Chapter
3
Marginal Analysis for
Optimal Decisions
H
I
G
G
S
After reading this chapter, you will, be able to:
3.1 Define several key concepts and terminology.
3.2 Use marginal analysis to find optimal activity levels in unconstrained
S
maximization problems and explain why sunk costs, fixed costs, and average
costs are irrelevant for decision
H making.
3.3 Employ marginal analysis to A
find the optimal levels of two or more activities in
constrained maximization and minimization problems.
N
I
C
aking optimal decisions about the levels of various business activities
is an essential skill forQall managers, one that requires managers to
analyze benefits and costs to make the best possible decision under
U
a given set of circumstances. When Ford Motor Company began producing a
A
redesigned and reengineered Explorer,
Ford’s CEO decided that the first 5,000
M
units rolling off assembly lines would not be delivered immediately to Ford
dealer showrooms, even as potential buyers waited anxiously to get the new
model. Instead, all of the new 1
vehicles were parked in lots outside factories
while quality control engineers1examined 100 of them for defects in assembly
and workmanship. The intense, 24-hour-a-day inspection process continued
for three months, delaying the 0launch of the highly profitable new Explorer.
While no one could blame Ford’s
5 executives for wanting to minimize costly
product recalls, many auto industry analysts, car buyers, and owners of Ford
T Ford was undertaking too much quality
dealerships nonetheless thought
control. Ford’s CEO assured his
S critics that choosing to add three months
of quality control measures was optimal, or best, under the circumstances.
Apparently, he believed the benefit of engaging in three months of quality
control effort (the savings attributable to preventing vehicle recalls) outweighed
88
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 89
the cost of the additional quality control measures (the loss and delay of profit
as 5,000 new Explorers spent three months in factory parking lots).
As you can see, Ford’s CEO, weighing costs and benefits, made a critical
decision that three months, rather than two months or four months, and a sample
of 100 vehicles, rather than 50 vehicles or 300 vehicles, were the optimal or best
levels of these two quality control decisions for launching the redesigned E
xplorer.
We don’t have enough information about Ford’s costs and benefits of quality
control to tell you whether the CEO succeeded in making the optimal decision for
Ford. We can, however, H
tell you that one year later Consumer Reports still ranked
the overall reliability of Ford automobiles dead last and Ford had a new CEO.
I effort would have been optimal.
Perhaps more quality control
A manager’s decisionG
is optimal if it leads to the best outcome under a given
set of circumstances. Finding the best solution involves applying the fundamental
G theory developed in this chapter. These analytical
principles of optimization
principles, which economists
refer to as “marginal analysis,” turn out to be
S
nothing more than a formal presentation of commonsense ideas you already apply,
, it, in your everyday life. Marginal analysis supplies
probably without knowing
the fundamental logic for making optimal decisions. Managers benefit from
understanding marginal analysis because it enables them to make better decisions
S
while avoiding some rather common errors in business decision making.
The idea behind marginal
H analysis is this: When a manager contemplates
whether a particular business activity needs adjusting, either more or less, to reach
A
the best value, the manager needs to estimate how changing the activity will affect
both the benefits the firmNreceives from engaging in the activity and the costs the
firm incurs from engaging
I in the activity. If changing the activity level causes
benefits to rise by more than costs rise, or, alternatively, costs to fall by more than
benefits fall, then the netCbenefit the firm receives from the activity will rise. The
manager should continueQadjusting the activity level until no further net gains are
possible, which means the activity has reached its optimal value or level.
U 1, managers face two general types of decisions:
As mentioned in Chapter
routine business practice
A or tactical decisions and strategic decisions that can
alter the firm’s competitive environment. Marginal analysis builds the essential
foundation for making everyday business decisions, such as choosing the
number of workers to hire,
1 the amount of output to produce, the amount to
spend on advertising, and so on. While strategic decision making relies heavily
1
on concepts from game theory,
strategic analysis nevertheless depends indirectly
on optimal decision making
as
the
means for computing or forecasting the payoffs
0
under various strategy options.
5
T
3.1 CONCEPTS AND TERMINOLOGY
Optimizing behavior on S
the part of a decision maker involves trying to maximize
objective function
The function the decision
maker seeks to maximize
or minimize.
tho21901_ch03_088-120.indd 89
or minimize an objective function. For a manager of a firm, the objective function
is usually profit, which is to be maximized. For a consumer, the objective function
is the satisfaction derived from consumption of goods, which is to be maximized.
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90 C H A P T E R 3 Marginal Analysis for Optimal Decisions
maximization problem
An optimization problem
that involves maximizing
the objective function.
minimization problem
An optimization problem
that involves minimizing
the objective function.
activities or choice
variables
Variables that determine
the value of the objective
function.
discrete choice
variables
A choice variable that
can take only specific
integer values.
continuous choice
variables
A choice variable that
can take any value
between two end points.
unconstrained
optimization
An optimization problem
in which the decision
maker can choose the
level of activity from
an unrestricted set of
values.
constrained
optimization
An optimization problem
in which the decision
maker chooses values
for the choice variables
from a restricted set of
values.
tho21901_ch03_088-120.indd 90
For a city manager seeking to provide adequate law enforcement services, the
objective function might be cost, which is to be minimized. For the manager of the
marketing division of a large corporation, the objective function is usually sales,
which are to be maximized. The objective function measures whatever it is that the
particular decision maker wishes to either maximize or minimize.
If the decision maker seeks to maximize an objective function, the optimization
problem is called a maximization problem. Alternatively, if the objective function
is to be minimized, the optimization problem is called a minimization problem.
As a general rule, when the objective
H function measures a benefit, the decision
maker seeks to maximize this benefit and is solving a maximization problem.
I a cost, the decision maker seeks to minimize
When the objective function measures
this cost and is solving a minimization
G problem.
The value of the objective function is determined by the level of one or more
G example, the value of profit depends on the
activities or choice variables. For
number of units of output produced
S and sold. The production of units of the good
is the activity that determines the value of the objective function, which in this
, controls the value of the objective function by
case is profit. The decision maker
choosing the levels of the activities or choice variables.
The choice variables in the optimization problems discussed in this text will
S
at times vary discretely and at other times vary continuously. A discrete choice
variable can take on only specified
H integer values, such as 1, 2, 3, . . . , or 10, 20,
30, . . . Examples of discrete choice variables arise when benefit and cost data
A
are presented in a table, where each row represents one value of the choice
N of discrete choice variables will be presented
variable. In this text, all examples
in tables. A continuous choice variable
can take on any value between two end
I
points. For example, a continuous variable that can vary between 0 and 10 can
C or any one of the infinite number of values
take on the value 2, 2.345, 7.9, 8.999,
between the two limits. Examples
Q of continuous choice variables are usually
presented graphically but are sometimes shown by equations. As it turns out, the
U in the discrete and continuous cases.
optimization rules differ only slightly
In addition to being categorized
A as either maximization or minimization
problems, optimization problems are also categorized according to whether the
decision maker can choose the values of the choice variables in the objective
function from an unconstrained
1 or constrained set of values. Unconstrained
optimization problems occur when a decision maker can choose any level of
1 to maximize the objective function. In this
activity he or she wishes in order
chapter, we show how to solve only
0 unconstrained maximization problems since
all the unconstrained decision problems we address in this text are maximization
5
problems. Constrained optimization problems involve choosing the levels of
T or minimize the objective function subject
two or more activities that maximize
to an additional requirement or constraint that restricts the values of A and B
S
that can be chosen. An example of such a constraint arises when the total cost
of the chosen activity levels must equal a specified constraint on cost. In this
text, we examine both constrained maximization and constrained minimization problems.
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 91
marginal analysis
Analytical technique for
solving optimization
problems that involves
changing values of
choice variables by small
amounts to see if the
objective function can be
further improved.
Now try Technical
Problem 1.
As we show later in this chapter, the constrained maximization and the
constrained minimization problems have one simple rule for the solution. Therefore,
you will only have one rule to learn for all constrained optimization problems.
Even though there are a huge number of possible maximizing or minimizing
decisions, you will see that all optimization problems can be solved using the
single analytical technique, mentioned at the beginning of this chapter: marginal
analysis. Marginal analysis involves changing the value(s) of the choice
variable(s) by a small amount to see if the objective function can be further
increased (in the case ofHmaximization problems) or further decreased (in the
case of minimization problems). If so, the manager continues to make
incremental adjustmentsI in the choice variables until no further improvements
are possible. Marginal
G analysis leads to two simple rules for solving
optimization problems, one for unconstrained decisions and one for constrained
decisions. We turn first G
to the unconstrained decision.
3.2 UNCONSTRAINED MAXIMIZATION
net benefit
The objective function
to be maximized:
NB 5 TB 2 TC.
S
,
Any activity that decision makers might wish to undertake will generate both
benefits and costs. Consequently, decision makers will want to choose the level
S
of activity to obtain the maximum
possible net benefit from the activity, where the
net benefit (NB) associated
with
a
specific
amount of activity (A) is the difference
H
between total benefit (TB) and total cost (TC) for the activity
A
NB 5 TB 2 TC
N
Net benefit, then, serves as the objective function to be maximized, and the amount
of activity, A, representsI the choice variable. Furthermore, decision makers can
choose any level of activity
C they wish, from zero to infinity, in either discrete or
continuous units. Thus, we are studying unconstrained maximization in this section.
Q
The Optimal Level of Activity
(A*)
U
We begin the analysis ofAunconstrained maximization with a rather typical set
of total benefit and total cost curves for some activity, A, as shown in Panel A of
Figure 3.1. Total benefit increases with higher levels of activity up to 1,000 units of
activity (point G); then total
1 benefit falls beyond this point. Total cost begins at a
value of zero and rises continuously as activity increases. These “typical” curves
1 rules for finding the best solution to all such unconallow us to derive general
strained problems, even0though specific problems encountered in later chapters
sometimes involve benefit and cost curves with shapes that differ somewhat from
5
those shown in Panel A. For example, total benefit curves can be linear. Total cost
curves can be linear or even
T S-shaped. And, as you will see in later chapters, total
cost curves can include fixed costs when they take positive values at zero units of
S
activity. In all of these variations, however, the rules for making the best decisions
do not change. By learning how to solve the optimization problem as set forth in
Figure 3.1, you will be prepared to solve all variations of these problems that come
later in the text.
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92 C H A P T E R 3 Marginal Analysis for Optimal Decisions
F I G U R E 3.1
The Optimal Level of Activity
TC
Total benefit and total cost (dollars)
4,000
G
TB
F
HD
I
G
D9
G
S
NB* 5 $1,225
,
3,000
B
2,310
2,000
C
1,085
1,000
B9
C9
0
200
350 5 A*
Net benefit (dollars)
Panel A — Total benefit and total cost curves
1,225
1,000
Panel B — Net benefit curve
1,000
c99
600
0
tho21901_ch03_088-120.indd 92
M
S
H
A
N600 700
Level of
I activity (A)
C
Q
U
A
200
350 5 A*
1d 99
f 99
1
0600
Level of activity (A)
5
T
S
1,000
NB
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 93
I L L U S T R AT I O N 3 . 1
Is Cost–Benefit Analysis Really Useful?
