Managerial Cost Analysis Techniques

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Managerial Cost Analysis and Estimation
Learning Objectives
A�er reading this chapter, you should be able to:
Dis�nguish between incremental costs and nonincremental
costs for the purposes of managerial
decision making.
Iden�fy incremental revenues as dis�nct from nonincremental
revenues.
Conduct contribu�on analysis in a range of decision-making
scenarios, including Project A versus
Project B decisions, make-or-buy decisions, and take-it-or-
leave-it decisions.
Es�mate unknown cost values for par�cular output levels using
known cost data from other output
levels using extrapola�on, interpola�on, and gradient analysis
techniques.
Use regression analysis to find which form of the cost func�on
(linear, quadra�c, or cubic) provides the
line of best fit to the data and thus the most reliable cost

es�mates for managerial decision making.
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Introduction
In this chapter, we con�nue with the general topic of cost
analysis for managers and will introduce several cost
es�ma�on and forecas�ng techniques that
allow managers to make profit-maximizing decisions. As we
know, profit is the difference between revenues and costs, and
if we can minimize costs rela�ng
to any par�cular decision that is expected to generate revenues,
this will allow the decision to be profit maximizing (or loss
minimizing). In this chapter, we
con�nue our focus on cost minimiza�on (for any given output
and quality level) and will turn our a�en�on to the profit-
maximizing issues in the following
chapter.
In the first half of this chapter, we will focus on "contribu�on
analysis" whereby decisions are evaluated based on the
financial contribu�on they make to
the firm’s overheads and profits a�er covering the variable
costs associated with those decisions. Contribu�on analysis
allows managers to make choices
among compe�ng alterna�ves—their decision choices—such
that the alterna�ve chosen is the one that contributes most to
the firm’s overheads and profits.
Thus, contribu�on analysis is concerned with accurately

es�ma�ng the incremental costs and incremental revenues of
each decision alterna�ve such that
the best (profit-maximizing) op�on can be chosen.
In the second half of this chapter, we will explore the
es�ma�on of cost curves. We begin with some simple
techniques and con�nue on to apply the
regression analysis technique that was introduced in Chapter 4.
Es�ma�ng cost values for the current period involves
collec�ng at least one data point on
total variable costs (TVC) or average variable costs (AVC) or
marginal costs (MC) from the current or past periods and
methodically projec�ng that/those
values forward or backward to predict the TVC, AVC, and MC
levels for any par�cular output level in the current produc�on
period.
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Fixed and variable costs are incremental to decisions made by
managers. Incremental costs represent the change in total
costs
resul�ng from a par�cular decision.
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6.1 Contribution Analysis

The contribu�on of a decision is defined as the excess of
incremental revenues over incremental costs, and it is called
the contribu�on because it
contributes to the firm’s fixed and unavoidable costs, and also
to profits if total revenues are more than total costs.
Contribu�on analysis is a form of cost–
benefit analysis where the costs are confined to incremental
costs and the benefits are confined to incremental revenues.
Incremental costs, you will recall
from the preceding chapter, are the costs that are consequen�al
to a decision made, while incremental revenues are the revenues
that are consequen�al to
the decision made. As an example, if the firm can sell any
amount of its product at a price $10 per unit, and average
variable cost (AVC) is constant at $6
per unit, then the contribu�on made towards fixed costs and
profit is $4 per unit. This difference between incremental costs
(which equal AVC in this case),
and incremental revenue (which equals price in this case), is
known as the contribu�on margin when it applies to the sale
of a single unit of the firm’s
output.
Incremental Costs and Revenues
In more complex cases, AVC will not be constant, as we saw in
the preceding chapter, so
incremental cost generally is not equal to AVC. Similarly,
incremental revenue is generally not
equal to price, as we shall see in the next chapter. Moreover,
decisions made by managers
usually cause both fixed and variable costs to change, since the
implementa�on of a new
decision (e.g., to expand produc�on) may cause not only extra
direct labor and direct material
costs but also require the purchase of addi�onal capital assets

and the employment of
salaried managers and other workers who must be treated as
fixed costs since their salaries
must be paid regardless of output levels. Thus, incremental
costs are defined as the change in
total costs that result from a par�cular decision.
But isn’t that the defini�on of marginal costs? No, as defined in
Chapter 5, the marginal cost
(MC) is the change in total costs (TC) for a one-unit change in
the output level (Q).
Incremental cost, on the other hand, is the change in TC that
results from a decision that may
or may not involve a change in the output level. For example,
the decision might be to
purchase a new machine or introduce a new produc�on method
that will allow produc�on of
the same output level at a lower AVC level.
Incremental costs must be accurately iden�fied for good
decision making. All costs that
change must be included, and costs that do not change must not
be included. For example,
capital assets that have been idle with no alterna�ve use do not
have an incremental cost and can be regarded as costless for the
decision at hand. On the
other hand, if capital assets are currently being used to
produce an alterna�ve product, and the decision at hand would
require them to be used elsewhere,
we have to include the foregone contribu�on as an opportunity
cost of the decision to be made and to treat the opportunity cost
as an incremental cost of
that decision. For example, a trucking firm u�lizing an idle
truck to complete a special delivery would include driver, fuel,
and toll costs, but would not
include any cost for the use of the truck in the calcula�on of

incremental costs. However, if the truck would have been used
to carry groceries to earn $200
during that �me, that foregone revenue would be the
opportunity cost of using the truck to make the special delivery
and should be included as an
incremental cost of the decision to use the truck that way
instead.
Incremental costs are o�en called relevant costs since they are
the costs that are relevant to the decision that is to be made, as
dis�nct from the irrelevant
costs that will be incurred regardless of the decision to be
made. Irrelevant costs are either sunk costs (fixed costs incurred
in the past) or unavoidable costs
(costs that must be incurred in the present or future period) as
discussed in Chapter 5.
Incremental Cost Categories
There are three main categories of relevant or incremental costs.
The first is present-period explicit costs. These are actual
outlays of cash to pay for the
variable and fixed inputs that are required to implement the
decision that is made. Of course, incremental costs will not
include unavoidable costs that must
be paid in the present period regardless of the decision to be
made.
The second category of incremental costs is opportunity costs.
Raw materials or components or finished goods
taken from inventory do not have a present-period explicit cost
but could presumably be sold to another producer
or an end user at a fair market value for the item, and that fair
market value is the opportunity cost and should
be accounted for as an incremental cost. Alterna�vely, if an
item in inventory has li�le or no market value (i.e.,

"dead stock") and would not be replaced in inventory then it
has no opportunity cost and, thus, its use does not
involve an incremental cost. The fact that there was previously
a historic cost of purchasing or manufacturing that
item is an irrelevant sunk cost for the purposes of the present
decision.
The third category is future costs. Many decisions will have
implica�ons for future costs, such as repairs and
maintenance to equipment, vehicles, or other capital assets
that will be necessitated as a result of their u�liza�on
for the decision to be made. Of course, as we saw in Chapters 1
and 2, future costs must be evaluated in present-
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Future costs of repairs and maintenance
must be taken into considera�on when
purchasing equipment.
value terms (if known for certain) or in expected present-value
terms (if there is uncertainty surrounding the
actual cost to be incurred in the future).
In Table 6.1, we can see a summary of the various costs that are
relevant or irrelevant for managerial decision
making. Note that by relevant and irrelevant we mean with
respect to the decision to be made. If a cost is a
consequence of the decision to be made, it is a relevant or

incremental cost. Some current period and future costs are
irrelevant because the firm is
commi�ed to them and they are, thus, unavoidable costs. No
prior expenditures (sunk costs) are incremental costs, unless
they have an opportunity cost of
being involved in the present decision to be made.
Table 6.1: Summary of cost concepts for decision making
Relevant costs = incremental costs Irrelevant costs =
nonincremental costs
Present-period explicit costs
Variable costs
Direct labor costs
Direct materials
Variable overheads
Fixed costs
New equipment needed
New salaried personnel needed
Opportunity costs
Contribu�on foregone on the best alterna�ve use of the
resources involved
Future-period incremental costs
EPV of probable costs to follow in the future as a
consequence of this decision
Unavoidable costs
Managers’ salaries
Payments on debt
Rental and lease costs
Salaries for ongoing workers
All other payments that must be made regardless of the
decision at hand
Sunk costs
Previously paid for purchases of assets including land,
buildings, plant and equipment, and deprecia�on

