Question

There are two factories in a small town. Both of them emit carbon dioxide into the air. Factory 1 currently emits 120 tons per month, whereas factory 2 currently emits 160 tons per month. The technology of each factory is different, so their costs of reducing emissions are different as well. The tables below show the costs of reducing emissions in increments of 20 tons per month for each factory:

Factory 1

Total cost of reducing emissions by 20 tons/month	$50
Total cost of reducing emissions by 40 tons/month	$150
Total cost of reducing emissions by 60 tons/month	$270
Total cost of reducing emissions by 80 tons/month	$410
Total cost of reducing emissions by 100 tons/month	$570

Factory 2

Total cost of reducing emissions by 20 tons/month	$20
Total cost of reducing emissions by 40 tons/month	$60
Total cost of reducing emissions by 60 tons/month	$110
Total cost of reducing emissions by 80 tons/month	$200
Total cost of reducing emissions by 100 tons/month	$300

The existing technology does not allow for reductions in emissions beyond 100 tons/month. That is, the most each factory could reduce its emissions by is 100 tons/month.

Suppose the government in this town would like to cut monthly emissions to half of the current level. To do that, the government has decided to impose a tax for every 20 tons of pollution per month emitted by a factory. To achieve its desired goal (but not exceed the goal), the tax would have to be set between $ _____________   and $ _________________   for every 20 tons/month. (The first number should be the lower end of the tax, and the second number should be the higher end of the tax.)

Incorrect answers: 450 and 720 respectively

Answer 1

Since total emissions produced without tax will be 120+160 = 280 units

Government would like to cut emissions to half i.e., 280/2 = 140 units

The factory will reduce the emissions if and only if for each additional 20 units of emissions it has to pay tax more than or equal to what it costs in reducing it by themselves.

So we will have to compare the cost incurred in additional 20 units of emissions reductions with tax per 20 unit of emission .

For 1st 20 additional unit of emissions reduction factory 1 has to spend $50 and factory 2 $20

For 2nd 20 additional units of emissions reduction factory 1 has to spend $ 150-$50 ie $ 100 and factory 2 has to spend $60 - $20 = $40

Similarly we get all other values .

Let us first draw a table which shows the additional cost incurred by each of the firm for 20 additional unit of emissions reduction. 2nd column and 3rd column shows the cost incurred by factory 1 and factory 2 in reducing additional 20 units of emissions .

	Additional cost incurred by factory 1for every 20 unit of emissions reduction (in $)	Additional cost incurred by Factory 2 in $ for every 20 units of emissions reduction
First 20 units	50	20
Second 20 units	100	40
Third 20 units	120	50
Fourth 20 units	140	90
Fifth 20 units	160	100

Now government aims to cut emissions by half which is to 140 tons of emissions per month and not more than that

If government sets the tax anywhere below $100 for every 20 tons per month, then factory 1 will only reduce first 20 emissions and would rather give tax than reduce emissions, since tax would cost lesser than cost incurred to reduce an additional 20 tons of emissions per month.

Now, when factory 1 reduces only 20 tons per month even if factory 2 reduces whole of the 100 units it can reduce, total emissions would only decrease by at most 120 tons per month

Therefore it is clear that tax has to be more than or equal to $100 units per 20 emissions.

When tax is $100 per 20 unit per month factory 1 compares $100 to the cost incurred in reducing 1st 20 tons of emissions (which is $50) and second 20 tons of emissions per month (which is $100 ) and finds it cost effective to reduce emissions by 40 tons per month

factory 2 similarly makes assesment by comparing tax for each 20 tons of emissions per month reduction and the cost it incurrs to reduce it . Doing so factory 2 would find it cost effective to reduce all 100 tons emissions per month( since cost incurred in reducing every 20 tons per month of emissions tax for each 20 tons per month of emissions ( which is $100))

Therefore at $100 tax per 20 tons of emissions per month , 140 tons/month of emissions are reduced.

But since the government does not aim at cutting more than 140 tons per month , it won't increase the tax so much so that factory 1 has to reduce any further emissions . ( Since factory 2 has already reduced as many emissions as it could , it isn't a concern anymore)

Therefore the government keeps the tax less than $120 per 20 tons per month, so that factory 1 isn't bothered enough to reduce any more emissions.

Answer is:

...........tax would have to be set between $ 100 and $120 for every 20 tons/month.