We have extolled the usefulness of marginal analysis
in optimal decision making—often referred to as cost–
benefit analysis—in business decision making as well as
decision making in everyday life. This process involves
weighing the marginal benefits and marginal costs of
an activity while ignoring all previously incurredH
or
sunk costs. The principal rule is to increase the levelI of
an activity if marginal benefits exceed marginal costs
G
and decrease the level if marginal costs exceed marginal
benefits. This simple rule, however, flies in the faceG
of
many honored traditional principles such as “Never
S
give up” or “Anything worth doing is worth doing
well” or “Waste not, want not.” So you might wonder
,
if cost–benefit analysis is as useful as we have said it is.
It is, at least according to an article in The Wall
Street Journal entitled “Economic Perspective Produces
S
Steady Yields.” In this article, a University of Michigan
research team concludes, “Cost–benefit analysis pays
H
off in everyday living.” This team quizzed some of
A
the university’s seniors and faculty members on such
questions as how often they walk out on a bad movie,
N
refuse to finish a bad novel, start over on a weak term
paper, or abandon a research project that no longer
I
looks promising. They believe that people who cut
optimal level of
activity
The level of activity that
maximizes net benefit (A*).
their losses this way are following sound economic
rules: calculating the net benefits of alternative courses
of action, writing off past costs that can’t be recovered,
and weighing the opportunity to use future time and
effort more profitably elsewhere.a
The findings: Among faculty members, those
who use cost–benefit reasoning in this fashion had
higher salaries relative to their age and departments.
Economists were more likely to apply the approach
than professors of humanities or biology. Among
students, those who have learned to use cost–benefit
analysis frequently are apt to have far better grades
than their SAT scores would have predicted. The
more economics courses the students had taken, the
more likely they were to apply cost–benefit analysis
outside the classroom. The director of the University
of Michigan study did concede that for many
Americans cost–benefit rules often appear to conflict
with traditional principles such as those we previously mentioned. Notwithstanding these probable
conflicts, the study provides evidence that decision
makers can indeed prosper by following the logic of
marginal analysis and cost–benefit analysis.
“Economic Perspective Produces Steady Yield,” The Wall
Street Journal, March 31, 1992.
a
C
Q
The level of activity that maximizes net benefit is called the optimal level of
U from other levels of activity with an asterisk: A*. In
activity, which we distinguish
Panel A of Figure 3.1, netAbenefit at any particular level of activity is measured by
the vertical distance between the total benefit and total cost curves. At 200 units
of activity, for example, net benefit equals the length of line segment CC9, which
happens to be $1,000 as shown
in Panel B at point c”. Panel B of Figure 3.1 shows
1
the net benefit curve associated with the TB and TC curves in Panel A. As you can
1
see from examining the net benefit curve in Panel B, the optimal level of activity,
0reaches its maximum value. At 350 units in Panel A, the
A*, is 350 units, where NB
vertical distance between TB and TC is maximized, and this maximum distance is
5
$1, 225 (5 NB*).1
1
T
S the optimal activity level was 700 units since two curves in
You might, at first, have thought
Panel A of Figure 3.1 intersect at point F, and this situation frequently identifies “correct” answers in
economics. But, as you can see in Panel B, choosing 700 units of activity creates no more net benefit
than choosing to do nothing at all (i.e., choosing A 5 0) because total benefit equals total cost at both
zero and 700 units of activity.
tho21901_ch03_088-120.indd 93
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94 C H A P T E R 3 Marginal Analysis for Optimal Decisions
Now try Technical
Problem 2.
Two important observations can now be made about A* in unconstrained maximization problems. First, the optimal level of activity does not generally result
in maximization of total benefits. In Panel A of Figure 3.1, you can see that total
benefit is still rising at the optimal point B. As we will demonstrate later in this
book, for one of the most important applications of this technique, profit maximization, the optimal level of production occurs at a point where revenues are
not maximized. This outcome can confuse managers, especially ones who believe
any decision that increases revenue should be undertaken. We will have much
more to say about this later in the
Htext. Second, the optimal level of activity in an
unconstrained maximization problem does not result in minimization of total cost.
In Panel A, you can easily verify Ithat total cost isn’t minimized at A* but rather at
zero units of activity.
G
Finding A* in Figure 3.1 seems easy enough. A decision maker starts with the
G in Panel A and subtracts the total cost curve
total benefit and total cost curves
from the total benefit curve to construct
the net benefit curve in Panel B. Then,
S
the decision maker chooses the value of A corresponding to the peak of the net
,
benefit curve. You might reasonably
wonder why we are going to develop an
alternative method, marginal analysis, for making optimal decisions. Perhaps
the most important reason for learning how to use marginal analysis is that
S
economists regard marginal analysis as “the central organizing principle of
2
economic theory.” The graphical
H derivation of net benefit shown in Figure 3.1
serves only to define and describe the optimal level of activity; it does not explain
A
why net benefit rises, falls, or reaches its peak. Marginal analysis, by focusing
N
only on the changes in total benefits
and total costs, provides a simple and
complete explanation of the underlying
forces causing net benefit to change.
I
Understanding precisely what causes net benefit to improve makes it possible
C when an activity needs to be increased,
to develop simple rules for deciding
decreased, or left at its current level.
Q
We are also going to show that using marginal analysis to make optimal deciU irrelevant information about such things as
sions ensures that you will not consider
fixed costs, sunk costs, or averageA
costs in the decision-making process. As you will
see shortly, decision makers using marginal analysis can reach the optimal activity
level using only information about the benefits and costs at the margin. For this
reason, marginal analysis requires
1 less information than would be needed to construct TB, TC, and NB curves for all possible activity levels, as shown in Figure 3.1.
1
There is no need to gather and process
information for levels of activity that will
never be chosen on the way to reaching
A*. For example, if the decision maker is
0
currently at 199 units of activity in Figure 3.1, information about benefits and costs
5
is only needed for activity levels from 200 to 351 units. The optimal level of activity
T
can be found without any information
about benefits or costs below 200 units or
above 351 units.
S
See Robert B. Ekelund, Jr., and Robert F. Hébert, A History of Economic Theory and Method, 4th ed.
(New York: McGraw-Hill, 1997), p. 264.
2
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 95
Marginal Benefit and Marginal Cost
marginal benefit (MB)
The change in total
benefit caused by an incremental change in the
level of an activity.
marginal cost (MC)
The change in total
cost caused by an
incremental change in
the level of an activity.
In order to understand and use marginal analysis, you must understand the two
key components of this methodology: marginal benefit and marginal cost. Marginal
benefit (MB) is the change in total benefit caused by an incremental change in
the level of an activity. Similarly, marginal cost (MC) is the change in total cost
caused by an incremental change in activity. Dictionaries typically define “incremental” to mean “a small positive or negative change in a variable.” You can think
of “small” or “incremental” changes in activity to be any change that is small relaH In most applications it is convenient to interpret an
tive to the total level of activity.
incremental change as a Ione-unit change. In some decisions, however, it may be
impractical or even impossible to make changes as small as one-unit. This causes
no problem for applyingG
marginal analysis as long as the activity can be adjusted
in relatively small increments.
G We should also mention that “small” refers only to
the change in activity level; “small” doesn’t apply to the resulting changes in total
S can be any size.
benefit or total cost, which
Marginal benefit and marginal
cost can be expressed mathematically as
,
Change in total benefit ____
_____________________
MB 5
5 DTB
DA
Change in activity
S
H
Change in total cost _____
DTC
A 5 __________________
MC
5
DA
Change in activity
N
where the symbol “D” means “the change in” and A denotes the level of an activI
ity. Since “marginal” variables
measure rates of change in corresponding “total”
variables, marginal benefit
and
marginal cost are also slopes of total benefit and
C
total cost curves, respectively.
Q 3.2 show how the total curves in Figure 3.1 are related
The two panels in Figure
to their respective marginal
U curves. Panel A in Figure 3.2 illustrates the procedure
for measuring slopes of total curves at various points or levels of activity. Recall
A classes or a pre-calculus course in college that the
from your high school math
and
slope of a curve at any particular point can be measured by first constructing a
line tangent to the curve at the point of measure and then computing the slope
1
of this tangent line by dividing
the “rise” by the “run” of the tangent line.3 Consider, for example, the slope
1 of TB at point C in Panel A. The tangent line at point
C rises by 640 units (dollars) over a 100-unit run, and total benefit’s slope at point
0
C is $6.40 (5 $640/100). Thus the marginal benefit of the 200th unit of activity is
5
When a line is tangent toT
a curve, it touches the curve at only one point. For smooth, continuous
curves, only one line can be drawn tangent to the curve at a single point. Consequently, the slope of
a curve at a point is unique andSequal to the slope of the tangent line at that point. The algebraic sign
3
of the slope indicates whether the variables on the vertical and horizontal axes are directly related
(a positive algebraic slope) or inversely related (a negative algebraic slope). For a concise review
of measuring and interpreting slopes of curves, see “Review of Fundamental Mathematics” in the
Student Workbook that accompanies this text.
tho21901_ch03_088-120.indd 95
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96 C H A P T E R 3 Marginal Analysis for Optimal Decisions
F I G U R E 3.2
Relating Marginals to Totals
TC
4,000
Total benefit and total cost (dollars)
3,000
100
520
B
100
2,000
640
1,000
C
B9
C9
100
340
200
350 5 A*
Marginal benefit and marginal cost (dollars)
Panel A — Measuring slopes along TB and TC
8
c
(200, $6.40)
6
5.20
b
4
c 9 (200, $3.40)
2
0
200
820
520
100
0
HD
ID 9
G
G100
S
,
TB
F
100
320
G
350 5 A*
S
H
A
600
800
N
Level of activity (A)
I
C
MC (5 slope of TC )
Q
U
Ad 9 (600, $8.20)
1
1 d (600, $3.20)
0
5
T
800
S600
1,000
MB (5 slope of TB)
g
1,000
Level of activity (A)
Panel B — Marginals give slopes of totals
tho21901_ch03_088-120.indd 96
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 97
$6.40, which means adding the 200th unit of activity (going from 199 to 200 units)
causes total benefit to rise by $6.40.4
You should understand that the value of marginal benefit also tells you that
subtracting the 200th unit (going from 200 to 199 units) causes total benefit to fall
by $6.40. Because the slope of TB at point C is $6.40 per unit change in activity,
marginal benefit at point c in Panel B is $6.40. You can verify that the same relation
holds for the rest of the points shown on total benefit (B, D, and G), as well as
for the points shown on total cost (C9, B9, and D9). We summarize this important
discussion in a principle:H
Principle Marginal benefit I(marginal cost) is the change in total benefit (total cost) per unit change in
the level of activity. The marginal benefit (marginal cost) of a particular unit of activity can be measured by
G
the slope of the line tangent to the total benefit (total cost) curve at that point of activity.
