Blank L., Tarquin A. Engineering Economy (McGraw-Hill Series in Industrial Engineering and Management)
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SECTION 9.3
Alternative Selection
Using Incremental B/C Analysis
There are several special considerations for
B/C
analysis that make it slightly
different from that for
ROR analysis.
As
mentioned earlier, all costs have a pos-
itive sign
in
the
B/C
ratio. Also, the ordering
of
alternatives
is
done on the basis
of
total costs in the denominator
of
the ratio. Thus,
if
two alternatives, A and B,
have equal initial investments and lives, but B has a larger equivalent annual
cost, then B must be incrementally justified against A. (This is illustrated in the
next example.)
If
this convention is not correctly followed, it is possible
to
get a
negative cost value in the denominator, which can incorrectly make
B/C
< 1 and
reject a higher-cost alternative that is actually justified.
Follow these steps to correctly perform a conventional
B/C
ratio analysis
of
two alternatives. Equivalent values can be expressed in
PW,
AW,
or FW terms.
l.
Determine the total equivalent costs for both alternatives.
2.
Order the alternatives by total equivalent cost; smaller first, then larger.
Calculate the incremental cost
(~C)
for the larger-cost alternative. This is
the denominator in
B/C.
3. Calculate the total equivalent benefits and any disbenefits estimated for
both alternatives. Calculate the incremental benefits
(~B)
for the larger-
cost alternative. (This is
~(B
-
D)
if
dis benefits are considered.)
4.
Calculate the incremental
B/C
ratio using Equation [9.2], (B -
D)/
c.
5.
Use the selection guideline
to
select the higher-cost alternative
ifB/C
:::::
1.0.
When the
B/C
ratio
is
determined for the lower-cost alternative, it is a compari-
son with the do-nothing (DN) alternative.
If
B/C
< l.0, then DN should be
selected and compared
to
the second alternative.
If
neither alternative has an
acceptable
B/C
value, the DN alternative must be selected. In public sector
analysis, the DN alternative
is
usually the current condition.
EXAMPLE
9.4
.
The city
of
Garden Ridge, Florida, has received designs for a new patient room wing to
the municipal hospital from two architectural consultants.
One
of
the two designs must
be accepted in order to announce it for construction bids. The costs and benefits are the
same
in
most categories, but the city financial manager decided that the three estimates
below should be considered
to
determine which design to recommend at the city council
meeting next week and to present to the citizenry
in
preparation for an upcoming bond
referendum next month.
Construction cost, $
Building maintenance cost, $/year
Patient usage cost, $/year
Design A
10,000,000
35,000
450,000
Design 8
15,000,000
55,000
200,000
The patient usage cost
is
an estimate
of
the amount paid by patients
over
the insurance
coverage generally allowed for a hospital room. The discount rate is 5%, and the life
of
325
Incremental ROR
326
CHAPTER 9 Benefit/Cost Analysis and Public Sector Economics
the building is estimated at
30 years.
(a) Use conventional BtC ratio analysis to select design A or B.
(b) Once the two designs were publicized, the privately owned hospital in the di-
rectly adjacent city
of
Forest Glen lodged a complaint that design A will reduce
its own municipal hospital's income by an estimated
$500,000 per year because
some
of
the day-surgery features
of
design A duplicate its services. Subsequently,
the Garden Ridge merchants' association argued that design B could reduce its
annual revenue by
an
estimated $400,000, because it will eliminate
an
entire
parking lot used by their patrons for short-term parking. The city financial man-
ager stated that these concerns would be entered into the evaluation
as
disbene-
fits
of
the respective designs. Redo the BtC analysis to determine if the economic
decision
is
still the same
as
when disbenefits were not considered.
Solution
(a) Since most
of
the cash flows are already annualized, the incremental BtC ratio
will use
AW
values. No disbenefit estimates are considered. Follow the steps
of
the procedure above:
1.
The
AW
of
costs is the sum
of
construction and maintenance costs.
A W
A
==
1
O,OOO,OOO(A/
P,5%,30) + 35,000 = $685,500
AWe
=
15,OOO,000(A/P
,
5%
,
30)
+ 55,000 = $1,030,750
2.
Design B has the larger
AW
of costs, so it
is
the alternative to
be
incre-
mentally justified. The incremental cost value is
ilC
=
AWe
-
AW
A = $345,250 per year
3.
