Chapter 3
Balancing Costs and Benefits
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.
Main Topics
Maximizing benefits less costs
Thinking on the margin
Sunk costs and decision-making
3-2
Maximizing Net Benefit
Net benefit: total benefit minus total cost
Total cost must include opportunity cost
Opportunity cost: the cost associated
with foregoing the opportunity to employ
a resource in its best alternative use
Right decision is the choice with the
greatest difference between total benefit
and total cost
3-3
Car Repair Example:
Benefit Schedule
Mechanic’s time is
available in one-hour
increments
Maximum repair time
is 6 hours
The more time the car
is repaired, the more
it is worth
Table 3.1: Benefits of Repairing Your
Car
Repair Time
(Hours)
Total Benefit
($)
0
0
1
615
2
1150
3
1600
4
1975
5
2270
6
2485
3-4
Car Repair Example:
Cost Schedule
Table 3.2: Costs of Repairing Your Car
Repair
Time
(Hours)
Cost of Mechanic and
Parts
($)
Lost Wages from
Pizza Delivery Job
($)
Total
Cost
($)
0
0
0
0
1
140
10
150
2
355
25
380
3
645
45
690
4
1005
75
1080
5
1440
110
1550
6
1950
150
2100
3-5
Car Repair Example:
Maximizing Net Benefit
How should you decide how many hours
is the “right” number to have your car
repaired?
Recall that every hour in the shop will
bring both benefits and costs
Choose the number of hours where
benefits exceed costs by the greatest
amount
3-6
Car Repair Example:
The Right Decision
Table 3.3: Total Benefit and Total Cost of
Repairing Your Car
Repair Time
(Hours)
Total Benefit
($)
Total Cost
($)
0
0
0
1
615
150
2
1150
380
3
1600
690
4
1975
1080
5
2270
1550
6
2485
2100
Net Benefit
($)
3-7
Car Repair Example:
The Right Decision
Table 3.3: Total Benefit and Total Cost of
Repairing Your Car
Repair Time
(Hours)
Total Benefit
($)
Total Cost
($)
Net Benefit
($)
0
0
0
0
1
615
150
465
2
1150
380
770
3
1600
690
910
4
1975
1080
895
5
2270
1550
720
6
2485
2100
385
3-8
Car Repair Example:
The Right Decision
Table 3.3: Total Benefit and Total Cost of
Repairing Your Car
Best
Choice
Repair Time
(Hours)
Total Benefit
($)
Total Cost
($)
Net Benefit
($)
0
0
0
0
1
615
150
465
2
1150
380
770
3
1600
690
910
4
1975
1080
895
5
2270
1550
720
6
2485
2100
385
3-9
Car Repair Example:
Graphical Approach
(Figure 3.1)
 Data from Table 3.3 are
shown in this graph
 Costs are in red; benefits
are in blue
 The best choice is where
benefits > costs and the
distance between them
is maximized
 This is at 3 hours, net
benefit = $910
Total
Benefit,
Total
Cost
($)
2400
2000
710
1600
910
1200
800
400
465
1
2
3
4
5
6
Repair
Hours
Best Choice
3-10
Maximizing Net Benefit:
Finely Divisible Actions
Many decisions involve actions that are
more finely divisible
E.g. mechanic’s time available by the
minute
In these cases can use benefit and cost
curves rather than points or a schedule
to make the best decision
Underlying principle is the same:
maximize net benefit
3-11
Sample Problem 1
Identifying costs
What are the costs of buying a concert ticket
if you have to wait in line for 2 hrs?
Ticket master
What is the cost of watching a movie for two
hours?
Car Repair Example:
Finely Divisible Benefit
(a): Total Benefit
Horizontal axis
measures hours of
mechanic’s time
Vertical axis
measures in dollars
the total increase in
your car’s value
B(H)=654H-40H2
Total Benefit
($)
B
2270
1602
614
0
1
2
3
4
5
6
Hours (H)
3-13
Car Repair Example:
Finely Divisible Cost
(b): Total Cost
Vertical axis
measures total cost
in dollars
Includes opportunity
cost
C(H)=110H+40H2
Total Cost
($)
C
1550
690
150
0
1
2
3
4
5
6
Hours (H)
3-14
Car Repair Example:
Finely Divisible Net Benefit
Best choice is 3.4
hours of repair,
maximizes net
benefit
Net benefit with finely
divisible choices is
greater than in
previous example;
more flexibility allows
you to do better
(c): Total Benefit versus Total
Cost
Total
Benefit
Total
Cost
($)
B
1761.20
924.80
C
836.40
0
1
2
3
Hours
(H)
4
5
6
3.4
3-15
Net Benefit Curve
(Figure 3.3)
Can also graph the
net benefit curve
Vertical axis shows
B-C, net benefit
Best choice is the
number of hours that
corresponds to the
highest point on the
curve, 3.4 hours
Net
Benefit
($)
924.80
B–C
0
1
2
3
Hours (H)
4
5
6
3.4
3-16
Sample Problem 2 (3.1)
 Suppose that the cost of hiring your
mechanic (including the cost of parts)
