Consider two firms with the following marginal abatement costs (MAC) as a function of emissions (E):
MAC_1 = 6 - 2E_1
MAC_2 = 4 - E_2.
If both firms cut their emissions to zero, total abatement costs (TAC) for the two firms is ____.

To find the total abatement costs (TAC) when both firms cut their emissions to zero, we'll need to integrate the marginal abatement cost (MAC) functions from the initial level of emissions to zero.

Given the marginal abatement cost functions:

- For Firm 1: \( MAC_1 = 6 - 2E_1 \)
- For Firm 2: \( MAC_2 = 4 - E_2 \)

### Step 1: Find the Total Abatement Cost for Firm 1

The total abatement cost (TAC) for Firm 1 is obtained by integrating the MAC function over the range of emissions from \( E_1 \) (initial level) to 0.

\[
TAC_1 = \int_{0}^{E_1} MAC_1 \, dE_1 = \int_{0}^{E_1} (6 - 2E_1) \, dE_1
\]

Perform the integration:

\[
TAC_1 = \left[ 6E_1 - \frac{2E_1^2}{2} \right]_0^{E_1} = 6E_1 - E_1^2
\]

### Step 2: Find the Total Abatement Cost for Firm 2

Similarly, the total abatement cost (TAC) for Firm 2 is:

\[
TAC_2 = \int_{0}^{E_2} MAC_2 \, dE_2 = \int_{0}^{E_2} (4 - E_2) \, dE_2
\]

Perform the integration:

\[
TAC_2 = \left[ 4E_2 - \frac{E_2^2}{2} \right]_0^{E_2} = 4E_2 - \frac{E_2^2}{2}
\]

### Step 3: Find the Total Abatement Costs when Emissions are Cut to Zero

If both firms cut their emissions to zero, \( E_1 = 3 \) and \( E_2 = 4 \) (these are the initial levels of emissions before any abatement).

Substitute these into the TAC formulas:

\[
TAC_1 = 6(3) - (3)^2 = 18 - 9 = 9
\]

\[
TAC_2 = 4(4) - \frac{(4)^2}{2} = 16 - 8 = 8
\]

### Step 4: Calculate the Total Abatement Costs (TAC) for Both Firms

\[
TAC_{total} = TAC_1 + TAC_2 = 9 + 8 = 17
\]

Thus, the total abatement costs (TAC) for the two firms when both cut their emissions to zero is **17**.

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Would you like further details or have any questions?

Here are some related questions:

1. How would TAC change if Firm 1's initial emissions were higher than 3?
2. What happens to the total abatement cost if Firm 2 increases its initial emissions level?
3. How do the initial emissions levels affect the efficiency of a cap-and-trade system?
4. What is the significance of marginal abatement cost functions in environmental economics?
5. How can these concepts be applied to a real-world pollution reduction scenario?

**Tip:** Remember, the integration of the marginal abatement cost curve gives the total cost of reducing emissions to any level, not just to zero. This is crucial for understanding the economic impact of environmental regulations.

Consider two firms with the following marginal abatement costs (MAC) as a function of emissions (E): MAC_1 = 6 - 2E_1, MAC_2 = 4 - E_2. If both firms cut their emissions to zero, total abatement costs (TAC) for the two firms is ____.

Learn how to calculate the total abatement costs (TAC) for two firms when both cut their emissions to zero using their marginal abatement cost (MAC) functions. This problem involves integration and the application of the fundamental theorem of calculus. Ideal for undergraduate students studying economics, environmental science, or mathematics.

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