G
At this point, you might
S be concerned that constructing tangent lines and
measuring slopes of the tangent lines presents a tedious and imprecise method of
finding marginal benefit,and marginal cost curves. As you will see, the marginal
benefit and marginal cost curves used in later chapters are obtained without
drawing tangent lines. It is quite useful, nonetheless, for you to be able to visualize
a series of tangent lines S
along total benefit and total cost curves in order to see
why marginal benefit and
Hmarginal cost curves, respectively, are rising, falling, or
even flat. Even if you don’t know the numerical values of the slopes at points C,
B, D, and F in Figure 3.2,Ayou can still determine that marginal benefit in Panel B
must slope downward because,
as you can tell by looking, the tangent lines along
N
TB get flatter (slopes get smaller) as the activity increases. Marginal cost, on the
I
other hand, must be increasing in Panel B because, as you can tell by looking, its
tangent lines get steeper C
(slope is getting larger) as the activity increases.
Q
U
As we stated earlier, the method of marginal analysis involves comparing
A cost to see if net benefit can be increased by making
marginal benefit and marginal
Finding Optimal Activity Levels with Marginal Analysis
an incremental change in activity level. We can now demonstrate exactly how
this works using the marginal benefit and marginal cost curves in Panel B of
Figure 3.2. Let’s suppose1the decision maker is currently undertaking 199 units of
activity in Panel B and wants
1 to decide whether an incremental change in activity
can cause net benefit to rise. Adding the 200th unit of activity will cause both total
benefit and total cost to 0
rise. As you can tell from points c and c9 in Panel B, TB
increases by more than TC
5 increases ($6.40 is a larger increase than $3.40). Consequently, increasing activity from 199 to 200 units will cause net benefit to rise by
4
T
S values for marginal benefit and marginal cost, remember that
When interpreting numerical
the values refer to a particular unit of activity. In this example, marginal benefit equals $6.40 for the
200th unit. Strictly speaking, it is incorrect, and sometimes confusing, to say “marginal cost is $6.40
for 200 units.” At 200 units of activity, the marginal benefit is $6.40 for the last unit of activity undertaken
(i.e., the 200th unit).
tho21901_ch03_088-120.indd 97
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98 C H A P T E R 3 Marginal Analysis for Optimal Decisions
F I G U R E 3.3
Using Marginal Analysis
to Find A*
Net benefit (dollars)
MB 5 MC
MB . MC
100
300
c0
0
200
MB , MC
H
I
G
G
350 5S
A*
,
M
100
d0
2500
600
Level of activity (A)
800
NB
1,000
S
$3 (5 $6.40 2 $3.40). Notice in Figure
H 3.3 that, at 200 units of activity (point c”), net
benefit is rising at a rate of $3 (5 $300/100) per unit increase in activity, as it must
A
since MB equals $6.40 and MC equals $3.40.
After increasing the activity toN200 units, the decision maker then reevaluates
benefits and costs at the margin to see whether another incremental increase in
I
activity is warranted. In this situation, for the 201st unit of activity, the decision
C MB is greater than MC, which indicates the
maker once again discovers that
activity should be further increased.
This incremental adjustment process
Q
continues until marginal benefit and marginal cost are exactly equal at point M
Uthe decision maker can make a single adjust(A* 5 350). As a practical matter,
ment to reach equilibrium, jumping
A from 199 units to 350 units in one adjustment
of A, or make a series of smaller adjustments until MB equals MC at 350 units of
activity. In any case, the number of adjustments made to reach A* does not, of
course, alter the optimal decision1or the value of net benefit at its maximum point.
Now let’s start from a position of too much activity instead of beginning with too
1 maker begins at 600 units of activity, which you
little activity. Suppose the decision
can tell is too much activity by looking
0 at the NB curve (in either Figure 3.1 or 3.3).
Subtracting the 600th unit of activity will cause both total benefit and total cost to
fall. As you can tell from points d5and d9 in Panel B of Figure 3.2, TC decreases by
more than TB decreases ($8.20 is a T
larger decrease than $3.20). C
onsequently, reducing
activity from 600 to 599 units will cause net benefit to rise by $5 (5 $8.20 2 $3.20). You
S
can now verify in Figure 3.3 that at 600 units of activity (point d”) net benefit is rising
at a rate of $5 per unit decrease in activity. Because MC is still greater than MB at 599
units, the decision maker would continue reducing activity until MB exactly equals
MC at 350 units (point M).
tho21901_ch03_088-120.indd 98
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 99
T A B L E 3.1
Marginal Analysis Decision
Rules
Increase activity
Decrease activity
MB . MC
NB rises
NB falls
MB , MC
NB falls
NB rises
Table 3.1 summarizes the logic of marginal analysis by presenting the relation
between marginal benefit, marginal cost, and net benefit set forth in the previous
H
discussion and shown in Figure 3.3. We now summarize in the following principle
the logic of marginal analysis
I for unconstrained maximization problems in which
the choice variable is continuous:
G
Principle If, at a given levelGof activity, a small increase or decrease in activity causes net benefit to
increase, then this level of the activity is not optimal. The activity must then be increased (if marginal benefit
S
exceeds marginal cost) or decreased (if marginal cost exceeds marginal benefit) to reach the highest net
benefit. The optimal level of the,activity—the level that maximizes net benefit—is attained when no further
increases in net benefit are possible for any changes in the activity, which occurs at the activity level for
which marginal benefit equals marginal cost: MB 5 MC.
S
Now tryTechnical
Problems 3–5.
While the preceding discussion of unconstrained optimization has allowed
H variable to influence net benefit, sometimes managers
only one activity or choice
will need to choose the A
levels of two or more variables. As it turns out, when
decision makers wish to maximize the net benefit from several activities, precisely
N
the same principle applies: The firm maximizes net benefit when the marginal
benefit from each activity
I equals the marginal cost of that activity. The problem
is somewhat more complicated mathematically because the manager will have to
C
equate marginal benefits and marginal costs for all of the activities simultaneously.
Q maker chooses the levels of two activities A and B to
For example, if the decision
maximize net benefit, then
U the values for A and B must satisfy two conditions at
once: MBA 5 MCA and MBB 5 MCB. As it happens in this text, all the unconstrained
A
maximization problems involve
just one choice variable or activity.
Maximization with Discrete Choice Variables
1
In the preceding analysis, the choice variable or activity level was a continuous
variable. When a choice1variable can vary only discretely, the logic of marginal
analysis applies in exactly
0 the same manner as when the choice variable is continuous. However, when choice variables are discrete, decision makers will not
5
usually be able to adjust the level of activity to the point where marginal benefit
exactly equals marginalTcost. To make optimal decisions for discrete choice
variables, decision makers must increase activity until the last level of activity is
S
reached for which marginal benefit exceeds marginal cost. We can explain this
rule for discrete choice variables by referring to Table 3.2, which shows a schedule
of total benefits and total costs for various levels of some activity, A, expressed in
integers between 0 and 8.
tho21901_ch03_088-120.indd 99
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100 C H A P T E R 3 Marginal Analysis for Optimal Decisions
T A B L E 3.2
Optimization with a
Discrete Choice Variable
(1)
Level of
activity
(A)
0
1
2
3
4
5
(2)
Total benefit
of activity
(TB)
$ 0
16
30
40
48
54
6
7
8
58
61
63
(3)
Total cost
of activity
(TC )
$ 0
2
6
11
20
30
H
I45
61
G
80
G
S
,
(4)
Net benefit
of activity
(NB)
$ 0
14
24
29
28
24
(5)
Marginal
benefit
(MB)
—
16
14
10
8
6
(6)
Marginal
cost
(MC )
—
2
4
5
9
10
13
0
217
4
3
2
15
16
19
Let’s suppose the decision maker is currently doing none of the activities
and wants to decide whether to undertake the first unit of activity. The marginal benefit of the first unit of S
the activity is $16, and the marginal cost is $2.
Undertaking the first unit of activity adds $16 to total benefit and only $2 to total
H
cost, so net benefit rises by $14 (from $0 to $14). The decision maker would choose
A to gain a higher net benefit. Applying this
to undertake the first unit of activity
reasoning to the second and third
N units of activity leads again to a decision to
undertake more activity. Beyond the third unit, however, marginal cost exceeds
marginal benefit for additional Iunits of activity, so no further increase beyond
three units of activity will add toCnet benefit. As you can see, the optimal level of
the activity is three units because the net benefit associated with three units ($29)
is higher than for any other levelQ
of activity. These results are summarized in the
following principle:
U
A an unconstrained maximization problem and must choose
Principle When a decision maker faces
among discrete levels of an activity, the activity should be increased if MB . MC and decreased if
MB , MC. The optimal level of activity is reached—net benefit is maximized—when the level of activity is
1 marginal cost.
the last level for which marginal benefit exceeds
1
Now try Technical
Problem 6.
tho21901_ch03_088-120.indd 100
Before moving ahead, we would like to point out that this principle cannot
0 the activity level where MB and MC are as
be interpreted to mean “choose
close to equal as possible.” To see
5 why this interpretation can lead to the wrong
decision, consider the fourth unit of activity in Table 3.2. At four units of activity,
T with MC (5 $9) than at the optimal level of
MB (5 $8) is much closer to equality
activity, where MB (5 $10) is $5 larger
S than MC (5 $5). Now you can see why the
rule for discrete choice variables cannot be interpreted to mean “get MB as close
to MC as possible.”
8/11/15 4:36 PM
C H A P T E R 3 Marginal Analysis for Optimal Decisions 101
Sunk Costs, Fixed Costs, and Average Costs Are Irrelevant
sunk costs
Costs that have previously been paid and cannot
be recovered.
fixed costs
Costs are constant and
must be paid no matter
what level of the activity
is chosen.
In our discussion of optimization problems, we never mentioned sunk costs or
fixed costs. Sunk costs are costs that have previously been paid and cannot be
recovered. Fixed costs are costs that are constant and must be paid no matter what
level of an activity is chosen. Such costs are totally irrelevant in decision making.
They either have already been paid and cannot be recovered, as in the case of sunk
costs, or must be paid no matter what a manager or any other decision maker
decides to do, as in the case of fixed costs. In either case, the only relevant decision
variables—marginal costHand marginal revenue—are in no way affected by the
levels of either sunk or fixed
I costs.
Suppose you head your company’s advertising department and you have just
G
paid $2 million to an advertising
firm for developing and producing a 30-second
television ad, which youG
plan to air next quarter on broadcast television networks
nationwide. The $2 million one-time payment gives your company full ownership
of the 30-second ad, and S
your company can run the ad as many times as it wishes
without making any further
, payments to the advertising firm for its use. Under
these circumstances, the $2 million payment is a sunk cost because it has already
been paid and cannot be recovered, even if your firm decides not to use the ad
after all.