The
AW
of
benefits is derived from the patient usage costs, since these are
consequences to the public. The benefits for the
BtC analysis are not the
costs themselves, but the
difference
if
design B is selected. The lower
usage cost each year
is
a positive benefit for design
B-
ilB = usage
A
-
usageB
= $450,000 - $200,000 = $250,000 per year
4.
The incremental BtC ratio
is
calculated by Equation [9.2].
BtC
==
. $250,000 = 0.72
$345,250
5.
The BtC ratio
is
less than 1.0, indicating that the extra costs associated
with design
Bare
not justified. Therefore, design A is selected for the con-
struction bid.
(b) The revenue loss estimates are considered disbenefits. Since the dis benefits of
design
Bare
$100,000 less than those of
A,
this positive difference
is
added to
the
$250,000 benefits
of
B to give it a total benefit of $350,000. Now
BtC = $350,000 = 1
OJ
$345,250 .
Design B
is
slightly favored.
In
this case the inclusion of disbenefits has reversed
the previous economic decision.
Thi.s
has probably made the situation more
di
f-
ficult politically. New disbenefits will surely be claimed in the near future
by
other special-interest groups.
SECTION 9.4 Incremental B/C Analysis
of
Multiple, Mutually Exclusive Alternatives
Like other methods, B/C analysis requires equal-service comparison
of
alter-
natives.
Usually, the expected useful life
of
a public project is long (25 or 30 or
more years), so alternatives generally have equal lives. However, when alterna-
tives do have unequal lives, the use
of
PW to determine the equivalent costs and
benefits requires that the
LCM
of
lives be used. This is an excellent opportunity
to use the A W equivalency
of
costs and benefits,
if
the implied assumption that
the project could be repeated is reasonable. Therefore, use AW-based analysis for
B/C ratios when different-life alternatives are compared.
9.4
INCREMENTAL SIC ANALYSIS
OF
MULTIPLE,
MUTUALLY EXCLUSIVE ALTERNATIVES
The procedure necessary to select one from three or more mutually exclusive
alternatives using incremental B/C analysis is essentially the same
as
that
of
the
last section. The procedure also parallels that for incremental
ROR analysis in
Section 8.6. The selection guideline
is
as
follows:
Choose the largest-cost alternative that is justified with an incremental
B/C
2:: 1.0 when this selected alternative has been compared with another
justified alternative.
There are two types
of
benefit
estimates-estimation
of
direct benefits, and
implied benefits based on usage cost estimates. Example 9.4 is a good illustra-
tion
of
the second type
of
implied benefit estimation. When direct benefits are
estimated, the
B/C
ratio for each alternative may be calculated first as an initial
screening mechanism to eliminate unacceptable alternatives. At least one alter-
native must have B/C
2:: 1.0 to perform the incremental B/C analysis.
If
all alter-
natives are unacceptable, the DN alternative
is
indicated as the choice. (This
is
the same approach as that
of
step 2 for "revenue alternatives only" in the
ROR
procedure
of
Section 8.6. However, the term "revenue alternative"
is
not applic-
able to public sector projects.)
As
in
the previous section comparing two alternatives, selection from multi-
ple alternatives by incremental B/C ratio utilizes total equivalent costs to initially
order alternatives from smallest to largest.
Pairwise comparison
is
then under-
taken. Also, remember that all costs are considered positive in B/C calculations.
The terms defender and challenger alternative are used in this procedure, as
in
a ROR-based analysis. The procedure for incremental B/C analysis
of
multiple
alternatives
is
as follows:
1.
Determine the total equivalent cost for all alternatives. (Use
AW,
PW, or
FW equivalencies for equal lives; use AW for unequal lives.)
2.
Order the alternatives by total equivalent cost, smallest first.
3.
Determine the total equivalent benefits (and any dis benefits estimated) for
each alternative.
4. Direct benefits estimation only: Calculate the
B/C
for the first ordered
alternative. (In effect, this makes
DN
the defender and the first alternative
the challenger.)
If
B/C <
1.0
, eliminate the challenger, and go to the next
327
Incremental
ROR
328
CHAPTER 9
Benefit/Cost Analysis and
Public Sector Economics
challenger. Repeat this until B/C ~ 1.0. The defender is eliminated, and the
next alternative is now the challenger. (For analysis
by
computer, deter-
mine the B/C for all alternatives initially and retain only acceptable ones
.)
