We multiply the carbon emissions by a factor of 1000 to express carbon intensities<fnref n="138" /> in gCO₂e/$ or equivalently tCO₂e/$ mn. Energy is the most polluting sector with 500 ktCO₂e, followed by materials and industrials with 200 ktCO₂e. Energy and services have respectively the highest and lowest carbon intensity, respectively 100 tCO₂e/$ mn and 10 tCO₂e/$ mn. We have:

$$
\begin{align*}
\mathcal{CI}_{\text{total}} &= \tilde{\mathcal{L}} \mathcal{CI}_1 \\
&= \begin{pmatrix}
1.1881 & 0.1678 & 0.1430 & 0.0715 \\
0.3894 & 1.2552 & 0.4110 & 0.1718 \\
0.4919 & 0.4336 & 1.6303 & 0.2993 \\
0.2884 & 0.1891 & 0.3044 & 1.6087
\end{pmatrix}
\begin{pmatrix}
100 \\
50 \\
25 \\
10
\end{pmatrix} \\
&= \begin{pmatrix}
131.49 \\
113.69 \\
114.62 \\
61.99
\end{pmatrix}
\end{align*}
$$

We can then decompose $\mathcal{CI}_{\text{total}}$ between $\mathcal{CI}_{\text{direct}}$ and $\mathcal{CI}_{\text{indirect}}$. Finally, we obtain we obtain the direct and indirect carbon intensities given in Table 8.46. While the Scope 1 carbon intensity of the energy sector is equal to 100 tCO₂e/$ mn, its total carbon intensity is equal to 131.49 tCO₂e/$ mn. The difference of 31.49 tCO₂e/$ mn corresponds to the indirect emissions. In the case of the energy sector, direct and indirect emissions represent 76.05% and 23.95% of the total emissions, respectively. In fact, this sector has the lowest ratio of indirect carbon emissions. On the contrary, for the services sector, 83.87% of the total emissions are indirect.

<table><caption>Table 8.46: Direct and indirect carbon intensities</caption><thead><tr><th rowspan="2">Sector</th><th rowspan="2">CI<sub>1</sub></th><th rowspan="2">CI<sub>total</sub></th><th rowspan="2">CI<sub>direct</sub></th><th rowspan="2">CI<sub>indirect</sub></th><th>CI<sub>direct</sub></th><th>CI<sub>indirect</sub></th><th>CI<sub>total</sub></th></tr><tr><th>(in %)</th><th>(in tCO<sub>2</sub>e/$ mn)</th><th>(in tCO<sub>2</sub>e/$ mn)</th></tr></thead><tbody><tr><td>Energy</td><td>100.00</td><td>131.49</td><td>100.00</td><td>31.49</td><td>76.05%</td><td>23.95%</td><td>1.31</td></tr><tr><td>Materials</td><td>50.00</td><td>113.69</td><td>50.00</td><td>63.69</td><td>43.98%</td><td>56.02%</td><td>2.27</td></tr><tr><td>Industrials</td><td>25.00</td><td>114.62</td><td>25.00</td><td>89.62</td><td>21.81%</td><td>78.19%</td><td>4.58</td></tr><tr><td>Services</td><td>10.00</td><td>61.99</td><td>10.00</td><td>51.99</td><td>16.13%</td><td>83.87%</td><td>6.20</td></tr></tbody></table>

We denote by $\mathcal{CI}_{(k)} = (A^{\top})^k \mathcal{CI}_1$ the indirect carbon intensity when considering the $k^{\text{th}}$ tier, and $\mathcal{CI}_{(1-k)} = \sum_{h=1}^k (A^{\top})^h \mathcal{CI}_1$ the cumulative indirect carbon intensity for the first $k$ tiers. The results are shown in Table 8.47. For the services sector, the first and second tiers add 18.50 and 13.50 tCO₂e/$ mn to the indirect carbon intensity, respectively. If we restrict the analysis to the first two tiers, the indirect carbon intensity is equal to 32. The tree presented in Figure 8.136 explains this computation<fnref n="139" />. To produce $1 of services, we must purchase 0.10$ of energy, $0.05$ of materials, and so on. It follows that the first-tier indirect carbon intensities for the services sector is equal to:

$$
\begin{align*}
\mathcal{CI}_{(1)}(S_4) &= 0.10 \times 100 + 0.05 \times 50 + 0.10 \times 25 + 0.35 \times 10 \\
&= 10 + 2.5 + 2.5 + 3.5 \\
&= 18.50
\end{align*}
$$