S
To decide how many times to run the ad next quarter, you call a meeting of
H
your company’s advertising department. At the meeting, the company’s media
A
buyer informs you that 30-second
television spots during American Idol will cost
$250,000 per spot. The marketing research experts at the meeting predict that the
N
24th time the ad runs it will generate $270,000 of additional sales, while running
I sales by $210,000. Using the logic of marginal analysis,
it a 25th time will increase
the marketing team decides
C running the new ad 24 times next quarter is optimal
because the 24th showing of the ad is the last showing for which the marginal
Q cost of showing the ad:
benefit exceeds the marginal
U
MB 5 $270,000 . 250,000 5 MC
It would be a mistake to A
go beyond 24 showings because the 25th showing would
decrease net benefit; the change in net benefit would be 2$40,000 (5 $210,000 2
$250,000).
1
Two days after this meeting,
you learn about a serious accounting error: Your
company actually paid 1
$3 million to the advertising firm for developing and
producing your company’s new television ad, not $2 million as originally re0
ported. As you consider how to handle this new information, you realize that you
don’t need to call another
5 meeting of the marketing department to reconsider its
decision about running the ad 24 times next quarter. Because the amount paid to
T
the advertising firm is a sunk cost, it doesn’t affect either the marginal benefit or
S the ad one more time. The optimal number of times
the marginal cost of running
to run the ad is 24 times no matter how much the company paid in the past to
obtain the ad.
tho21901_ch03_088-120.indd 101
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102 C H A P T E R 3 Marginal Analysis for Optimal Decisions
average (unit) cost
Cost per unit of activity
computed by dividing
total cost by the number
of units of activity.
Now try Technical
Problem 7.
Converting this example to a fixed cost, suppose that two days after your
meeting you find out that, instead of making a sunk payment to buy the ad, your
company instead decided to sign a 30-month contract leasing the rights to use the
television ad for a monthly lease payment of $10,000. This amount is a fixed payment in each of the 30 months and must be paid no matter how many times your
company decides to run the ad, even if it chooses never to run the ad. Do you need
to call another meeting of the marketing department to recalculate the optimal
number of times to run the ad during American Idol? As before, no new decision
needs to be made. Because the fixed
H monthly loan payment does not change the
predicted gain in sales (MB) or the extra cost of running the ad (MC), the optimal
I
number of times to run the ad remains
24 times.
While you should now understand
that
things over which you have no control
G
should not affect decisions, some economic experiments do, surprisingly, find that
many people fail to ignore fixedG
or sunk costs when making decisions. They say
things such as, “I’ve already got S
so much invested in this project, I have to go on
with it.” As you are aware, they should weigh the costs and benefits of going on
,
before doing so. Then, if the benefits
are greater than the additional costs, they
should go on; if the additional costs are greater than the benefits, they should not
go on. As Illustration 3.1 shows, failing to ignore fixed or sunk costs is a bad policy
S
even in everyday decision making.
Another type of cost that should
H be ignored in finding the optimal level of
an activity is the average or unit cost of the activity. Average (or unit) cost is the
A
cost per unit of activity, computed by dividing total cost by the number of units
N decisions, decision makers should not be
of activity. In order to make optimal
concerned about whether their Idecision will push average costs up or down.
The reason for ignoring average cost is quite simple: The impact on net benefit of
Cactivity depends only on marginal benefit and
making an incremental change in
marginal cost (DNB 5 MB 2 MC),
Qnot on average benefit or average cost. In other
words, optimal decisions are made at the margin, not “on the average.”
U the decision in Table 3.2 once again. The
To illustrate this point, consider
average cost of two units of activity
A is $3 (5 $6y2) and average cost for three units
of activity is $3.67 (5 $11y3). Recall from our earlier discussion, the decision to
undertake the third unit of activity is made because the marginal benefit exceeds
the marginal cost ($10 . $5), and1net benefit rises. It is completely irrelevant that
the average cost of three units of activity is higher than the average cost of two
1
units of activity. Alternatively, a decision
maker should not decrease activity from
three units to two units just to achieve
a
reduction
in average cost from $3.67 to $3
0
per unit of activity; such a decision would cause net benefit to fall from $29 to $24.
5
The following principle summarizes the role of sunk, fixed, and average costs in
T
making optimal decisions:
S
Principle Decision makers wishing to maximize the net benefit of an activity should ignore any sunk
costs, any fixed costs, and the average costs associated with the activity because none of these costs affect
the marginal cost of the activity and so are irrelevant for making optimal d ecisions.
tho21901_ch03_088-120.indd 102
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 103
3.3 CONSTRAINED OPTIMIZATION
On many occasions a manager will face situations in which the choice of
activity levels is constrained by the circumstances surrounding the maximization
or minimization problem. These constrained optimization problems can be solved,
as in the case of unconstrained maximization, using the logic of marginal analysis.
As noted in Section 3.1, even though constrained optimization problems can be
either maximization or minimization problems, the optimization rule is the same
for both types.
H
A crucial concept for solving
constrained optimization problems is the concept
of marginal benefit per Idollar spent on an activity. Before you can understand
how to solve constrained optimization problems, you must first understand how
G marginal benefit of an activity divided by the price of
to interpret the ratio of the
the activity.
G
S Spent on an Activity
Marginal Benefit per Dollar
,
Retailers frequently advertise that their products give “more value for your
money.” People don’t usually interpret this as meaning the best product in its class
or the one with the highest value. Neither do they interpret it as meaning the
S
cheapest. The advertiser wants to get across the message that customers will get
more for their money orH
more value for each dollar spent on the product. When
product rating services A
(such as Consumer Reports) rate a product a “best buy,”
they don’t mean it is the best product or the cheapest; they mean that consumers
will get more value per N
dollar spent on that product. When firms want to fill a
position, they don’t necessarily
hire the person who would be the most productive
I
in the job—that person may cost too much. Neither do they necessarily hire the
C for the lowest wages—that person may not be very
person who would work
productive. They want the
Q employee who can do the job and give the highest
productivity for the wages paid.
U such as “most value for your money,” “best buy,”
In these examples, phrases
and “greatest bang per buck”
A mean that a particular activity yields the highest
marginal benefit per dollar spent. To illustrate this concept, suppose you are the
office manager for an expanding law firm and you find that you need an extra
copy machine in the office—the
one copier you have is being overworked. You
1
shop around and find three brands of office copy machines (brands A, B, and C)
1
that have virtually identical features. The three brands do differ, however, in price
0 the machines will make before they wear out. Brand
and in the number of copies
A’s copy machine costs $2,500 and will produce about 500,000 copies before it
5
wears out. The marginal benefit of this machine is 500,000 (MBA 5 500,000) since
the machine provides theTlaw office with the ability to produce 500,000 additional
copies. To find the marginal
S benefit per dollar spent on copy machine A, marginal
benefit is divided by price (PA 5 2,500):
MBA/PA 5 500,000 copies/2,500 dollars
5 200 copies/dollar
tho21901_ch03_088-120.indd 103
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104 C H A P T E R 3 Marginal Analysis for Optimal Decisions
You get 200 copies for each of the dollars spent to purchase copy machine A.
Now compare machine A with machine B, which will produce 600,000 copies
and costs $4,000. The marginal benefit is greater, but so is the price. To determine
how “good a deal” you get with machine B, compute the marginal benefit per
dollar spent on machine B:
MBB/PB 5 600,000 copies/4,000 dollars
5 150 copies/dollar
Even though machine B provides
Ha higher marginal benefit, its marginal benefit
per dollar spent is lower than that for machine A. Machine A is a better deal than
I
machine B because it yields higher marginal benefit per dollar. The third copy
machine produces 580,000 copiesG
over its useful life and costs $2,600. Machine C is
neither the best machine (580,000 , 600,000 copies) nor is it the cheapest machine
G
($2,600 . $2,500), but of the three machines, machine C provides the greatest
marginal benefit per dollar spent:S
MBC/PC 5,580,000 copies/2,600 dollars
5 223 copies/dollar
Now try Technical
Problem 8.
On a bang per buck basis, you would
S rank machine C first, machine A second, and
machine B third.
H
When choosing among different activities, a decision maker compares the
A on each of the activities. Marginal benefit (the
marginal benefits per dollar spent
“bang”), by itself, does not provide sufficient information for decision-making
N
purposes. Price (the “buck”), by itself, does not provide sufficient information for
I
making decisions. It is marginal benefit
per dollar spent (the “bang per buck”) that
matters in decision making.
C
Q
U
In the general constrained maximization
problem, a manager must choose the
levels of two or more activities inA
order to maximize a total benefit (objective) funcConstrained Maximization
tion subject to a constraint in the form of a budget that restricts the amount that can
be spent.5 Consider a situation in which there are two activities, A and B. Each unit
of activity A costs $4 to undertake,1and each unit of activity B costs $2 to undertake.
The manager faces a constraint that allows a total expenditure of only $100 on ac1
tivities A and B combined. The manager
wishes to allocate $100 between activities
A and B so that the combined total
benefit
from both activities is maximized.
0
The manager is currently choosing 20 units of activity A and 10 units of activity
5
B. The constraint is met for the combination 20A and 10B since ($4 3 20) 1 ($2 3
10) 5 $100. For this combination T
of activities, suppose that the marginal benefit of
S
It may look like constrained maximization problems no longer use net benefit as the objective
function to be maximized. Note, however, that maximizing total benefit while total cost must remain
constant in order to meet a budget constraint does indeed result in the maximum possible amount of
net benefit for a given level of total cost.
5
tho21901_ch03_088-120.indd 104
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 105
the last unit of activity A is 40 units of additional benefit and the marginal benefit
of the last unit of B is 10 units of additional benefit. In this situation, the marginal
benefit per dollar spent on activity A exceeds the marginal benefit per dollar spent
on activity B:
MBB
MBA ___
_____
5 40 5 10 . 5 5 ___
10 5 _____
2
PA
PB
4
Spending an additional dollar on activity A increases total benefit by 10 units,
while spending an additional dollar on activity B increases total benefit by
5 units. Because the marginal
benefit per dollar spent is greater for activity
H
A, it provides “more bang per buck” or is a better deal at this combination of
I
activities.
To take advantage ofG
this fact, the manager can increase activity A by one
unit and decrease activity B by two units (now, A 5 21 and B 5 8). This combiG $100 [($4 3 21) 1 ($2 3 8) 5 $100]. Purchasing one
nation of activities still costs
more unit of activity A causes
S total benefit to rise by 40 units, while purchasing
two less units of activity B causes total benefit to fall by 20 units. The combined
, A and B rises by 20 units (5 40 2 20) and the new
total benefit from activities
combination of activities (A 5 21 and B 5 8) costs the same amount, $100, as
the old combination. Total benefit rises without spending any more than $100
S
on the activities.
Naturally, the manager
H will continue to increase spending on activity A and
reduce spending on activity B as long as MBA/PA exceeds MBB/PB. In most
A
situations, the marginal benefit of an activity declines as the activity increases.6
N A is increased, MBA gets smaller. As activity B is
Consequently, as activity
decreased, MBB gets larger.
