Cost Curves and How They Relate
J. Wilson Mixon, Jr. and Soumaya Tohamy*

 
 

Introduction

In a supplemental note to a reprint of the classic "Cost Curves and Supply Curves," Jacob Viner relates his confusion about how long-run and short-run cost curves relate. 1 He asked a draftsman to make the long-run curve a U-shaped envelope that consisted of the minimum points on all short-run curves. The draftsman pointed out the impossibility of this construction, causing Viner to realize that what he really wanted was that the long-run curve consist of points on the the average cost curves at the minimum-cost scale for each quantity, the well-established envelope. Despite more than a half-century of fine-tuning, imprecision about how the various curves—long-run and short-run, average and marginal—relate remains a source of potential confusion, at least when drafting is involved.

This paper presents a set of Excel workbooks (see footnote 4 for information on downloading the workbooks) that produce graphs of long-run and short-run cost curves that are consistent. That is, the long-run curve is the proper envelope of the short-run curves and the average and marginal curves (both long-run and short-run) are consistent. Before presenting the workbooks, we review the relationships that must hold if graphs are to represent economic relationships accurately.

The paper is organized as follows. The next section reviews textbook materials, how the various long-run and short-run curves relate to each other. It is followed by a section that discusses the specific functional forms we use to illustrate these relationships. Then a section describes how a set of Excel workbooks provides a graphic representation of these curves and describes the options the user has when employing these workbooks. Finally, we extend the analysis briefly to include revenue as well as costs, for both price-taking and price-making firms.
 
 

The Cost Curves and How They Bend

The typical microeconomics textbook and classroom development of cost curves consists of two parts. One shows how the per-unit cost curves (average and marginal) relate to total costs. The second shows how long-run cost curves (total, average, and marginal) relate to their short-run counterparts.

First consider the relationships between average and marginal curves. When the average value involves a linear relationship, the representation is simple: the marginal curve is half the horizontal distance to the average curve. For nonlinear cost curves, however, drawing the marginal curves so that they correspond to the average curve (or vice versa) can be tedious. Too often we simply sketch a marginal cost curve that cuts the average cost curve at its minimum point and assume that this is good enough. Even textbook authors commit this error fairly frequently. (This is not an exercise in textbook bashing. We do not cite the textbooks that commit the errors noted below. Readers may contact the authors for examples.)

Failure to draw the curves consistently causes at least two inconsistencies. One is that the quantity at which marginal revenue equals marginal cost will not be the quantity at which profit—(price less average cost) times quantity—is, in fact, maximized. The other is that profit as defined above will not equal profit, defined as the area between the marginal revenue and marginal cost curves. One of the leading principles textbooks contains a graph in which the area between the marginal revenue curve and the marginal cost curve is roughly two thirds larger than the area defined in terms of price, average cost, and quantity. Such a discrepancy is large enough to confuse students.

The representation of long-run vs. short-run curves contains the above difficulties and at least one more. When drawing the long-run/short-run per-unit curves, a common error is to forget that, when the average curves are tangent, the marginal curves must intersect. Generally textbook authors are careful to get this right (or to avoid it by drawing sections of the long-run marginal curves that do not approach the point of necessary intersection), but the occasional error occurs. Certainly in hand-drawing examples for classroom or for examinations, it is easy to overlook this necessary relationships.

To summarize: For representations of cost curves to be consistent, the conditions below must pertain.

One other condition must be imposed if the long-run average (or total) cost curve is to be an envelope consisting of minimum points on a series of short-run average (or total) curves:


Functional Form

Our task is to define a family of long-run cost curves and short-run cost curves such that the above interrelations can be ensured. Each of the restrictions cited above provides two conditions at some specified quantity (one for the average function's average value and the other for its derivative, the marginal value). This requires that the functional form used be defined by two parameters. For tractability, we use the polynomial form. The third condition establishes limits on the orders of polynomials that can be employed. After considerable experimentation, we determined that the following curves perform quite well.

The long-run total cost curve has the form: 2
 

1. LTC = aQ2 + bQ0.5. 3

The short-run total cost curve’s form is:
 

2. STC = mQ3 + n.

This form for the STC has one drawback: The average variable cost curve approaches zero as quantity decreases. Circumventing this difficulty involves arbitrary impositions on the functional form. Rather than encumber the analysis with such impositions (which can result in absurd results like downward-sloping total cost curves), we add a set of worksheets for short-run curves, based on the functional form:
 

2'. SAC = p + q(Q – Q*)2 (or equivalently SAC = rQ2 + sQ + t).

Implementing this form requires two arbitrary impositions, that SAC approach a specific finite value as Q approaches zero and that fixed cost be a specified fraction of total cost at a specified value of Q.

The Spreadsheets

To make the workbooks as useful as possible, we allow considerable flexibility. 4   Consider first the workbook that relates to costs alone. The user provides two pieces of data that establish the long-run cost curves. These are the quantity at which the long-run cost curve is minimized and the long-run average cost at that quantity. The first two sheets in the workbook report the cost curves consistent with this information. First, the total cost curve (both the graph and a table of points on the graph) appears; then, the next sheet shows the long-run total cost and the associated per-unit (average and marginal) curves.

