Lecture 8: Cost Functions
Cost Minimization and the Cost Function
Because any profit maximizing firm wants to minimize the cost of producing any given level of
output, we start with the problem of minimizing a firm’s total cost. In terms of our single product dual input example, the problem is to
Isocost Lines
Isocost Lines give combinations of labor and capital representing
equal cost. While similar to budget lines, there is an obvious
difference: firms have an infinite number of Isocost Lines.
minimize pKK + pLL
subject to the constraint that f(K,L) = qo
To follow historical convention, we denote the prices by p K = r and pL = w. Thus our problem becomes
min.
s.t.
rK + wL
f(K,L) = qo
Rather than solve the problem mathematically, however, we are going to do a graphical
derivation. In this simple case, we can graph isocost lines, that represent equal expenditures on labor
and capital, much like those plotted.
The isocost lines are obviously straight lines of the form rK + wL = c o. The further the line
from the origin, the higher the cost. Ideally we would like to be at the least cost, but we must also
achieve our output objective. In the case illustrated, we must operate at the point K 0, L0, where the
isocost line is just tangent to the isoquant.
Cost-Minimizing Factor Inputs
Co is the minimum cost of producing Qo units of output. It means
using Ko and Lo to do so. Similarly, C1 is the minimum cost of
producing Q1 units of output. Obviously these values depend on r
and w.
By repeating this operation for each different level of output, we can come up with a general
cost function C(q,r,w).
Properties of the Cost Function with Respect to Output
Because the general cost function contains more than we want to use at the moment, let us
hold r and w constant and concentrate on the relation between cost and quantity. We simplify our cost
function to C = C(q).
Total Cost
The following illustrates three cost functions that embody specific assumptions about the
relation between cost and quantity. In all three cases, total cost is an increasing function of quantity
produced. (How could it be otherwise?) However costs increase at different rates. In the first case (A),
total costs increase at a faster rate than the quantity produced. Here, we save that we have decreasing
returns to scale. In the second case (B), the function is a straight line going through the origin. Here,
costs increase in proportion to output. This case is called constant returns to scale. (You should see
that this is the case with either the Cobb-Douglas or the Leontief production functions). In the third
case, (C), costs increase less than in proportion to quantity. This is the case of increasing returns to
scale.
Three Cost Functions
Case A
Decreasing Returns to
Scale
Case B
Constant
Returns to Scale
Case C
Increasing Returns to
Scale
Here, cost increases
faster than output
Cost increases in
proportion to output
Cost increases slower
than output
Average Cost
To further the picture, consider the corresponding average cost curves is presented. In Case A,
average cost is increasing as a function of output; in Case B, average cost is constant; in Case C,
average cost is decreasing as a function of output.
Three Cases of Average Cost Functions
Case A
Decreasing Returns to
Scale
Here, average cost rises
with output
Case B
Constant
Returns to Scale
Case C
Increasing Returns to
Scale
Here, average costs is
constant
Here, average cost falls
with output
Marginal Cost
Still a third property of interest is the marginal cost curve. Mathematically this is defined as
the derivative of the total cost function (or, if you will the cost of producing an incremental unit of
output.
There is a relationship between the two:

