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Unformatted text preview: CHAPTER 2 1 COST
CU RVES In the last chapter we described the costminimizing behavior of a ﬁrm.
Here we continue that investigation through the use of an important geo
metric construction, the cost curve. Cost curves can be used to depict
graphically the cost function of a ﬁrm and are important in studying the
determination of optimal output choices. 21.1 Average Costs Consider the cost function described in the last chapter. This is the function
C(w1,w2,y) that gives the minimum cost of producing output level y when
factor prices are (101,1.U2). In the rest of this chapter we will take the factor
prices to be ﬁxed so that we can write cost as a function of y alone, c(y). Some of the costs of the ﬁrm are independent of the level of output of
the ﬁrm. As we’ve seen in Chapter 20, these are the ﬁxed costs. Fixed
costs are the costs that must be paid regardless of what level of output the
ﬁrm produces. For example, the ﬁrm might have mortgage payments that
are required no matter What its level of output. Figure
21.1 368 COST CURVES (Ch. 21) Other costs change when output changes: these are the variable costs.
The total costs of the ﬁrm can always be written as the sum of the variable
costs, cv(y), and the ﬁxed costs, F: C(y) : Cut”) + F The average cost function measures the cost per unit of output. The
average variable cost function measures the variable costs per unit of
output, and the average ﬁxed cost function measures the ﬁxed costs
per unit output. By the above equation: AC(y) = 5% = Cf”) + g : AVC(y) + AFC(y) where AVC(y) stands for average variable costs and AF C(y) stands for
average ﬁxed costs. What do these functions look like? The easiest one is
certainly the average ﬁxed cost function: when y = 0 it is inﬁnite, and as
y increases the average ﬁxed cost decreases toward zero. This is depicted
in Figure 21.1A. AC AC AC
‘ AFC ‘ AVC  AC
y Y y A B _ c Construction of the average cost curve. (A) The average
ﬁxed costs decrease as output is increased. (B) The average vari
able costs eventually increase as output is increased. (C) The
combination of these two effects produces a U—shaped average
cost curve. ' Consider the variable cost function. Start at a zero level of output and
consider producing one unit. Then the average variable costs at y = 1 is
just the variable cost of producing this one unit. New increase the level
of production to 2 units. We would expect that, at worst, variable costs
would double, so that average variable costs would remain constant. If MARGINAL COSTS 369 we can organize production in a more efﬁcient way as the scale of output
is increased, the average variable costs might even decreaSe initially. But
eventually we would expect the average variable costs to rise. Why? If ﬁxed
factors are present, they will eventually constrain the production process. For example, suppose that the ﬁxed costs are due to the rent or mortgage
payments on a building of ﬁxed size. Then as production increases, average
variable costs—the perunit production costs—may remain constant for a
while. But as the capacity of the building is reached, these costs will rise
sharply, producing an average variable cost curve of the form depicted in
Figure 21.1B. The average cost curve is the sum of these two curves; thus it will have
the Ushape indicated in Figure 21.10. The initial decline in average costs
is due to the decline in average ﬁxed costs; the eventual increase in average
costs is due to the increase in average variable costs. The combination of
these two effects yields the Ushape depicted in the diagram. 21.2 Marginal Costs There is one more cost curve of interest: the marginal cost curve. The
marginal cost curve measures the change in costs for a given change in
output. That is, at any given level of output y, we can ask how costs will
change if we change output by some amount Ag: We could just as well write the deﬁnition of marginal costs in terms of
the variable cost function: MC(y) .: Acoly) C'u(y + Ay) _ 611 (y) . Ay Ay This is equivalent to the ﬁrst deﬁnition, since C(y) : cv(y) + F and the
ﬁxed costs, F, don’t change as y changes. Often we think of Ag as being one unit of output, so that marginal
cost indicates the change in our costs if we consider producing one more
discrete unit of output. If we are thinking of the production of a discrete
good, then marginal cost of producing 3,; units of output is just C(y) —
C(y — 1). This is often a convenient way to think about marginal cost,
but is sometimes misleading. Remember, marginal cost measures a rate of
change: the change in costs divided by a change in output. If the change
in output is a single unit, then marginal cost looks like a simple change
in costs, but it is really a rate of change as we increase the output by one
unit. 370 COST CURVES (Ch. 21) How can we put this marginal cost curve on the diagram presented above?
