Problem:
This problem, adapted from Watkins (1996) [2], illustrates how a Benefit Cost Analysis might be applied to a project such as a highway widening. The improvement of the highway saves travel time and increases safety (by bringing the road to modern standards). But there will almost certainly be more total traffic than was carried by the old highway. This example excludes external costs and benefits, though their addition is a straightforward extension. The data for the “No Expansion” can be collected from off-the-shelf sources. However the “Expansion” column’s data requires the use of forecasting and modeling.
Table 1: Data
No Expansion Expansion Peak
Passenger Trips (per hour) 18,000 24,000
Trip Time (minutes) 50 30
Off-peak
Passenger Trips (per hour) 9,000 10,000
Trip Time (minutes) 35 25
Traffic Fatalities (per year) 2 1
Note: the operating cost for a vehicle is unaffected by the project, and is $4.
Fundamentals of Transportation/Evaluation 71
Table 2: Model Parameters
Peak Value of Time ($/minute) $0.15 Off-Peak Value of Time ($/minute) $0.10 Value of Life ($/life) $3,000,000
What is the benefit-cost relationship?
Solution:
Benefits
Figure 1: Change in Consumers' Surplus
A 50 minute trip at $0.15/minute is
$7.50, while a 30 minute trip is only
$4.50. So for existing users, the expansion saves $3.00/trip. Similarly in the off-peak, the cost of the trip drops from $3.50 to $2.50, saving
$1.00/trip.
Consumers’ surplus increases both for the trips which would have been taken without the project and for the trips which are stimulated by the project
(so-called “induced demand”), as illustrated above in Figure 1. Our analysis is divided into Old and New Trips, the benefits are given in Table 3.
Table 3: Hourly Benefits
TYPE Old trips New Trips Total
Peak $54,000 $9000 $63,000
Off-peak $9,000 $500 $9,500
Note: Old Trips: For trips which would have been taken anyway the benefit of the project equals the value of the time saved multiplied by the number of trips. New Trips: The project lowers the cost of a trip and public responds by increasing the number of trips taken. The benefit to new trips is equal to one half of the value of the time saved multiplied by the increase in the number of trips. There are 250 weekdays (excluding holidays) each year and four rush hours per weekday. There are 1000 peak hours per year. With 8760 hours per year, we get 7760 offpeak hours per year. These numbers permit the calculation of annual benefits (shown in Table 4).
Fundamentals of Transportation/Evaluation 72
Table 4: Annual Travel Time Benefits
TYPE Old trips New Trips Total
Peak $54,000,000 $9,000,000 $63,000,000
Off-peak $69,840,000 $3,880,000 $73,720,000
Total $123,840,000 $12,880,000 $136,720,000
The safety benefits of the project are the product of the number of lives saved multiplied by the value of life. Typical values of life are on the order of $3,000,000 in US transportation analyses. We need to value life to determine how to trade off between safety investments and other investments. While your life is invaluable to you (that is, I could not pay you enough to allow me to kill you), you don’t act that way when considering chance of death rather than certainty. You take risks that have small probabilities of very bad consequences. You do not invest all of your resources in reducing risk, and neither does society. If the project is expected to save one life per year, it has a safety benefit of $3,000,000. In a more complete analysis, we would need to include safety benefits from non-fatal accidents.
The annual benefits of the project are given in Table 5. We assume that this level of benefits continues at a constant rate over the life of the project.
Table 5: Total Annual Benefits
Type of Benefit Value of Benefits Per Year
Time Saving $136,720,000
Reduced Risk $3,000,000
Total $139,720,000
Costs
Highway costs consist of right-of-way, construction, and maintenance. Right-of-way includes the cost of the land and buildings that must be acquired prior to construction. It does not consider the opportunity cost of the right-of-way serving a different purpose. Let the cost of right-of-way be $100 million, which must be paid before construction starts. In principle, part of the right-of-way cost can be recouped if the highway is not rebuilt in place (for instance, a new parallel route is constructed and the old highway can be sold for development). Assume that all of the right-of-way cost is recoverable at the end of the thirty-year lifetime of the project. The $1 billion construction cost is spread uniformly over the first four-years. Maintenance costs $2 million per year once the highway is completed.
The schedule of benefits and costs for the project is given in Table 6.
Fundamentals of Transportation/Evaluation 73
Table 6: Schedule Of Benefits And Costs ($ millions)
Time (year) Benefits Right-of-way costs Construction costs Maintenance costs
0 0 100 0 0
1-4 0 0 250 0
5-29 139.72 0 0 2
30 139.72 -100 0 2
Conversion to Present Value
The benefits and costs are in constant value dollars. Assume the real interest rate (excluding inflation) is 2%. The following equations provide the present value of the streams of benefits and costs.
To compute the Present Value of Benefits in Year 5, we apply equation (2) from above.
To convert that Year 5 value to a Year 1 value, we apply equation (1)
The present value of right-of-way costs is computed as today’s right of way cost ($100 M) minus the present value of the recovery of those costs in Year 30, computed with equation (1):
The present value of the construction costs is computed as the stream of $250M outlays over four years is computed with equation (2):
Maintenance Costs are similar to benefits, in that they fall in the same time periods. They are computed the same way, as follows: To compute the Present Value of $2M in Maintenance Costs in Year 5, we apply equation (2) from above.
To convert that Year 5 value to a Year 1 value, we apply equation (1)
As Table 7 shows, the benefit/cost ratio of 2.5 and the positive net present value of $1563.31 million indicate that the project is worthwhile under these assumptions (value of time, value of life, discount rate, life of the road). Under a different set of assumptions, (e.g. a higher discount rate), the outcome may differ.
Fundamentals of Transportation/Evaluation 74
Table 7: Present Value of Benefits and Costs ($ millions)
Present Value
Which is a more appropriate decision criteria: Benefit/Cost or Benefit - Cost? Why?
Is it only money that matters?
Problem
Is money the only thing that matters in Benefit-Cost Analysis? Is "converted" money the only thing that matters? For example, the value of human life in dollars?
Solution
Absolutely not. A lot of benefits and costs can be converted to monetary value, but not all. For example, you can put a price on human safety, but how can you put a price on, say, asthetics--something that everyone agrees is beneficial.
What else can you think of?
Can small units of time be given the same value of time as larger units of time?
In other words, do 60 improvements each saving a traveler 1 minute equal 1 improvement saving a traveler 60 minutes? Similarly, does 1 improvement saving a 1000 travelers 1 minute equal the value of time of a single traveler of 1000 minutes. These are different problems, one is intra-traveler and one is inter-traveler, but related.
Several issues arise.
A. Is value of time linear or non-linear? To this we must conclude the value of time is surely non-linear. I am much more agitated waiting 3 minutes at a red light than 2, and I begin to suspect the light is broken. Studies of ramp meters show a similar phenomena[3] .
B. How do we apply this in a benefit-cost analysis? If we break one project into 60 smaller projects, each with a smaller value of travel time saved, and then we added the gains, we would get a different result than the what obtains with a single large project. For analytical convenience, we would like our analyses to be additive, not sub-additive, otherwise arbitrarily dividing the project changes the result. In particular many smaller projects will produce an undercount that is quite significant, and result in a much lower benefit than if the projects were bundled.
As a practical matter, every Benefit/Cost Analysis assumes a single value of time, rather than assuming non-linear value of time. This also helps avoiding biasing public investments towards areas with people who have a high value of time (the rich)