If marginal cost is above average cost, then average cost is rising. If marginal cost is below
average cost, then it is falling.
This is a sufficiently important relationship that we will give two proofs, one not using
calculus, and another using calculus.
The non-calculus proof
Let C(qo) be the cost of producing qo units of output. The marginal cost of producing the q oth
unit of output is C(qo) - C(qo-1). The average cost of producing qo units of output is C(qo)/qo, while the
average cost of producing qo-1 units of output is C(qo-1)/(qo-1). Suppose marginal cost is greater than
average cost. Then
C(qo) -C(qo-1) > C(qo)/qo
In turn this means that
C(qo) > C(qo-1) + C(qo)/qo
implying that
C(qo)/qo > [C(qo-1)/(qo-1)][(qo-1)/qo] + C(qo)/qo2
Rearranging terms
[C(qo)/qo][1-1/qo] > [C(qo-1)/(qo-1)][(qo-1)/qo]
The two right most terms on both sides of the inequality are equal. Thus, canceling them, we get
C(qo)/qo > C(qo-1)/(qo-1)
In short, if marginal cost is above average cost, then average cost is rising.
The Calculus Proof
To see the relation between marginal and average cost, recall that average cost is given by
C(q)/q, and the derivative of average cost with respect to q is
C’(q)/q -C(q)/q2
or, with a little rewriting
(1/q)(C’(q)-C(q)/q).
The intuitive reasoning is simple: average cost can be rising only when the cost of incremental
units exceeds the average cost of the previous units.
Some Graphs
The following gives graphs of the marginal cost curve corresponding to the three cases of the
average cost curve. In case A, the average cost curve is upward sloping, indicating MC > AC. In Case
B, marginal cost is constant and equal to average cost. Finally, in case C, average cost is decreasing so
that MC < AC. Note that, even in this case marginal cost never becomes negative. Why?
Three Cases of Marginal Cost Functions
Case A
Decreasing Returns to
Scale
Marginal cost is above
average cost
Case B
Constant
Returns to Scale
Case C
Increasing Returns to
Scale
Marginal cost and
average cost are equal
Marginal cost is below
average cost
As you will note, I have drawn marginal cost curves that have constant slope. This need not
be the case, as we will see in the case of the U-shaped average cost curve.
The U-Shaped Average Cost Curve
As a practical matter, most economists believe that the typical competitive firm is
characterized by a U-shaped average cost curve, where average cost initially falls as a function of the
level of output, reaches a minimum and then begin to rise. At the minimum level of AC, MC = AC. At
the minimum AC, the slope of the AC curve is zero, which is sufficient to prove that MC = AC. Finally
note that the MC cost curve must cut the AC curve from below. Thus we may draw the average cost
curve and the marginal cost curve.
The U-Shaped Average Cost Curve
Here average cost first falls, and then rises. As you can see the
marginal cost curve "cuts" the average cost curve at the minimum
average cost.
Note that I have drawn the MC curve so that it is initially falling. This need not be the case,
and we have no special interest in this issue.
Fixed and Variable Cost
It is sometimes useful to divide the firm’s costs into fixed and variable components. Fixed
costs, FC, are the costs the firm would incur if it had no output. That is, FC = C(0). Variable costs are
those that vary with output. They may either be defined as the difference between total and fixed
costs,
VC(q) = C(q) - C(0).
It should be obvious that what is defined as “fixed” and “variable” depends on what time
period we are talking about. In the long run, there truly are no fixed costs, but in the short run, there
may well be.
Here is a relation between the average total cost and average variable cost for a U-shaped
average cost curve. As you will note





The marginal cost curve cuts the ATC and the AVC costs at their minimum
The minimum of AVC occurs at a lower level of output that the minimum of the ATC cost
curve
Average Variable Cost is not equal to marginal cost
Average Variable Cost is less than average cost
As output grows infinitely large, AVC approaches ATC
Average Variable Cost and Average Total Cost for the U-Shaped
Average Cost Curve
Average fixed cost is continually declining, and the minimum of
AVC occurs at a lower level of output that the minimum of ATC.
MC cuts both the ATC and AVC curve at their respective
minimum.
Other Factors Affecting Cost
We originally wrote the cost function as C(q,r,w) to indicate that cost was affected by things
other than output. Here are some trick questions:
First Trick Question

Acme Widgets’ costs are now $100, of which $50 represents factor payments to workers
and $50 represents payments to capital. (To make life easier, assume that Acme Widgets
leases all plant and equipment.). Now suppose that wage rates are increase by 100%.
Will total costs go from $100 to $150
Clearly not. If Acme continued to use the same amounts of capital and labor, the answer
would be yes. But, given the new slope of the isocost lines (the situation is illustrated in Figure 8-8)
Acme will substitute capital for labor and reduce the cost increase to less than 50%.
Isocost Lines with a Factor Price Shift
A change in factor prices causes isocost lines to rotate. As this
graph shows,, the minimum cost of producing Qo units is not 50%
higher. Rather than stay at (Ko, Lo), the firm will move to another
point such as (K1, L1) on an isocost line below C=$150.
Second Trick Question