First we note the following. The variable costs are zero when zero units
of output are produced, by deﬁnition. Thus for the ﬁrst unit of output
produced c.(1)+ F — am) — F as) MC<1)=***r——”= 1 = AVC'(1). Thus the marginal cost for the ﬁrst small unit of amount equals the average
variable cost for a single unit of output. Now suppose that we are producing in a range of output where average
variable costs are decreasing. Then it must be that the marginal costs are
less than the average variable costs in this range. For the way that you
push an average down is to add in numbers that are less than the average. Think about a sequence of numbers representing average costs at differ—
ent levels of output. If the average is decreasing, it must be that the cost
of each additional unit produced is less than average up to that point. To
make the average go down, you have to be adding additional units that are
less than the average. Similarly, if we are in a region where average variable costs are rising,
then it must be the case that the marginal costs are greater than the average
variable costsfit is the higher marginal costs that are pushing the average
up. Thus we know that the marginal cost curve must lie below the average
variable cost curve to the left of its minimum point and above it to the
right. This implies that the marginal cost curve must intersect the average
variable cost curve at its minimum point. Exactly the same kind of argument applies for the average cost curve. If
average costs are falling, then marginal costs must be less than the average
costs and if average costs are rising the marginal costs must be larger than the average costs. These observations allow us to draw in the marginal cost
curve as in Figure 21.2.
To review the important points: 0 The average variable cost curve may initially slope down but need not.
However, it will eventually rise, as long as there are ﬁxed factors that
constrain production. 0 The average cost curve will initially fall due to declining ﬁxed costs but
then rise due to the increasing average variable costs. 0 The marginal cost and average variable cost are the same at the first
unit of output. o The marginal cost curve passes through the minimum point of both the
average variable cost and the average cost curves. MARGINAL COSTS AND VARIABLE COSTS 371 AC
AVC Y Cost curves. The average cost curve (AC), the average vari«
able cost curve (AVG), and the marginal cost curve (M C'). 21.3 Marginal Costs and Variable Costs There are also some other relationships between the various curves. Here is
one that is not so obvious: it turns out that the area beneath the marginal
cost curve up to y gives us the variable cost of producing y units of output.
Why is that? The marginal cost curve measures the cost of producing each additional
unit of output. If we add up the cost of producing each unit of output we
will get the total costs of production—except for ﬁxed costs. This argument can be made rigorous in the case where the output good
is produced in discrete amounts. First, we note that My) = lady)  cv(y — 1)] + [My  1)  My — 2l+
'+ [60(1)  6140)].
This is true since cv(0) = 0 and all the middle terms cancel out; that is, the
second term cancels the third term, the fourth term cancels the ﬁfth term, and so on. But each term in this sum is the marginal cost at a different
level of output: my) = MC(y — 1) + MC(y — 2) +    + MC(0). Figure
21.3 372 COST CURVES (Ch, 21) Thus each term in the sum represents the area of a rectangle with height
M C (y) and base of 1. Summing up all these rectangles gives us the area
under the marginal cost curve as depicted in Figure 21.3. Y Marginal cost and variable costs. The area under the
marginal cost curve gives the variable costs. EXAMPLE: Specific Cost Curves Let’s consider the cost function C(y) = y2 + 1. We have the following
derived cost curves: 0 variable costs: cv(y) : y2 a ﬁxed costs: cf(y) =_ 1 0 average variable costs: AVC (y) = y2/y = y
0 average ﬁxed costs: AFC(y) = 1/y 2
y +1 1
y y 0 average costs: AC(y) = I marginal costs: M C(y) 2 2y MARGINAL COSTS AND VARIABLE COSTS 373 These are all obvious except for the last one, which is also obvious if you
know calculus. If the cost function is C(y) 2 y2 + F, then the marginal
cost function is given by M C (y) = 2y. If you don’t know this fact already,
memorize it, because you’ll use it in the exercises. What do these cost curves look like? The easiest way to draw them is
ﬁrst to draw the average variable cost curve, which is a straight line with
slope 1. Then it is also simple to draw the marginal cost curve, which is a
straight line with slope 2. The average cost curve reaches its minimum where average cost equals marginal cost, which says 1
y+—=2y,
3/ which can be solved to give ymin : 1. The average cost at y = 1 is 2, which
is also the marginal cost. The ﬁnal picture is given in Figure 21.4. AC
MC
AVC Cost curves. The cost curves for 'c(y) = y‘2 + 1. EXAMPLE: Marginal Cost Curves for Two Plants Suppose that you have two plants that have two different cost functions,
c1(y1) and 02(y2). You want to produce 3; units of output in the cheapest 374 COST CURVES (Ch. 21) way. In general, you will want to produce some amount of output in each
plant. The question is, how much should you produce in each plant?
