these are called variable costs. As your bakery produces more cakes, you need more flour, sugar, and other ingredients, along with labor hours. You also needs more packaging and electricity for ovens and refrigerators. Suppose this table represents your variable cost function:

Units of Output Q	Total Variable Costs
0	0
125	1800
250	2500
375	4300
500	9300

Take your ggplot from part A and add a plot of the total variable costs: use geom_point and geom_line. Then use lm to estimate a model of the variable cost function (C = 0 + I(Q^3) + I(Q^2) + Q). Use stat_function to add the line of best fit to your plot.

3. Variable cost functions often have that S-shape you found in part B. Explain why that might be the case in the context of a bakery.

4. The total short-run cost curve is equal to the fixed costs plus the variable costs. Add a total cost curve to your plot in part B. What is the total cost equation (use the parameters you found by using lm along with what you know about the bakery’s fixed costs)?

5. The average variable costs (AVC) are equal to the total variable costs divided by \(Q\). Show that another formula for AVC is \(\text{AVC} = \frac{p_1}{\text{average product}}\), where \(p_1\) refers to the price of the variable input. Then interpret: average variable costs (increase/decrease) with the price of inputs, and they (increase/decrease) with the average amount of product you can produce using your inputs. Because AVC has an inverse relationship with AP, the two curves have opposite patterns. We know AP generally (increases then decreases / decreases then increases). Therefore, AVC will generally (increase then decrease / decrease then increase).

6. The marginal cost curve measures the slope of the total variable cost (TVC) curve (rise over run): \(\text{MC} = \frac{\Delta \text{TVC}}{\Delta Q}\). Note that the change in TVC is equal to the price of variable inputs (\(p_1\)) times the change in units of input (\(\Delta X_1\)). That is, \(\Delta \text{TVC} = p_t \times \Delta X_1\). Recall that the marginal product is \(\text{MP} = \frac{\Delta Q}{\Delta X_1}\). Show that \(\text{MC} = \frac{p_1}{\text{MP}}\). Then interpret: marginal costs (increase/decrease) with the price of variable inputs and (increase/decrease) with the marginal productivity of inputs. We know that marginal product usually (increases and then decreases/decreases and then increases). Since marginal cost has an inverse relationship with marginal product, we’d expect marginal costs will usually (increase and then decrease/decrease and then increase).