<footnote n="138">¹³⁸Remember that the unit of carbon intensity is tCO₂e/$ mn. When we divide by 1 million, we get an equivalent unit of gCO₂e/$.</footnote>

<footnote n="139">¹³⁹To make the graph easier to read, we use the following correspondences: energy ← $S_1$, materials ← $S_2$, industrials ← $S_3$ and services ← $S_4$.</footnote>
<figure><img src="image_417.png" /><figcaption>Figure 8.136: Upstream tree of the first- and second-tier rounds for the services sector</figcaption></figure>
We can continue the analysis and consider the second tier. In fact, the companies involved in the first-tier round also purchase goods and services that generate new indirect emissions. We have:

$$
\begin{aligned}
\mathcal{CI}_{(2)}(S_4) &= 0.10 \times \underbrace{(0.10 \times 100 + 0.10 \times 50 + 0.05 \times 25 + 0.02 \times 10)}_{\text{Indirect emissions from energy sector companies}} + \\
&\quad 0.05 \times \underbrace{(0.20 \times 100 + 0.10 \times 50 + 0.20 \times 25 + 0.05 \times 10)}_{\text{Indirect emissions from materials sector companies}} + \\
&\quad 0.10 \times \underbrace{(0.20 \times 100 + 0.20 \times 50 + 0.30 \times 25 + 0.10 \times 10)}_{\text{Indirect emissions from industrials sector companies}} + \\
&\quad 0.35 \times \underbrace{(0.10 \times 100 + 0.05 \times 50 + 0.10 \times 25 + 0.35 \times 10)}_{\text{Indirect emissions from services sector companies}} \\
&= 13.495
\end{aligned}
$$

We can continue the analysis, and we verify that $\mathcal{CI}_{(3)}(S_4) = 8.45$, $\mathcal{CI}_{(4)}(S_4) = 4.98$, etc. Finally, the cumulative sum converges to $\mathcal{CI}_{(1-\infty)}(S_4) = 51.99$.

<table><caption>Table 8.47: Tier decomposition of carbon intensities</caption><thead><tr><th></th><th>Sector</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>10</th><th>15</th><th>&infin;</th></tr></thead><tbody><tr><td rowspan="4">CI<sub>(k)</sub></td><td>Energy</td><td>16.45</td><td>6.99</td><td>3.60</td><td>1.97</td><td>1.09</td><td>0.06</td><td>0.00</td><td>0.00</td></tr><tr><td>Materials</td><td>30.50</td><td>14.97</td><td>8.13</td><td>4.47</td><td>2.48</td><td>0.14</td><td>0.01</td><td>0.00</td></tr><tr><td>Industrials</td><td>38.50</td><td>22.79</td><td>12.58</td><td>6.96</td><td>3.88</td><td>0.21</td><td>0.01</td><td>0.00</td></tr><tr><td>Services</td><td>18.50</td><td>13.50</td><td>8.45</td><td>4.98</td><td>2.86</td><td>0.16</td><td>0.01</td><td>0.00</td></tr><tr><td rowspan="4">CI<sub>(1-k)</sub></td><td>Energy</td><td>16.45</td><td>23.44</td><td>27.04</td><td>29.02</td><td>30.11</td><td>31.41</td><td>31.48</td><td>31.49</td></tr><tr><td>Materials</td><td>30.50</td><td>45.47</td><td>53.59</td><td>58.06</td><td>60.55</td><td>63.52</td><td>63.68</td><td>63.69</td></tr><tr><td>Industrials</td><td>38.50</td><td>61.29</td><td>73.87</td><td>80.83</td><td>84.71</td><td>89.35</td><td>89.61</td><td>89.62</td></tr><tr><td>Services</td><td>18.50</td><td>32.00</td><td>40.44</td><td>45.43</td><td>48.29</td><td>51.79</td><td>51.98</td><td>51.99</td></tr></tbody></table>

The previous analysis concerns the carbon intensity. To estimate total emissions, we simply multiply by the output and we have the following identities:

$$ \frac{\mathcal{CE}_{\text{total}}}{\mathcal{CE}_1} = \frac{\mathcal{CI}_{\text{total}}}{\mathcal{CI}_1} \Leftrightarrow \mathcal{CE}_{\text{total}} = \mathcal{CI}_{\text{total}} \odot \frac{\mathcal{CE}_1}{\mathcal{CI}_1} = x \odot \mathcal{CI}_{\text{total}} \quad (8.38) $$