I Thus as spending on A rises and spending on B falls,
MBA/PA falls and MBB/PB rises. A point is eventually reached at which activity A
is no longer a better dealCthan activity B; that is, MBA/PA equals MBB/PB. At this
point, total benefit is maximized
subject to the constraint that only $100 is spent
Q
on the two activities.
U of spending on activities A and B had been one,
If the original allocation
where
A
MBB
MBA _____
_____
,
PA
PB
1
the manager would recognize that activity B is the better deal. In this case,
1
total benefit could be increased by spending more on activity B and less on
0
activity A while maintaining
the $100 budget. Activity B would be increased
by two units for every 5
one-unit decrease in activity A (in order to satisfy the
T
Decreasing marginal benefit is quite common. As you drink several cans of Coke in
S amounts of additional satisfaction from successive cans. As you
succession, you get ever smaller
6
continue studying for an exam, each additional hour of study increases your expected exam grade
by ever smaller amounts. In such cases, marginal benefit is inversely related to the level of the
activity. Increasing the activity causes marginal benefit to fall, and decreasing the activity level
causes marginal benefit to rise.
tho21901_ch03_088-120.indd 105
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106 C H A P T E R 3 Marginal Analysis for Optimal Decisions
$100 spending constraint) until the marginal benefit per dollar spent is equal
for both activities:
MBB
MBA _____
_____
5
PA
PB
If there are more than two activities in the objective function, the condition is
expanded to require that the marginal benefit per dollar spent be equal for all
activities.
H
Principle To maximize total benefits subject to a constraint on the levels of activities, choose the level
I
of each activity so that the marginal benefit per dollar spent is equal for all activities
MB
MB GMB
MB
____
5 ____
5 ____
5 . . . 5 ____
P GP
P
P
and at the same time, the chosen level of activities
S must also satisfy the constraint.
,
A
Now try Technical
Problems 9–10.
C
B
A
B
C
Z
Z
Optimal Advertising Expenditures: An Example of Constrained Maximization
To illustrate how a firm can use the technique of constrained maximization to
S
allocate its advertising budget, suppose a manager of a small retail firm wants
to maximize the effectiveness (in
H total sales) of the firm’s weekly advertising
budget of $2,000. The manager has the option of advertising on the local
A
television station or on the local AM radio station. As a class project, a marketing
N the impact on the retailer’s sales of varying
class at a nearby college estimated
levels of advertising in the two different
media. The manager wants to maximize
I
the number of units sold; thus the total benefit is measured by the total number of units sold. The estimatesCof the increases in weekly sales (the marginal
benefits) from increasing the levels
Q of advertising on television and radio are
given in columns 2 and 4 below:
(1)
Number of ads
1
2
3
4
5
6
(2)
MB TV
U
A
(3)
MB
PTV
TV
_____
(4)
MBradio
(5)
MB
Pradio
______
radio
400
300
280
260
240
200
1.0
360
1.2
1
0.75
270
0.9
1
0.7
240
0.8
0.65
225
0.75
0
0.6
150
0.5
5
0.5
120
0.4
T
Television ads are more “powerful” than radio ads in the sense that the
S
marginal benefits from additional TV ads tend to be larger than those for more
radio ads. However, since the manager is constrained by the limited advertising
budget, the relevant measure is not simply marginal benefit but, rather, marginal
benefit per dollar spent on advertising. The price of television ads is $400 per ad,
tho21901_ch03_088-120.indd 106
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 107
and the price of radio ads is $300 per ad. Although the first TV ad dominates the
first radio ad in terms of its marginal benefit (increased sales), the marginal benefit
per dollar’s worth of expenditure for the first radio ad is greater than that for the
first television ad:
Ad 1
Marginal benefit/price
Television
Radio
400/400 5 1.00
360/300 5 1.2
This indicates that salesH
rise by 1 unit per dollar spent on the first television ad
and 1.2 units on the firstIradio ad. Therefore, when the manager is allocating the
budget, the first ad she selects will be a radio ad—the activity with the larger
G spent. Following the same rule and using the MByP
marginal benefit per dollar
values in columns 3 and 5Gabove, the $2,000 advertising budget would be allocated
as follows:
Decision
Buy radio ad 1
Buy TV ad 1
Buy radio ad 2
Buy radio ad 3
Buy TV ad 2
Buy radio ad 4
S
, MB/P
360/300 5 1.20
400/400 5 1.00
270/300 5 0.90
240/300 5 0.80
300/400 5 0.75
225/300 5 0.75
Ranking
of MB/P
1
2
3
4
Cumulative
expenditures
$ 300
700
1,000
1,300
1,700
2,000
S
H
A
}
5 (tie)
N
By selecting two television ads and four radio ads, the manager of the firm has
maximized sales subject Ito the constraint that only $2,000 can be spent on advertising activity. Note that C
for the optimal levels of television and radio ads (two TV
and four radio):
Q
MBradio
MBTV _______
5
5 0.75
U _____
PTV
Pradio
A
The fact that the preceding application used artificially simplistic numbers
Now try Technical
Problems 11–12.
shouldn’t make you think that the problem is artificial. If we add a few zeros to the
prices of TV and radio ads, we have the real-world situation faced by advertisers.
1
Constrained Minimization
1
Constrained minimization
0 problems involve minimizing a total cost function (the
objective function) subject to a constraint that the levels of activities be chosen
such that a given level of5total benefit is achieved. Consider a manager who must
minimize the total cost T
of two activities, A and B, subject to the constraint that
3,000 units of benefit are to be generated by those activities. The price of activity A
Sof activity B is $20 per unit. Suppose the manager is curis $5 per unit and the price
rently using 100 units of activity A and 60 units of activity B and this combination
of activity generates total benefit equal to 3,000. At this combination of activities,
the marginal benefit of the last unit of activity A is 30 and the marginal benefit of
tho21901_ch03_088-120.indd 107
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108 C H A P T E R 3 Marginal Analysis for Optimal Decisions
I L L U S T R AT I O N 3 . 2
Seattle Seahawks Win on “Bang Per Buck”
Defense
Behind every professional sports team, a team of
business decision makers is constantly at work—there
is no off-season for the business team—trying to figure
out how to put together the most profitable team of
players. In the NFL, the team-building process is a
constrained optimization problem because the football
league imposes restrictions on the amount each team
can spend on players in a season, as well as the number
of players the team can carry on its roster. Currently,
NFL teams are limited to 53 players and a salary cap
of $85 million per season. While teams can, and do,
structure cash bonuses to players in ways that allow
them to exceed the salary caps in any single year, the
NFL spending constraint nonetheless restricts the total
amount a team can spend on its players. Based on what
you have learned in this chapter about constrained
optimization, it should come as no surprise to you that,
in the business of sports, finding and keeping players
who can deliver the greatest bang for the buck may be
the most important game a team must play. History has
shown that most teams making it to the Super Bowl
have played the “bang for the buck” game very well.
To see how personnel directors of NFL teams
follow the principles of constrained maximization in
choosing their teams’ rosters, we can look at the story
of the Seattle Seahawks, who played (and lost to) the
Pittsburgh Steelers in Super Bowl XL. According to a
recent article in The Wall Street Journal, the Seahawks’
personnel director, Tim Ruskell, who enjoyed wide
acclaim for building the highly regarded defenses
at Tampa Bay and Atlanta, faced a particularly harsh
salary-cap constraint in Seattle for the 2005 foot
ball season. The Seahawks’ salary cap in 2005 was
penalized by $18 million of “dead money”—the term
used for money paid by previous contract to players
no longer on the team—so Ruskell began with only
$67 million to spend on players. Making matters even
worse for the team’s chief business decision maker was
the fact that Seattle had signed giant contracts, even
by NFL standards, to keep its biggest stars on offense.
Obviously, this left Ruskell with very little money to
spend on building the Seahawks’ defense. Compared
tho21901_ch03_088-120.indd 108
with its Super Bowl rival, Seattle spent $11 million less
on defense than did the Steelers. Ruskell’s strategy for
hiring defensive players, then, had to be extremely
effective if Seattle was to have any chance of going to
the Super Bowl in 2005.
The way that Ruskell built Seattle’s defense, subject
to H
a very tight spending constraint, drew high praise
from others in the league: “They did it (built a defense)
I breaking the bank at Monte Carlo, and I think
without
that’s
G extremely impressive,” remarked Gil Brandt, a
former personnel director for the Dallas Cowboys.a As
Gknow from our discussion in this chapter, Ruskell
you
must have been very successful at finding defensive
S
players who could deliver the highest possible marginal
, benefits per dollar spent. To accomplish this, he
recruited only inexpensive draft picks and young free
agents, who were also likely to play with “exuberance”
S and perform defensive tasks well enough to get
to the Super Bowl. We must stress that Ruskell’s strategyHdepended crucially on both the numerator and denominator in the MB/MC ratio. He understood that
A
simply hiring the cheapest players would not produce
a Super
N Bowl team. Team scouts had to find players
who would also deliver high marginal benefits to the
I defensive squad by making lots of tackles and
team’s
intercepting
lots of passes. Perhaps the best example of
C
Ruskell’s success at getting the most “bang for the buck”
in Q
2005 was Lofa Tatapu, a rookie linebacker. Tatapu,
who was thought by many team scouts to be too small
U
to be a great linebacker, became a star defensive player
forA
Seattle and cost the team only $230,000—one-tenth
the amount paid on average for linebackers in the NFL.
As you can see from this Illustration, making
optimal
1 constrained maximization decisions in practice
takes not only skill and experience, it sometimes
1 a bit of luck! History shows, however, that
involves
NFL personnel directors who spend their salary caps to
0
get either the very best players (the greatest bang) or the
very
5 cheapest players (the smallest buck), don’t usually
make it to the Super Bowl. Thus, on Super Bowl game
Tfans can generally expect to see the two NFL teams
day,
with
S the highest overall MB/MC ratios. Of course
winning the Super Bowl is just a betting matter.
Gil Brandt as quoted in Walker, “Holding the Line.”
a
8/11/15 4:36 PM
C H A P T E R 3 Marginal Analysis for Optimal Decisions 109
the last unit of activity B is 60. In this situation, the marginal benefit per dollar
spent on activity A exceeds the marginal benefit per dollar spent on activity B:
MBB
MBA ___
_____
5 30 5 6 . 3 5 ___
60 5 _____
5
PA
20
PB
Because the marginal benefit per dollar spent is greater for activity A than for
activity B, activity A gives “more for the money.”
To take advantage of activity A, the manager can reduce activity B by one unit,
causing total benefit to fall by 60 units and reducing cost by $20. To hold total
H
benefit constant, the 60 units of lost benefit can be made up by increasing activity
I
A by two units with a marginal
benefit of 30 each. The two additional units of
activity A cause total cost
to
rise
by $10. By reducing activity B by one unit and
G
increasing activity A by two units, the manager reduces total cost by $10 (5 $20 2
G benefit.