Set up the minimization problem: mincl(yl) + 02(92)
ylyy2 such that yl + y2 = y. Now how do you solve it? It turns out that at the optimal division of
output between the two plants we must have the marginal cost of producing
output at plant 1 equal to the marginal cost of producing output at plant
2. In order to prove this, suppose the marginal costs were not equal; then
it would pay to shift a small amount of output from the plant with higher
marginal costs to the plant with lower marginal costs. If the output division
is optimal, then switching output from one plant to the other can’t lower
costs. Let C(y) be the cost function that gives the cheapest way to produce
y units of output—that is, the cost of producing y units of output given
that you have divided output in the best way between the two plants. The
marginal cost of producing an extra unit of output must be the same no
matter which plant you produce it in. We depict the two marginal cost curves, M 01(y1) and M 02(y2), in Fig
ure 21.5. The marginal cost curve for the two plants taken together is just
the horizontal sum of the two marginal cost curves, as depicted in Figure 21.5C. MAR
GINA].
COST MAR~
GINA
COST MAR~
GINAL
COST Yi‘tYi' Y1+Y2
B C Marginal costs for a ﬁrm with two plants. The overall
marginal cost curve on the right is the horizontal sum of the
marginal cost curves for the two plants shown on the left. LONGRUN COSTS 375 For any ﬁxed level of marginal costs, say 0, we will produce yf and y;
such that M01 (yi‘) : MC(y§) = c, and we will thus have yf + y; units of
output produced. Thus the amount of output produced at any marginal
cost c is just the sum of the outputs where the marginal cost of plant 1
equals c and the marginal cost of plant 2 equals c: the horizontal sum of
the marginal cost curves. 21.4 LongRun Costs In the above analysis, we have regarded the ﬁrm’s ﬁxed costs as being the
costs that involve payments to factors that it is unable to adjust in the short
run. In the long run a ﬁrm can choose the level of its “ﬁxed” factorsithey
are no longer ﬁxed. Of course, there may still be quasi—ﬁxed factors in the long run. That
is, it may be a feature of the technology that some costs have to be paid
to produce any positive level of output. But in the long run there are no
ﬁxed costs, in the sense that it is always possible to produce zero units of
output at zero costsAthat is, it is always possible to go out of business. If
quasi—ﬁxed factors are present in the long run, then the average cost curve
will tend to have a U—shape, just as in the short run. But in the long run
it will always be possible to produce zero units of output at a zero cost, by
deﬁnition of the long run. Of course, what constitutes the long run depends on the problem we are
analyzing. If we are considering the ﬁxed factor to be the size of the plant,
then the long run will be how long it would take the ﬁrm to change the
size of its plant. If we are considering the ﬁxed factor to be the contractual
obligations to pay salaries, then the long run would be how long it would
take the ﬁrm to change the size of its work force. Just to be speciﬁc, let’s think of the ﬁxed factor as being plant size and
denote it by k. The ﬁrm’s short—run cost function, given that it has a plant
of is square feet, will be denoted by c5(y, k), where the s subscript stands
for “short run.” (Here k: is playing the role of E2 in Chapter 20.) For any given level of output, there will be some plant size that is the
optimal size to produce that level of output. Let us denote this plant size
by My). This is the ﬁrm’s conditional factor demand for plant size as a
function of output. (Of course, it also depends on the prices of plant size
and other factors of production, but we have suppressed these arguments.)