Therefore, the indirect emissions are given by:

$$ \mathcal{CE}_{\text{indirect}} = \mathcal{CE}_{\text{total}} - \mathcal{CE}_{\text{direct}} = (\mathcal{CI}_{\text{total}} - \mathcal{CI}_{1}) \odot \frac{\mathcal{CE}_1}{\mathcal{CI}_1} \quad (8.39) $$

The breakdown of the total carbon emissions is provided in Table 8.48. We notice that indirect carbon emissions are double counted. Indeed, the total direct carbon emissions are equal to 1025 ktCO₂e and the indirect emissions add 1779 ktCO₂e. Based on direct emissions, we have the following distribution: 49% for energy, 20% for materials, 20% for industrials and 12% for the services sector. If we include the indirect emissions, we get a different picture. For instance, the services sector represents more than 25% of the total emissions because the direct emissions have been multiplied by a factor of 6.2, while the contribution of energy is now less than 25%.

**Remark 91** It would be wrong to directly diffuse the carbon emissions instead of the carbon intensities: $\mathcal{CE}_{\text{total}} = (I_n - A^\top)^{-1} \mathcal{CE}_1$. Indeed, carbon emissions are not comparable from one sector to another sector, because they are not normalized and monetary input-output tables give the technical coefficients for $1 of output from each sector.
<table><caption>Table 8.48: Decomposition of carbon emissions</caption><thead><tr><th rowspan="2">Sector</th><th colspan="3">(in ktCO<sub>2</sub>e)</th><th colspan="3">(in %)</th></tr><tr><th><b>C</b><b>E</b><sub>direct</sub></th><th><b>C</b><b>E</b><sub>indirect</sub></th><th><b>C</b><b>E</b><sub>total</sub></th><th><b>C</b><b>E</b><sub>direct</sub></th><th><b>C</b><b>E</b><sub>indirect</sub></th><th><b>C</b><b>E</b><sub>total</sub></th></tr></thead><tbody><tr><td>Energy</td><td>500</td><td>157.44</td><td>657.44</td><td>48.78</td><td>8.85</td><td>23.45</td></tr><tr><td>Materials</td><td>200</td><td>254.76</td><td>454.76</td><td>19.51</td><td>14.32</td><td>16.22</td></tr><tr><td>Industrials</td><td>200</td><td>716.97</td><td>916.97</td><td>19.51</td><td>40.30</td><td>32.70</td></tr><tr><td>Services</td><td>125</td><td>649.92</td><td>774.92</td><td>12.20</td><td>36.53</td><td>27.64</td></tr><tr><td>Total</td><td>1 025</td><td>1 779.10</td><td>2 804.10</td><td>100.00</td><td>100.00</td><td>100.00</td></tr></tbody></table>

## Upstream vs. downstream analysis

The previous analysis is an output-based analysis. This is obvious if we look at Figure 8.136, which illustrates the impacts of requirement to produce $1 in a sector. Once we have produced $1 in a given sector, we may wonder how it is used by the value chain. In this case, we get an input-based analysis. In fact, instead of moving up the supply chain, we move down the value chain (Figure 8.137). Therefore, this approach is also known as downstream analysis while the output-based approach is known as upstream analysis.

To perform a downstream analysis, we must first define the technical coefficients for $1 input (and not output):

$$ \check{A}_{i,j} = \frac{Z_{i,j}}{x_i} $$

$\check{A}_{i,j}$ indicates the proportion of $1 produced by sector $i$ that is used by sector $j$. We denote the matrix of input impacts by $\check{A} = (\check{A}_{i,j}) = \text{diag}(x)^{-1} Z$. We note that:

$$ \check{A}_{i,j} = \frac{Z_{i,j}}{x_j} \cdot \frac{x_j}{x_i} = A_{i,j} \cdot T_{i,j} $$

