# Cost-Benefit Analysis (Rosen Chapter 12 )

2. Suppose Mary uses BART to get to work at a cost of \$1.25 per trip, but would ride a bus if the price were any higher. Her next best alternative is a bus that takes five minutes longer, but costs only \$1. She makes 10 trips per week. The city is considering renovations of the BART system that would reduce the trip by 10 minutes, but fares would rise by \$.40 per trip to cover the costs. The fare increase and reduced travel time both take effect in one year and last forever. The interest rate is 10 percent.

Below is an example of how you might respond to the questions:

a. What is the approximate present value of the projects benefits to Mary? What is the approximate present value of the projects costs to Mary?

a. Mary is willing to pay \$0.25 to save 5 minutes, so she values commute time at \$.05 per minute.

The subway would save her 10 minutes per trip, or \$0.50.

Since she makes10 trips per week this is probably her commute. Most people work about 50 weeks per year. So she would save time on about 500 trips per year. Thus her benefit is about \$250 per year.

The cost of each trip is \$0.40, or \$200 per year.

The annual net benefit to Mary is therefore \$50.

Note Formula for PV of a perpetuity = A/r

The present value to Mary of the benefits = \$250/0.10 = \$2,500;

the present value to Mary of the costs is \$200/0.10 = \$2,000.

b. If there are 10,000 people like Mary, and 10,000 people who do not use any form of public transportation. What are the total benefits and costs of the project? What is the net present value of the project?

b. Total PV benefits = \$2,500 x 10,000 =\$25,000,000.

Total PV costs \$2,000 x 10,000 = \$20,000,000

Net PV = \$5,000,000

Formula for PV of a value in a single year = A/(1+r)t

c. Theres another alternative project that would benefit the other 10,000 people each by a net of \$10 per year for two years. (Assume these benefits are received at the end of each of the years.) What is the present value of the project?

c. PV of net benefits = (10,000x\$10)/1.1 + (10,000x\$10)/(1.1) 2

PV of net benefits = 100,000/1.1 + 100,000/1.21

PV of net benefits = 90,909.09 + 82,644.63 = 173,553.7

d. If you must choose between the projects, assuming both groups of people are weighted equally in the social welfare function, which project do you recommend?

d. Subway project has a higher net present value so choose the subway project.

e. What "distributional weight" would make you indifferent between the projects?

e. Let X = distributional weight.

Set NPV project 1 = NPV of project 2 with weight on benefits to the second group

\$5,000,000 = 173,553.7 X

X = \$5,000,000/173,553.7

X = 28.8