Then, as we’ve seen in Chapter 20, the longrun cost function of the ﬁrm
will be given by c3(y, k(y)). This is the total cost of producing an output
level 1;, given that the ﬁrm is allowed to adjust its plant size optimally.
The longrun cost function of the ﬁrm is just the shortrun cost function
evaluated at the optimal choice of the ﬁxed factors: 6(9) = My, kW) 376 COST CURVES (01.21} Let us see how this looks graphically. Pick some level of output y*, and
let k* = k(y*) be the optimal plant size for that level of output. The short
run cost function for a plant of size 14* will be given by c5(y, k“), and the
longrun cost function will be given by C(y) = cs(y, k:(y)), just as above. Now, note the important fact that the shortrun cost to produce output
y must always be at least as large as the long—run cost to produce y. Why?
In the short run the ﬁrm has a ﬁxed plant size, while in the long run the
ﬁrm is free to adjust its plant size. Since one of its longrun choices is
always to choose the plant size 16*, its optimal choice to produce y units of
output must have costs at least as small as C(y, 16*). This means that the ﬁrm must be able to do at least as well by adjusting plant size as by having
it ﬁxed. Thus C(y) S cs(y,k*) for all levels of y.
In fact, at one particular level of 3;, namely y*, we know that CW) = cs(y*,k*) Why? Because at y* the optimal choice of plant size is 19*. So at y*, the
long—run costs and the short—run costs are the same. )b y _ ' y Shortrun and iong—run average costs. The shortrun av
erage cost curve must be'tangent to the longrun average cost
curve. DISCRETE LEVELS OF PLANT SIZE 377 If the shortrun cost is always greater than the longrun cost and they
are equal at one level of output, then this means that the shortrun and the
longrun average costs have the same property: AC(y) g A0431, k*) and
AC(y*) = AC5 (y*, k’“). This implies that the shortrun average cost curve
always lies above the longrun average cost curve and that they touch at
one point, y*. Thus the longrun average cost curve (LAC) and the short
run average cost curve (SAC) must be tangent at that point, as depicted
in Figure 21.6. We can do the same sort of construction for levels of output other than
31*. Suppose we pick outputs y1,y2, . . . ,yn and accompanying plant sizes
k1 = k(y1),k2 = k(y2), . . . , kn = My"). Then we get a picture like that in
Figure 21.7. We summarize Figure 21.7 by saying that the long—run average
cost curve is the lower envelope of the shortrun average cost curves. AC Shortrun average
cost curves Longrun average
cost curve y“ y
Shortrun and longrun average costs. The longrun av—
erage cost curve is the envelope of the short~run average cost
curves. 21.5 Discrete Levels of Plant Size In the above discussion we have implicitly assumed that we can choose
a continuous number of different plant sizes. Thus each different level of
output has a unique optimal plant size asSOCiated with it. But we can also 378 COST CURVES (Ch. 21) consider what happens if there are only a few different levels of plant size
to choose from. Suppose, for example, that we have four different Choices, 161, 192, 193, and
.164. We have depicted the four different average cost curves associated with
these plant sizes in Figure 21.8. AC Shortrun average
cost curves up Long—run average
cost curve Y Discrete levels of plant size. The long—run cost curve is the
lower envelope of the short—run curves, just as before. How can we construct the lOngrun average cost curve? Well, remember
the longrun average cost curve is the cost curve you get by adjusting k
optimally. In this case that isn’t hard to do: since there are only four
different plant sizes, we just see which one has the lowest costs associated
with it and pick that plant size. That is, for any level of output y, we just
choose the plant size that gives us the minimum cost of producing that
output level. Thus the longrun average cost curve will be the lower envelope of the
short—run average costs, as depicted in Figure 21.8. Note that this ﬁgure has
qualitatively the same implications as Figure 21.7: the short—run average
costs always are at least as large as the longrun average costs, and they
are the same at the level of output where the long—run demand for the ﬁxed
factor equals the amount of the ﬁxed factor that you have. LONGRUN MARGINAL COSTS 379 21.6 LongRun Marginal Costs We’ve seen in the last section that the long—run average cost curve is the
lower envelope of the short—run average cost curves. What are the impli—
cations of this for marginal costs? Let’s ﬁrst consider the case where there
