Question

A firm that produces electricity using coal that emits pollution. For every part per million, P, the firm is fined $10,000. However, it can invest in scrubbers, S, that cost $25,000 each and reduce pollution according to the formula P=10/S

1. Explain how the firm should approach this problem.

2. The cost of the fine is given by ($10,000)(P) or ($10,000)(10/S). So, each additional scrubber reduces the cost of the fine by $100,000/S². Find the optimal number of scrubbers to invest in, what the amount of pollution will be, and what the total cost to the firm will be.

3. Suppose the fine doubles. Find the optimal number of scrubbers to invest in, what the amount of pollution will be, and what the total cost to the firm will be. (Round to the nearest whole scrubber) Explain what is happening.

4. Keeping the fine at $10,000, suppose the cost of scrubbers doubles. Find the optimal number of scrubbers to invest in, what the amount of pollution will be, and what the total cost to the firm will be. (Round to the nearest whole scrubber)) Explain what is happening.

5. Does it ever make sense for the firm to not invest in scrubbers?

6. Does it make sense for the government to try to eliminate pollution?

Thanks

Answer 1

1) In the original case with fine 10,000 and cost 25,000

Since increase of S by one costs 25000 and the fine reduces by 100,000/S^2

We get optimal amount by equating the two that is

25000=100000/S^2 or S=2

So the optimal number of scrubbers is 2,the pollution will be 10/2=5 and the total cost to the firm will be 25000*2+10000*5=100,000$

In this case the scrubbers reduce the pollution to half and so the scrubbers cost 50,000

2) if the fine remains the same 10,000 however the cost of scrubbers doubles to 50,000

Now equating the reduction in fine to the marginal cost of scrubbers we get

100000/S^2 =50,000

S=2^(0.5)=1.41

In this case the optimal amount of scrubbers is less than 2 which means beyond 1,thhe increase in scrubbers don't help reducing the fine. So the optimal number of scrubbers after rounding will be 1,the amount of pollution will be 10/1=10 and the cost to the firm will be 50,000 + 100,000 (cost of 1 scrubber + fine of 10,000P) = 150,000$

In this case the firm invests in 1 scrubber so the pollution is 10 and the fine is 100,000$

3) No,it never makes sense for the firm to not invest in scrubbers as otherwise the pollution would be very high(tending to infinite from the equation). Therefore the firm should invest in atleast 1 scrubber so that the pollution is limited to a maximum of P=10

4) It makes sense for the government to try to eliminate pollution as it creates negative externalities and a social cost for rest of the community. It is important for government to fine since these external costs are not reflected in the private costs of the firm and so these fines incentivise pollution control which is necessary for an optimal amount of production.

Hope it helps. Do ask for any clarifications if required.