5.
Calculate incremental costs
(~C)
and benefits
(~B)
using the relations
~c
= challenger cost - defender cost
~B
= challenger benefits - defender benefits
[9.4]
[9.5]
If
relative usage costs are estimated for each alternative, rather than direct
benefits,
~B
may be found using the relation
~B
= defender usage costs - challenger usage costs [9.6]
6.
Calculate the incremental B/C for the first challenger compared
to
the
defender.
B/C
=
~B/~C
[9.7]
If
incremental B/C
~
1.0 in Equation [9.7], the challenger becomes the
defender and the previous defender is eliminated. Conversely, if incremen-
tal B/C < 1.0, remove the challenger and the defender remains against the
next challenger.
7.
Repeat steps 5 and 6 until only one alternative remains. It is the selected one.
In all the steps above, incremental disbenefits may be considered by replacing
~B
with
~(B
- D),
as
in
the conventional B/C ratio, Equation [9.2].
EXAMPLE
9.5
..
"
The Economic Development Corporation (EDC) for the city
of
Bahia, California, and
Moderna County
is
operated
as
a not-for-profit corporation. It
is
seeking a developer that
wi
ll
place a major water park
in
the city or county area. Financial incentives will
be
awarded. In response to a request for proposal (RFP) to the major water park developers
in
the country, four proposals have been received. Larger and more intricate water rides and
in
creased size
of
the park will attract more customers, thus different levels
of
initial ince
n-
tives are requested
in
the proposals. One
of
these proposals will be accepted by the EDC
and recommended to the Bahia City Council and Moderna County Board
of
Trustees for
approval.
Approved and in-place economic incentive guidelines allow entertainment industry
prospects to receive up to
$500,000 cash as a first-year incentive award and
10
%
of
this
amount each year for 8 years
in
property tax reduction. All the proposals meet the require-
ments for these two incentives. Each proposal includes a provision that residents
of
the city
or county
wi
ll
benefit from reduced entrance (usage) fees when using the park. This fee re-
duction will be in effect
as
long
as
the property tax reduction incentive continues. The EDC
has estimated the annual total entrance fees with the reduction included for local residents.
Also, EDC estimated the extra sales tax revenue expected for the four park designs. These
estimates and the costs for the initial incentive and annual 10% tax reduction are summa-
rized
in
the top section
of
Table 9-1.
SECTION 9.4
Incremental B/C Analysis
of
Multiple, Mutually Exclusive Alternatives
TABLE
9-1
Estimates
of
Costs and Benefits, and the Incremental BIC Analysis for Four Water
Park Proposals, Example 9.5
Proposal 1
Proposal 2 Proposal 3
Initial incentive, $ 250,000 350,000
500,000
Tax incentive cost, $/year 25,000 35,000
50,000
Resident entrance fees, $/year 500,000 450,000 425,000
Extra sales taxes, $/year
310,000 320,000
320,000
Study
period, years
8 8
8
A W of total costs, $
66,867 93,614 l33,735
Alternatives compared
2-to-l 3-to-2
Incremental costs
ilC
, $/year 26,747
40
,1
20
Entrance fee reduction, $/year 50,000
25,000
Extra sales tax, $/year 10,000
0
Incremental benefits ilB, $/year 60,000
25,000
Incremental B/C ratio 2.24
0.62
Increment justified?
Yes
No
Alternative selected 2
2
Utilize hand and computer analysis
to
perform
an
incremental B/C study to determine
which park proposal is the best economically. The discount rate used
by
the EDC
is
7% per
year. Can the
ClUTent
incentive guidelines be used to accept the winning proposal?
Solution
by
Hand
The viewpoint taken for the economic analysis is that
of
a resident
of
the city or county.
The first-year cash incentives and annual tax reduction incentives are real costs to the res-
idents. Benefits are derived from two components: the decreased entrance fee estimates
and the increased sales tax receipts. These will benefit each citizen indirectly through the
increase
in
money available to those who use the park and through the city and county
budgets where sales tax receipts are deposited.