expenses based on these
All prepaid and nonrecoverable expenses
Incremental Revenues
Similarly on the revenue side, there are some revenues that are
relevant to the decision to be made and some that are irrelevant.
Incremental revenues are
those that will be received as a result of the decision, so these
are the relevant revenues. Irrelevant revenues (for the purpose
of the decision to be made)
are those that would be received or lost regardless of the
decision to be made.
So, any decision to be made may cause some costs to be
incurred (incremental costs) and may also cause some revenues
to be earned (incremental
revenues). Note that some decisions impact only incremental
costs (such as replacing broken equipment) while other
decisions may impact only incremental
revenues (such as selling an unwanted asset). Most decisions
have both incremental costs and incremental revenues to
consider. We shall proceed to work
through some realis�c business examples using contribu�on
analysis and will examine three main types of contribu�on
analysis with these sugges�ve
names: Project A versus Project B, make-or-buy decisions, and
take-it-or-leave-it decisions.
Project A Versus Project B Decisions
Managers o�en have to choose between two or more projects
that are both poten�ally profitable because they do not have the
produc�ve capacity or the
funding to handle both projects at the same �me. A profit-

maximizing firm would want to undertake the most profitable
project first, and defer the less
profitable project to a later period when it would be compared
with other poten�al projects that were available for
implementa�on at that �me. The
appropriate method for choosing between compe�ng projects is
contribu�on analysis—profits will be maximized by choosing
the project that contributes
the most towards overheads and profits.
Suppose a firm is considering implementa�on of either Project
A or Project B as detailed in Table 6.2. Project A promises sales
of 10,000 units at $2 each,
with materials, labor, variable overhead, and allocated
overhead costs as shown, and so apparently makes a profit of
$2,000. Project B promises sales
revenue of $18,000 with $14,000 of direct and allocated costs,
and, thus, apparently makes a profit of $4,000. It might seem
that Project B is superior to
Project A, because it seems to make higher profits. But, what
do we know about relevant and irrelevant costs?
Table 6.2: Income statements for Project A and Project B
Project A Project B
Revenues
Costs
$20,000 Revenues
Costs
$18,000
Materials
Direct labor
Variable overhead

Allocated overhead
Total costs
$2,000
6,000
4,000
6,000 $18,000
Materials
Direct labor
Variable overhead
Allocated overhead
Total costs
$5,000
3,000
3,000
3,000 $14,000
Profit $2,000 Profit $4,000
When contribu�on analysis is applied to the above choice
situa�on the result may be surprising. In Table 6.3, we show
only the incremental costs and
revenues and see that the contribu�on of Project A actually
exceeds that of Project B. For each project we include only the
materials, direct labor, and
variable overhead costs, presuming that these costs would not
be incurred unless the project is undertaken. We exclude
allocated overhead charges since
these relate to the sunk costs of previously purchased capital
assets or the salaries of management and other workers who
must be paid whether or not the

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Companies are frequently faced with the
decision to make or buy. This model can
also be applied to a firm deciding
between hiring an external cleaning crew
or doing cleaning and maintenance work
itself.
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project is undertaken. Thus, we see that Project A promises the
larger contribu�on to overheads and profit and should be
chosen for implementa�on by the
profit-maximizing firm.
Table 6.3: Contribu�on analysis for Project A and Project B
decision
Project A Project B
Incremental revenues
Incremental costs
$20,000 Incremental revenues
Incremental costs
$18,000
Materials
Direct labor
Variable overhead
Incremental costs

$2,000
6,000
4,000
$12,000
Materials
Direct labor
Variable overhead
Incremental costs
$5,000
3,000
3,000
$11,000
Contribu�on $8,000 Contribu�on $7,000
The danger of using an arbitrary rule to allocate fixed overhead
costs is illustrated in this example. In Table 6.2, the overhead
charge is set equal to the cost
of direct labor for both projects, implying that the manager
who produced the data in Table 6.2 used an alloca�on rule of
"100% of direct labor costs."
Simple rules like that almost certainly do not correctly reflect
the relevant costs of undertaking any project. Contribu�on
analysis allows an incisive look at
the actual changes in the costs and revenues that would follow
the decision to choose one project over the other.
Also, note that in this simple example we implicitly assumed
there were no opportunity costs or opportunity
revenues1

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�o
ns/sec6.1#footernote1) and no future costs or revenues
associated
with either project. In this case, there is only $1,000 difference
in the contribu�ons of the two projects, so the
decision to implement Project A is very sensi�ve to the
assump�on of zero opportunity and future costs and
revenues. This calls for sensi�vity analysis, an analysis of the
impact of inaccurate data on the desirability of the
decision made. A decision is called sensi�ve to the
assump�ons on which it is based if a rela�vely small change
in
those assump�ons would cause a different decision to be made.
It is useful to express the degree of sensi�vity in
terms of the percentage varia�on in costs that it would take to
reverse the decision; in this case, the change in costs
of Project A that would reverse the decision is the propor�on
(or percentage) by which the incremental costs of
Project A could increase without reducing its contribu�on to
that of the next-best alterna�ve (Project B). In this case
it is $1,000/$12,000 = 8.33%, assuming the costs of Project B
are calculated accurately. In prac�ce, decision makers
must be careful to assess whether there are any opportunity or
future implica�ons of their decision. If they cannot
easily es�mate these addi�onal costs they should consider, if
such addi�onal costs are thought to exist, whether
they are likely to be more than 8.33% (in this case) higher than
the ini�al es�mate. Where the sensi�vity percentage
is rela�vely high, for example 40%, we say the decision is
rela�vely insensi�ve to the accuracy of the cost es�mates
of incremental costs, unless the costs involved are likely to be
highly vola�le (e.g., the pump price of gasoline).
Make-or-Buy Decisions
The next category of decision that involves incremental costs

and revenues is the make-or-buy decision. Such
decisions are required when the firm could either manufacture
the product in-house (i.e., make) or outsource the
manufacture from another firm (i.e., buy). Similarly, the firm
might consider doing its own cleaning and maintenance
work (using current employees and purchasing the necessary
equipment and supplies) and compare the incremental
cost of this with the alterna�ve solu�on of having an outside
firm supply the maintenance and cleaning work.
In Table 6.4, we consider the make-or-buy problem facing
Wilson Tools. Wilson Tools manufactures high-quality power
tools such as drills, jigsaws, and
sanders. All these tools require the same roller-bearing unit,
which the company manufactures in its own bearing department.
Table 6.4 shows the costs
data for the past month for the bearing department.
Table 6.4: Wilson Tool Company—Bearing department costs,
July 2012
Cost category Total Per unit
Direct materials
Direct labor
Allocated overhead
Total costs
$38,640
$126,390
$252,780
$417,810
$0.56
$1.81
$3.63
$6.00

Now suppose that Wilson Tools has an opportunity to expand
the sales of its power tools by an addi�onal 7,500 units a month
by supplying its tools to a
chain of hardware stores in another state. Wilson could produce
the addi�onal 7,500 bearings in its bearing department, but this
addi�onal output would
congest opera�ons somewhat, so management is considering
having the addi�onal roller-bearing units supplied by a
specialist bearing manufacturer. It is
es�mated that it will require an addi�onal 15% in direct labor
costs and an addi�onal 12% in total materials costs to make the
bearings in-house. No
addi�onal capital expenditures will be required as all
machines have excess capacity currently. A specialist bearing
manufacturer has been asked to submit a
quote to produce and supply the 7,500 bearings per month and
has studied the specifica�ons and submi�ed a proposal to
provide the bearings at a total
cost of $30,000 per month, or $4 each. So, should Wilson make
or buy the addi�onal bearing units?
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To answer this we must find the incremental cost of producing
the bearings in-house, to compare this with the incremental cost
of buying them from
outside, which is $30,000 per month. The es�mated incremental

cost of direct labor is 15% of $126,390, or $18,959 per month.
The es�mated incremental
cost of direct materials is 12% of $38,640, or $4,637 per
month. Wilson expects no change in overhead costs, so the
incremental cost of producing the
bearings in-house totals $23,596 per month, or about $3.15 per
unit. The decision to make rather than to buy the bearings
would, thus, appear to save
Wilson about $6,404 per month, or about $0.85 per unit.
Variability of Overheads
The above analysis assumes no variability at all in overhead
costs as a result of the decision to make the bearings in-house.
It is likely, however, that some
costs that are treated as fixed overheads, such as electricity
expenses, office and administra�on expenses, and equipment
maintenance expenses, might
actually increase as a result of this decision to make the extra
7,500 bearings in-house each month. But the changes in these
overhead costs are likely to be
hard to measure. Rather than undertake expensive search costs
or make arbitrary assump�ons, the manager should first apply
sensi�vity analysis to the
assump�on that overhead costs will remain unchanged—that
is, the manager should ask by how much could the overhead
costs actually change without
causing the decision (to make the bearings) to be the wrong
decision. In this case, the percentage varia�on in overhead
costs would need to be
$6,404/$252,780, which equates to about 2.5%. This is a very
slim margin for error, so if the manager thinks that the
overhead expenses are indeed likely to
change by that amount or more, the decision should be reversed
and the extra bearings should be bought from the outside
supplier.

