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The most useful "shape" for an investigation into the relationship between short-run
average cost and long-run average cost is the so-called
"U-shape." Although this is a
bit misleading since we do not want the upward-pointing arms of the curve to ever be perfectly
vertical, it is this general shape for the **average cost curve** that
corresponds to an "S-shaped"
**total cost curve**, as shown in the two charts below.

In the paired charts above, the upper chart shows a total cost curve, while the lower chart shows the associated average cost and marginal cost curves. Note that the total cost curve has an "S-shape," or "backwards-S" shape, while each of the average cost and marginal cost curves has a "U" or "bowl" shape.

[ *Question*: Is the total cost curve a long-run total cost curve or a
short-run total cost curve; can we determine this visually?

*Hint:* According to the total cost curve, what is the total cost to produce zero units of output? ]

Each of the charts has a diamond-shaped data marker located at an output of
*q* = 30 and each chart has a circular marker located at an output of
*q* = 20. Interesting things occur at these output levels!

**In the chart showing the average and marginal cost curves only**, reproduced
below by itself, each of the two data markers is located at a point that is "special"
for an obvious reason: the circular marker occurs at the minimum point on the marginal cost
curve and the diamond-shaped marker occurs at the minimum point on the average cost curve.
In addition, marginal cost is equal to average cost at the minimum point
on the average cost curve.

**In the chart showing the total cost curve only**, reproduced below by itself,
note the thin, solid line that begins at the lower-left corner of the chart and passes through
the diamond-shaped marker. Observe that the entire total cost curve
*lies above* this thin, solid line except at the diamond-shaped marker and where the
total cost curve begins at *q* = 0. Average cost at any level of
output, *AC* = *TC*/*q*, has the geometric interpretation of being
the slope of a line drawn from the beginning point of the total cost curve at
*q* = 0 to any point on the total cost curve. Since the thin, solid line
that passes through the diamond-shaped marker at *q* = 30 is the
"flattest" or "least steep" such line from the beginning point at
*q* = 0 to any point on the total cost curve, average cost reaches its
minimum point here, at *q* = 30!

**Still analyzing the total cost curve**, the circular data marker occurs where
*q* = 20 and there is a thin, dashed line that passes through the circular data
marker. To understand what is happening here, suppose that we were to draw a line that is
*tangent* to the total cost curve at *q* = 10. Would this line be
"flatter" or "steeper" than the thin, dashed line that passes through
the circular data marker at *q* = 20?

Clearly, the line tangent to the total cost curve at *q* = 10 would be
**steeper** than the thin, dashed line that passes through the circular data marker
at *q* = 20; see the chart below with the thick, solid tangent line passing
through the point on the total cost curve where output is *q* = 10 and
*TC* = $750.

Suppose we were to draw a line that is *tangent* to the total cost curve at
*q* = 30. Would this line be "flatter" or "steeper" than
the thin, dashed line that passes through the circular data marker at *q* = 20?

Actually, we do not even have to imagine this tangent line: the thin, solid line that
passes through the *diamond-shaped* marker at *q* = 30 *is tangent to
the total cost curve* at *q* = 30; see the original chart for
total cost, which is reproduced below. Clearly, this tangent line is steeper than the thin,
dashed line that passes through the circular data marker at *q* = 20.

Indeed, every line that would be tangent to the total cost curve at an output less than
*q* = 20 would be steeper than the thin, dashed line passing through
the circular marker at *q* = 20. Further, every line that would be tangent to
the total cost curve at an output greater than *q* = 20 would be
steeper than the thin, dashed line passing through the circular marker at
*q* = 20. Therefore, the thin, dashed line in the upper chart is the
"flattest" such tangent line. Since marginal cost at any level of output has the
geometric interpretation of being the slope of a line that is tangent to the total cost
curve at that level of output, the fact that the thin, dashed line through the circular marker
at *q* = 20 is tangent to the total cost curve at that point and
it is the "flattest" such tangent line affirms that
marginal cost reaches its minimum point at *q* = 20!

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