A factory is emitting 12 tons of pollutant. The plot below shows the marginal abatement cost (Q is in tons) with an old technology MC(t=1) and a new technology MC(t=2).

The government is seeking to improve air quality and is imposing an emissions tax.

If the tax is set at Tb, what is the total cost (abatement + taxes) the factory will incur with the old technology?

Question

A factory is emitting 12 tons of pollutant. The plot below shows the marginal abatement cost (Q is in tons) with an old technology MC(t=1) and a new technology MC(t=2).

The government is seeking to improve air quality and is imposing an emissions tax.

If the tax is set at Tb, what is the total cost (abatement + taxes) the factory will incur with the old technology?

BlackTom AI · Accepted Answer

We start by restating what the question asks and identifying the components involved in the total cost under the old technology when a tax Tb is imposed. The factory emits 12 tons of pollutant initially. With the old technology, the marginal abatement cost curve is MC(t=1). The tax Tb per ton creates an external cost per unit abated, and the total cost to the firm (abatement cost plus taxes paid) depends on the quantity of abatement chosen by the firm, which in turn is determined by where the MC(t=1) curve intersects the tax level Tb. In the diagram, the regions labeled with letters (A, B, C, … S, O, P, Q, R, S, etc.) correspond to specific areas under or between the MC curves and the tax lines. The four answer choices propose different sums of these lettered regions as representing the total cost the firm incurs under the old technology when the tax Tb is in place.

Option 1: C+D+F
- This set comprises only three areas. If you picture abatement cost under MC(t=1) up to the tax quantity and add the tax revenue (or tax cost) incurred for the abated quantity, this small combination likely omits many portions of the abatement cost and the taxbetaling that should be included. In other words, it seems to undercount both the area under MC(t=1) up to the chosen abatement level and the corresponding tax paid on those abated units. Therefore, conceptually this option appears insufficient to represent the full abatement-plus-tax total.

Option 2: C+H+O+D+F+I+J+K+M+P+Q+R+S
- This option includes a large collection of regions spread across the lower, middle, and right portions of the graph. If the tax Tb is binding, you would expect the total cost to include the entire abatement cost up to the chosen abatement quantity (the area under MC(t=1) from 0 to Q_abate) plus the total tax paid on the abated quantity (Tb times Q_abate). Depending on the labeling, many of these regions could correspond to exactly those components (areas under MC(t=1) up to Q_abate plus the tax-area Tb×Q_abate). However, because the diagram also includes regions that are not directly part of what is paid (e.g., regions under MC(t=2) or outside the abatement interval), this option might be overinclusive if it accidentally includes areas not incurred under the old technology with the tax. Careful mapping of each letter to its physical meaning in the cost decomposition is required to confirm.

Option 3: H+I+J+K+M+O+P+Q+R+S
- This set is also quite extensive and spans a broad region of the graph, similar in scope to Option 2 but with a different combination. If the tax Tb applies and the firm chooses an abatement level based on MC(t=1) = Tb, the total cost should sum the abatement cost (areas under MC(t=1) up to the abatement quantity) and the tax paid on that abated amount. Depending on how the letters map to those specific components, this option could plausibly represent the right decomposition. Yet, like Option 2, it risks including areas that do not belong to the old-tech-abatement-plus-tax total if any of the included letters lie outside the 0 to Q_abate interval or outside the under-MC1 area relevant to abatement up to the tax level.

Option 4: H+I+P+O
- This option again selects a small, distinct subset of regions. If H, I, P, and O correspond to core components of the abatement cost under MC(t=1) up to the tax level and the tax payment for that abated quantity, this could reflect the essential pieces. However, excluding other potentially relevant regions (such as portions near A, B, C, D, E, F, etc.) risks omitting parts of the abatement cost or tax payments, leading to an undercount. Conversely, if those letters include pieces not actually incurred under the old tech-plus-tax scenario, it could overcount.

Cross-cutting considerations to guide interpretation
- The correct decomposition hinges on two pieces: (1) the total abatement cost under MC(t=1) up to the chosen abatement level (Q_abate where MC(t=1) = Tb), which is the area under MC(t=1) from 0 to Q_abate, and (2) the total tax paid, Tb times Q_abate.
- Any option that omits portions of the area under MC(t=1) up to Q_abate or omits Tb×Q_abate will understate the total cost. Conversely, any option that includes areas outside the 0 to Q_abate interval or outside the under-MC1 region will overstate the total cost.
- Without a precise mapping of each labeled region (A, B, C, … S, O, P, Q, R, S) to its physical meaning in terms of abatement cost and tax payments, it is not possible to definitively pick the correct sum among the options from the diagram alone.

In sum, each option represents a different aggregation of regions from the stylized diagram. The key to choosing the correct one is to include exactly the areas that comprise (i) the abatement cost up to the tax-implied abatement level under the old technology and (ii) the taxes paid on that abated quantity, and to exclude regions that lie outside that 0 to Q_abate abatement interval or that pertain to the wrong technology curve. Given only the lettered diagram and without a legend mapping each letter to its meaning, a definitive determination cannot be made from the information provided here.

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