Question

A steel plant produces S tons of steel, at a total cost of S2 + X2 - 8X dollars, where X denotes the number of gallons of pollution that the plant dumps into a lake. On the same lake, a fish farm produces F tons of fish, at a total cost of F2 + X2 dollars. The market prices per ton are fixed at $4 for steel and $10 for fish.

A. Assuming that the two producers do not negotiate an agreement on the amount of pollution, find the equilibrium level of X.

B. Derive the socially optimal level of X.

C. Suppose that the government imposes a tax of t dollars per gallon of pollution dumped by the steel plant. Calculate the value of t that leads to the socially optimal level of X.

Answer 1

For the solution to parts a, b, and c, refer to the images below:

Hence, the solution to the above parts are 4 gallons, 2 gallons, and t = 4.