INDR 202
ENGINEERING ECONOMICS
CHAPTER 8
BENEFIT-COST ANALYSIS
SPRING 2015
BENEFIT-COST ANALYSIS
Evaluation of Public Projects
Benefit-Cost Ratio
Incremental B/C Analysis
Profitability Index
EVALUATION OF PUBLIC PROJECTS
Social benefits vs. social costs
Nonmonetary benefits
EVALUATION OF PUBLIC PROJECTS
TYPICAL GOALS
Maximize benefits for given costs
BENEFITS & COSTS
USERS’ BENEFITS & DISBENEFITS
Primary (New businesses) & Secondary (economic growth)
B = Benefits - Disbenefits
SPONSOR’ S COSTS
Expenditures & Savings
C = Capital Costs + Annual Operating Costs – Revenues (toll)
SOCIAL DISCOUNT RATE
(analogue of MARR)
Ø Government borrowing rate if no private counterpart Ø Earning rate from private sector if there are private
BENEFITS & COSTS
𝐵 = #
𝑏
%1 + 𝑖
% )%*+
𝐶 = #
𝑐
%1 + 𝑖
% )%*+
𝑏% benefits at the end of period 𝑛 𝑐% expenses at the end of period 𝑛
𝑁 project life
BENEFITS & COSTS
𝐼 = #
𝑐
%1 + 𝑖
% 1%*+
𝐶
2= #
𝑐
%1 + 𝑖
% )%*134
𝐶 = 𝐼 + 𝐶
2Here, we assume that a series of initial investments is required during the first 𝐾
BENEFIT-COST RATIO (B/C)
𝐵𝐶 𝑖 =
𝐵
𝐶
=
𝐵
𝐼 + 𝐶
2Well-defined if 𝐶 > 0
If 𝐵𝐶 𝑖 > 1, then 𝑃𝑊 𝑖 > 0.
A project is acceptable if 𝐵𝐶 𝑖 > 1.
BENEFIT-COST ANALYSIS
9
STEP 1: All Users’ Benefits & Disbenefits
Quantify all users’ benefits & disbenefits in same unit. Compute users’ net benefit.
STEP 2: Sponsor’s Costs
Identify sponsor’s recurring & non-recurring costs. Determine annual revenues, e.g., toll revenues. Compute sponsor’s net annual cost.
STEP 3: Benefit-Cost Analysis Specify the interest rate.
Calculate users’ equivalent net benefit. Calculate sponsor’s equivalent net cost. Compute benefit-cost ratio 𝐵𝐶 𝑖 .
EXAMPLE 1: BENEFIT-COST ANALYSIS
𝑛 𝑏% 𝑐%
0 $10
1 $10
2 $20 $5
3 $30 $5
4 $30 $8
5 $20 $8
𝐵 = $71.98
𝐶 = $37.41
𝐵𝐶 10% = 1.92 > 1
Accept the project.
𝑖 = 10%
In this example,
INCREMENTAL B/C ANALYSIS
STEP 1. Eliminate alternatives with B/C < 1.
STEP 2. Rank remaining alternatives in increasing order of 𝐼 + 𝐶′.
STEP 3. Compute differences for 𝑗, 𝑘 with 𝐼J + 𝐶J2 > 𝐼K + 𝐶K2.
∆𝐵 = 𝐵J − 𝐵K, ∆𝐼 = 𝐼J − 𝐼K, ∆𝐶2 = 𝐶J2 − 𝐶K2
STEP 4. Compute B/C ratio for 𝑘 − 𝑗.
𝐵𝐶 𝑖 JMK = ∆𝐵 ∆𝐼 + ∆𝐶2
INCREMENTAL B/C ANALYSIS
𝐵𝐶 𝑖
JMK=
∆𝐵
∆𝐼 + ∆𝐶
2If ∆𝐼 + ∆𝐶2 = 0, select the alternative with largest 𝐵.
Calculate 𝑩 & 𝑪 on annual basis for repeatable
EXAMPLE 2: INCREMENTAL B/C ANALYSIS
A1 A2 A3
𝐵 $12,000 $35,000 $21,000
𝐼 $5,000 $20,000 $14,000
𝐶′ $4,000 $8,000 $1,000
EXAMPLE 2: INCREMENTAL B/C ANALYSIS
A1 A2 A3
𝑃𝑊 𝑖 $3,000 $7,000 $6,000
𝐵𝐶 𝑖 1.33 1.25 1.40
Independent projects: all acceptable
EXAMPLE 2: INCREMENTAL B/C ANALYSIS
Ranking
Base A1 A3 A2
𝐼 + 𝐶2 $9,000 $15,000 $28,000
A1 vs. A3: Prefer A3 over A1 since
𝐵𝐶 10% PM4 = $21,000 − $12,000
$15,000 − $9,000 = 1.50 > 1.
A3 vs. A2: Prefer A2 over A3 since
PROFITABILITY INDEX (PI)
𝑃𝐼 𝑖 =
𝐵 − 𝐶
2
𝐼
BC>(<)1 ↔ PI>(<)1
BC: to see whether the benefits outweight the costs
PI: to quantify the amount of value created per unit of investment
Well-defined if 𝐼 > 0
If 𝑃𝐼 𝑖 > 1, then 𝑃𝑊 𝑖 > 0.
EXAMPLE 3: PROFITABILITY INDEX
𝑛 𝑏% 𝑐%
0 $10
1 $20
2 $35 $5
3 $30 $5
4 $20 $7
5 $10 $7
𝐵 − 𝐶2 = $58.66
𝐼 = $28.69
𝑃𝐼 7% = 2.04 > 1
Accept the project.
𝑖 = 7%
EXAMPLE 4: INCREMENTAL PI ANALYSIS
A1 A2 A3
𝐵 $12,000 $35,000 $21,000
𝐼 $5,000 $20,000 $14,000
𝐶′ $4,000 $8,000 $1,000
𝑃𝐼 𝑖 1.60 1.35 1.43
EXAMPLE 2: INCREMENTAL B/C ANALYSIS
A1 A2 A3
𝑃𝑊 𝑖 $3,000 $7,000 $6,000
𝑃𝐼 𝑖 1.60 1.35 1.43
Independent projects: all acceptable
EXAMPLE 2: INCREMENTAL PI ANALYSIS
23
Ranking
Base A1 A3 A2
𝐼 $5,000 $14,000 $20,000
A1 vs. A3: Prefer A3 over A1 since
𝑃𝐼 10% PM4 = $20,000 − $8,000
$14,000 − $5,000 = 1.33 > 1.
A3 vs. A2: Prefer A2 over A3 since
𝑃𝐼 10% RMP = $27,000 − $20,000
$20,000 − $14,000 = 1.17 > 1.
SUMMARY
Benefit-cost analysis is typically used in evaluation of public projects.
Challenges in public-project analysis include identifying all the users distinctly from the project sponsor, the benefits & disbenefits of all involved parties, quantifying nonmonetary benefits & disbenefits, choosing the appropriate interest rate.
Benefit-cost ratio (B/C) of a project with 𝐼 + 𝐶2 > 0 is:
𝐵𝐶 𝑖 = 𝐵
𝐶 =
𝐵
SUMMARY
Profitability index (PI) is the analogue of B/C in private sector. It measures capital efficiency. For 𝐼 > 0:
𝑃𝐼 𝑖 = 𝐵 − 𝐶
2
𝐼 .
A project is acceptable if 𝑃𝐼 𝑖 > 1.