Since these benefits must be calculated
in-
directly from these two components, the initial proposal B/C values cannot be calculated to
initially eliminate any proposals. A B/C analysis incrementally comparing two alternatives
at a time must be conducted.
Table 9- 1 includes the results
of
applying the procedure above. Equivalent
AW
values
are used for benefit and cost amounts per year.
Since the benefits must be derived indirectly
from the entrance fee estimates and sales tax receipts, step 4 is not used.
1.
For each alternative, the capital recovery amount over 8 years
is
determined and
added to the annual property tax incentive cost. For proposal #1,
AW
of
total costs = initial incentive(A/P,7%,8) + tax cost
= $250,000(A/ P,7%,8) + 25,000 = $66,867
Proposal 4
800,000
80,000
250,000
340,000
8
2l3,976
4-to-2
120,360
200,000
20,000
220,000
1.83
Yes
4
329
330
~
E-Solve
CHAPTER 9 Benefit/Cost Analysis and Public Sector Economics
2.
The alternatives are ordered
by
the
AW
of
total costs in Table 9-1.
3.
The annual benefit
of
an
alternative
is
the incremental benefit
of
the entrance fees and
sales tax amounts. These are calculated in step 5.
4.
This step
is
not used.
5.
Table
9-1
shows incremental costs calculated
by
Equation [9.4]. For the 2-to-l
comparison,
~c
= $93,614 - 66,867 = $26,747
Incremental benefits for
an
alternative are the sum
of
the resident entrance fees com-
pared
to
those
of
the next-lower-cost alternative, plus the increase in sales tax receipts
over those
of
the next-lower-cost alternative. Thus, the benefits are determined incre-
mentally for each pair of alternatives. For example, when proposal #2
is
compared
to
proposal #1, the resident entrance fees decrease
by
$50,000 per year and the sales
tax
receipts increase by $10,000. Then the total benefit is the sum
of
these, that is,
~B
= $60,000 per year.
6.
For the 2-to-l comparison, Equation
[9
.
7]
results in
B/C = $60,000/$26,747 = 2.24
Alternative #2 is clearly incrementally justified. Alternative
#1
is eliminated, and
alternative
#3
is the new challenger
to
defender #2.
7.
This process
is
repeated for the 3-to-2 comparison, which has an incremental B/C
of
0.62 because the incremental benefits are substantially less than the increase in costs.
Therefore, proposal #3
is
eliminated, and the 4-to-2 comparison results in
B/C = $220,000/$120,360 = 1.83
Since
B/C > 1.0, proposal #4 is retained. Since proposal #4 is the one remaining
alternative, it
is
selected.
The recommendation for proposal #4 requires an initial incentive
of
$800,000, which
exceeds the
$500,000 limit
of
the approved incentive limits. The EDC will have
to
request
the City Council and County Trustees
to
grant
an
exception to the guidelines.
If
the excep-
tion
is
not approved, proposal #2 is accepted.
Solution by
Computer
Figure
9-2
presents a spreadsheet using the same calculations
as
those in Table 9-1. Row 8
cells include the function PMT(7%,8,
-initial
incentive) to calculate the capital recovery
for each alternative, plus the annual
tax
cost. These
AW
of
total cost values are used
to
order the alternatives for incremental comparison.
The cell tags for rows 10 through
13
detail the formulas for incremental costs and
benefits used in the incremental
B/C computation (row 14). Note the difference in row
11
and 12 formulas, which find the incremental benefits for entrance fees and sales tax, re-
spectively. The order
of
the subtraction between columns
in
row
11
(e.g., =
B5
- C5, for
the
2-to-1 comparison) must be correct
to
obtain the incremental entrance fees benefit. The
IF operators
in
row
15
accept or reject the challenger, based upon the size
of
B/C. After
the 3-to-2 comparison with
B/C = 0.62
in
cell D14, alternative #3 is eliminated. The final
selection is alternative #4, as in the solution
by
hand.
SECTION 9.4 Incremental B/C Analysis
of
Multiple, Mutually Exclusive Alternatives
X Microsoft
Excel-
Example
9_5
.
I!I~Ei
Figure
9-2
Spreadsheet so
lu
ti
on
for an incremental B/C analysis
of
four mutually exclusive alternatives,
Example 9.5.