are discrete levels of plant size. In this situation the long—run marginal
cost curve consists of the appropriate pieces of the shortrun marginal cost
curves, as depicted in Figure 21.9. For each level of output, we see which
shortrun average cost curve we are operating on and then look at the
marginal cost associated with that curve. AC Longrun
average
costs Use Use Use _ y
AC, AC2 AC3 Long—run marginal costs. When there are discrete levels of
the ﬁxed factor, the ﬁrm will choose the amount of the ﬁxed
factor to minimize average costs. Thus the longrun marginal
cost curve will consist of the various segments of the shortrun marginal cost curves associated with each different level of the
ﬁxed factor. This has to hold true no matter how many different plant sizes there are,
so the picture for the continuous case looks like Figure 21.10. The long—run
marginal cost at any output level y has to equal the shortrun marginal
cost associated with the optimal level of plant size to produce 3;. 380 COST CURVES (Ch. 2]) LMC Longrun marginal costs. The relationship between the
long~run and the short~run marginal costs with continuous levels of the ﬁxed factor. Summary 1. Average costs are composed of average variable costs plus average ﬁxed
costs. Average ﬁxed costs always decline with output, while average vari
able costs tend to increase. The net result is a Ushaped average cost
curve. 2. The marginal cost curve lies below the average cost curve when average
costs are decreasing, and above when they are increasing. Thus marginal
costs must equal average costs at the point of minimum average costs. 3. The area under the marginal cost curve measures the variable costs. 4. The longrun average cost curve is the lower envelope of the shortrun
average cost curves. APPENDIX 381 REVIEW QUESTIONS 1. Which of the following are true? (1) Average ﬁxed costs never increase
with output; (2) average total costs are always greater than or equal to
average variable costs; (3) average cost can never rise while marginal costs
are declining. 2. A ﬁrm produces identical outputs at two different plants. If the marginal
cost at the ﬁrst plant exceeds the marginal cost at the second plant, how
can the ﬁrm reduce costs and maintain the same level of output? 3. True or false? In the long run a ﬁrm always operates at the mini—
mum level of average costs for the optimally sized plant to produce a given
amount of output. APPENDIX In the text we claimed that average variable cost equals marginal cost for the
ﬁrst unit of output. In calculus terms this becomes lim Ody) = lim cl(y).
y>0 y y—+0 The lefthand side of this expression is not deﬁned at y : 0. But its limit is
deﬁned, and we can compute it using l’Hopital’s rule, which states that the limit
of a fraction whose numerator and denominator both approach zero is given by
the limit of the derivatives of the numerator and the denominator. Applying this
rule, we have hm cue) : limysodcuwdy 2 6(0)
y—»0 y limyno dy/dy 1 ’ which establishes the claim.
We also claimed that the area under the marginal cost curve gave us variable
cost. This is easy to show using the fundamental theorem of calculus. Since = dcvly) we know that the area under the marginal cost curve is The discussion of long~run and shortrun marginal cost curves is all pretty clear
geometrically, but what does it mean economically? It turns out that the calculus
argument gives the nicest intuition. The argument is simple. The marginal cost 382 COST CURVES (Ch. 21) of production is just the change in cost that arises from changing output. In the
short run we have to keep plant size (or whatever) ﬁxed, while in the long run
we are free to adjust it. So the longrun marginal cost will consist of two pieces:
how costs change holding plant size ﬁxed plus how costs change when plant size
adjusts. But if the plant size is chosen optimally, this last term has to be zero!
Thus the long—run and the shortsrun marginal costs have to be the same. The mathematical proof involves the chain rule. Using the deﬁnition from the
text: C(y) E Cs(y, 16(10) Differentiating with respect to y gives dc(y) = 8c3(y,k) + 365(y,k) 6k(y)
63?! 33/ 8k By ' If we evaluate this at a speciﬁc level of output y’“ and its associated optimal
plant size k" = k(y*), we know that Beau/*7 15“) _ 8k 0 because that is the necessary ﬁrst—order condition for k" to be the costminimizing
plant size at y". Thus the second term in the expression cancels out and all that
we have left is the shortrun marginal cost: doe”) = at33(y*.k’“)
dy By ‘ ...
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