In a matrix form, we have $\check{A} = A \odot T$ where $T = (T_{i,j}) = (x_i^{-1} x_j)$. Using the same reasoning as in the previous paragraphs, we can show that:

$$ \mathcal{C}\mathcal{I}_{\text{total}}^{\text{down}} = (I_n - \check{A})^{-1} \mathcal{C}\mathcal{I}_1 \qquad (8.40) $$

where $\mathcal{C}\mathcal{I}_{\text{total}}^{\text{down}}$ is the vector of downstream (direct plus indirect downstream) carbon intensities. It follows that the indirect downstream carbon intensities are given by:

$$ \mathcal{C}\mathcal{I}_{\text{indirect}}^{\text{down}} = \mathcal{C}\mathcal{I}_{\text{total}}^{\text{down}} - \mathcal{C}\mathcal{I}_1 = \left( (I_n - \check{A})^{-1} - I_n \right) \mathcal{C}\mathcal{I}_{\text{direct}} \qquad (8.41) $$

In particular, we can decompose $\mathcal{C}\mathcal{I}_{\text{indirect}}^{\text{down}}$ as follows:

$$ \mathcal{C}\mathcal{I}_{\text{indirect}}^{\text{down}} = \underbrace{\check{A}\mathcal{C}\mathcal{I}_1}_{\text{First-tier}} + \underbrace{\check{A}^2\mathcal{C}\mathcal{I}_1}_{\text{Second-tier}} + \dots + \underbrace{\check{A}^k\mathcal{C}\mathcal{I}_1}_{\text{$k^{th}$-tier}} + \dots $$

and we have:

$$ \mathcal{C}\mathcal{I}_{\text{total}}^{\text{down}} = \underbrace{\mathcal{C}\mathcal{I}_1}_{\text{Scope 1}} + \underbrace{\check{A}\mathcal{C}\mathcal{I}_1}_{\text{First-tier}} + \underbrace{\check{A}^2\mathcal{C}\mathcal{I}_1}_{\text{Second-tier}} + \dots + \underbrace{\check{A}^k\mathcal{C}\mathcal{I}_1}_{\text{$k^{th}$-tier}} + \dots $$
<figure><img src="image_418.png" /><figcaption>Figure 8.137: Downstream tree of the first- and second-tier rounds for the services sector</figcaption></figure>
Again, we use the proportionality rule to calculate carbon emissions. We have:

$$ \mathcal{C}\mathcal{E}_{\text{total}}^{\text{down}} = \mathcal{C}\mathcal{I}_{\text{total}}^{\text{down}} \odot \frac{\mathcal{C}\mathcal{E}_1}{\mathcal{C}\mathcal{I}_1} $$

and:

$$ \mathcal{C}\mathcal{E}_{\text{indirect}}^{\text{down}} = \mathcal{C}\mathcal{E}_{\text{total}}^{\text{down}} - \mathcal{C}\mathcal{E}_{\text{direct}}^{\text{down}} = (\mathcal{C}\mathcal{I}_{\text{total}}^{\text{down}} - \mathcal{C}\mathcal{I}_1) \odot \frac{\mathcal{C}\mathcal{E}_1}{\mathcal{C}\mathcal{I}_1} $$

Using our previous example, the downstream matrix $\tilde{A}$ is equal to:

$$ \begin{aligned} \tilde{A} &= \operatorname{diag}(x)^{-1} Z \\ &= \begin{pmatrix} 0.10000 & 0.16000 & 0.32000 & 0.25000 \\ 0.12500 & 0.10000 & 0.40000 & 0.15625 \\ 0.03125 & 0.10000 & 0.30000 & 0.15625 \\ 0.00800 & 0.01600 & 0.06400 & 0.35000 \end{pmatrix} \end{aligned} $$

We deduce the downstream multiplier matrix:

$$ \tilde{\mathcal{L}} = \begin{pmatrix} 1.18811 & 0.31149 & 0.78705 & 0.72104 \\ 0.20970 & 1.25525 & 0.86717 & 0.59085 \\ 0.08938 & 0.20550 & 1.63035 & 0.47568 \\ 0.02859 & 0.05497 & 0.19156 & 1.60872 \end{pmatrix} $$

In Tables 8.49, 8.50 and 8.51, we report the downstream carbon intensities and emissions. These figures can be compared with those obtained in the case of the upstream analysis (Tables 8.46, 8.47 and 8.48).