$10) without reducing total
As long as MBA/PA . S
MBB/PB, the manager will continue to increase activity A
and decrease activity B at the rate that holds TB constant until
,
MBB
MBA _____
_____
5
PA
PB
If there are more than two S
activities in the objective function, the condition is e xpanded
to require that the marginal
H benefit per dollar spent be equal for all activities.
A total costs subject to a constraint on the levels of activities, choose the
Principle In order to minimize
level of each activity so that theN
marginal benefit per dollar spent is equal for all activities
MB
MB
MB
MB
I ____
5 ____
5 . . . 5 ____
5 ____
P
P
P
P
C
and at the same time, the chosen level of activities must also satisfy the constraint.
Q
As you can see, this isUthe same condition that must be met
constrained maximization.
A
A
A
Now try Technical
Problem 13.
3.4 SUMMARY
■
Formulating
1
an optimization problem involves
specifying three things: (1) the objective function to1
be
either maximized or minimized, (2) the activities or
0
choice variables that determine the value of the objective function, and (3) any constraints that may restrict
5
the range of values that the choice variables may take.
T
Choice variables determine the value of the objective
function and may be either discrete or continuous. A disS
crete choice variable can take on only specified integer
values. A continuous choice variable can take on any
value between two end points. Marginal analysis is an
analytical technique for solving optimization problems
by changing the value of a choice variable by a small
tho21901_ch03_088-120.indd 109
■
C
B
B
C
Z
Z
in the case of
amount to see if the objective function can be further increased (in the case of maximization problems) or further
decreased (in the case of minimization problems). (LO1)
Net benefit from an activity (NB) is the difference between
total benefit (TB) and total cost (TC) for the activity: NB 5
TB 2 TC. The net benefit function is the objective function to be maximized in unconstrained maximization
problems. The optimal level of the a ctivity, A*, is the level
of activity that maximizes net benefit. Marginal benefit
(marginal cost) is the change in total benefit (total cost) per
unit change in the level of activity. The marginal benefit
(marginal cost) of a particular unit of activity can be measured by the slope of the line tangent to the total benefit
8/11/15 4:36 PM
110 C H A P T E R 3 Marginal Analysis for Optimal Decisions
(total cost) curve at that point of activity. The optimal
level of the activity is attained when no further increases
in net benefit are possible for any changes in the activity.
This point occurs at the activity level for which marginal
benefit equals marginal cost: MB 5 MC. Sunk costs are
costs that have previously been paid and cannot be recovered. Fixed costs are costs that are constant and must
be paid no matter what level of activity is chosen. Average (or unit) cost is the cost per unit of activity. Decision
makers should ignore any sunk costs, any fixed costs,
and the average costs associated with the activity because
they are irrelevant for making optimal decisions. (LO2)
■
The
ratio of marginal benefit divided by the price
of an activity (MB/P) tells the decision maker the
additional benefit of that activity per additional
dollar spent on that activity, sometimes referred to
informally as “bang per buck.” In constrained optimization problems, the ratios of marginal benefits to
prices of the various activities are used by decision
makers to determine how to allocate a fixed number
of dollars among activities. To maximize or minimize
an objective function subject to a constraint, the ratios
H
of the marginal benefit to price must be equal for all
Iactivities. (LO3)
G
G
S
,
KEY TERMS
activities or choice variables
average (unit) cost
constrained optimization
continuous choice variables
discrete choice variables
fixed costs
marginal analysis
marginal benefit (MB)
marginal cost (MC)
maximization problem
minimization problem S
net benefit
TECHNICAL PROBLEMS
1.
objective function
optimal level of activity
sunk costs
unconstrained optimization
H
A
N
For each of the following decision-making problems, determine whether the problem
I
involves constrained or unconstrained
optimization; what the objective function is and,
for each constrained problem, what the constraint is; and what the choice variables are.
C
a. We have received a foundation grant to purchase new PCs for the staff. You decide
Q
what PCs to buy.
b. We aren’t earning enough profits.
U Your job is to redesign our advertising program
and decide how much TV, direct-mail, and magazine advertising to use. Whatever
A very well.
we are doing now isn’t working
c. We have to meet a production quota but think we are going to spend too much
doing so. Your job is to reallocate the machinery, the number of workers, and the
raw materials needed to meet1the quota.
2. Refer to Figure 3.2 and answer the
1 following questions:
a. At 600 units of activity, marginal benefit is
(rising, constant, positive, neg0 at D is sloping
ative) because the tangent line
(downward, upward).
b. The marginal benefit of the 5
600th unit of activity is $
. Explain how this
value of marginal benefit can be computed.
T
c. At 600 units of activity, decreasing
the activity by one unit causes total benefit
to
(increase, decrease)
by
$
. At point D, total benefit changes at
S
a rate
times as much as activity changes, and TB and A are moving in the
(same, opposite) direction, which means TB and A are
(directly,
inversely) related.
d. At 1,000 units of activity, marginal benefit is
. Why?
e. The marginal cost of the 600th unit of activity is $
. Explain how this value
of marginal cost can be computed.
tho21901_ch03_088-120.indd 110
8/11/15 4:36 PM
C H A P T E R 3 Marginal Analysis for Optimal Decisions 111
f. At 600 units of activity, decreasing the activity by one unit causes total cost to
(increase, decrease) by $
. At point D’, total cost changes at a
rate
times as much as activity changes, and TC and A are moving in the
(same, opposite) direction, which means TC and A are
(directly,
inversely) related.
g. Visually, the tangent line at point D appears to be
(flatter, steeper) than the
tangent line at point D’, which means that
(NB, TB, TC, MB, MC) is larger
than
(NB, TB, TC, MB, MC).
h. Because point D lies above point D’,
(NB, TB, TC, MB, MC) is larger than
(NB, H
TB, TC, MB, MC), which means that
(NB, TB, TC, MB,
MC) is
(rising,
falling,
constant,
positive,
negative,
zero).
I
3. Fill in the blanks below. In an unconstrained maximization problem:
Gincreased if
a. An activity should be
exceeds
.
b. An activity should be
decreased
if
exceeds
.
G
c. The optimal level of activity occurs at the activity level for which
equals
S
.
d. At the optimal level, of activity,
is maximized, and the slope of
equals the slope of
.
e. If total cost is falling faster than total benefit is falling, the activity should
S
be
.
f. If total benefit is rising
H at the same rate that total cost is rising, the decision maker
should
.
A then total benefit must be rising at a rate
g. If net benefit is rising,
(greater
than, less than, equal
(rising, falling).
Nto) the rate at which total cost is
4. Use the graph below to answer the following questions:
I
C
Q
U
A
Marginal benefit and marginal cost (dollars)
10
9
8
7
6
1
1
0
5
T
S
5
4
3
2
1
0
MC
20
40
60
MB
80 100 120 140 160 180 200 220 240 260 280 300
Level of activity
tho21901_ch03_088-120.indd 111
8/11/15 4:36 PM
112 C H A P T E R 3 Marginal Analysis for Optimal Decisions
a. At 60 units of the activity, marginal benefit is $
and marginal cost is
$
.
b. Adding the 60th unit of the activity causes net benefit to
(increase,
decrease) by $
.
c. At 220 units of the activity, marginal benefit is $
and marginal cost is
$
.
d. Subtracting the 220th unit of the activity causes net benefit to
(increase,
decrease) by $
.
e. The optimal level of the activity is
units. At the optimal level of
H
the activity, marginal benefit is $
and marginal cost is $
.
I statement:
5. Fill in the blanks in the following
If marginal benefit exceeds marginal
G cost, then increasing the level of activity by one
unit
(increases, decreases)
(total, marginal, net) benefit by more than
G
it
(increases, decreases)
(total, marginal) cost. Therefore,
(increasing, decreasing) the level of activity by one unit must increase net benefit. The
S
manager should continue to
(increase, decrease) the level of activity until
marginal benefit and marginal cost
(zero, equal).
, are
6. Fill in the blanks in the following table to answer the questions below.
a.
b.
c.
S TC
NB
MB
MC
$
$0
H
1
27
$35
$
A
2
65
10
3
85
30
N
4
51
14
I 60
5
8
6
5
20
C
Q in the table above?
What is the optimal level of activity
What is the value of net benefit
U at the optimal level of activity? Can net benefit be
increased by moving to any other level of A? Explain.
A
Using the numerical values in the table, comment on the statement, “The optimal
A
TB
0
$ 0
level of activity occurs where marginal benefit is closest to marginal cost.”
7. Now suppose the decision maker in Technical Problem 6 faces a fixed cost of $24. Fill in
1
the blanks in the following table to answer the questions below. AC is the average cost
per unit of activity.
1
A
TB
0
$ 0
1
2
3
4
5
6
tho21901_ch03_088-120.indd 112
65
85
0
5
T
S
54
TC
NB
MB
MC
$35
$
AC
2$24
3
$32
10
27
___
14
8
5
16.80
20
8/11/15 4:36 PM
C H A P T E R 3 Marginal Analysis for Optimal Decisions 113
8.
9.
10.
11.
a. How does adding $24 of fixed costs affect total cost? Net benefit?
b. How does adding $24 of fixed cost affect marginal cost?
c. Compared to A* in Technical Problem 6, does adding $24 of fixed cost change the
optimal level of activity? Why or why not?
d. What advice can you give decision makers about the role of fixed costs in finding A*?
e. What level of activity minimizes average cost per unit of activity? Is this level also
the optimal level of activity? Should it be? Explain.
f. Suppose a government agency requires payment of a one-time, nonrefundable
license fee of $100 toHengage in activity A, and this license fee was paid last month.
What kind of cost is this? How does this cost affect the decision maker’s choice of
I
activity level now? Explain.
You are interviewing three
G people for one sales job. On the basis of your experience
and insight, you believe Jane can sell 600 units a day, Joe can sell 450 units a day, and
Joan can sell 400 units G
a day. The daily salary each person is asking is as follows: Jane,
$200; Joe, $150; and Joan, $100. How would you rank the three applicants?
S
Fill in the blanks. When choosing the levels of two activities, A and B, in order to maxi, a given budget:
mize total benefits within
a. If at the given levels of A and B, MByP of A is
MByP of B, increasing A and
decreasing B while holding expenditure constant will increase total benefits.
b. If at the given levelsS
of A and B, MByP of A is
MByP of B, increasing B and
decreasing A while H
holding expenditure constant will increase total benefits.
c. The optimal levels of A and B are the levels at which
equals
.
A
A decision maker is choosing the levels of two activities, A and B, so as to maximize
total benefits under a given
N budget. The prices and marginal benefits of the last units
of A and B are denoted PA, PB, MBA, and MBB.