Other Considera�ons
Quality. A number of other considera�ons should also enter
the make-or-buy decision. First, the manager should be
concerned about the quality of the
bearings supplied by the outside manufacturer. A make-or-buy
decision that ignores the compara�ve quality of the bearings
and simply focuses on
incremental costs might be a very bad decision if the outside
supplier’s bearings are of poorer quality. Low-quality bearings
in the power tools could lead to
product failure, increased warranty claims, unhappy customers,
and a reduc�on in the market reputa�on enjoyed by Wilson
Tools. On the other hand, if the
quality of the specialist manufacturer’s bearing was
significantly be�er, Wilson should consider paying extra to get
bearings of higher quality than can be
produced in-house and, perhaps, begin making and selling a
premium line of power tools.
Longer-term supplier rela�ons. Wilson Tools may be nearing
its full capacity output level and, if demand con�nues to
grow, may need to expand its plant in
the near future. An alterna�ve strategy would be to establish a
rela�onship with a specialist supplier of bearings to allow it to
meet demand for its power
tools in the future. The "buy" alterna�ve gives it the
opportunity to both test out a poten�al longer term supplier and
to start building a mutually beneficial
long-term supply rela�onship or work towards a poten�al
strategic alliance that might be desirable in the future.
Labor rela�ons. If Wilson management chooses the "make"
decision, the extra workers required will lead to greater
crowding of the factory floor, cafeteria,

toilets, and the parking lot, and may, as a consequence, reduce
job sa�sfac�on. As a consequence, worker produc�vity
(marginal and average product) may
fall resul�ng in a rise in average variable and marginal costs
per unit of output, thus, poten�ally nullifying the cost
advantage of the "make" op�on.
Conversely, if Wilson takes the "buy" decision and contracts
with the outside firm for ongoing supply of bearings, workers in
the bearing department (in
par�cular) may fear that they might lose their jobs if demand
for power tools later falls. This might seriously hurt
management-labor workplace rela�ons.
Of course, these addi�onal considera�ons may be difficult (or
costly) to quan�fy. Thus, the manager will need to exercise
judgment and, perhaps, make a
calculated gamble to decide whether to make or buy,
par�cularly when the incremental costs of both op�ons are
rela�vely close together.
Take-It-or-Leave-It Decisions
In other situa�ons, the manager might be faced with an offer
that is non-nego�able and must decide whether to accept or
decline that offer. For example, a
prospec�ve buyer might offer a fixed sum of money for a
par�cular capital asset, such as land, buildings, or piece of
equipment owned by the firm, or for a
par�cular quan�ty of the firm’s output. Or, the purchasing
agent for a chain of discount stores might approach your firm
and ask for a special deal on a bulk
purchase (e.g., 10,000 units) of your firm’s output. Or, a
poten�al customer might say, "I can get this (e.g., car) for $X
from another supplier. If you can beat
that price you have a deal." The manager’s task is to evaluate

the contribu�on of the offer and compare it with the status
quo—if the deal offers a posi�ve
contribu�on to overheads and profits it will be profit-
maximizing to take the offer since it will contribute addi�onal
funds towards the firm’s profit (or
reduce the firm’s loss if revenues are insufficient to fully cover
its overhead costs).
Let us work through an example to demonstrate how take-it-or-
leave-it analysis works. Suppose Idaho Instruments Ltd. makes
hand-held and dashboard-
mounted GPS (global posi�oning satellite) devices that allow
pedestrians and drivers to navigate unfamiliar streets or
highways and to find their way to
specific des�na�ons. Normally, the company manufactures
these devices and sells them to a distributor at an agreed
distributor price. That distributor, in
turn, sells the product to retailers at the wholesale price, and the
retail stores then sell the product to end-user customers at the
retail price.2
ns/sec6.1#footernote2) But, yesterday, the purchasing agent of a
large retail store (that does not currently stock the
Idaho GPS device) has come directly to the manufacturer and
says she wants to "cut out the middle man" and buy 20,000 units
of Idaho Instrument’s X1
model for $40 each, which is $10 less per unit than the
distributor price. Idaho’s sales manager knows that the present
produc�on level of the X1 model is
nearly at full capacity at 160,000 units, but he could supply
the addi�onal 20,000 units by foregoing produc�on and sale of
5,000 units of the more
sophis�cated and more expensive X2 model. Per�nent data
rela�ng to these two models is shown in Table 6.5. Because of

the automated produc�on
process, the per unit variable costs (AVC) of both units is
constant at those levels over a wide range of output levels. The
sales manager is reluctant to sell
the X1 model for $40 per unit when he normally gets $50,
par�cularly since he will have to sacrifice 5,000 units of sales
of the more expensive X2 model.
He also thinks that about 20% of the X1 units that would go to
the new retailer customer (i.e., 4,000 units) will simply
replace sales to customers who
6.1#footernote2
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would have purchased the device through other stores that
already stock Idaho’s GPS devices. He has tried to nego�ate for
a be�er price but the purchasing
agent is adamant and insists that $40 is her only offer. Should
Idaho Instruments take it or leave it?
Table 6.5: Per-unit cost-price-profit data for Idaho Instruments
GPS devices
Cost item Model X1$
Model X2
$
Direct materials
Direct labor

Variable overhead
Average variable cost (AVC)
Allocated overhead
Short-run average cost (SAC)
Profit margin (20%)
Price to distributor
6.88
9.67
4.29
20.83
20.83
41.67
8.33
50.00
7.79
12.58
4.63
25.00
25.00
50.00
10.00
60.00
Because AVC is expected to be constant over a wide range of
output levels for both products, we can calculate the
incremental costs on the basis of the
AVC data shown in Table 6.5, where AVC is equal to the sum
of the direct materials, direct labor, and variable overheads.
Note that Idaho Instruments
appears to have a very simple rule for the alloca�on of
overhead costs: It simply adds 100% of the AVC for each
product to arrive at the short-run average
cost (SAC). Subsequently, the company marks up its SAC by
20% to find the normal price to their distributor, and the profit

margin is the difference between
the SAC and the price it normally receives from its distributor.
Note also that we can deduce from Table 6.5 that the
contribu�on per unit to overheads and
profit (i.e., the contribu�on margin) for the X1 model is the
sum of the allocated overhead charge ($20.83) plus the profit
margin ($8.33) equals $29.17
(rounded), and for the X2 model it is $25 + $10 = $35.
So, to produce 20,000 more units of model X1 will cause an
incremental cost of 20,000 �mes the AVC of $20.83, or
$416,667 in total. But this is not the
total incremental costs of taking the deal, because there are
also opportunity costs associated with sacrificing the
produc�on and sale of 5,000 units of the
X2 model and 4,000 units of the X1 model. To calculate that
opportunity cost we note that if 5,000 units of X2 are not
produced, the firm will not spend
5,000 �mes the $25 AVC of the X2 model, that is $125,000,
but neither will the firm receive the foregone sales revenue of
5,000 �mes the $60 price, that is
$300,000. Thus, taking the deal will also cause a net reduc�on
of $175,000 (i.e., $300,000 minus $125,000) in the
contribu�on of the X2 model towards the
overheads and profits of the company. Alterna�vely, and more
quickly, we could mul�ply X2’s contribu�on margin ($35) by
5,000 units to find the $175,000
figure for the contribu�on foregone if the sales manager
decides to take the deal. Similarly, the foregone contribu�on
from the 4,000 units of the X1 model
that is expected to be lost is 4,000 �mes its contribu�on
margin of $29.17, or $116,667 in total. In Table 6.6, we have
assembled the data to allow the sales
manager to make a decision.
Table 6.6: Contribu�on analysis of the take-it-or-leave-it offer

