A paper company dumps nondegradable waste into a river that flows by the firm's plant. The firm estimates its production function to be:

Q = 6KP,
where Q = annual paper production measured in pounds, K = machine hours of capital, and P = gallons of polluted water dumped into the river per year. The firm currently faces no environmental regulation in dumping waste into the river. Without regulation, it costs the firm $7.50 per gallon dumped. The firm estimates a $30 per hour rental rate on capital. The firm produces 600 million pounds of paper per year. For this problem, consider the long-run production of output.
a. Determine the firm's optimal ratio of wastewater to capital.
b. Given the firm's output of 600 million lbs, how much capital and wastewater should the firm employ?
c. How much will it cost the firm to produce the 600 million lbs of paper?
d. The state environmental protection agency plans to impose a $7.50 fee for each gallon that is dumped (this is in addition to the current cost of $7.50). Assuming that the firm intends to maintain its same output level, how much capital and wastewater should the firm employ?
e. How much will the firm pay in fees? What happens to the firm's cost as a result of the fee?

a. MRTS = K/P
w/r = 7.5/30 = 1/4
At the optimal point, K/P = 1/4 (MRTS = w/r)
b. From part (a), K = P/4
600,000,000 = 6(P/4)P P2 = 400,000,000 P* = 20,000
K* = 5,000
c. C(6mill) = 7.5 ∗ 20,000 + 30 ∗ 5,000 = $300,000
d. w/r = 15/30 = 1/2
K/P = 1/2 K = P/2
600,000,000 = 6(P/2)P P2 = 200,000,000 P* = 14,142
K* = 7,071
e. The firm will pay 7.5P* = $106,066 in fees
The costs have increased by $124,260
(The costs have increased by more than the fees collected. The firm has moved to a sub-optimal isocost line, so part of the higher cost is not collected in fees but reflects the increased cost from moving from the optimal ratio.)