I
a. If PA 5 $20, PB 5 $15, MBA 5 400, and MBB 5 600, what should the decision maker do?
b. If PA 5 $20, PB 5 $30,C
MBA 5 200, and MBB 5 300, what should the decision maker do?
c. If PA 5 $20, PB 5 $40,Q
MBA 5 300, and MBB 5 400, how many units of A can be obtained
if B is reduced by one unit? How much will benefits increase if this exchange is made?
d. If the substitution inU
part c continues to equilibrium and MBA falls to 250, what will
MBB be?
A
A decision maker wishes to maximize the total benefit associated with three activities,
X, Y, and Z. The price per unit of activities X, Y, and Z is $1, $2, and $3, respectively.
The following table gives
1 the ratio of the marginal benefit to the price of the activities
for various levels of each activity:
1
0
5
T
S
Level of
activity
1
2
3
4
5
6
7
8
tho21901_ch03_088-120.indd 113
MB
____x
Px
MBy
____
Py
MB
____z
Pz
10
22
14
9
8
7
6
5
4
3
18
12
10
6
4
2
1
12
10
9
8
6
4
2
8/11/15 4:36 PM
114 C H A P T E R 3 Marginal Analysis for Optimal Decisions
a. If the decision maker chooses to use one unit of X, one unit of Y, and one unit of Z,
the total benefit that results is $
.
b. For the fourth unit of activity Y, each dollar spent increases total benefit by
$
. The fourth unit of activity Y increases total benefit by $
.
c . Suppose the decision maker can spend a total of only $18 on the three activities.
What is the optimal level of X, Y, and Z? Why is this combination optimal? Why is
the combination 2X, 2Y, and 4Z not optimal?
d. Now suppose the decision maker has $33 to spend on the three activities. What is
the optimal level of X, Y, and Z? If the decision maker has $35 to spend, what is the
H
optimal combination? Explain.
12. Suppose a firm is considering two
I different activities, X and Y, which yield the total
benefits presented in the schedule below. The price of X is $2 per unit, and the price of
G
Y is $10 per unit.
Level of
activity
0
1
2
3
4
5
6
13.
benefit
GTotal
of activity X
S(TB )
$ 0
,
30
Total benefit
of activity Y
(TBY)
X
$ 0
100
190
270
340
400
450
54
72
84
92
98
S
H
A
a. The firm places a budget constraint
of $26 on expenditures on activities X and Y.
N
What are the levels of X and Y that maximize total benefit subject to the budget
I
constraint?
b. What is the total benefit associated
C with the optimal levels of X and Y in part a?
c. Now let the budget constraint increase to $58. What are the optimal levels of X and
Q
Y now? What is the total benefit when the budget constraint is $58?
U problem, P 5 $10, P 5 $10, MB 5 600, and
a. If, in a constrained minimization
MB 5 300 and one unit of B is taken away, how many units of A must be added to
A
keep benefits constant?
A
B
A
B
b. If the substitution in part a continues to equilibrium, what will be the equilibrium
relation between MBA and MBB?
APPLIED PROBLEMS
1.
1
1
0
Using optimization theory, analyze the following quotations:
5
a. “The optimal number of traffic deaths in the United States is zero.”
T
b. “Any pollution is too much pollution.”
c. “We cannot pull U.S. troopsSout of Afghanistan. We have committed so much
already.”
d. “If Congress cuts out the International Space Station (ISS), we will have wasted all the resources that we have already spent on it. Therefore, we must continue funding the ISS.”
tho21901_ch03_088-120.indd 114
8/11/15 4:36 PM
C H A P T E R 3 Marginal Analysis for Optimal Decisions 115
e. “Since JetGreen Airways has experienced a 25 percent increase in its insurance premiums, the airline should increase the number of passengers it serves next quarter
in order to spread the increase in premiums over a larger number of tickets.”
2. Appalachian Coal Mining believes that it can increase labor productivity and, therefore,
net revenue by reducing air pollution in its mines. It estimates that the marginal cost
function for reducing pollution by installing additional capital e quipment is
MC 5 40P
where P represents a reduction of one unit of pollution in the mines. It also feels that
H reduction the marginal increase in revenue (MR) is
for every unit of pollution
3.
I
MR 5 1,000 2 10P
G
How much pollution reduction
should Appalachian Coal Mining undertake?
Two partners who ownG
Progressive Business Solutions, which currently operates out of
an office in a small town near Boston, just discovered a vacancy in an office building in
downtown Boston. OneSof the partners favors moving downtown because she believes
the additional business gained by moving downtown will exceed the higher rent at the
,
downtown location plus the cost of making the move. The other partner at PBS opposes
moving downtown. He argues, “We have already paid for office stationery, business
cards, and a large sign that cannot be moved or sold. We have spent so much on our
S afford to waste this money by moving now.” Evaluate the
current office that we can’t
second partner’s advice
Hnot to move downtown.
4. Twentyfirst Century Electronics has discovered a theft problem at its warehouse and
A guards. The firm wants to hire the optimal number of secuhas decided to hire security
rity guards. The following
N table shows how the number of security guards affects the
number of radios stolen per week.
I
C
Q
U
A
Number of
security guards
0
Number of radios
stolen per week
50
1
2
3
30
20
14
4
5
8
6
1
a. If each security guard is paid $200 a week and the cost of a stolen radio is $25, how
many security guards
1 should the firm hire?
b. If the cost of a stolen radio is $25, what is the most the firm would be willing to pay
0
to hire the first security guard?
5 is paid $200 a week and the cost of a stolen radio is $50, how
c. If each security guard
many security guards should the firm hire?
T
5. U.S. Supreme Court Justice Stephen Breyer’s book Breaking the Vicious Circle: Toward
S examines government’s role in controlling and managing
Effective Risk Regulation
the health risks society faces from exposure to environmental pollution. One major
problem examined in the book is the cleanup of hazardous waste sites. Justice Breyer
was extremely critical of policymakers who wish to see waste sites 100 percent clean.
tho21901_ch03_088-120.indd 115
8/11/15 4:36 PM
116 C H A P T E R 3 Marginal Analysis for Optimal Decisions
a. Explain, using the theory of optimization and a graph, the circumstances under
which a waste site could be made “too clean.” (Good answers are dispassionate and
employ economic analysis.)
b. Justice Breyer believes that society can enjoy virtually all the health benefits of
cleaning up a waste site for only a “small fraction” of the total cost of completely
cleaning a site. Using graphical analysis, illustrate this situation. (Hint: Draw MB
and MC curves with shapes that specifically illustrate this situation.)
6. In Illustration 3.1, we noted that the rule for maximization set forth in the text
contradicts some honored traditional principles such as “Never give up,” “Anything
H or “Waste not, want not.” Explain the contradiction
worth doing is worth doing well,”
for each of these rules.
I
7. Janice Waller, the manager of the customer service department at First Bank of J efferson
G a high school diploma for $20,000 annually and
County, can hire employees with
employees with a bachelor’s degree for $30,000. She wants to maximize the number
G
of customers served, given a fixed payroll. The following table shows how the total
number of customers served varies
S with the number of employees:
Number of
employees
1
,
Total number of customers served
High school diploma
120
Bachelor’s degree
100
S 220
190
300
270
H
370
330
A 430
380
470
410
N
a. If Ms. Waller has a payroll of I$160,000, how should she allocate this budget in order
to maximize the number of customers
served?
C
b. If she has a budget of $150,000 and currently hires three people with high school
Q degrees, is she making the correct decision? Why
diplomas and three with bachelor’s
or why not? If not, what should
U she do? (Assume she can hire part-time workers.)
c. If her budget is increased to $240,000, how should she allocate this budget?
A
Bavarian Crystal Works designs and produces lead crystal wine decanters for export
2
3
4
5
6
8.
to international markets. The production manager of Bavarian Crystal Works estimates
total and marginal production costs to be
1
1
and
0
MC 5 40 1 0.005Q
5
where costs are measured in U.S. dollars and Q is the number of wine decanters proT Crystal Works is only one of many crystal producers
duced annually. Because Bavarian
in the world market, it can sellS
as many of the decanters as it wishes for $70 apiece.
TC 5 10,000 1 40Q 1 0.0025Q2
Total and marginal revenue are
TR 5 70Q and MR 5 70
where revenues are measured in U.S. dollars and Q is annual decanter production.
tho21901_ch03_088-120.indd 116
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C H A P T E R 3 Marginal Analysis for Optimal Decisions 117
a. What is the optimal level of production of wine decanters? What is the marginal
revenue from the last wine decanter sold?
b. What are the total revenue, total cost, and net benefit (profit) from selling the optimal number of wine decanters?
c. At the optimal level of production of decanters, an extra decanter can be sold for
$70, thereby increasing total revenue by $70. Why does the manager of this firm not
produce and sell one more unit?
9. Joy Land Toys, a toy manufacturer, is experiencing quality problems on its assembly
line. The marketing division estimates that each defective toy that leaves the plant
H
costs the firm $10, on average,
for replacement or repair. The engineering department
recommends hiring quality
inspectors
to sample for defective toys. In this way many
I
quality problems can be caught and prevented before shipping. After visiting other
G team derives the following schedule showing the approxicompanies, a management
mate number of defective
G toys that would be produced for several levels of inspection:
S Number of
, inspectors
0
1
Average number
of defective toys
(per day)
92
62
2
42
S
3
27
H 4
17
10
A 5
6
5
N
The daily wage of inspectors is $70.
I
a. How many inspectors should the firm hire?
C to a be if the wage rate is $90?
b. What would your answer
c . What if the averageQcost of a defective toy is $5 and the wage rate of inspectors
is $70?
U
A
MATHEMATICAL APPENDIX A Brief Presentation of Optimization Theory
Theory of Unconstrained Maximization
1
This section sets forth a mathematical analysis
of unconstrained maximization. We begin with1a
single-variable problem in its most general form. An
0
activity, the level of which is denoted as x, generates both
5
benefits and costs. The total benefit function is B(x) and
the total cost function is C(x). The objective is to maximize
T
net benefit, NB, defined as the difference between total
benefit and total cost. Net benefit is itself a function of the
S
level of activity and can be expressed as
(1)
tho21901_ch03_088-120.indd 117
NB 5 NB(x) 5 B(x) 2 C(x)
The necessary condition for maximization of net benefit
is that the derivative of NB with respect to x equal zero
dNB(x) _____
dB(x) _____
dC(x)
(2) _______
5
2
5 0
dx
dx
dx
Equation (2) can then be solved for the optimal level of x,
denoted x*. Net benefit is maximized when
dB(x) _____
dC(x)
(3) _____
5
dx
dx
Since dBydx is the change in total benefit with respect to
the level of activity, this term is marginal benefit. Similarly
for cost, dCydx is the change in total cost with respect to
8/11/15 4:36 PM
118 C H A P T E R 3 Marginal Analysis for Optimal Decisions
the level of activity, and this term is marginal cost. Thus,
net benefit is maximized at the level of activity where
marginal benefit equals marginal cost.