Incremental revenue
Sale of 20,000 units of X1 at $40 each
Incremental costs
Variable costs of 20,000 units of X1 at $20.83 each
Foregone contribu�on of 5,000 units of X2 at $35.00 each
Foregone contribu�on of 4,000 units of X1 at $29.17 each
Total incremental costs
Contribu�on to overheads and profit
$800,000
$416,667
$175,000
$116,667
$708,333
$91,667
You can see that there will be a net contribu�on of $91,667 to
the overheads and profit of Idaho Instruments if the sales
manager takes the deal—despite
the fact that ini�ally the take-it-or-leave-it offer looked like a
bad deal, being $10 less than the normal distributor’s price and
requiring the sacrifice of 5,000
units of sales of the more expensive X2 model, plus the
probable loss of contribu�on from 4,000 units of the X1 model.
But this demonstrates the beauty of
contribu�on analysis: It cuts through arbitrary overhead cost
alloca�ons and pricing rules to focus only on what costs and
revenues actually change as a
result of making a par�cular decision.
Other Considera�ons
The preceding analysis is subject to some simplifying

assump�ons of course, and the sales manager must consider
these before ge�ng back to the
purchasing manager. The first issue is the sales manager’s
assump�on that 4,000 units of the X1 model that will be placed
directly into this retail store will
simply replace or cannibalize exis�ng sales, that is, will be
sold via this new retailer instead of through Idaho’s regular
distribu�on channel and by the
exis�ng retailers of Idaho’s GPS devices. The sales manager
must carefully consider the extent to which sales via this
retailer will be at the expense of sales
via its normal distribu�on channels. Applying sensi�vity
analysis to his assump�on he could make a few simple
calcula�ons and find that the cri�cal ra�o of
sales that replace normal sales in this example is about 35%.
For example, if more than 35% of sales (7,000 units) via this
new channel replace sales via the
normal channel, the deal will give almost exactly the same
result as con�nuing to supply the market through the regular
distribu�on channels, and so, the
decision should be reversed.3
ns/sec6.1#footernote3) The sales manager must find out where
the extra units will be
retailed. Will this be in a new geographic area where the firm
currently has li�le or no sales? Or, perhaps the new retail
stores (that currently do not carry
Idaho’s product line) do carry a rival manufacturer’s GPS
devices, and Idaho’s X1 model would then be accessible to
poten�al new customers who would
poten�ally buy it instead of the rival’s product. So, once again,
faced by the need to get data that is hard or expensive to get,
the sales manager must

6.1#footernote3
9/23/2019 Print
When a business is able to en�ce new customers to its store
by
selling products at discount prices while s�ll realizing a
profit,
customers are likely to come back to buy more products in
the
future.
exercise his judgment and hypothesize about the propor�on of
this 20,000-unit deal that will simply cannibalize exis�ng sales
and make the decision
accordingly.
The second main issue is the rela�onship with the distributor
and the exis�ng retailers.
Undoubtedly, the distributor and the exis�ng retailers will
become aware that a new
store has entered the market to sell Idaho’s X1 GPS devices.
Having bought them at 20%
below the normal wholesale price, the new outlet is likely to set
a somewhat lower retail
price on this product, and perhaps also promote them as a
special, and will consequently
a�ract customers away from the exis�ng marke�ng channel.
Losing sales (and share of

the market) will possibly cause the distributor and exis�ng
retailers to become unhappy
with what they might feel is unfair compe��on, and they may
take their business
elsewhere in the future. This would cause a future reduc�on in
demand through the
regular marke�ng channels and poten�al loss of future income.
If this is likely to occur,
the sales manager must include it as another opportunity cost of
taking the deal.
A third issue is the prospect of future (or repeat) business with
the new retail store. If
indeed this store is able to reach mostly new customers, and,
especially if it sells at lower
(discount store) prices, it is likely to come back to buy more
product from Idaho in future
periods, perhaps also to expand purchases to include the X2
device and other items in
Idaho’s product line. In the above analysis, we have treated the
deal as a "once-off" deal,
which is the most conserva�ve assump�on to make. But if this
new customer were to
repeat or increase this purchase periodically in the future, the
expected present value of such future contribu�ons must be
considered by the sales manager
before making the decision. Of course, it would be prudent for
the sales manager to set the purchasing agent’s expecta�ons at
an appropriate level by
sta�ng that this is to be understood as a once-off deal to kick
start a business rela�onship and that future dealings would be
expected to allow Idaho a
be�er profit margin.
In Chapters 1 and 2, we discussed decisions that had cost and

revenue outcomes over mul�ple periods into the future, and we
saw that we needed to
express those future outcomes in expected present value terms.
In those chapters, we spoke of profits in the first and
subsequent �me periods that had to
be discounted back to present value. We now know that it is
the contribu�on, rather than an accoun�ng measure of profits
(which might reflect an
inaccurate alloca�on of overhead costs), that is important for
managerial decision making.
1. Just as an opportunity cost is a revenue (or in this case a
contribu�on) that must be foregone if a decision is taken, an
opportunity revenue is a cost (or a nega�ve contribu�on) that
is
avoided if a decision is taken. [return
ns/sec6.1#return1) ]
2. The difference between the distribu�on price and the
wholesale price provides the contribu�on margin for the
distributor, and the difference between the wholesale price and
the
retail price provides the contribu�on margin for the retailer,
assuming no other incremental costs—all other costs of the
distributor and the retailer are fixed salary or other
unavoidable costs. For example, the distributor’s price might be
$50 and the distributor might mark this up by 50% to sell it to
the retailer at the wholesale price of $75. The retailer
might then mark up the wholesale price by 100% to sell it to the
end user at the retail price of $150. [return
3. That is, 7,000 units x $29.17 means $204,167 addi�onal

opportunity cost that nearly wipes out the $208,333
contribu�on to be made by the deal if there is no replacement
of exis�ng
sales. [return
6.1#return1
6.1#return2
6.1#return3
9/23/2019 Print
6.2 Cost Estimation Methods
In this sec�on, we will examine several methods for
es�ma�ng the level of costs per unit based on data collected
from the firm’s prior produc�on
experience. In the short-run context of business decision
making, we are primarily concerned with the behavior of
variable costs, but we also know that
changes in a fixed cost category might be necessitated by a
par�cular decision. We start with simple extrapola�on and
later proceed to more complex, but
probably more accurate, measures.
Extrapolation of Prior Observations
Extrapola�on means to impute values to a variable outside the

range of previous data observa�ons. Extrapola�on is achieved
by projec�ng (or extending)
the rela�onship that is iden�fied between the output level and
the cost level inside the range of data observa�ons to output
levels outside the range of
previously observed output levels.
So, if average variable cost (AVC) was observed to be constant
over a range of output levels, we could make the simple
assump�on that it will remain
constant at that level for rela�vely small changes in output
levels that are both higher and lower than the range of
observa�ons for which we have prior
data. In the le�-hand side in Figure 6.1, we show a situa�on
where prior data indicates that AVC was about $4 at produc�on
rates of both 3,000 and 4,000
units of output. These observa�ons are indicated by the stars.
The broken lines to the le� and to the right of the observed data
points indicate our
extrapola�on of AVC to both higher and lower levels of output.
Another example of extrapola�on is shown on the right-hand
side in Figure 6.1. Suppose we have (in a different produc�on
process) observed that marginal
cost (MC) increased from $4 to $5 when the output rate was
increased from 3,000 to 4,000 units per day. Accordingly, we
can extrapolate this data to
es�mate that MC is likely to increase by another dollar to $6
per unit if output is increased to 5,000 per day.
Figure 6.1: Examples of cost extrapola�on given prior cost
observa�ons
No�ce that the interrela�onships of the cost concepts mean
that, if we have data on AVC at par�cular output levels, we
can deduce the value of TVC; or

conversely, if we know TVC and the output level, we can
deduce the value of AVC. If we have two or more values of
AVC or TVC we can deduce the value of
MC since MC is equal to ΔTVC/ΔQ, or the rate of change of
TVC. Similarly, if we have total cost (TC) data points we can
deduce the value of short-run
average costs (SAC) and MC, and also by subtrac�on of TVC
from TC, we can find total fixed costs (TFC) and average fixed
costs (AFC). Deducing the shape of
the costs curves from a limited amount of data generated by past
produc�on experience will allow the manager to make more
accurate es�mates of the
incremental cost associated with Project A versus Project B,
make-or-buy, and take-it-or-leave-it decisions.
Interpolation Between Prior Observations
While extrapola�on means making es�mates outside the data
range, interpola�on means making es�mates inside the data
range. In Figure 6.1, we implicitly
interpolated between the data points by drawing a straight line
between the data points. In other situa�ons it will be clear the
rela�onship cannot be a
linear one but must be curvilinear. We saw in Chapter 5 that the
law of variable propor�ons, also known as the law of
diminishing returns, will cause cost
curves to bend in predictable ways. Suppose we have a
produc�on situa�on where data has been collected twice. The
first data point, when the output rate
was 1,600 units per period, measured TVC as $6,400 and
deduced AVC to be $4, and a second data collec�on at output
rate 3,800 measured TVC as
$15,200 and deduced that AVC was s�ll $4. A�er the second
data collec�on, however, MC was es�mated to be TVC/Q =
$6,230/100 = $6.23 by calcula�ng
the costs of direct materials, direct labor, and variable