is $200 an hour up to four hours (she
charges for her time in one-hour
increments). Your benefits and other
costs are the same as in tables 3.1
and 3.2. Construct a table like Table
3.3. What is your best choice?
Thinking on the Margin
Thinking like an economist
Another approach to maximizing net
benefits
Capture the way that benefits and costs
change as the level of activity changes
just a little bit
For any action choice X, the marginal units
are the last DX units, where DX is the
smallest amount you can add or subtract
3-18
Marginal Cost
The marginal cost of an action at an activity
level of X units is equal to the extra cost
incurred due to the marginal units, divided by
the number of marginal units
DC C ( X )  C ( X  DX )
MC 

DX
DX
3-19
Car Repair Example:
Marginal Cost
Marginal cost measures the additional
cost incurred from the marginal units
(DH) of repair time
If C(H) is the total cost of H hour of repair
work, the extra cost of the last DH hours
is DC = C(H) – C(H-DH)
To find marginal cost, divide this extra
cost by the number of extra hours of
repair time, DH
3-20
Car Repair Example:
Marginal Cost
So the marginal cost of an additional hour of
repair time is:
DC C ( H )  C ( H  DH )
MC 

DH
DH
Using the data from Table 3.2, if H= 3, we see:
C (3)  C (2)
MC 
 690  380  310
1
3-21
Car Repair Example:
Marginal Cost Schedule
Table 3.5: Total Cost and Marginal
Cost of Repairing Your Car
Repair
Time
(Hours)
Total Cost
($)
Marginal Cost (MC)
($/hour)
0
0
-
1
150
150
2
380
230
3
690
310
4
1080
390
5
1550
470
6
2100
550
3-22
Marginal Benefit
The marginal benefit of an action at an
activity level of X units is equal to the extra
benefit produced due to the marginal units,
divided by the number of marginal units
DB B( X )  B( X  DX )
MB 

DX
DX
3-23
Car Repair Example:
Marginal Benefit
Marginal benefit measures the additional
benefit gained from the marginal units
(DH) of repair time
This parallels the definition and formula
for marginal cost
3-24
Car Repair Example:
Marginal Benefit
The marginal benefit of an additional hour of
repair time is:
DB B ( H )  B ( H  DH )
MC 

DH
DH
Using the data from Table 3.1, if H= 3, we see:
B (3)  B(3  1)
MC 
 1600  1150  450
1
3-25
Car Repair Example:
Marginal Benefit Schedule
Table 3.6: Total Benefit and Marginal Benefit
of Repairing Your Car
Repair Time
(Hours)
Total Benefit
($)
Marginal Benefit (MB)
($/hour)
0
0
-
1
615
615
2
1150
535
3
1600
450
4
1975
375
5
2270
295
6
2485
215
3-26
Marginal Analysis and Best
Choice
Comparing marginal benefits and
marginal costs can show whether an
increase or decrease in a level of an
activity raises or lowers the net benefit
Increase level if MB of doing so is greater
than MC; if MC of last increase was
greater than MB, decrease level
At the best choice, a small change in
activity level can’t increase the net
benefit
3-27
Marginal Analysis and Best
Choice
Table 3.7: Marginal Benefit and Marginal
Cost of Repairing Your Car
Best
Choice
Repair
Time
(Hours)
Marginal
Benefit (MB)
($/hour)
Marginal
Cost (MC)
($/hour)
0
-
-
1
615
>
150
2
535
>
230
3
450
>
310
4
375
<
390
5
295
<
470
6
215
<
550
3-28
Marginal Analysis with
Finely Divisible Actions
Can conduct the same analysis if choices are
finely divisible by using marginal benefit and
marginal cost curves
Derive marginal benefit and marginal cost from
total benefit and total cost curves
Marginal benefit at H hours of repair time is
equal to the slope of the line drawn tangent to
the total benefit function at point
Usually called simply the “slope of the total
benefit curve” at point D
3-29
Car Repair Example:
Finely Divisible Marginal Benefit
Let DH' = the smallest possible change in hours
of car repair
Adding the last DH‘ of repairs increases total
benefit from point F to point D in Figure 3.4 (on
the next slide), this equal to:
DB  B( H )  B( H  DH )
Recall that marginal benefit is DB' /DH'
Since the vertical axis measures hours of work
and the horizontal axis measures total benefit,
then marginal benefit equals “rise” over “run”
between points F and D.
