Cost Curves and How They Relate |

**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.

- For both long-run and short-run curves, the average and marginal curves must derive from the same total cost curve;
- At the quantity for which long-run and short-run average cost curves are tangent, the accompanying marginal cost curves must intersect.

- The short-run average cost curve must exhibit more curvature than its long-run counterpart (or equivalently the short-run marginal cost curve must intersect its long-run counterpart from below).

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 = aQ^{2} + bQ^{0.5}. ^{ 3
} |

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

2. | STC = mQ^{3} + 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 = rQ^{2} + 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 (C_{0}), 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.

* |
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 . |

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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