overheads for the last batch of 100 units of output produced.
This observa�on, that MC is
substan�ally above AVC, should immediately ring alarm bells
in the manager’s mind. If the data is accurate, it must mean
that AVC is rising, and if AVC is s�ll
at the same level ($4) as it was before then AVC must have
fallen and then risen between the known data points. In Figure
6.2, we use our knowledge of
the law of variable propor�ons to interpolate between the
known data points and show curved lines represen�ng our
es�mates of the AVC and MC values
between the known data points.
Figure 6.2: Interpola�on of cost data between known data
points
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ientToken=8b70ed6… 10/19
As with extrapola�on, interpola�on provides a best es�mate
given the data available but may not be especially accurate. For
example, in Figure 6.2, the AVC
curve might be more or less U-shaped than we have shown, and
the lowest point of the AVC curve (where the MC curve
intersects) might be nearer to the
1,600 output rate or nearer to the 3,800 output rate. Thus, the
manager must exercise judgment, u�lizing other sources of
informa�on perhaps, to sketch in
the AVC and MC curves. But the most important message to the
manager from this cost es�ma�on exercise is that MC is rising
rapidly, pulling AVC upwards.

But, as we saw in Chapter 5, the average fixed cost (AFC) will
be falling so that the sum of AVC and AFC (i.e., short-run
average cost, SAC) might s�ll be
falling at the higher output rate. The manager must pay close
a�en�on to this issue to decide whether higher output rates
should be avoided or whether
considera�ons should be given to installing a larger size of
plant if economies of scale are indeed available.
Gradient Analysis Using Known Data Points
A gradient is the slope at which the ver�cal eleva�on of a
line or surface changes over a horizontal distance. That is a
fancy way of saying that "slope equals
rise over run." In the context of cost curves, the rise will be
the change of a cost value (e.g., TC or TVC) and the run will
be the change in the output rate.
Marginal cost is a gradient, of course, in the case where the
horizontal change is only one unit of output (or the average MC
per unit over a rela�vely small
DQ). More commonly in cost es�ma�on we find that output
data will not be available as con�nuous data (i.e., not
available for one-unit increments of
output), but that data will be collected periodically at different
�mes when output levels are discretely different and we need to
interpolate the cost values
in between the known data points, as we did above. Gradient
analysis involves (possibly nonlinear) interpola�on between
mul�ple data points.
Suppose weekly data has been collected for total fixed and
variable costs for various output levels over five weeks as
shown in Table 6.7. To calculate the
gradient of the cost curves these data must first be rearranged in
ascending order of output, that is, from the smallest to the
largest. It is a simple ma�er to

calculate the average cost levels for SAC, AVC, and AFC by
simply dividing the TC, TVC, or TFC figures by the relevant
output level. For decision-making
purposes, the manager will be most interested in the behavior
of marginal costs and will want to derive an es�mate of MC at
various output rates. We
know that MC is the change in TC (or TVC) for a one-unit
change in output, but the changes in output are much larger in
our data. Accordingly, we must
es�mate MC as the average change in TC over the output
interval by taking the gradient of TC or TVC with respect to
output. In Table 6.7, we es�mate MC
at four output rates by evalua�ng the ra�o ΔTVC/ΔQ for each
output interval, where Δ (as usual) symbolizes a discrete
change in the variable concerned.
Table 6.7: Gradient analysis to es�mate marginal cost levels
Produc�on period Output rate (Q) TVC($)
AVC
($/Q)
ΔTVC
($)
ΔQ
(Q)
MC=ΔTVC/ΔQ
($/Q)
Week 4 4,500 27,000 6.00
6,600
3,775

4,625
6,750
1,500
500
500
500
4.40
7.55
9.25
13.50
Week 3 6,000 33,600 5.60
Week 5 6,500 37,375 5.75
Week 1 7,000 42,000 6.00
Week 2 7,500 48,750 6.50
No�ce that the four es�mates of MC in Table 6.7 are shown in
the middle of the intervals between the five observa�ons. This
is more evident in Figure 6.3
where we show the AVC data points and the es�mated MC
curve as the broken line joining the four gradient values that
were calculated in Table 6.7. Note
that the loca�on of the MC curve is more reliable in this case,
compared with our earlier interpola�on exercise where the

curvature of the AVC and MC
curves was chosen arbitrarily. By placing the es�mated MC
value in the middle of the output interval, we can gain a more
accurate es�mate of the MC
values for all output rates between the known data points.
Figure 6.3: Gradient es�ma�on of the marginal cost curve from
known TVC data
9/23/2019 Print
Most businesses output levels vary up and
down from week to week as orders come
in from retailers to replenish their stocks.
Cost Estimation Using Regression Analysis
Note that for both interpola�on and gradient analysis, we have
essen�ally sketched in a line of best fit to join the
calculated or es�mated data points. As we saw in Chapter 4,
we can u�lize regression analysis to find the line of
best fit to data points, and this is especially useful when we
have a larger number of output and cost observa�ons
and it is not so easy to see the shape of the rela�onship
between output and costs. Suppose we have 12 weeks of
data on the weekly output and total variable costs of an ice
cream factory, as presented in Table 6.8.

Table 6.8: Data on output and TVC levels for an ice cream
factory
Week Ending Output (gallons) Total variable costs ($)
Sept. 7 7,300 5,780
Sept. 14 8,450 7,010
Sept. 21 8,300 6,550
Sept. 28 9,500 7,620
Oct. 5 6,700 5,650
Oct. 12 9,050 7,100
Oct. 19 5,450 5,060
Oct. 26 5,950 5,250
Nov. 2 5,150 4,490
Nov. 9 10,050 7,520
Nov. 16 10,300 8,030
Nov. 23 7,750 6,350
No�ce that the output levels vary up and down from week to
week as orders come in from retailers to replenish their stocks
of ice cream. We can see that
output and TVC are posi�vely related, but is this posi�ve
rela�onship a linear rela�onship (implying constant MC) or a
curvilinear rela�onship (implying
falling and/or rising MC)? Knowing what we know about the
law of variable propor�ons, namely that for equal increments of

the variable inputs the output
level will increase first at an increasing rate and later at a
decreasing rate, our default assump�on ought to be that the line
of best fit to the TVC data is
most likely to be a cubic func�on, taking the form:
TVC = α + β1Q + β2Q2 + β3Q3 (6-1)
9/23/2019 Print
Where the parameter α represents the unknown factors not
explained by the independent variables (Q, Q2 and Q3), and
the β’s are the es�mated
regression coefficients to the independent variables. If the line
of best fit does prove to be a cubic func�on, then we can
es�mate the MC func�on as the
rate of change of the TVC curve, which is mathema�cally
equivalent to the first deriva�ve of the TVC func�on, or:
MC = δTVC/δQ = β1 + 2β2Q + 3β3Q (6-2)
As you can see, and consistent with our analysis in Chapter 5, a
cubic TVC func�on will give rise to a quadra�c MC func�on,
which will be U-shaped, falling
at first due to increasing returns to the variable inputs and later
rising due to diminishing returns to the variable inputs.4
ns/sec6.2#footernote4) The U-shape means we should expect a

nega�ve value for the β2 coefficient and posi�ve
values for the β1 and β3 coefficients—these would cause the
MC curve to start with a ver�cal axis intercept of β1, fall as
output levels increase (at first) due
to the nega�ve β2 coefficient, which is rela�vely large
compared to the β3 coefficient, and then rise later as the
rela�vely large Q-squared values dominate
the equa�on. Regression analysis will allow us to find
es�mates of these β coefficients in the TVC func�on and, thus,
we will be able to calculate the
es�mated MC at any value of Q by plugging that value of Q
into the MC expression given by equa�on 6-2.
In Chapter 4, when we introduced mul�ple regression analysis
in the context of es�ma�ng the demand func�on, the
independent variables were different
drivers of demand. Here, in the context of cost es�ma�on, the
independent variables are different variants of the same driver
of costs—namely the output
level. So, the three independent variables on the right-hand
side of the regression equa�on for TVC will be (i) the output
level Q; (ii) the output level
squared (Q2); and (iii) the output level cubed (Q3). To
conduct the regression analysis we first need to enter these data
into columns in an Excel
spreadsheet. To avoid huge numbers we will express the data in
thousands of units, as shown in Table 6.9.
Table 6.9: Data set up for the regression of TVC against output
TVC ($000s) Q (000s) Q2 (000s) Q3 (000s)
5.78 7.30 53.29 389.02

