Consider two firms with the following marginal abatement costs (MAC) as a function of emissions (E):

MAC_1 = 22 - 2E_1

MAC_2 = 10 - E_2,

and assume marginal external damages (MED) from the aggregate emissions of both firms (E_Agg = E_1 + E_2) is:

MED = .5E_Agg.

To achieve the socially efficient level of aggregate emissions (E*_Agg), the government institutes a cap and trade policy and sets the cap equal to the socially efficient level of emissions (E*_Agg).  If firm 2 is initially grandfathered 2 permits, it will buy ____ more permits.  

(Note: your answer can be positive or negative; if it is positive, the firm is buying additional permits; if it is negative, the firm sells permits. So, for example, an answer of -10 implies that the firm sells 10 permits)

To solve this problem, we need to follow these steps:

1. **Find the Socially Efficient Level of Aggregate Emissions** (\(E^*_{Agg}\)) by equating the Marginal Abatement Costs (MAC) of both firms to the Marginal External Damage (MED) of aggregate emissions.
   
2. **Determine the emissions for each firm** at the socially efficient level of aggregate emissions.
   
3. **Determine how many permits firm 2 needs or sells** by comparing its efficient emissions to its grandfathered permits.

### Step 1: Socially Efficient Emissions (\(E^*_{Agg}\))

The condition for the socially efficient level of emissions is that the sum of the marginal abatement costs (MAC) across both firms equals the marginal external damage (MED) of the aggregate emissions. This is because at the socially efficient level, the costs of reducing one additional unit of emissions should equal the damages avoided by reducing that unit.

The aggregate emissions are:

\[
E_{Agg} = E_1 + E_2
\]

The marginal external damage (MED) is:

\[
MED = 0.5E_{Agg}
\]

For the socially efficient level of emissions, we set:

\[
MAC_1 = MAC_2 = MED
\]

#### MACs:
\[
MAC_1 = 22 - 2E_1
\]
\[
MAC_2 = 10 - E_2
\]

Equating the MACs to the MED:

\[
22 - 2E_1 = 0.5E_{Agg}
\]
\[
10 - E_2 = 0.5E_{Agg}
\]

### Step 2: Solve for Efficient Emissions

Substitute \(E_{Agg} = E_1 + E_2\) into both equations.

First, solve the first MAC equation for \(E_1\):

\[
22 - 2E_1 = 0.5(E_1 + E_2)
\]

Multiply both sides by 2 to eliminate the fraction:

\[
44 - 4E_1 = E_1 + E_2
\]

Rearrange:

\[
44 = 5E_1 + E_2
\quad \text{(Eq. 1)}
\]

Now solve the second MAC equation for \(E_2\):

\[
10 - E_2 = 0.5(E_1 + E_2)
\]

Multiply by 2:

\[
20 - 2E_2 = E_1 + E_2
\]

Rearrange:

\[
20 = E_1 + 3E_2
\quad \text{(Eq. 2)}
\]

### Step 3: Solve the System of Equations

We have two equations:

1. \(44 = 5E_1 + E_2\)
2. \(20 = E_1 + 3E_2\)

#### Solving for \(E_1\) and \(E_2\):

From equation (2), solve for \(E_1\):

\[
E_1 = 20 - 3E_2
\]

Substitute this into equation (1):

\[
44 = 5(20 - 3E_2) + E_2
\]

Expand:

\[
44 = 100 - 15E_2 + E_2
\]

Simplify:

\[
44 = 100 - 14E_2
\]

Rearrange:

\[
14E_2 = 100 - 44
\]
\[
14E_2 = 56
\]
\[
E_2 = 4
\]

Now substitute \(E_2 = 4\) into the expression for \(E_1\):

\[
E_1 = 20 - 3(4) = 20 - 12 = 8
\]

Thus, the socially efficient emissions are:

\[
E_1 = 8, \quad E_2 = 4
\]

### Step 4: Determine How Many Permits Firm 2 Buys or Sells

Firm 2 is initially grandfathered 2 permits. At the socially efficient level, firm 2 emits 4 units. Therefore, firm 2 needs \(4 - 2 = 2\) additional permits.

### Conclusion:

Firm 2 will buy **2** more permits.

---

Would you like more details or further explanation on any part of this problem? Here are 5 related questions to explore:

1. What happens to the efficient emissions if the MAC of one firm changes?
2. How would the total cost for both firms change if no trading were allowed?
3. What are the welfare benefits of a cap-and-trade system compared to uniform emission reductions?
4. How would firm 1's need for permits change if the cap were stricter?
5. How do marginal abatement cost curves reflect the firms' differences in abatement technology?

**Tip**: Always ensure that the marginal external damage is equal to the marginal abatement cost across all firms for an efficient emissions level.

Consider two firms with the following marginal abatement costs (MAC) as a function of emissions (E):

MAC_1 = 22 - 2E_1

MAC_2 = 10 - E_2,

and assume marginal external damages (MED) from the aggregate emissions of both firms (E_Agg = E_1 + E_2) is:

MED = 0.5E_Agg.

To achieve the socially efficient level of aggregate emissions (E*_Agg), the government institutes a cap and trade policy and sets the cap equal to the socially efficient level of emissions (E*_Agg). If firm 2 is initially grandfathered 2 permits, it will buy ____ more permits.

(Note: your answer can be positive or negative; if it is positive, the firm is buying additional permits; if it is negative, the firm sells permits. So, for example, an answer of -10 implies that the firm sells 10 permits)

Solve the problem of calculating the socially efficient level of emissions under a cap-and-trade policy. Understand how to balance marginal abatement costs (MAC) and marginal external damages (MED) using linear equations and economic equilibrium conditions. Suitable for undergraduate economics and mathematics students.

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