<table><caption>Table 8.49: Direct and indirect downstream carbon intensities</caption><thead><tr><th rowspan="2">Sector</th><th rowspan="2">CI<sub>1</sub></th><th colspan="2">CI<sub>total</sub></th><th colspan="2">CI<sub>direct</sub></th><th colspan="2">CI<sub>indirect</sub></th><th rowspan="2">CI<sub>total</sub><br>CI<sub>1</sub></th></tr><tr><th colspan="2">(in tCO<sub>2</sub>e/$ mn)</th><th colspan="2">(in %)</th><th colspan="2">(in %)</th></tr></thead><tbody><tr><td>Energy</td><td>100.00</td><td>161.27</td><td>100.00</td><td>61.27</td><td>62.01%</td><td>37.99%</td><td>1.61</td></tr><tr><td>Materials</td><td>50.00</td><td>111.32</td><td>50.00</td><td>61.32</td><td>44.92%</td><td>55.08%</td><td>2.23</td></tr><tr><td>Industrials</td><td>25.00</td><td>64.73</td><td>25.00</td><td>39.73</td><td>38.62%</td><td>61.38%</td><td>2.59</td></tr><tr><td>Services</td><td>10.00</td><td>26.48</td><td>10.00</td><td>16.48</td><td>37.76%</td><td>62.24%</td><td>2.65</td></tr></tbody></table>

<table><caption>Table 8.50: Tier decomposition of downstream carbon intensities</caption><thead><tr><th></th><th>Sector</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>10</th><th>15</th><th>&infin;</th></tr></thead><tbody><tr><td rowspan="4">CI<sup>down</sup><sub>(k)</sub></td><td>Energy</td><td>28.50</td><td>14.68</td><td>8.00</td><td>4.45</td><td>2.48</td><td>0.14</td><td>0.00</td><td>0.00</td></tr><tr><td>Materials</td><td>29.06</td><td>14.39</td><td>7.92</td><td>4.39</td><td>2.45</td><td>0.13</td><td>0.01</td><td>0.00</td></tr><tr><td>Industrials</td><td>17.19</td><td>10.00</td><td>5.54</td><td>3.09</td><td>1.72</td><td>0.09</td><td>0.01</td><td>0.00</td></tr><tr><td>Services</td><td>6.70</td><td>4.14</td><td>2.44</td><td>1.40</td><td>0.79</td><td>0.04</td><td>0.00</td><td>0.00</td></tr><tr><td rowspan="4">CI<sup>down</sup><sub>(1-k)</sub></td><td>Energy</td><td>28.50</td><td>43.17</td><td>51.18</td><td>55.63</td><td>58.11</td><td>61.10</td><td>61.26</td><td>61.27</td></tr><tr><td>Materials</td><td>29.06</td><td>43.45</td><td>51.37</td><td>55.76</td><td>58.21</td><td>61.15</td><td>61.31</td><td>61.32</td></tr><tr><td>Industrials</td><td>17.19</td><td>27.19</td><td>32.73</td><td>35.82</td><td>37.54</td><td>39.61</td><td>39.72</td><td>39.73</td></tr><tr><td>Services</td><td>6.70</td><td>10.84</td><td>13.27</td><td>14.67</td><td>15.46</td><td>16.43</td><td>16.48</td><td>16.48</td></tr></tbody></table>

We find that the results of the downstream analysis are different. While the energy and materials sectors have the lowest upstream indirect emissions, they have the highest downstream emissions.
This is due to the structure of the supply chain. Most of the output from the energy and materials sectors is destined to be used by the value chain to produce goods and services. In contrast, the industrials and services sectors require a lot of output from the value chain to produce goods and services directly. In this context, carbon emissions generally move downward for the energy and materials sectors, while they move upward for the industrials and services sectors. Finally, the indirect downstream emissions are equal to 1075 ktCO₂e, while the indirect upstream emissions are equal to 1779 ktCO₂e. To better understand the difference between the downstream and the upstream, we also plot the downstream tree of the first two tiers for the services sector in Figure 8.137. If we compare this tree with Figure 8.137, we see that the downstream trees grow to the right, while the upstream trees grow to the left.

<table><caption>Table 8.51: Decomposition of downstream carbon emissions</caption><thead><tr><th rowspan="2">Sector</th><th colspan="3">down</th><th colspan="3">up</th></tr><tr><th>direct</th><th>indirect</th><th>total</th><th>direct</th><th>indirect</th><th>total</th></tr><tr><td></td><td colspan="3">(in ktCO<sub>2</sub>e)</td><td colspan="3">(in %)</td></tr></thead><tbody><tr><td>Energy</td><td>500</td><td>306.36</td><td>806.36</td><td>48.78</td><td>28.49</td><td>38.39</td></tr><tr><td>Materials</td><td>200</td><td>245.28</td><td>445.28</td><td>19.51</td><td>22.81</td><td>21.20</td></tr><tr><td>Industrials</td><td>200</td><td>317.83</td><td>517.83</td><td>19.51</td><td>29.55</td><td>24.65</td></tr><tr><td>Services</td><td>125</td><td>206.04</td><td>331.04</td><td>12.20</td><td>19.16</td><td>15.76</td></tr><tr><td>Total</td><td>1025</td><td>1075.50</td><td>2100.50</td><td>100.00</td><td>100.00</td><td>100.00</td></tr></tbody></table>