7.01 8.45 71.40 603.35
6.55 8.30 68.89 571.79
7.62 9.50 90.25 857.38
5.65 6.70 44.89 300.76
7.10 9.05 81.90 741.22
5.06 5.45 29.70 161.88
5.25 5.95 35.40 210.65
4.49 5.15 26.52 136.59
7.52 10.05 101.00 1,015.08
8.03 10.30 106.09 1,092.73
6.35 7.75 60.06 465.48
To do the regression analysis of this data, check that Statpro (or
other sta�s�cs add-in program) has been added to your Excel
so�ware by pulling down the
Add-Ins tab to find it. (If it is not there, you will have to do
an Internet search and download a copy.) When it is
downloaded click on the Statpro name and
select Regression. You will then need to iden�fy which is the
dependent variable (TVC) and the independent variables you
want to enter into the regression
equa�on, that is, Q, Q2 and Q3. Indicate where you want the
results to be posted—below or adjacent to the data columns or
in a separate worksheet. A�er

you have indicated which of the independent variables are to be
entered, allow the program to make the calcula�ons. A table
showing the results will
appear in the chosen area of the spreadsheet. This will include
the value of the a and the various β coefficients, as well as the
coefficient of determina�on
(R2), the standard error of es�mate (Se) and the standard
errors of the coefficients (Sβ) sta�s�cs. Your results table will
look something like Table 6.10.
Table 6.10: Results from the regression analysis to es�mate a
cubic TVC func�on
Variable Coefficient Std err of coeffic. t-sta�s�c P-value
Intercept α = 2.8318 8.3176 0.3405 0.7423
Output (Q) β1 = 0.0377 3.839 0.0111 0.9914
Output squared (Q2) β2 = 0.0802 0.4462 0.1798 0.8612
Output cubed (Q3) β2 = −0.0035 0.0191 −0.1825 0.8598
6.2#footernote4
9/23/2019 Print
Adjusted R2 0.9676

Standard error of es�mate 0.2031
These results indicate a very high adjusted R2 (i.e., adjusted
for degrees of freedom), which might seem like a good result,
but the rela�vely high standard
errors of the coefficients, the rela�vely low t-sta�s�cs, and
the very high P-values indicate that the cubic form of the line
of best fit does not fit the data
very well at all, and that using this func�on to predict TVC,
AVC, and MC values would be poten�ally unreliable.5
ns/sec6.2#footernote5) So we can conclude that the line of best
fit is perhaps not a cubic func�on, and that the law
of variable propor�ons does not seem to be opera�ng over this
range of output levels.
Thus, we need to find a line of best fit that is a be�er fit to the
data. To find whether only diminishing returns are evident, we
would repeat the regression
analysis to es�mate TVC as a quadra�c func�on of output,
namely:
TVC = α + β1Q + β2Q2 (6-3)
If this quadra�c func�on is an acceptably-reliable line of best
fit, this would imply a linear MC func�on of the form:
MC = β1 + 2β2Q (6-4)
Repea�ng the regression analysis, but this �me entering only
Q and Q2 as independent variables, we find the results shown
in Table 6.11. Again we get a
strong R2 result but it is unreliable since the t-sta�s�cs are

all too low and the P-values are too high. We note that the
0.0956 P-value for the Output (Q)
variable indicates that output is a "marginally significant"
determinant of TVC in this form of the TVC equa�on,
indica�ng that we could be confident at the
90.44% level that TVC is a func�on of output, but the
unreliability of the other independent variable (Q2) renders the
quadra�c es�mate unreliable as well.
quadra�c TVC func�on
Intercept α = 1.3354 1.3060 1.0225 0.3332
Output (Q) β1 = 0.6514 0.3499 1.8614 0.0956
Output squared (Q2) β2 = 0.0011 0.0226 −0.0468 0.9637
Adjusted R2 0.9711
So once more we ask Excel to calculate a regression equa�on,
this �me using a simple bivariate equa�on of the form:
TVC = α + βQ (6-5)
which, if reliable, would mean that
MC = β (6-6)
The regression results for this simple linear TVC func�on are
shown in Table 6.12. At last, this form of the TVC func�on

provides a reliable es�mate of the
coefficients, and we can be highly confident (above the 99%
level, according to the P-values) that TVC is a simple linear
func�on of output. Note that the
explanatory power (adjusted R2) is slightly be�er than it was
for the other two forms of the regression equa�on, and also that
the standard error of
es�mate is smaller than it was for the other two forms of the
TVC func�on that we es�mated.6
ns/sec6.2#footernote6)
linear TVC func�on
Intercept α = 1.3953 0.2505 5.5703 0.0002
Output β = 0.6351 0.0313 20.3020 0.0000
6.2#footernote5
6.2#footernote6
9/23/2019 Print

Adjusted R2 0.9763
So what does all this mean for the ice cream factory? The
regression results in Table 6.12 indicate that the TVC = 1.3953
+ 0.6351 Q. That is, the line of best
fit to the TVC func�on intersects the ver�cal axis at $1,395.30
and slopes upward at $0.6351 per gallon of ice cream
manufactured. Do not be concerned
about the posi�ve intercept value for the TVC curve—our data
had no output values anywhere near zero—the intercept value
simply serves to li� up the
TVC curve so it passes through the data points at the correct
height. We are more concerned with the slope of the TVC curve
in the relevant range of our
data observa�ons, which provides our es�mated value for
marginal costs, and which we have es�mated to be constant
across the range of output values
contained in the data at about 63 cents per gallon. Perhaps
diminishing returns will later set in (at higher output levels)
but they are not evident in the
output range represented by the data we have.
So, the manager of the ice cream factory now knows that she
can reliably es�mate the marginal cost of ice cream at $0.63
per gallon for any volume of
output within the observed data range (i.e., interpola�on) or
for rela�vely small extrapola�ons outside the observed data
range (i.e., less than 5,150 gallons
or more than 10,300 gallons; see Table 6.8). Pricing, make-or-
buy, and take-it-or-leave-it decisions can be made based on this
es�mate of the marginal cost
(which is also the incremental cost of an extra gallon of ice
cream in this case because there were no varia�ons in fixed

costs associated with the varia�ons
in the output levels).
4. In case your math is rusty, we have used the power rule to
find the deriva�ve of the TVC curve, because the independent
variables included variables that were raised to the power 2
(squared) and 3 (cubed). The power rule says that the deriva�ve
of aXb = baXb-1. [return
5. The t-sta�s�cs need to be somewhere close to 2.0 (or larger)
to allow us to be confident (at the 95% level of confidence or
be�er) that the variable is a sta�s�cally significant
determinant of TVC. The P-values indicate the level of
significance for each independent variable; for example, a P-
value of 0.05 would indicate that we could be confident at the
95%
level—the confidence level is given by 1 minus the P-value. As
you can see in Table 6.10, the P-values are way too high to
allow us to hold any reasonable level of confidence in this
par�cular es�mate of the TVC curve. [return
6. Another form of the TVC func�on that we might have
considered is TVC = a + β2Q2, which would imply MC = 2β2Q.
I did run that regression equa�on to find that while the Q2
variable
is sta�s�cally significant above the 99% level of confidence,
the R2 was marginally lower and the standard error of es�mate
was higher, so that the best line of best fit is the linear TVC
equa�on given by equa�on 6-5. [return