Is it possible that the higher price of labor will lead Acme to lower total costs for the given
level of output?
This is a good trick question for examinations, and sometimes people do claim that higher
factor payments can lead to lower costs. But this is nonsense. Suppose, for instance, that Acme could
find a new solution involving $60 of capital payments and $30 of labor costs (at the new wage rate). At
the old wage rate, costs would only have been $80 = $60 + $20, which would violate our assumption
that $100 represented the least cost. Any manager claiming that higher wager rates lead to lower
costs is simply admitting that he wasn’t doing his job in the first place.
Third Trick Question

Does an increase in total cost mean that average cost has gone up, and that marginal cost
has gone up?
Why a Rise in Total Cost Doesn't Mean a Rise in Marginal Cost
Here, a rise in total cost leads to a different production technology
with the lower marginal cost.
Another trick question. A possible illustration of why not is given. The average cost curve
shifts up and to the right from AC to AC*, and the marginal cost curve shifts from MC to MC*. So MC
has declined.
The intuition behind this is quite simple. A rise in labor cost, say, may lead a firm to purchase
a new piece of capital. Thus the average cost will rise (see the previous example to see why it cannot
fall), but, equipped with the new machine, marginal cost may well be less.
Fourth Trick Question

Does technological change reduce costs?
We normally mean by technological change the ability to get more output from a given
combination of labor and capital. In our terms, technological change shifts the isoquants out. And, if
all of the isoquant (or at least the portion we are currently using) is shifted out, then technological
change does reduce costs. But some caveats are in order.
Don’t confuse changes in technology with changes in technique. In the example given
above, a 50% increase in wage rates leads to a change in technique. There are always several different
ways of achieving the same objective, using what we call techniques. We all know how to mow a lawn
by hand, or with a standard lawn mower, or with one of the larger mowers that commercial services
use. Switching from one to another is switching from one technique to another. The switch in
technology comes when a new type of mowing machine is invented.
For reasons we will get to later, it is important to distinguish between firm specific
technologies and industry technologies. A firm specific technology is one that only one firm can
use, protected either by something like trade secrets or patent rights, while industry technology
represents something that affects all firms in the industry.
An example: when Apple developed the Macintosh, it patented its software, and thus froze out
competitors. When IBM came out with the PC in the 1980’s it allowed for open architecture, including
the DOS system. Other firms quickly adopted DOS, and any advantage that IBM had quickly
disappeared.
Technological change occurs gradually. Two examples will make the point. The original
IBM PC, not yet 15 years old, had 64 k of memory, a 64k floppy and an 8086 chip. The current system
is more likely to use 250 times as much memory, a floppy disk 20 times as large (not to speak of hard
disks), and a Pentium chip, with many times the power. Yet this change occurred gradually, and in
fact represented a continuation of the computer revolution of the last 40 years.
Fifth Trick Question

Last year, Acme Widgets produced widgets at $5 each. This year the plant manager
reports that he can cut production cost to $4 each. Does this mean he simply was not
doing his job last year?
While it is always possible that the manager was not doing his job last year, it may also mean
something more complicated. In World War II, the Pentagon contracted large production runs of
planes, tanks and the like to specific plants. They discovered over time that the cost of producing a
plane or tank at a specific plant declined, because, over time, the plant learned all sorts of production
tricks. Economists and production specialists labeled this phenomenon as the learning curve,
reflecting the commonsense notion of learning by doing. That is, the more we do of something, the
better we get at it. Thus a firm's cost function might be a function C(q t, Qt) the number of units it
produces this period, qt, as well as the cumulative number of units it has produced, Qt. Thus over time,
the firm's average cost function shifts down because of learning by doing.
Figure 8-10 shows how the production function could shift.
function is
Imagine a firm whose cost
Ct = ctqt
That is, at any time, the marginal cost of producing the product is constant. However, over
time, marginal cost declines as a function of Qt, the cumulative number of units produced.
Learning by Doing: How the Cost Function Shifts
Imagine a firm whose cost function is Ct = ctqt, but where ct is a
declining function of the cumulative number of units produced. ct
might look like this.
Sixth Trick Question

