Suppose that a given district has two massive aluminium smelters, smelter 1 and smelter 2, that transform "alumina" (extracted from bauxite) into aluminium. So the smelters use alumina as an input, but its use leads to emissions of aluminium flourides (F) as particulates into the air, creating health problems for some residents in the district's nearby city.

Suppose next that the district government cannot reliably observe the individual F emissions of the two smelters, only the overall F_Total. However, the government can reliably track the individual smelter's use of the alumina input, A₁and A₂, in metric tonnes.

1. Suppose aggregate MAC =600000 - 1000F_Total, and MAB = 500 F_Total.

Please identify the socially optimal level of F_Total. Show your work.

2. If the two smelters just look to their own profits, what will F_Total be? 

3. Suppose that government scientists have a good estimate of the relationship between alumina use and total F air pollution: F_Total = .001(A₁ + A₂)     

What must the combined level of alumina inputs (A₁ + A₂) be capped at to achieve the socially optimal F_Total from part (a)?

4. We can express the two smelters' aggregate MAC over F as their aggregate MAC over the use of A. If MAC = 600000 - 1000F_Total = 600000 - 1000*.001(A₁ + A₂) = 600000-1(A₁ + A₂)=600000-A_Total.

At what tax per metric tonne of A would the two smelters cut back to a combined purchase of the socially optimal level of A_Total*?

5. Suppose that smelter 1's newer production techniques are such that it doesn't need nearly as much alumina as smelter 2 to be viable, but finds it far more costly to reduce its use of the input: 

MAC₁ = 600000 - 6 A₁

MAC₂ = 600000 - 1.2 A₂

How would the two smelters respond to a tax of $200,000 per tonne of A?

6. What is the A reduction cost borne by the two smelters individually as a result of the tax, and thus the total reduction cost?

7. Next suppose that the district government thinks a tax on purchases of A is too complicated to administer, or unpopular with the smelters. Instead, it just says that "neither smelter may use more than 200,000 tonnes of alumina."

What would the resulting total level of A use be from the two smelters, given their particular MAC curves? What would the total A reduction cost be of this policy?