6.2#return4
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Summary
In this chapter, we have been concerned with cost es�ma�ons
and calcula�ons for decision-making purposes. We began with
contribu�on analysis, which
required the accurate iden�fica�on of incremental costs and
incremental revenues. We applied contribu�on analysis to three
types of business decision
problems, namely Project A versus Project B decisions, make-
or-buy decisions; and take-it-or-leave-it decisions. To
maximize the firm’s contribu�on to
overheads and profits, the decision maker must consider all
costs and revenues that vary as a result of the decision but only
those that vary as a result of
the decision. When the decision involves a varia�on in the
output level, we start with changes in the costs of the variable
inputs—how is TVC expected to
change as a result of the proposed change in the output level?
Then we must consider whether there will be any changes in

the fixed costs (also known as
overheads) that are a consequence of the decision to be made. If
so, these are also incremental costs to be included in the
contribu�on analysis. These are
the present-period explicit costs of the decision to be made. We
must also consider implicit costs such as the opportunity cost
of resources u�lized, which
may include contribu�on foregone on other sales that cannot be
made as a result of the decision; the cost of replacing owned
items (to be used because of
the decision) in inventory; or the market value of owned items
or resources that might have alterna�vely been sold. Future
implica�ons of the decision
must also be considered. The decision at hand might cause
future costs or future revenues to be incurred or received, such
as loss or gain of future business
due to changing the firm’s rela�onship with exis�ng customers
(known as ill will or goodwill, respec�vely); impacts on the
rela�onship between
management and workers (known as employment rela�ons),
which may affect future labor produc�vity; impacts on the
business rela�onship with suppliers
(i.e., supplier rela�ons); and impacts on future demand due to
changes in the actual or perceived quality of the firm’s product.
In the second half of this chapter, we examined techniques to
es�mate the shape and placement of cost curves, or, more
importantly, to es�mate the values
of cost categories at par�cular output rates. We began with
simple extrapola�on of known data points, a method that is
appropriate for es�ma�ng cost
values that are outside the range of our data observa�ons.
Extrapola�on means to simply extend the values of the
observed data points in the direc�on
they seem to be heading. We noted that extrapola�on becomes
increasingly more unreliable the further one extrapolates

outside the observed data points,
due to the law of variable propor�ons (or diminishing returns)
causing the extrapola�on to be inaccurate. We next considered
interpola�on, or the
es�ma�on of data points between known data points. A linear
interpola�on is the simplest assump�on unless we have data to
suggest a curvilinear
interpola�on is more appropriate, such as diminishing returns
to the variable inputs being evidenced by rising MC
observa�ons. We then considered
gradient analysis, which is simply interpola�on between data
points when we have several data points. This allows us to
more accurately sketch in a line of
best fit to the data observa�ons.
With more data points, regression analysis can be used to find
the sta�s�cal rela�onship between costs and output levels, but
as we saw the choice of
func�onal form and the reliability of the results obtained are of
paramount importance. Because the law of variable propor�ons
is likely to be present in
any produc�on process, it makes sense to start the regression
analysis of observed total cost (TC) or total variable cost (TVC)
data by including squared and
cubic output quan�ty terms in the regression equa�on.
Observa�on of the regression sta�s�cs (the adjusted R2, the
standard error of es�mate, the standard
error of the coefficient, the t-sta�s�cs and the P-values) will
allow us to judge whether the form of the regression equa�on is
sufficiently reliable. Regardless
of the adjusted R2 value, if any of the independent variables
(Q, Q2 and/or Q3) are not significant at the 95% level of
confidence (i.e., do not have t-
sta�s�cs close to or above 2, or P-values less than 0.05) we

cannot be confident that they explain the varia�on in the
dependent variable (TVC) and thus,
they should not be used in the predic�ve equa�on to es�mate
levels of TVC for future levels of output.
If the cubic regression equa�on does not provide a reliable
explana�on of the varia�on in TVC then we would revert to a
quadra�c regression equa�on,
effec�vely assuming that the range of data observed does not
include the ini�al increasing returns to the variable inputs but
only observes diminishing
returns to the variable inputs. Re-running the regression
analysis in the form TVC = α + β1Q + β2Q2 will provide new
es�mates of the regression parameters
(β1 and β2) and new regression sta�s�cs to indicate whether
the quadra�c form offers a more reliable es�mate of the
rela�onship between TVC and the
output level. Again we scru�nize the t-sta�s�cs and/or the P-
values to see which of the independent variables are reliable
determinants at the 95%
confidence level. If all the independent variables included (Q
and Q2) are found to be reliable, we can stop there, assuming
the adjusted R2 is sufficiently
high (say, above 0.7). If either of those independent variables
is not significant at the 95% level, it behooves us to check for
a simple linear rela�onship
between TVC and Q (as we did in this chapter), and if we find
this to be the most reliable explanatory equa�on then this is
the form we should use for
predic�ng future values of TVC for proposed output levels.
Chapters 5 and 6 have been concerned with the cost side of the

firm’s opera�ons, just as Chapters 3 and 4 were concerned with
the demand (or revenue)
side of the firm’s business. As you know, profit is the excess of
revenues over costs. In the following two chapters, we will
u�lize the concepts learned in the
preceding chapters to consider the firm’s pricing decision on the
presump�on that the firm’s objec�ve is to maximize profit.
Ques�ons for Review and Discussion
Click on each ques�on to reveal the answer.
1. List out all the categories of incremental cost that you can
recollect from your reading of this chapter and provide
examples of each one.
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Incremental costs may be present-period explicit costs, future-
period explicit costs, or implicit costs (opportunity costs) in
the present or future periods.
For example, if I were to buy a new car, I might incur
explicit costs of $30,000 today, $1,000 a year for scheduled
maintenance services, and $2,000 a
year for fuel. Implicit costs would include loss of salvage
value (due to deprecia�on), of say $5000 the first year, $2500
the second year, $1250 the third
year, and so on. Another implicit cost is the opportunity cost of
the foregone interest on my $30,000 which could have earned
(say) 5% interest,
compounding annually.
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2. Accountants are concerned with historical costs of resources
and the alloca�on of some of these against current period
revenues and consequently
derive an accoun�ng measure of the firm’s profit. The
economic profit of the firm is likely to differ from this. Please
explain.
Generally Agreed Accoun�ng Principles require that the
historical cost of assets should be charged against revenues in
subsequent years according to a
straight-line or reducing balance deprecia�on method. This is

necessary to allow investors to compare the profitability of
different firms before making
their investment decisions. The economic profit might be more
or less than the accoun�ng profit depending on the difference
between the
deprecia�on charge and the actual salvage value of the asset (in
the market for used equipment, for example). Economic profit
would also consider
other implicit and future costs that the accoun�ng conven�ons
would ignore (but investors should consider).
3. Why does contribu�on analysis ignore the fixed overhead
costs that financial accountants would want to include in the
full cost of the firm’s product?
Contribu�on analysis is concerned with the es�ma�on of
incremental economic costs and revenues to determine the
change in profits due to a specific
decision to be made. It essen�ally asks whether each decision
will increase profit (or not) given that the firm has previously
invested in par�cular assets
and other resources. Accoun�ng profits are found a�er
considering all explicit costs and revenues across all
decisions over a specific produc�on period
and seek to measure the return on the firm's investment for
comparison with other investment opportuni�es.
4. How should future costs and revenues be included in the
calcula�on of the contribu�on of a decision?

Future costs and revenues that are incremental to the decision
should be included in expected net present value (ENPV) terms
in order to weight these
future period and uncertain cash flows appropriately so they can
be added to and compared with present-period certain costs.
5. Under what circumstances would a manager make a decision
that ignores the future cost and revenue implica�ons of that
decision? (There are many
reasons so your thinking may range widely on this one.)
If the chances of a future event were judged to be extremely
small, and thus the ENPV of that event is expected to be
trivial, the manager might
ra�onally ignore that possible event rather than spend funds on
search costs. Also, if the firm is expected to be bankrupt, or
the manager expects to be
re�red, promoted, or working elsewhere in the future, the
manager might ignore the future cost and revenue implica�ons
of a decision, focusing
instead on the near-term consequences.
6. When is extrapola�on a sa�sfactory method of cost
es�ma�on and when is it not?
Extrapola�on is safe enough—i.e., the margin for error is
rela�vely small—if the distance between the known data point
and the es�mated data point
is rela�vely small. The greater this distance, the greater the
probability that the assumed rela�onship between the dependent

variable and the
independent variable(s) will not hold true.
7. Gradient analysis interpolates between known data points.
This interpola�on may be linear or curvilinear. How do we
know when we should fit a
curvilinear line of best fit to the gradient data points?
Linear interpola�on between a series of pairs of data points
would result in a series of straight lines, joining with kinks at
the known data points (if the
overall rela�onship is not exactly linear). Such abrupt changes
in the rela�onship between the dependent variable and the
independent variable(s) are
not at all likely, due to the con�nuous rela�onship (and the
smooth rate of change) we expect between input and output
rela�onships. Thus, rather
than use straight lines that result in kinks, we bend the lines to
depict a smoothly changing rela�onship between the variables.
8. Regression analysis of cost data does not interpolate between
known data points—instead it es�mates a line of best fit to the
observed data points,
allowing for poten�al devia�ons from the line of best fit.
Please explain.
Whereas interpola�on joins the data points, assuming them to
be error free, regression analysis assumes that the data points
may each contain random
error and looks for a single line of best fit across all data

