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 exchange 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 of annual real 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%.

For your answers, please attach a spreadsheet or other calculations that show how you got your answer, including any formulas that you used.

a) Assuming that the yearly net benefits accrue at the end of the year, calculate the net benefits of leasing the land. (This is your total CBA)

b) How sensitive is your answer to the discount rate? Use a graph to show how your answer to part a) changes as you vary the discount rate from 0-10%. For what values of the discount rate are the net benefits positive?

c) Some analysts in the agency argue that the annual real benefits will grow by 2% per year due to increases in population and income. Recalculate the net benefits assuming that they are correct.

d) Now imagine that the owner is willing to sell the land for the price of $2 million. Assume that this price is the social opportunity cost of the land. Repeat parts a), b) and c) in this scenario.

e) What is the maximum prices that the government should pay to buy the land (assuming a discount rate of 4%)?

Answer 1

=> The present value of the real yearly benefits is most easily calculated using the formula for the present value of an annuity

PV(benefits) = ($110,000)[1-(1+.05)^-20]/(.05)

= $1,370,843

NPV = $1,370,843 - $1,100,000

= $270,843

=> In this case we can use the present value of an annuity with a growth rate in benefits of 2 percent:

First, calculate (.05 - .02)/(1+.02) = .0294

PV(benefits) = [($110,000)/(1+.02)][1-(1+dg)^-20]/dg] = $1,613,370

NPV = $1,613,370 - $1,100,000 = $513,370

=> The benefit stream can now be viewed as a perpetuity. If the growth rate of benefits isassumed 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.

=> The maximum prices that the government should pay to buy the land (assuming a discount rate of 4%) is

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

4% discount rate at $2,750,000

$2,750,000 - ($2,750,000 x .04) = $2,640,000