The next three sheets involve the short run. The user provides a quantity at which long-run total cost equals short-run total cost. The first of this set of sheets returns the long-run and short-run total cost curves (and associated tabled values). The next sheet does the same for per-unit curves. The third sheet shows the per-unit short-run curves: short-run average cost, average variable cost, short-run marginal cost, and average fixed cost.

The figure below is representative. The user specifies the quantity at which long-run average cost is minimized (Q 0), the long-run average cost at that quantity (C0), and the quantity at which long-run and short-run average cost curves are tangent (Q 1). (Also, for purposes of controlling appearance, the user may change the size of the increments between adjacent observed quantities). The resulting short-run average cost at Q 1 is reported along with the graphs of the pertinent average and marginal relationships. The user may type a chosen value for the variables or may use the scroll bars. The "Reset Values" button returns the values to their default values. The other two buttons provide navigation, to the "Definitions" sheet or to the table on this sheet on which the graphs are built.


Figure 1. Worksheet from CostCurves_Basic.
Click on the title to download the workbook.







The sheets described above provide an accurate drawing of long-run and short-run cost curves, depicting the pertinent relationships among them. As noted above and as observed in Figure 1, the choice of functional form for the long-run curves dictates that AVC and SMC achieve their minimum at a quantity of zero. Inter alia, this implies that the firm’s short-run supply curve begins at the origin. To allow more flexibility, an additional set of graphs based on a cubic short-run cost curve is appended. Figure 2 shows one such curve.
 
 


Figure 2. Worksheet from CostCurves_quadratic_AVC
Click on the title to download the workbook.

Costs and Revenue

While the primary purpose of this article is to provide an easy way to depict cost curves accurately, we add two more workbooks that show revenue as well as cost. The first depicts a price-taking firm, the second a price-making firm. In the former, the user specifies a price; in the latter, the price intercept for the demand curve. The sheets that previously showed total cost now also show total revenue and profits along with total cost.5    The sheets that show per-unit costs now also show the demand curve and the marginal revenue curve. In each case, the profit-maximizing quantity (and, for the price-making firm, the price) and the maximum profit level are shown. Figure 3 shows one of the graphs, this one for a price-making firm. The graph shows the short run only, but the table reminds the user that the firm will behave differently given more time to adjust.


Figure 3. Worksheet from CostCurves_w_revenue
Click on the title to download the workbook.
 

Conclusion

It is important that we represent economic relationships accurately. Failure to do so can confuse students, whose grasp of graphical representations is often tenuous at best. This paper provides a means to draw total and per-unit cost curves for either the short run or the long run and to depict the relationship between costs in the short run and the long run. The analysis also incorporates revenue relationship, thereby showing how profits relate to production.

The main purpose of the paper is to present the instructor a way to draw these relationships accurately using Microsoft Excel. The spreadsheets can be used for developing displays in classroom instruction and for handouts. Instructors may also use the workbooks as the basis for homework assignments. Students can explore how changes in the model's parameters affect efficient output levels and the resulting levels of profits.
 



Footnotes
*
Mixon: Dana Professor of Economics, Berry College (wmixon@berry.edu).
Tohamy: Economist, Consumer Products Safety Commission (stohamy@cpsc.gov).
This is part of a larger set of material, mostly related to placing microeconomic models in Microsoft Excel. For a set of related material, see http://www.campbell.berry.edu/faculty/economics/index.shtml.
1. Irwin (1991 ) provides background for this account as well as a general overview of Viner's work. See also Samuelson (1998 ). Return to text .
2. We also provide a set of worksheets in which the long-run average cost has a "bathtub" shape. Return to text .
3. One advantage of this functional form is that it avoids the necessity of imposing nonnegativity conditions. See Chiang ( 1967 ) for the requisite conditions for a cubic total cost curve.  Return to text .
4. The workbooks are as follows. The cost curves are developed in  CostCurves_Basic. Revenue for price-taking and price-making firms are incorporated into the analysis in  CostCurves_w_revenue . Finally, the short-run configuration that allows U-shaped average variable costs is in the workbook  CostCurves_quadratic_AVC .  Click on titles to download workbooks.  Return to text .
5. The profit maximizing quantities are determined numerically rather than analytically. Accordingly, the reported values are only (very close) approximations. Return to text .

 



References  (Back to Footnotes)

Chiang, Alpha C. 1967. Fundamental Methods of Mathematical Economics . New York: McGraw Hill.

Irwin, Douglas A. 1991. "Introduction" in Viner (1991), 6 - 7.

Samuelson, Paul A. 1998. "How Foundations Came To Be," Journal of Economic Literature, Vol 36, 1375 - 1386.

Viner, Jacob 1932. "Cost Curves and Supply Curves." Zeitschrift für Nationalökonomie. Vol. 3, 23 - 46.

Viner, Jacob 1991. Essays on the Intellectual History of Economics (Douglas A. Irwin, editor). Princeton NJ: Princeton University Press.
 
 

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