Suppose Stanislaus is thinking about enhancing park facilities with an initial cost of \$200,000 to install and \$1,000 a year to maintain (which must be paid at the end of each year).  About 5,000 people currently visit these parks about 2 times each year and their enjoyment would be increased by about \$1.00 on each visit.  Another thousand 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 project’s costs?

### PV (costs) =  installation+ maintenance

\$200,000 now + 1,000 a year forever              Note formula for a perpetuity PV=A/r

\$200,000 now + 1,000/.05

\$200,000 + 20,000

# PV (costs) =  \$220,000

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

NPV = PV (benefits) – PV(costs)

Each year benefits =  5,000 people x 2 visits x \$1   + 1,000 people x 10 visits x \$1

Each year benefits =  \$20,000

PV (benefits) = \$20,000/.05

### PV (benefits) = \$400,000

\ NPV = \$400,000 – 220,000

# NPV = \$180,000

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 3% inflation.  How would your estimates of the approximate present value of the project’s costs, benefits, and net present value change?

Real interest rate = Nominal interest rate – inflation rate

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

PV (costs) =  \$200,000 now + 1,000/.02 = \$250,000

PV (benefits) = \$20,000/.02  = \$1,000,000

\ NPV = \$1,000,000 – 250,000 = \$750,000

c)    Suppose the facilities cost \$200,000 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 project’s costs? What would be the approximate present value of the project’s benefits? Is the project admissible?

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

## PV (costs) =  \$200,000 + 980 + 961+ 942 =  \$202,883

PV (benefits) =  20,000/(1+.02) + 20,000/(1+.02)2 + 20,000/(1+.02)3   =  \$57,677

\ NPV = \$57,677– \$202,883= -\$145,206

d)   There’s another project that would benefit a different group of 1,000 people \$100 each immediately at a total cost of \$50,000.  They argue since their project is less expensive and has a positive net present value that you should under take it.  Unfortunately, due to insufficient budget you can only do one of the projects. What "distributional weight" would make you indifferent between the projects?

All happens now so no discounting

1,000 people x \$100 benefit = \$100,000 benefit

net benefit = \$100,000 – 50,000 = \$50,000

Compared to a) need weight X such that \$50,000 X =  \$180,000

\ X = \$180,000/ 50,000 = 3.6

Compared to b) need weight X such that \$50,000 X =  \$750,000

\ X = \$750,000/ 50,000 = 15

Compared to c) no weight is needed since the first project is inadmissible you would already prefer this project.