When the lives
of
alternatives are so long that they can be considered infinite,
the capitalized cost
is
used to calculate the equivalent PW or
AW
values for costs
and benefits. Equation [5.3], A = P(i), is used to determine the equivalent
AW
values in the incremental
B/C
analysis.
If
two or more independent projects are evaluated using
B/C
analysis
and there is
no
budget limitation,
no
incremental comparison is necessary.
The only comparison is between each project separately with the do-nothing
alternative. The project B/C values are calculated, and those with B/C
:2::
1.0
are accepted. This is the same procedure as that used to select from indepen-
dent projects using the
ROR
method (Chapter 8). When a budget limitation is
imposed, the capital budgeting procedure discussed in Chapter
12
must be
applied.
I
Sec.5.5
I
Capitalized cost
331
332
CHAPTER
9 Benefit/Cost Analysis and Public Sector Economics
The Army Corps
of
Engineers wants to construct a
dam
on a flood-prone river. The
estimated construction cost and average annual dollar benefits are listed below.
(a)
If
a 6%
per
year rate applies and dam life is infinite for analysis purposes, select the one best location
using the
B/C method.
If
no site is acceptable, other sites will
be
determined later.
(b)
If
more
than one dam site can be selected, which sites are acceptable, using the
B/C method?
Construction Cost,
Annual
Site
$ millions Benefits, $
A 6 350,000
B 8 420,000
C
3
125,000
D
10
400,000
E 5
350,000
F
11
700,000
Solution
(a) The capitalized cost A =
Pi
is used to obtain AW values for annual capital recovery
of
the construction cost, as shown in the first row
of
Table
9-2.
Since benefits are
estimated directly, the site
B/C ratio can
be
used for initial screening. Only sites E
and F have
B/C > 1.0, so they are evaluated incrementally.
The
E-to-DN comparison
is performed because
it is not required that
one
site must be selected. The analysis be-
tween the mutually exclusive alternatives in the lower portion
of
Table
9-2
is based
on Equation [9.7].
Incremental
B/C =
_Ll_a_n_n_ua_l_b_e_n_efi_t_s
Ll
annual costs
Since only site E is incrementally justified, it is selected.
(b) The dam site proposals are now independent projects. The site B/C ratio is used to se-
lect from none to all six sites. In Table
9-2,
B/C > 1.0 for sites E and F only; they are
acceptable, the rest are not.
TABLE
9-2
Use
of
Incremental B/C Ratio Analysis for Example 9.6 (Values
in
$ I 000)
C
E A B D F
Capital recovery cost, $
180 300 360
480
600
660
Annual benefits, $ 125
350
350
420
400
700
Site
B/C
0.69 1.17 0.97 0.88 0.67
l.06
Decision No Retain
No No No
Retain
Comparison E-to-DN F-to-E
L1
Annual cost, $
300
360
L1
Annual benefits, $
350 350
L1
(B/C)
ratio 1.17
0.97
Increment justified? Yes
No
Site selected E E
PROBLEMS
Comment
In part (a), suppose that site G is added with a construction cost
of
$10 million and an
annual benefit
of
$700,000. The site B/C is acceptable at B/C =
700/600
= 1.17. Now, in-
crementally compare
G-to-E; the incremental B/C =
350/300
= 1.17, in favor
of
G.
In this
case, site F must be compared with G. Since the annnal benefits
are the same ($700,000),
the B/C ratio
is
zero and the added investment is not justified. Therefore, site G
is
chosen.
CHAPTER
SUMMARY
The benefit/cost method is used primarily to evaluate projects and to select from
alternatives
in
the public sector. When one
is
comparing mutually exclusive al-
ternatives, the incremental B/C ratio must be greater than or equal to
l.0
for the
incremental equivalent total cost to be economically justified. The
PW,
AW,
or
FW
of the initial costs and estimated benefits can be used to perform an incre-
mental B/C analysis.
If
alternative lives are unequal, the
AW
values should be
used, provided the assumption
of
project repetition is not unreasonable. For inde-
pendent projects, no incremental B/C analysis is necessary. All projects with
B/C
2:
1.0 are selected provided there is no budget limitation.
Public sector economics are substantially different from those
of
the private
sector. For public sector projects, the initial costs are usually large, the expected
life is long (25,
35
, or more years), and the sources for capital are usually a combi-
nation
of
taxes levied on the citizenry, user fees, bond issues, and private lenders.
