Question

The environmental protection agency of a county would like to preserve a piece of land as a wilderness area. The current owner has offered to lease the land to the county for 20 years in return for a lump sum payment of $1.1 million, which would be paid at the beginning of the 20- year period. The agency has estimated that the land would generate $110,000 per year in benefits to hunters, bird watchers, and hikers. Assume that the lease price represents the social opportunity cost of the land and that the appropriate real discount rate is 4 percent.

a. Assuming that the yearly benefits, which are measured in real dollars, accrue at the end of each of the 20 years, calculate the net benefits of leasing the land.

b. Some analysts in the agency argue that the annual real benefits are likely to grow at a rate of 2 percent per year due to increasing population and county income. Recalculate the net benefits assuming that they are correct.

c. Imagine that the current owner of the land in the previous exercise was willing to sell the land for $2 million. Assuming this amount equaled the social opportunity cost of the land, calculate the net benefits if the county were to purchase the land as a permanent wildlife refuge. In making these calculations, first assume a zero annual growth rate in the $110,000 of annual real benefits; then assume that these benefits grow at a rate of 2 percent per year.

Answer 1

4.a. The present value of the real yearly benefits is most easily calculated using the formula for the present value of an annuity presented in Appendix 6a:

PV(benefits) = ($110,000)[1-(1+.04) -20]/(.04) = $1,494,936

NPV = $1,494,936 - $1,100,000 = $394,936 4.

b. In this case we use the formula for the present value of an annuity with a growth rate in benefits of 2 percent:

First, calculate dg = (.04-.02)/(1+.02) = .01961

PV(benefits) = [($110,000)/(1+.02)][1-(1+dg) -20]/dg] = $1,770,045

NPV = $1,770,045-$1,100,000 = $670,045

c.The benefit stream can now be viewed as a perpetuity. If the growth rate of benefits is assumed to be zero, then

PV(benefits with zero growth rate) = ($110,000)/(.04) = $2,750,000 NPV(zero growth rate) = $2,750,000 - $2,000,000 = $750,000

PV(benefits with 2% growth rate) = ($110,000)/(.04-.02) = $5,500,000

NPV(benefits with 2% growth rate) = $5,500,000 - $2,000,000 = $3,500,000

Thus, the land should be purchased whether the growth rate is zero or 2 percent.