3-30
Relationship between Total
Benefit and Marginal Benefit
(Figure 3.4)
Total
Benefit
($)
Slope = MB
D
B( H )
Slope = MB = DB' / DH'
DB' '
E
B( H  DH' ' )
DB'
B( H  DH' )
F
H  DH'
Slope
= MB = DB' ' / DH' '
H  DH' '
H
Hours (H)
DH' '
DH'
3-31
Relationship between Total
Benefit and Marginal Benefit
Tangents to the total benefit function at
three different numbers of hours (H = 1,
H = 3, H = 5)
Slope of each tangent equals the
marginal benefit at each number of hours
Figure (b) shows the MB curve: note how
the MB varies with the number of hours
Marginal benefit curve is described by
the function MB(H)= 654-80H
3-32
Relationship between Total
Benefit and Marginal Benefit
(Figure 3.5)
(b): Marginal Benefit
(a): Total Benefit
Slope = MB = 254
Total
Benefit ($)
Marginal Benefit
($/hour)
Slope = MB = 414
654
2270
B
574
Slope = MB = 574
1602
414
MB
254
614
0
1
2
3
Hours (H)
4
5
6
0
1
2
3
4
5
6
Hours (H)
3-33
Relationship between Total Cost
and Marginal Cost
Parallels relationship between total benefit curve
and marginal benefit
When actions are finely divisible, the marginal
cost when choosing action X is equal to the
slope of the total cost curve at X
3-34
Relationship between Total Cost
and Marginal Cost
Tangents to the total cost curve at three
different numbers of hours (H = 1, H = 3,
H = 5)
Slope of each tangent equals the
marginal cost at each number of hours
Figure (b) shows the MC curve: note how
the MC varies with the number of hours
Marginal cost curve is described by the
function MC(H)= 110+80H
3-35
Relationship between Total Cost
and Marginal Cost
(Figure 3.6)
(a): Total Cost
(b): Marginal Cost
Marginal Cost
($/hour)
Total Cost
($)
C
MC
510
1550
350
Slope = MC = 510
Slope = MC = 190
690
190
110
Slope = MC = 350
150
0
1
2
3
Hours (H)
4
5
6
0
1
2
3
4
5
6
Hours (H)
3-36
Marginal Benefit Equals Marginal
Cost at a Best Choice
At the best choice of 3.4 hours, the No
Marginal Improvement Principle holds so
MB = MC
At any number of hours below 3.4, MB >
MC, so a small increase in repair time will
improve the net benefit
At any number of hours above 3.4, MC >
MB, so that a small decrease in repair
time will improve net benefit
3-37
Marginal Benefit Equals Marginal
Cost at a Best Choice (Figure 3.7)
Marginal
Benefit,
Marginal
Cost
($/hour)
654
MC
382
MB
110
0
1
2
Hours (H)
3
4
5
6
3.4
3-38
Slopes of Total Benefit and Total
Cost Curves at the Best Choice
MC = MB at the best choice of 3.4 hours
of repair
Therefore, the slopes of the total benefit
and total cost curves must be equal at
this point
Tangents to the total benefit and total
cost curves show this relationship
3-39
Slope of Total Benefit and Total
Cost Curves
(Figure 3.8)
Total
Benefit
Total
Cost
($)
B
924.80
0
1
2
3
Hours (H)
4
C
5
6
3.4
3-40
Sample Problem 3 (3.8)
Suppose you can hire your mechanic for
up to six hours. The total benefit and total
cost functions are B(H)= 400H1/2 and
C(H) = 100H. The corresponding
formulas for marginal benefit and
marginal cost are MB(H) = 200/H1/2 and
MC(H) = 100. What is your best choice?
Sunk Costs and Decision Making
A sunk cost is a cost that the decision
maker has already incurred, or
A cost that is unavoidable regardless of
what the decision maker does.
Sunk costs affect the total cost of a
decision
Sunk costs do not affect marginal costs
So sunk costs do not affect the best
choice
3-42
Car Repair Example: Best Choice
with a Sunk Cost
Figure 3.9 shows a cost-benefit
comparison for two possible cost
functions with sunk fixed costs: $500 and
$1100.
In both cases, the best choice is H = 3.4:
the level of sunk costs has no effect on
the best choice
Notice that the slopes of the two total
cost curves, and thus the marginal costs,
are the same
3-43
Best Choice with a Sunk Cost
(Figure 3.9)
C´
Total
Benefit,
Total
Cost
($)
B
-175.20
C
424.80
1100
500
0
1
2
3
Hours (H)
4
5
6
3.4
3-44
Sample Problem 4
Let’s say you run a bakery. If you are
month-to-month on your lease, is your
next month’s rent a sunk cost? What if
you are currently in the middle of a oneyear lease?
It will cost you $5,000 to purchase a new
oven. Is this a sunk cost?