SME
Engineering Economics · Problem type
Incremental benefit–cost
Use this for “Plan 1: PW of benefits $4,200,000, of costs $3,000,000. Plan 2: $5,600,000 and $4,500,000. Which plan do you choose by incremental B/C?”
Chart
This is an example. Type your own values to start a new problem.
ΔB
ΔC
=
$
−
$
$
−
$
=
ΔB/ΔC
PWB and PWC are each plan’s present worths of benefits and of costs; the bars show the differences, plan 2 less plan 1, whose ratio is ΔB/ΔC.
B/C₁
Ratio of 1
calculated
B/C₂
Ratio of 2
calculated
ΔB
Added benefits
calculated · $
ΔC
Added cost
calculated · $
Formulas
B/C₁
=
PWB1
PWC1
1.4
=
4,200,000
3,000,000
B/C₂
=
PWB2
PWC2
1.244
=
5,600,000
4,500,000
ΔB = PWB2 − PWB1
1,400,000 = 5,600,000 − 4,200,000
ΔC = PWC2 − PWC1
1,500,000 = 4,500,000 − 3,000,000
ΔB/ΔC
=
ΔB
ΔC
0.9333
=
1,400,000
1,500,000
Assumptions
•
Alternative 1 is the one that costs less, and its own B/C₁ is at least 1.
•
Take alternative 2 only if ΔB/ΔC ≥ 1, whatever the two ratios alone say.
•
Each PW is worked at the same rate and study period (the Benefit–cost ratio page finds them).
Step-by-step
WE KNOW
PWB1 = $4,200,000, PWC1 = $3,000,000, PWB2 = $5,600,000, PWC2 = $4,500,000
FIND
ratio of 1 (B/C₁), ratio of 2 (B/C₂), added benefits (ΔB), added cost (ΔC), incremental ratio (ΔB/ΔC)
Step 1 · Find ratio of 1 (B/C₁)
Use B/C₁ = PWB1 ÷ PWC1
Alternative 1’s own ratio: its PW of benefits over its PW of costs.
B/C₁
=
4,200,000
3,000,000
B/C₁ = 1.4
Step 2 · Find ratio of 2 (B/C₂)
Use B/C₂ = PWB2 ÷ PWC2
Alternative 2’s own ratio: its PW of benefits over its PW of costs.
B/C₂
=
5,600,000
4,500,000
B/C₂ = 1.244
Step 3 · Find added benefits (ΔB)
Use ΔB = PWB2 − PWB1
The benefits the dearer alternative adds.
ΔB = 5,600,000 − 4,200,000
ΔB = $1,400,000
Step 4 · Find added cost (ΔC)
Use ΔC = PWC2 − PWC1
The extra cost of the dearer alternative; above 0 when 1 costs less.
ΔC = 4,500,000 − 3,000,000
ΔC = $1,500,000
Step 5 · Find incremental ratio (ΔB/ΔC)
Use ΔB/ΔC = ΔB ÷ ΔC
What each added dollar of cost buys in added benefits: 1 or more, take alternative 2.
ΔB/ΔC
=
1,400,000
1,500,000
ΔB/ΔC = 0.9333
Check
Put the numbers back in. Both sides should match.
1.4
=
4,200,000
3,000,000
✓
1.244
=
5,600,000
4,500,000
✓
1,400,000 = 5,600,000 − 4,200,000 ✓
1,500,000 = 4,500,000 − 3,000,000 ✓
0.9333
=
1,400,000
1,500,000
✓
Topic check
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