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Question: Suppose that a given district has two massive aluminium smelters, smelter 1 and smelter 2, that t...
Suppose that a given district has two massive aluminium smelters, smelter 1 and smelter 2, that transform âaluminaâ (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 districtâs nearby city.
Suppose next that the district government cannot reliably observe the individual F emissions of the two smelters, only the overall FTotal. However, the government can reliably track the individual smelterâs use of the alumina input, A1 and A2, in metric tonnes.
a. Suppose aggregate MAC =600000 â 1000FTotal, and MAB = 500 FTotal.
Please identify the socially optimal level of FTotal. Show your work.
Ftotal=400
b. If the two smelters just look to their own profits, what will FTotal be?
MAC=0 600,000-1000Ftotal=0 Ftotal=600
c. Suppose that government scientists have a good estimate of the relationship between alumina use and total F air pollution: FTotal = .001(A1 + A2)
What must the combined level of alumina inputs (A1 + A2) be capped at to achieve the socially optimal FTotal from part (a)?
400=0.001(A1+A2) A1+A2=400,000
d. We can express the two smeltersâ aggregate MAC over F as their aggregate MAC over the use of A. If MAC = 600000 â 1000FTotal = 600000 â 1000*.001(A1 + A2) = 600000-1(A1 + A2)=600000-ATotal.
At what tax per metric tonne of A would the two smelters cut back to a combined purchase of the socially optimal level of ATotal*?
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e) Suppose that smelter 1âs newer production techniques are such that it doesnât 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 â 6A1
MAC2 = 600000 â 1.2 A2
How would the two smelters respond to a tax of $200,000 per tonne of A?
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f) what is the A reduction cost borne by the two smelters individually as a result of the tax, and thus the total reduction cost?
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g) 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 âneither smelter may use more than 200,000 tonnes of alumina.â
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?
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2a. FTotal* = _________________.
2b. If both firms pay nothing for emissions, then when maximising profit FTotal = ______________.
2c. To achieve the FTotal* found in 2a), ATotal* = _________________.
2d. The two smelters would cut back to ATotal* at a tax of $ per tonne.
2e. At the tax found in 2d), Smelter 1 would choose A1* = ___________ tonnes of alumina.
At the tax found in 2d), Smelter 2 would choose A2* = ___________ tonnes of alumina.
2f. The cost of reducing A1 use to Smelter 1 under this tax is $ ______ . (This does not
include the tax paid by Smelter 1 to the government for the A it continues to use.)
The cost of reducing A2 use to Smelter 2 under this tax is $ . (This does not
include the tax paid by Smelter 2 to the government for the A it continues to use.)
So the total A reduction cost from the tax is $ ____ .
2g. With the uniform rule of a 200,000 cap in A use, Smelter 1 would use A1 = ___________.
Smelter 2 would use A2 = ______________. So the total A use would be ______________.
The cost of reducing A1 use to Smelter 1 under this rule is $______________.
The cost of reducing A2 use to Smelter 2 under this rule is $______________.
So the total A reduction cost from this rule is $________________.