points that minimizes the sum of the squared devia�ons (of the
actual data point from the
line of best fit) for the en�re data set.
9. How do we know that the func�onal form of the regression
equa�on (i.e., a linear, quadra�c, or cubic func�on) is the best
form of the regression
equa�on for predic�ng cost levels at future output levels?
The func�onal form (linear, quadra�c or cubic) that best fits
the data set is the one that has the highest R2 and each
independent variable significant at
the 95% (or higher) confidence level (as evidenced by the t-
sta�s�cs having a value around 2 or greater) or the p-value
being 0.05 or lower, which
should be accompanied by rela�vely low standard errors of
es�mate and standard errors of the coefficients.
10. When the regression equa�on predicts an es�mated value
of TVC at a par�cular level of Q, how do we calculate the 95%
confidence interval around that
predicted value of TVC? (You may need to refer to Chapter 4 to
refresh your memory about the standard error of es�mate.)
The predicted value of TVC for any value of Q is obtained by
inser�ng that value of Q into the regression equa�on (including
Q2 and Q3 if the
regression equa�on is quadra�c or cubic) and solving for TVC.
Given that the sample data probably contains error terms, we
find the 95% confidence

interval by adding and subtrac�ng twice the standard error of
es�mate (to and from the predicted value of TVC) to indicate a
range of values into
which the actual value of TVC is likely to fall 95% of the �me.
Decision Problems
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1. The Muscle-Man Company (MMC) manufactures and
assembles forkli� tractors and supplies parts to other forkli�
manufacturers. It fabricates most of the
component parts but buys the engines, hydraulic systems,
wheels, and �res from suppliers. Demand es�mates indicate
that MMC should increase
produc�on level from 60 units to 70 units monthly. Sufficient
capacity exists in most departments to allow this increase,
except that produc�on of 10 extra
chassis assemblies could be a�ained only by realloca�ng labor
and equipment from fork assembly to chassis assembly. The
fork assembly department

currently produces 90 units monthly and supplies the extra 30
units to other forkli� manufacturers at $1,880 each. This
department could only produce 10
more fork assemblies if the remainder of its labor and
equipment is to be reallocated to build the extra 10 chassis
assemblies, so the sale of fork assemblies
to other manufacturers must be forgone. Alterna�vely, the extra
10 chassis could be purchased from a supplier, and the lowest
quote is from Fenton
Fabricators, for $3,050 per unit. The costs for the Chassis and
Fork departments for a representa�ve month are as follows:
Costs Chassis department Fork department
Produc�on level
Direct materials
Direct labor
Deprecia�on
Allocated overheads (200% of direct labor)
Total
60
46,500
63,000
7,500
126,000
243,500
90
20,700
40,500
5,000
81,000
147,200

a. Should MMC make or buy the 10 addi�onal chassis
assemblies?
b. What qualifica�ons would you add to your decision?
2. The Rakita Racquets Company restrings tennis racquets, a
business with highly seasonal demand. Given this seasonality,
Rakita tries to keep its overheads
low and uses largely casual labor. The owner-manager has kept
a record over the past 12 months, as shown in the following
table. During that �me the costs
of casual labor and of other variable inputs (stringing materials,
energy, and packaging) have remained constant, and because of
the con�nual turnover of
casual labor the produc�vity of labor has also remained more or
less constant.
Month TVC ($) Racquets restrung
June
July
August
September
October
November
December
January
February
March
April
May
35,490
42,470

48,980
52,530
37,480
33,510
31,850
27,860
22,160
19,520
25,960
32,980
4,500
5,575
6,300
6,525
5,325
4,050
2,850
2,450
1,525
925
1,925
3,500
a. Derive average variable cost (AVC) data from the data in this
table.
b. Use gradient analysis to provide an es�mate of 11 data
points that seem to represent the MC curve over this range of
outputs. Plot these data points
and sketch in es�mated MC and AVC curves that seem to best
fit these data points.
c. Suppose that demand is es�mated to move from its present
(May) level of 3,500 units to 5,000 units next month (June).
What is the incremental cost

of mee�ng this demand?
d. Assuming that Rakita’s price to restring a racquet has been
constant at $15 over the past year, and will remain at that level,
what contribu�on to
overheads and profit can it expect in June?
3. The Tico Taco Company has es�mated its weekly TVC
func�on from data collected over the past several months, as
TVC = 435.85 – 1.835Q2 + 3.658Q3
where TVC represents thousands of dollars and Q represents
thousands of boxes of tacos produced per week. The company is
currently producing 2,000
boxes weekly and is considering expanding its output to 2,200
boxes weekly. To do this, it will have to hire another taco
machine operator ($400 per week)
and lease another taco machine ($200 per week).
a. Derive an expression for the marginal cost (MC) curve.
b. Es�mate the incremental costs of the extra 200 boxes per
week.
c. Should Tico Taco expand its output? Why or why not? State
all assump�ons and qualifica�ons which underlie your
recommenda�on.
4. Scruples Footwear Design is a bou�que manufacturer of
designer loafer shoes. The TVC func�on has been es�mated as
TVC = 20Q + 0.00782Q2 and the
demand func�on has been es�mated as Q = 1,346.55 – 27.495P
where Q represents pairs of shoes and P is the price Scruples
receives per pair of shoes. The
coefficients of determina�on for these two regression equa�ons
were 0.9638 and 0.9422, respec�vely. The standard error of
es�mate was 286.22 for the
cost func�on and 30.967 for the demand func�on. Its current

price is $32.50 per pair (wholesale price) and it has been
producing well below full capacity
output levels, and its inventory levels are at the desired level of
100 pairs.
Today the purchasing agent of a high-class chain store has
asked for a special deal for what would be Scruples’ largest
single order ever, namely 400 pairs of
shoes. This represents a large opportunity for Scruples, since
this order would allow its shoes to reach a na�onal market and
would most likely cause
substan�al growth of sales. The purchasing agent has offered
only $28 per pair, however, and says "Take it or leave it!"
a. From the es�mated cost func�on, and given that fixed costs
are $2,000 per week, calculate and plot the per unit cost curves
that Scruples faces.
b. What are the profit-maximizing price and output levels for
Scruples shoes, in the absence of the deal offered by the chain
store?
9/23/2019 Print
c. What is the contribu�on from the chain-store deal, presuming
that this deal is over and above the profit-maximizing price and
output level?
d. What do you recommend Scruples do, with respect to the
proposed price change and the chain-store deal?
e. What assump�ons and qualifica�ons underlie your
recommenda�ons?

5. Over the past 12 months the Four Winds Novelty Company
firm has recorded its Internet sales (equals its monthly output
levels) and its monthly total
variable costs (TVC) for a par�cular novelty item as shown in
the following table. Sales have grown over this period with
rela�vely few shocks due to
uncontrollable weather, poli�cal and spor�ng events. This
online retailer carries no inventories; when it receives a pre-
paid online order from a customer, it
simply buys the product from a supplier and ships it out to the
customer.
Sales = Output TVC ($)
102,813
176,163
196,121
222,885
226,356
296,416
378,446
450,666
579,696
607,082
624,680
636,133
201,953
340,608
377,940
432,863
441,714
629,267
867,596

1,103,807
1,701,125
1,917,861
2,195,352
2,479,195
a. Using regression analysis, find an equa�on that best fits the
data to represent the TVC func�on.
b. At what sales/output level will average variable costs (AVC)
reach a minimum?
c. At what sales/output level will marginal costs (MC) reach a
minimum?
d. Es�mate the value of TVC for sales/output level 250,000
units and calculate the 95% confidence interval for your
es�mate.
Key Terms
Click on each key term to see the defini�on.
cannibalize exis�ng sales
.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU
A situa�on where sales of a product, via a new distribu�on
channel or new retail outlet, will replace or eat into the sales
of the product through the pre-
exis�ng channels and retailers.
contribu�on
The excess of incremental revenues over incremental costs,

rela�ng to a par�cular decision, is called the contribu�on
because it contributes to pay for the
firm’s fixed and unavoidable costs and also to profits if total
revenues are more than total costs.
contribu�on analysis
A process of assessing the incremental costs and incremental
revenues associated with a decision to determine whether the
la�er will exceed the former
and thus whether the decision should in fact be made by a
profit-maximizing firm.
es�ma�on of cost curves
A process of es�ma�ng the values of costs, in par�cular
categories of costs, at various output rates.
extrapola�on
To es�mate a data value (e.g., a cost level) that lies outside
the range of previous data observa�ons by projec�ng, or
extending, the rela�onship observed
within the range of data points to higher or lower level of the
independent variable (e.g., output).
future costs

The costs a firm might expect to incur in one or more future
produc�on periods as a result of a decision made in the present
or prior produc�on periods.
gradient
A measure of the steepness (slope) between two points on a cost
curve, calculated by the ra�o of the rise (increase in cost) over
the run (increase in output
level).
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Managerial Cost Analysis Techniques

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