**Remark 92** As noted by Desnos et al. (2023), we must be careful with the upstream and downstream concepts of input-output analysis because they do not correspond to the upstream and downstream concepts of the GHG Protocol. There are several reasons for this. First, an input-output analysis does not distinguish between Scope 2 and Scope 3 emissions. Both are embedded in indirect emissions. Second, the GHG Protocol divides Scope 3 emissions into 8 upstream and 7 downstream categories. Downstream in the GHG Protocol refers to carbon emissions after goods and services have been produced. It includes their use by other sectors as well as final demand. Input-output analysis does not take into account downstream carbon emissions due to final demand (e.g., recycling or waste management). Thus, the downstream concept in input-output analysis is not consistent with the GHG Protocol definition. Third, we can see that there is a lot of double counting in the two analyses. For example, suppose that the matrix A is diagonal. In this case, we can show that $\mathcal{C}\mathcal{E}_{\text{indirect}}^{\text{up}} = \mathcal{C}\mathcal{E}_{\text{indirect}}^{\text{down}}$. In this particular case, the upstream and downstream analyses refer to the same carbon emissions, and we do not really know whether these emissions are upstream or downstream in the value chain.

## Mathematical properties

The mathematical properties derived in Section 8.4.1 on page 807 remain valid when we consider the upstream or downstream analysis of carbon intensities. Indeed, we reiterate that the properties require only that A is a doubly substochastic matrix. This is also the case for the matrices $A^\top$ (upstream multiplier matrix $\tilde{\mathcal{L}}$) and $\tilde{\mathcal{A}}$ (downstream multiplier matrix $\tilde{\mathcal{L}}$). In particular, since the carbon intensities are not negative, the partial monotony property is satisfied. This means that there exists an index $k^*$ such that $\mathcal{CI}_{(k)} \succeq \mathcal{CI}_{(k+1)}$ for the upstream analysis and $\mathcal{CI}_{(k)}^{\text{down}} \succeq \mathcal{CI}_{(k+1)}^{\text{down}}$ for the downstream analysis.

We recall that $\mathcal{CI}_{\text{total}} = (I_n - A^\top)^{-1} \mathcal{CI}_1 = \sum_{k=0}^\infty (A^\top)^k \mathcal{CI}_1 = \sum_{k=0}^\infty \mathcal{CI}_{(k)}$. Let $w_{(k)}$ be the relative contribution vector of the $k^{\text{th}}$ tier. We have:

$$w_{(k),j} = \frac{\mathcal{CI}_{(k),j}}{\sum_{h=0}^{\infty} \mathcal{CI}_{(k),j}}$$
Box 8.20: Calculating the upstreamness index

Let M be a square matrix of dimension n × n. We have:

$$ \frac{\partial (I_n - M)^{-1}}{\partial M} = (I_n - M)^{-1} (I_n - M)^{-1} $$

and:

$$ \frac{\partial \sum_{k=0}^{\infty} M^k}{\partial M} = \sum_{k=0}^{\infty} k M^{k-1} $$

It follows that:

$$ \begin{align*} \sum_{k=0}^{\infty} k M^k &= M \sum_{k=0}^{\infty} k M^{k-1} \\ &= M \frac{\partial \sum_{k=0}^{\infty} M^k}{\partial M} \\ &= M \frac{\partial (I_n - M)^{-1}}{\partial M} \\ &= M (I_n - M)^{-1} (I_n - M)^{-1} \end{align*} $$

Let z be a vector. Since we have $(I_n - M)^{-1} z = \sum_{k=0}^{\infty} M^k z = \sum_{k=0}^{\infty} z_{(k)}$ with $z_{(k)} = M^k z$, we deduce that:

$$ \sum_{k=0}^{\infty} k \cdot z_{(k)} = \left( \sum_{k=0}^{\infty} k M^k \right) z = M (I_n - M)^{-1} (I_n - M)^{-1} z $$

The upstreamness index of sector j is then equal to:

$$ \boldsymbol{\kappa}_{j}^{\text{up}} := \frac{\left(\sum_{k=0}^{\infty} k \cdot z_{(k)}\right)_j}{\left(\sum_{k=0}^{\infty} z_{(k)}\right)_j} = \frac{\left(M (I_n - M)^{-1} (I_n - M)^{-1} z\right)_j}{\left((I_n - M)^{-1} z\right)_j} $$

This expression is not exactly the formula proposed by Antràs et al. (2012), because they do not weight the tiers in the same way.