Last year, Acme Widgets produced widgets for $5 each. This year Acme's Costs will be $4.
A new competitor, Wonder Widgets, figures that it can produce widgets for $4.50 each.
Does this mean that Acme was inefficient? Does it mean that Wonder Widgets stole trade
secrets from Acme?
Again, Acme may have been inefficient, and Wonder Widgets may have stolen trade secrets
from Acme. One can learn a lot by simply observing competitor. If Wonder Widgets simply purchases
a widget from Acme, and turns it over to its production specialists for study, there will be a chorus of
"So that is what they do!" It is, if you will, just like looking at instructor's old exams. The knowledge
will pay off.
In fact, there is a general principle: much technological change can confer only a
temporary advantage on its developer. We rightly have patent laws to protect new technologies,
for else there would be little reason to develop new technology. But the “high tech” firms clearly
recognize that the result of new technology hitting the market is for it to be copied, sometimes illegally,
but often legally. Thus a high tech firm cannot rest on its laurels, but must continually innovate to
keep a cost advantage.
Applications of Cost Functions to Some Problems of the Firm
We can show how some of this material applies by considering two problems that a firm faces
in making production decisions.
Allocating output between two different plants
The first involves allocating output between two different plants. Suppose a manufacturer
has two plants, "A" and "B". The average and marginal cost functions for the two plants are given in
Figure 8-11. As you can see, plant "A" has a higher average cost and a higher marginal cost for each
and every level of output than plant "B". While this makes for a nice graph, it is not relevant for
management decision making.
Marginal and Average Cost for Two Different Plants
The example is constructed so that plant "A" has a higher
marginal and average cost than plant "B" for every level of
output. While this makes for a nice graph, it is irrelevant for
decision making.
To see why, suppose management is considering producing q* units of output from the two
plants. Then it wants to find the allocation of output between the two plants that minimizes total cost
of production. Here, the only cost that is relevant is the marginal cost, and the least cost solution is
achieved when the marginal cost of each plant is equal. Even though "B" appears to be a more efficient
plant, it still pays to allocate some of its production to other the older plant.
The graphical solution is shown. Here, the marginal cost of plant "A" is measured from the
left-hand side, while the marginal cost of plant "B" is measured from the right hand side. As you can
see the two marginal costs are equated when qA is produced at plant "A" and qB = q*-qA is produced at
plant "B". Any other point would involve higher cost.
Allocating Production Between Two Plants
The least cost solution requires that the marginal cost of
operation be the same. If the firm is going to produce q* units of
output, then optimization occurs at qA.
Allocating output between a new type of plant and old plants
To see a similar problem, suppose a firm has several plants of type "A" such as illustrated in
Figure 8-13. It now becomes possible to build a new type of plant, type "B". As you can see, type B
represents a technological innovation, representing lower cost. What should the firm do? (The first
thing to recognize is that long run marginal cost is now given by the minimum average cost of plant
"B". The firm can build as many of those as it wishes, and operate them at the minimum of the LRAC.
Marginal and Average Cost for Two Different Plants
A firm is operating a number of plants of type "A". and it now
becomes possible to build new, more efficient plants of type "B".
This does not mean the firm should shut down its existing plants,
merely change the scale of operation. The existing plants should
not begin producing qA units of output.
That does not mean that it should junk its existing plants. It should continue to operate them
where the marginal cost equals the new long run marginal cost.
Choosing Plant Size
Now consider a different problem. A firm is considering building one of three different plants,
whose average costs are as depicted in Figure 8-14. Its long run average cost curve is then a peculiar