It is very difficult to make accurate estimates
of
benefits
for
a public sector project.
The interest rates, called the discount rates in the public sector, are lower than
those for corporate capital financing. Although the discount rate is
as
important to
establish as the MARR, it can be difficult
to
establish, because various government
agencies qualify for different rates. Standardized discount rates are established
for some federal agencies.
PROBLEMS
333
Public Sector Economics
9.1
State the difference between public and
private sector alternatives with respect
to
the following characteristics.
9.2 Indicate whether the following character-
istics are primarily associated with public
sector or private sector projects.
(a) Size
of
investment
(b)
Life
of
project
(c) Funding
(d)
MARR
(a) Profits
(b)
Taxes
(c) Disbenefits
(d)
Infinite life
(e)
User fees
(f)
Corporate bonds
334
CHAPTER
9
Benefit/Cost Analysis and
Public Sector Economics
9.3 Identify
each
cash flow as a benefit, dis-
bene
fit
,
or
cost.
(a) $
500
,
000
annual income from
tourism created by a freshwater
reservoir
(b)
$
700,000
per
year
maintenance by
container ship port authority
(c)
Expenditure
of
$45 million for tun-
nel construction
on
an interstate
hi
ghway
(d)
Elimination
of
$1.3 million in
salaries for
co
unty residents based
on reduced international trade
(e) Reduction
of
$375,000 per year in
car
accident repairs
because
of
im-
proved lighting
(f)
$700,000 per year loss
of
revenue
by
farmers because
of
highway
right-of-
way purchases
9.4 During its
20
years
in
business,
Deware
Co
nstruction
Company
has always devel-
oped its contracts under a fixed-fee
or
cost-plus arrangement.
Now
it has been
offered an opportunity to participate in a
proj
ect
to provide cross-country
highway
transportation in an international
se
tting,
speci fically, a
country
in Africa.
If
ac-
ce
pted, D
eware
will
work
as subcontractor
to a lar
ge
r European corporation, and the
BOT
form
of
contracting will
be
used with
the African country government.
Describe
for the president of
Dew
are at least
four
of
the signi ficant
diff
erences that may be ex-
pected when the
BOT
format is utilized in
I ieu
of
its more traditional forms
of
con-
tract maki ng.
9.5
If
a corporation accepts the
BOT
form
of
contracting,
(a)
identify two risks taken by
a corporation and
(b) state how these risks
can be reduced by the
government
partner.
Project
B/C
Value
9.6
The
estimated annual cash flows for a pro-
posed city
gove
rnment
project are costs
of
$450,000
per year, benefits
of
$600,000
per
year, and disbenefits
of
$100,000
per
year.
Determine
the (a) B/C ratio and
(b)
value
of
B -
C.
9.7 Use spreadsheet
software
such as Excel,
PW
analysis, and a
discount
rate
of
5% per
year
to determine
that
the B/C value for
the following estimates is 0.375, making
the project not acceptable us
in
g the ben
efi
t!
cost method. (a)
Enter
the values and
equations on the spreadsheet so they may
be
changed
for the
purpose
of
sensitivity
analysis.
First
cost
= $8 million
Annual
cost
=
$800,000
per year
Benefit
=
$550,000
per
year
Disbenefit =
$100
,
000
per
year
(b)
Do
the following sensitivity analysis
by changing only
two
cells on
your
spreadsheet.
Change
the
discount rate to
3%
per year,
and
adjust
the annual
cost
es-
timate until
B/C =
l.023
.
This
make
s the
project
just
acceptable using benefit!cost
analysis.
9.8 A proposed regulation regarding the re-
moval
of
arsenic
from
drinking water
is
expected to
have
an annual
cost
of
$200
per household
per
year.
If
it
is assumed
that there are
90
million households
in
the
country and that the regulation could save
12
lives per year,
what
would the value
of
a human life have to
be
for the B/C ratio to
be equal to
1.0?
9.9
The
U.S. Environmental Protection Agency
has established that 2.5%
of
the median
household
income
is a
reasonable
amount
to pay for safe drinking water.
The
median
hou
se
hold
income
is
$30,000
per year. For
a regulation that
would
affect the health
of
people in 1 %
of
the households,
what
would the health benefits have to equal
in
dollars per
household
(for that 1 %
of