Following Antràs et al. (2012), we define the upstreamness index as the weighted average of the tiers with respect to their relative contributions:

$$ \boldsymbol{\kappa}_{j}^{\text{up}} = \sum_{k=0}^{\infty} k \cdot w_{(k),j} = 0 \times \frac{\mathcal{CI}_{(0),j}}{\mathcal{CI}_{\text{total},j}} + 1 \times \frac{\mathcal{CI}_{(1),j}}{\mathcal{CI}_{\text{total},j}} + 2 \times \frac{\mathcal{CI}_{(2),j}}{\mathcal{CI}_{\text{total},j}} + \dots = \frac{\left(\sum_{k=0}^{\infty} k \cdot \mathcal{CI}_{(k)}\right)_j}{\left(\sum_{k=0}^{\infty} \mathcal{CI}_{(k)}\right)_j} $$

In Box 8.20, we show that<fnref n="140" />:

$$ \boldsymbol{\kappa}_{j}^{\text{up}} = \frac{\left( A^{\top} (I_n - A^{\top})^{-2} \mathcal{CI}_1 \right)_j}{\left( (I_n - A^{\top})^{-1} \mathcal{CI}_1 \right)_j} $$

<footnote n="140">¹⁴⁰We set $M = A^{\top}$ and $M = \tilde{A}$, respectively.</footnote>
and:

$$ \kappa_j^{\text{down}} = \frac{\left( \check{A} (I_n - \check{A})^{-2} \mathcal{CI}_1 \right)_j}{\left( (I_n - \check{A})^{-1} \mathcal{CI}_1 \right)_j} $$

If we consider our previous example, we obtain the following results:

<table><thead><tr><th></th><th></th><th>Energy</th><th>Materials</th><th>Industrials</th><th>Services</th></tr></thead><tbody><tr><td>M = A</td><td>z = y</td><td>2.01</td><td>1.92</td><td>1.40</td><td>0.88</td></tr><tr><td>M = A<sup>T</sup></td><td>z = <b>CI</b><sub>1</sub></td><td>0.49</td><td>1.21</td><td>1.79</td><td>2.13</td></tr><tr><td>M = A&#x0303;</td><td>z = <b>CI</b><sub>1</sub></td><td>0.84</td><td>1.20</td><td>1.40</td><td>1.48</td></tr></tbody></table>

The upstreamness index of the energy sector is equal to 0.49, while its downstreamness index is equal to 0.84. In the case of the services sector, we obtain a higher upstreamness index ($\kappa_4^{\text{up}} = 2.13$). This indicates that the generation of carbon emissions in this sector affects higher tiers.

## Comparison of upstream emissions between Exiobase, Trucost and WIOD

We report here some calculations made by Desnos et al. (2023). We recall that the total carbon intensity and emission vectors are equal to $\mathcal{CI}_{\text{total}} = (I_n - A^{\top})^{-1} \mathcal{CI}_1$ and $\mathcal{CE}_{\text{total}} = x \odot \mathcal{CI}_{\text{total}}$. For the $k^{\text{th}}$ tier, the formulas become $\mathcal{CI}_{(k)} = (A^{\top})^k \mathcal{CI}_1$ and $\mathcal{CE}_{(k)} = x \odot \mathcal{CI}_{(k)}$. The dimension of all these vectors is $n \times 1$, where $n$ is the number of countries times the number of industries. For Trucost, carbon emissions and intensities are directly available. Direct emissions from input-output models can be compared to Scope 1 emissions of Trucost, while the total emissions correspond to Scope 1 plus Scope 2 plus upstream Scope 3 emissions from Trucost. We can also compare the direct plus first-tier indirect emissions of Trucost with the first-tier cumulative emissions $\mathcal{CE}_{(0-1)}$ calculated from input-output models.

