# Benefit Cost Example

Suppose Modesto is thinking about enhancing park facilities with an initial cost of \$340,000 to install and \$3,000 a year to maintain (which must be paid at the end of each year). About 10,000 people currently visit these parks about once each year and their enjoyment would be increased by about \$1.00 on each visit. Another 5,000 that were reluctant to visit without these facilities will now also visit the parks about 10 times a year and receive about \$1.00 worth of pleasure each time. The interest rate is 5 percent.

a) What is the approximate present value of the projects costs?

PV (costs) = installation + maintenance

PV (costs) = \$340,000 now + 3,000 a year forever

Note formula for a perpetuity PV=A/r

\$340,000 now + 3,000/.05

\$340,000 + 60,000

PV (costs) = \$400,000

What is the approximate net present value of the project based on the data you have?

NPV = PV (benefits) PV(costs)

Each year benefits = 10,000 people x 1 visits x \$1 + 5,000 people x 10 visits x \$1

Each year benefits = \$60,000

These benefits presumably occur each year so again we us the formula for the present value of a perpetuity: PV=A/r

PV (benefits) = \$60,000/.05

PV (benefits) = \$1,200,000

NPV = \$1,200,000 400,000

NPV = \$800,000

Is the project admissible? Yes since NPV is positive.

b) Suppose the maintenance cost estimates and benefit estimates were in current dollars and the interest rate of 5% was in nominal terms. Someone points out they expect 2% inflation. How would your estimates of the approximate present value of the projects costs, benefits, and net present value change?

Real interest rate = Nominal interest rate inflation rate

Real interest rate = 5% - 2% = 3%

PV (costs) = \$340,000 now + 3,000/.03 = \$440,000

PV (benefits) = \$60,000/.03 = \$2,000,000

NPV = \$2,000,000 440,000 = \$1,560,000

c) Suppose the facilities cost \$340,000 to install, but even with the maintenance after three years they would have to be scrapped. The salvage value at that point is just equal to the costs of removal. What would be the approximate present value of the projects costs?

Note formula for a costs in a single period in the future PV=A/(1+r)t

PV (costs) = \$340,000 now + 3,000/(1+.03)+3,000/(1+.03)2 +3,000/(1+.03)3

PV (costs) = \$340,000 + 2,912.62 + 2,827.78 + 2,745.43 = \$348,485.83

d) Theres another project that would benefit a different group of 10,000 people \$100 each immediately at a total cost of \$600,000. Unfortunately, due to insufficient budget you can only do one of the projects. What "distributional weight" would make you indifferent between the original project in part a) and this new project?

Since this new project has benefits of \$100 immediately x 10,000 people, PV(benefits)= 1,000,000

And since the costs occur now the PV(costs) = 600,000 so the NPV = 400,000. Thus normally you would prefer the first project since it had a NPV=800,000.

To find the distributional weight that would make you indifferent between the projects look for the weight X that would make the NPVs equal.

So compared to a) need weight X such that

\$400,000 X = \$800,000

X = \$800,000/ 400,000

X= 2.