scallop-shaped figure, representing the minimum of each of these average cost curves.
Long Run Average Cost when Different Sized plants are possible
A firm can choose between three different plants. Its long run
average cost curve is now the scallop-shaped figure, the minimum
of the average cost for each of the three plants.
It is also possible to extend this analysis to the case of many different sized plants, so that the
firm can select from a continuum of plant sizes. While, this discussion is much beloved by professors of
economics, we will not go into it here.1
Cost functions
In talking about cost functions, we have presupposed a simple cost function C = C(q). While
the precise definition of the cost function will depend on the exact nature of the firm's production
function, we often illustrate the properties of a cost function with one of two simple standard functions:
Jacob Viner, who originally did the work, was then at the University of Chicago and later at Princeton University.
Viner stumbled on this discussion by accident. He did it wrong, and only discovered his error when one of his Ph.D.
students, assigned the task of drafting the graphs reported that he could not draw them the way Viner wanted
them. For this, Viner got an important article published and the Ph.D. student got a footnote. It should serve as a
reminder to readers of these notes to check the logic carefully behind each graph.
1
The Linear Cost Function
C=a+bq
The Quadratic Cost Function
C = a + b q + c q2
For example, the cost function
C = 3+ 5q
is a linear cost function, while
C = 4 + 5q + q2
is a quadratic cost function. The difference between the two functions is straightforward. The first
cost function simply contains q, while the second term contains q 2.
Let us illustrate some of our basic notions using these cost functions.
Fixed Cost
Fixed cost is, quite simply the costs when q = 0. It is a simple matter to show that, in the case
of both our linear and quadratic cost functions, the first term, a, represents fixed cost.
Variable Cost
Variable cost is the cost varying with output, or, if you will, total cost less fixed cost. In the
case of the linear cost function, variable cost will be the term bq; in the case of the quadratic cost
function, variable cost will be bq + cq2.
Average Cost
Average cost is total cost divided by quantity, or C(q)/q. In the case of the linear cost function,
average cost will be
a/q + b.
In the case of the quadratic cost function, average cost will be
a/q + b + cq.
We could also compute average fixed cost and average variable cost.
Marginal Cost
Marginal cost is simply the incremental cost of producing an additional unit of the product.
The idea is the incremental cost C, associated with producing an incremental unit of output, Q.
When we are dealing with simple production functions such as linear or quadratic functions, we
customarily compute the marginal cost by taking the derivative of the cost function. That is
In words, marginal cost is the derivative of the cost function with respect to output. For those rusty on
elementary differential calculus, remember that, in the case of our linear cost function
C'(q) = b
and in the case of the quadratic cost function
C'(q)= b + 2cq
Two Problems
Here are two simple exercises. One is worked, and one is left for you to do on your own.
Suppose a firm has a cost function
C = 10 +20q + 4q2.

Compute total cost, total variable cost, total fixed cost, average cost, average variable cost,
average fixed cost, and marginal cost when q=10.

Find the value of q at which MC = AC.

What level of output minimizes AC?

At what level of output is marginal cost equal 60?

Total cost is 10 + 20 (10) + 4(10)2 = 610. Total variable cost is the last two terms, or 600.
Average cost is 10. To compute average cost, average variable cost, and average foxed
cost, just divide by 10 to get 61,60, and 1 respectively. Marginal cost is given by 20 + 8q =
100 when q = 10.

Average cost is 10/q + 20 + 4q. We want to find the value at which
Answers
10/q + 20 + 4q = 20 + 8q.
Simplifying, we get that
10/q = 4q
and we can solve that equation to get q2 = 2.5, or q = 1.58

We know that MC = AC at the minimum of the AC function. Thus, from the answer
above, we know that the minimum occurs at 1.58.

Recall that MC = 20 + 8q. That equals 60 when q = 5.
A Second Problem
Suppose the cost function is
C = 5+ 10q2.
Compute the same answers asked for above.