Question

The lake near Grassy Narrows is contaminated with mercury from an old paper mill, which has been shut down for many years. Some of the people who eat fish from the lake are showing signs of mercury poisoning. Mercury is still leaking into the lake but no one knows the exact location of the leak. Marginal damages per year are given by MD = 0.02E (t = 0, …) and emissions are E = 4000. The cost to locate the leak and clean up the lake is $2 million (t = 0).

a) If r = 2% and you use a 100 year time frame (t = 0, …, 99) calculate the NPV to clean the lake.

b) If r = 8% and you use a 100 year time frame (t = 0, …, 99) calculate the NPV to clean the lake.

c) If r = 2% and you use a 50 year time frame (t = 0, …, 49) calculate the NPV to clean the lake.

d) If r = 8% and you use a 50 year time frame (t = 0, …, 49) calculate the NPV to clean the lake. e) Why are the present values for the damages using the 50 year time frame (half the time) not half of the present values using the 100 year time frame?

Answer 1

Please chedk if MD has been correctly reported by you in the question above as = 0.02E. I have a strong feeling that some formatting is lost while copying and pasting the question, because the answers we are getting below are all negative. If MD has been reported correctly then please see the solution below. Otherwise please post the question again stating MD properly.

Initial investment, C₀ = $ 2 mn = $ 2 x 10⁶

Annual damages saved = C = 0.02E = 0.02 x 4,000 = 80

NPV = - C₀ + C / r x [1 - (1 + r)^-n]

Part (a)

r = 2%, n = 99

NPV = - 2 x 10⁶ + 80 / 2% x [1 - (1 + 2%)^-99] = - $ 1,996,563

Part (b)

r = 8%, n = 99

NPV = - 2 x 10⁶ + 80 / 8% x [1 - (1 + 8%)^-99] = - $ 1,999,000

Part (c)

r = 2%, n = 49

NPV = - 2 x 10⁶ + 80 / 2% x [1 - (1 + 2%)^-49] = - $ 1,997,516

Part (d)

r = 8%, n = 49

NPV = - 2 x 10⁶ + 80 / 8% x [1 - (1 + 8%)^-49] = - $ 1,999,023

Part (e)

The present values for the damages using the 50 year time frame (half the time) not half of the present values using the 100 year time frame, because the PV calculation is not a linear calculation. The present value function is not linearly related to time frame t. The relationship is far more convoluted. PV varies with inverse power of t. Hence, the PV value is not halved if time frame is halved.