Chapter 7, Cost Curves
John Eckalbar
We’ve taken D & S about as far as we can. To get more insight, we need more tools. Here’s the
sort of thing we are interested in:
1. Do the firms make a profit?
2. What happens in the long run?
3. What if the market is not competitive?
In a sense, we make a fresh start.
We make a big assumption: Firms try to maximize their profits.
When they select a price, or enter a market, or hire a worker, they are trying to make max
profit? Is that the whole truth and nothing but the truth? Probably not.
B = TR - TC,
where TR = P x Q and TC is total cost
There are many different breakdowns or types of costs:
TC = explicit cost + implicit cost
explicit costs are payments for resources...look in the firm’s checkbook
implicit cost are the opportunity cost of resources already owned by the firm...
like wear and tear due to use of the owner’s car
accounting profit = TR - explicit cost.....
economic profit = TR - (explicit cost + implicit cost)
when we discuss B, we mean economic profit
when B = 0, we call that a “normal” profit
We are going to begin by discussing cost in the short run....meaning some factor of production,
maybe the factory, cannot be changed, but some other factors can be changed. The factors that
can be changed are called the variable inputs. To make things simple, we are going to look at a
case where there is only one variable input, labor, L.
The production function shows how output, Q, might be related to variable input, L.
Here’s a sample production function, F. Note that when L is lower than about 5, Q grows rapidly
with any increase in L. This is due to assumed benefits from co-operation, specialization,
division of labor. Note also that when L greater than about 20, Q falls as we add labor. Why?
Because we are crowding the fixed facility with variable input, and this is pulling output down.
F is not a cost curve, but it “lies beneath” the cost curves, and we can begin deriving cost curves
from F. Here’s how. Suppose we had to pay each unit of L $8/hr. Then our total variable cost
would look like the following:
The above is one version of a total variable cost curve, TVC, but it is not very handy. It would fit
with D better if we had Q on the horizontal axis and $ on the vertical. If we did that, here’s what we
would get:
Notice that I have “chopped off” the back-bending segment of TVC at the top right.
Let fixed cost be $25, then TC = FC + TVC, and the total cost curve would look like this:
Our other cost curves derive from these.
Perhaps the most important derived curve is MC, the marginal cost, the cost of producing one more
unit ... )TC/)Q...the slope of the TC curve. Look at the picture on the next page.
TC is in the upper figure. MC is in the lower figure. Red tangent lines are drawn at points A, B, and
C. Where the tangent lines are steep, cost rises quickly as output goes up–in other words, MC is high
at Q’s where TC is steep, as at points A and C. When TC is flattest, MC is lowest, as at point B.
MC is very important for determining optimal output.
Define ATC to be TC/Q. There is a geometrical trick for deriving the shape of the ATC curve. Look
at the above figure. If Q = Q0, then TC = TC0. So the ATC when Q = Q0 is TC0/Q0. Notice that
TC0/Q0 is the slope of the green dashed ray from the origin through point A. What this shows is that
if you draw a ray from the origin through the TC, the slope of the ray gives the ATC for the quantity
below where the ray hits the TC. I should be obvious that as Q goes from 0 to Q0, the slope of the
ray, and therefore the ATC falls. Further, as Q moves past Q0, ATC continues to fall until we reach
the lowest possible ray that touches (i.e., is tangent to) TC at point B and quantity Q1.. That
identifies the minimum ATC. As Q goes past Q1, ATC begins to rise.
ATC is very important for determining profitability.
Here we combine ATC and MC. The is something important to notice: ATC hits its minimum when
the green ray from the origin is tangent to TC in the upper figure. MC is always found with a
tangent. There is only one tangent line at any point on a smooth curve, so MC = ATC at the
minimum value for ATC, or MC always comes through the bottom point on ATC. This isn’t hard
to prove logically: If MC < ATC, ATC must be falling, and if MC > ATC, ATC must be rising. I’ll
give lots of examples of this in class...quiz scores, batting averages, fat people getting on a bus, etc.
In general, all average curves come from rays through the origin, and all marginal curves come from
tangent lines. Now that we know this, it is easy to derive the remaining curves.
The Average Variable Cost, AVC, is TVC/Q. Like all average curves it is determined by the slope
of a ray from the origin, as in the above. See if you can convince yourself that geometry forces MC
to run through the low point of AVC, just as it does for ATC.
AVC is especially important for determining whether or not a money losing firm will shut down.
It should be getting easier now. AFC, average fixed cost, is defined to be FC/Q. As an average
curve, it comes from a ray from the origin through the FC curve as above. Since the rays get
progressively flatter as we check them at higher and higher Q’s, AFC falls steadily.
Total cost, TC, is fixed cost plus total variable cost, or
TC = FC + TVC
so
TC/Q = FC/Q +TVC/Q, which is to say
ATC = AFC + AVC. Putting all this together, we have the following:
Note that we can “see” AFC as the vertical difference between ATC and AVC, and since AFC goes
to zero as Q increases, AVC gets closer and closer to ATC at Q rises.
In the figure below, we derive the slope of the production function.
What would the lower curve tell us? The slope of the upper curve is )Q/)L, i.e., it is the change
in output as you change labor input, or it is the amount of extra stuff produced as you hire an extra
unit of labor. This is important, and we will use it later. It is called the “Marginal Productivity of
Labor” or MPL. Other variable inputs also have MPs. Note that it peaks under the inflection point
on the production function, which is the point of diminishing returns.
At the moment, MPL is important for the following reason:
This has several important implications:
1. If w, or any variable input price rises, MC shifts up.
2. If MPL is falling, MC is rising, and vice versa. That is why MC looks like c and MPL looks like
1.
Other factors that shift the curves:
1. If FC rises, then MC and AVC remain in the same place, but ATC goes up.
Here’s a good way to establish this fact. MC is the cost of hiring enough variable input to produce
one more unit of Q. Fixed costs, like rent, property taxes, insurance, have nothing to do with the cost
of making one more. Or you could look at it like this: MC = w/MPL. FC simply isn’t in the
equation. In the same way, TVC = w@L. There is no FC in this equation either. But ATC = AFC
+ AVC, or ATC = FC/Q + TVC/Q, so when FC rises, ATC rises. What it does is “slide” up the MC
so it still gets hit at its minimum point.
2. If w (or any variable input price rises), then MC, AVC, and ATC rise. Why? Look at the
equations:
MC = w/MPL
TVC = w@L, so AVC = w@L/Q
Finally, here is how we use these curves. In the figure below, we see that if Q = Q0, the cost to make
one more is $49, and the average cost is $30. If Q = Q1, the total cost would be the area of the blue
shaded rectangle,
since the area is
ATC x Q =
(TC/Q) x Q =
TC.