This unconstrained optimization problem can be easily expanded to more than one choice variable or kind of
activity. To this end, let total benefit and total cost be functions of two different activities, denoted by x and y. The
net benefit function with two activities is expressed as
(4)
NB 5 NB(x, y) 5 B(x, y) 2 C(x, y)
Maximization of net benefit when there are two activities affecting benefit and cost requires both of the partial
derivatives of NB with respect to each of the activities to
be equal to zero:
−NB(x, y) _______
−B(x, y) _______
−C(x, y)
(5a) _________
5
2
50
−x
−x
−x
−NB(x, y) _______
−B(x, y) _______
−C(x, y)
(5b) _________
5
2
50
−y
−y
−y
Equations (5a) and (5b) can be solved simultaneously for
the optimal levels of the variables, x* and y*. Maximization of net benefit thus requires
−B 5 ___
(6a) ___
−C
−x
−x
and
To find the optimal value of x, take the derivative of
the net benefit function with respect to x and set it equal
to zero:
dNB 5 a 2 2bx 2 c 1 2dx 2 3ex2
(10) ____
dx
5 (a 2 c) 2 2(b 2 d)x 2 3ex2 5 0
This quadratic equation can be solved using the quadratic
formula or by factoring.a
H
Suppose the values of the parameters are a 5 60,
b 5I0.5, c 5 24, d 5 2, and e 5 1. The net benefit function is
(11)G NB 5 NB(x) 5 60x 2 0.5x2 2 24x 1 2x2 2 x3
G
Now take the derivative of NB [or substitute parameterSvalues into equation (10)] to find the condition for
optimization:
,
(12) (60 2 24) 2 2(0.5 2 2)x 2 3(1)x2 5 36 1 3x 2 3x2 5 0
This
Sequation can be factored: (12 2 3x)(3 1 x) 5 0. The
solutions are x 5 4, x 5 23. (Note: The quadratic e quation
canH
also be used to find the solutions.) The value of x that
maximizes net benefit is x* 5 4.b To find the optimal, or
A
maximum, value of net benefit, substitute x* 5 4 into
equation
N (11) to obtain
INB* 5 60(4) 2 0.5(4) 2 24(4) 1 2(4) 2 (4) 5 104
C
Theory
Q of Constrained Maximization
In aU
constrained maximization problem, a decision maker
determines the level of the activities, or choice variables,
A to obtain the most benefit under a given cost
in order
2
−B 5 ___
(6b) ___
−C
−y
−y
For each activity, the marginal benefit of the activity equals the
marginal cost of the activity. The problem can be expanded to
any number of choice variables with the same results.
Turning now to a mathematical example, consider the
following specific form of the total benefit and total cost
functions:
(7)
B(x) 5 ax 2 bx2
2
3
constraint. In a constrained minimization problem,
a decision maker determines the levels of the choice
1 in order to obtain the lowest cost of achieving
variables
1
0 The solution to a quadratic equation yields two values for
x. The maximization, rather than minimization, solution is the
value
5 of x at which the second-order condition is met:
NB 5 22(b 2 d) 2 6ex , 0
_____
ddx
T
SThis value of x is the one that satisfies the second-order
a
and
(8)
C(x) 5 cx 2 dx2 1 ex3
2
where the parameters, a, b, c, d, and e are all positive.
Now the net benefit function can be expressed as
(9)
NB 5 NB(x) 5 B(x) 2 C(x)
5 ax 2 bx2 2 cx 1 dx2 2 ex3
tho21901_ch03_088-120.indd 118
2
b
condition for a maximum in the preceding footnote:
2
____
d NB 5 22(21.5) 2 6(1)(4) 5 221 , 0
dx2
8/11/15 4:36 PM
C H A P T E R 3 Marginal Analysis for Optimal Decisions 119
a given level of benefit. As we showed in the text, the
solutions to the two types of problems are the same. We
first consider constrained maximization.
Constrained optimization
We first assume a general total benefit function with two
choice variables, the levels of which are denoted x and y:
B(x, y). The partial derivatives of this function represent
the marginal benefit for each activity:
H
−B(x, y)
−B(x, y) I
MBx 5 _______
and MBy 5 _______
−x
−y
G
The constraint is that the total cost function must equal a
G
specified level of cost, denoted as C:
(13)
S
,
C(x, y) 5 Pxx 1 Pyy 5 C
where Px and Py are the prices of x and y. Now the Lagrangian function to be maximized can be written as
S
(14)
+ 5 B(x, y) 1 l(C 2 P x 2 P y)
H
A
where l is the Lagrangian multiplier. The first-order condition for a maximum requires the partial derivatives of the
N
Lagrangian with respect to the variables x, y, and l to be zero:
I
−+ 5 ___
(14a) ___
−B 2 lP 5 0
−x
−x
C
−+
−B
___
___
(14b) 5 2 lP 5 0
Q
−y
−y
U
−+ 5 C 2 P x 2 P y 5 0
(14c) ___
−l
A
Notice that satisfaction of the first-order condition (14c)
x
y
y
y
tho21901_ch03_088-120.indd 119
y
y
x
____
5 ____
Px
Py
+ 5 Pxx 1 Pyy 1 l[B 2 B(x, y)]
The first-order conditions are
−+ 5 P 2 l ___
−B 5 0
___
x
−x
−x
−+ 5 P 2 l ___
−B 5 0
(17) ___
y
−y
−y
−+ 5 [B 2 B(x, y)] 5 0
___
−l
MBx __
−B 5 __
(18) ___
1 P or ____
5 1
−x l x
l
Px
1
1
−x
0
MB
___
−B 5 lP or ____
5 l
−y
P
5
It therefore follows that the levels of x and y must be choT
sen so that
S
MB
MB
(15)
(16)
y
MBx
or ____
5l
Px
y
For the constrained minimization problem, we want to
choose the levels of two activities, x and y, to obtain a
given level of benefit at the lowest possible cost. Therefore, the problem is to minimize C 5 Pxx 1Pyy, subject to
B 5 B(x, y), where B is the specified level of benefit. The
Lagrangian function is
As in the constrained maximization problem, the first two
equations can be rearranged to obtain
requires that the cost constraint be met.
Rearranging the first two equations (14a) and (14b):
x
Constrained minimization
x
x
−B 5 lP
___
The marginal benefits per dollar spent on the last units of
x and y must be equal.
The three equations in (14) can be solved for the
equilibrium values x*, y*, and l* by substitution or by
Cramer’s rule. Therefore, x* and y* give the values of the
choice variables that yield the maximum benefit possible
at the given level of cost.
MBy 1
−B 5 __
___
1 P or ____
5 __
−y l y
l
Py
Once again the marginal benefits per dollar spent on the
last units of x and y must be the same, because, from (18),
MBy
MBx ____
____
5
Px
Py
The three equations in (17) can be solved for the
equilibrium values x*, y*, and l* by substitution or
by Cramer ’s rule. These are the values of the choice
variables that attain the lowest cost of reaching the
given level of benefit.
8/11/15 4:36 PM
120 C H A P T E R 3 Marginal Analysis for Optimal Decisions
MATHEMATICAL EXERCISES
1. Assume the only choice variable is x. The total benefit function is B(x) 5 170x 2 x2, and
the cost function is C(x) 5 100 2 10x 1 2x2.
a. What are the marginal benefit and marginal cost functions?
b. Set up the net benefit function and then determine the level of x that maximizes net
benefit.
c. What is the maximum level of net benefit?
2. The only choice variable is x. The total benefit function is B(x) 5 100x 2 2x2, and the
total cost function is C(x) 5 _ 13 x3 2H6x2 1 52x 1 80.
a. What are the marginal benefitI and marginal cost functions?
b. Set up the net benefit function and then determine the level of x that maximizes net
G
benefit. (Use the positive value of x.)
c. What is the maximum level ofG
net benefit?
3. A decision maker
wishes to maximize
__
S total benefit, B 5 3x 1 xy 1 y, subject to the cost
constraint, C 5 4x 1 2y 5 70. Set up the Lagrangian and then determine the values of
x and y at the maximum level of, benefit, given the constraint. What are the maximum
benefits?
4. A decision maker wishes to minimize the cost of producing a given level of total benSC 5 6x 1 3y and the total benefit function is B 5 xy.
efit, B 5 288. The cost function is
Set up the Lagrangian and then determine levels of x and y at the minimum level of
H
cost. What is the minimum value of cost?
A total cost curves are represented by the following
5. In Figure 3.1, the total benefit and
mathematical functions:
N
I
and
C
TC 5 TC(A) 5 A 1 0.006A2
Q
a. Find the marginal benefit function.
U Verify that points c, b, and d in Figure 3.2 lie on
the marginal benefit curve.
A Verify that points c’, b, and d’ in Figure 3.2 lie on the
b. Find the marginal cost function.
TB 5 TB(A) 5 8A 2 0.004A2
marginal cost curve.
c. Derive the net benefit function. Verify the slopes of net benefit at points M, c”,
1
and d” in Figure 3.3.
d. Find the optimal level of activity
1 and the maximum value of net benefit. Does your
answer match Figure 3.3?
0
5
T
S
tho21901_ch03_088-120.indd 120
8/11/15 4:36 PM
Instructions: This assignment must be done in APA format. This part of the assignment is
questions based off of two case studies to be answered with a minimum word count of 1600 for
the overall assignment (without references included). A minimum of four (4) references along
with in-text citations is required for this assignment.
Please note: The word count for the overall assignment is
1600 words (not including
the questions and references). The required reference count for the overall assignment is 4
references. Also, although this assignment is in APA format; please keep the question and
answer line up (see example below).
For example: Question: XYZ
Answer:XYZ
Reference: XYZ
1. Use the graph below to answer the following questions:
•
•
•
•
•
At 200 units of the activity, marginal benefit is $__________ and
marginal cost is $__________.
Adding the 200th unit of the activity causes net benefit to __________
(increase, decrease) by $__________.
At 700 units of the activity, marginal benefit is $__________ and
marginal cost is $__________.
Subtracting the 700th unit of the activity causes net benefit to
__________ (increase, decrease) by $__________.
The optimal level of the activity is __________units. At the optimal
level of the activity, marginal benefit is $__________ and marginal
cost is $__________.
2. You are interviewing three people for one sales job. On the basis of
your experience and insight, you believe Jack can sell 64 widgets per
day, Jill can sell 50 widgets per day, and John can sell 100 widgets
per day. The daily salary each person is asking is as follows: Jack,
$100; Jill, $100; and John, $300. How would you rank the three
applicants?
Part Two
Instructions: Please write a 150 word response with one reference each.
1.
The theory of optimization revolves around a cost-benefit analysis. A cost-benefit
analysis is something that businesses use to calculate the benefits and cost of a
project or a production of something (Bukvic, 2018). For any business, this means
there is an optimal point of activity or production where the benefit is the highest
above the cost. This also means that there are times when the activity or
production is not optimal, and the cost outweighs the benefit. Both of these things
are graphed together in two different curves one being Total Benefit and the other
Total Cost. At some point in the graph, the two lines w...
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