Three firms put 4 tons of noxious gases into the atmosphere...

Three firms put 4 tons of noxious gases into the atmosphere each in a year. The total abatement costs for each firm are $$ \begin{aligned} & T A C_1=5 e_1^2, \\ & T A C_2=10 e_2+5 e_2^2, \\ & T A C_3=20 e_3+5 e_3^2, \end{aligned} $$ where $e_i$ is the amount of the gas emitted by firm $i$. (a) Suppose the government wants to cut the total emissions from 12 units to 6 units by direct regulation, requiring each firm to cut back on two units of emission each. How much will this plan cost overall? (b) Suppose the government wants to cut the total emissions from 12 units to 6 units. What is the socially efficient (i.e., that costs the least to society) way to cut back on emissions? How much will each firm cut back in this case? (c) Suppose the government wants to cut the total emissions from 12 units to 6 units using an emissions tax. What level of tax will bring this about? How much will the tax revenue collected be?

Three firms put 4 tons of noxious gases into the atmosphere each in a year. The total abatement costs for each firm are
$$
\begin{aligned}
& T A C_1=5 e_1^2, \\
& T A C_2=10 e_2+5 e_2^2, \\
& T A C_3=20 e_3+5 e_3^2,
\end{aligned}
$$
where $e_i$ is the amount of the gas emitted by firm $i$.
(a) Suppose the government wants to cut the total emissions from 12 units to 6 units by direct regulation, requiring each firm to cut back on two units of emission each. How much will this plan cost overall?
(b) Suppose the government wants to cut the total emissions from 12 units to 6 units. What is the socially efficient (i.e., that costs the least to society) way to cut back on emissions? How much will each firm cut back in this case?
(c) Suppose the government wants to cut the total emissions from 12 units to 6 units using an emissions tax. What level of tax will bring this about? How much will the tax revenue collected be?

Key Concepts

Emissions Tax

An emissions tax is a market-based instrument that imposes a per-unit cost on emissions. By setting the tax at a level equal to the marginal social cost of emissions, firms are incentivized to reduce emissions up to the point where their individual marginal abatement cost matches the tax. This mechanism not only helps to achieve the desired level of overall emission reductions, but it also generates government revenue, reflecting the external cost of pollution.

Socially Efficient Emissions Reduction

This concept entails minimizing the total cost of reducing emissions across all firms while achieving the desired overall pollution reduction. The socially efficient solution is reached when the marginal abatement costs are equalized across firms, meaning that reallocating abatement efforts would not lower the overall cost. This optimization ensures that each unit of emission reduction is achieved at the lowest possible cost to society.

Marginal Abatement Cost

The marginal abatement cost is the incremental cost of reducing one additional unit of emissions. In the context of achieving cost-effective pollution reduction, it is crucial for each firm to reduce emissions up to the point where its marginal cost equals that of others. This equality condition is the basis for identifying the socially efficient allocation of emission reductions.

Total Abatement Cost Functions

These functions represent the cost incurred by each firm as it reduces its emissions. They are formulated as mathematical expressions that quantify the cost of abating a given amount of pollution. In environmental economics, such cost functions help to compare the efficiency of different firms in reducing emissions and are essential in evaluating the overall cost of pollution control strategies.

Direct Regulation

Direct regulation involves the government prescribing specific emission reductions for each polluting entity, regardless of individual differences in abatement costs. This approach mandates a fixed reduction target per firm, which can lead to inefficiencies if the cost of abatement varies significantly among firms, as each firm may face higher or lower costs than others in achieving the mandated reduction.

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