If we want to aggregate the results such that $i \in \Omega$, we have:

$$ \mathcal{CE}_{\text{total}}(\Omega) = \sum_{i \in \Omega} \mathcal{CE}_{\text{total},i} = \omega^{\top} \mathcal{CE}_{\text{total}} $$

where $\omega = (\omega_i)$ is a vector of dimension $n \times 1$ with $\omega_i = 1$ if $i \in \Omega$ and $\omega_i = 0$ otherwise. We deduce that the carbon intensity of $\Omega$ is equal to:

$$ \mathcal{CI}_{\text{total}}(\Omega) = \frac{\sum_{i \in \Omega} \mathcal{CE}_{\text{total},i}}{\sum_{i \in \Omega} x_i} = \frac{\sum_{i \in \Omega} x_i \mathcal{CI}_{\text{total},i}}{\sum_{i \in \Omega} x_i} = \sum_{i \in \Omega} w_i \mathcal{CI}_{\text{total},i} $$

where $w_i = (\sum_{j \in \Omega} x_j)^{-1}$ $x_i$ is the weight of item $i$ in the set $\Omega$. The carbon intensity of $\Omega$ is then equal to its weighted average carbon intensity (WACI). These calculations also apply to $\mathcal{CE}_{(k)}$, $\mathcal{CE}_{(0:k)}$, $\mathcal{CI}_{(k)}$ and $\mathcal{CI}_{(0:k)}$.

We perform a global analysis by setting $\omega = \mathbf{1}_n$. In Table 8.52, we report the multiplication coefficients $m_{(k)} = \mathcal{CE}_{(k)} / \mathcal{CE}_1$ and $m_{(0-k)} = \mathcal{CE}_{(0-k)} / \mathcal{CE}_1$, and we also compute the contribution ratio $c_{(0-k)} = \mathcal{CE}_{(0-k)} / \mathcal{CE}_{\text{total}}$. For the WIOD table, the direct plus indirect emissions are 3.14 times the Scope 1 emissions, which means that the indirect emissions are more than twice the direct emissions. In the case of the Exiobase tables, the ratio $m_{(0-\infty)}$ is equal to 2.76 in 2014 and 2.75 in 2022. We note that the convergence is rapid, as more than 90% of the total emissions are located in the first five tiers.
<table><caption>Table 8.52: Ratio of upstream carbon emissions (global analysis)</caption><thead><tr><th rowspan="2">Tier</th><th colspan="3">WIOD 2014</th><th colspan="3">Exiobase 2014</th><th colspan="3">Exiobase 2022</th></tr><tr><th>m<sub>(k)</sub></th><th>m<sub>(0-k)</sub></th><th>c<sub>(0-k)</sub></th><th>m<sub>(k)</sub></th><th>m<sub>(0-k)</sub></th><th>c<sub>(0-k)</sub></th><th>m<sub>(k)</sub></th><th>m<sub>(0-k)</sub></th><th>c<sub>(0-k)</sub></th></tr></thead><tbody><tr><td>0</td><td>1.00</td><td>1.00</td><td>31.8%</td><td>1.00</td><td>1.00</td><td>36.2%</td><td>1.00</td><td>1.00</td><td>36.4%</td></tr><tr><td>1</td><td>0.77</td><td>1.76</td><td>56.1%</td><td>0.72</td><td>1.72</td><td>62.5%</td><td>0.73</td><td>1.73</td><td>62.9%</td></tr><tr><td>2</td><td>0.50</td><td>2.26</td><td>71.9%</td><td>0.43</td><td>2.15</td><td>78.0%</td><td>0.42</td><td>2.15</td><td>78.3%</td></tr><tr><td>3</td><td>0.32</td><td>2.58</td><td>82.1%</td><td>0.25</td><td>2.40</td><td>87.0%</td><td>0.25</td><td>2.40</td><td>87.3%</td></tr><tr><td>4</td><td>0.20</td><td>2.78</td><td>88.6%</td><td>0.15</td><td>2.55</td><td>92.3%</td><td>0.14</td><td>2.54</td><td>92.5%</td></tr><tr><td>5</td><td>0.13</td><td>2.91</td><td>92.7%</td><td>0.09</td><td>2.63</td><td>95.5%</td><td>0.08</td><td>2.62</td><td>95.5%</td></tr><tr><td>6</td><td>0.08</td><td>3.00</td><td>95.4%</td><td>0.05</td><td>2.69</td><td>97.3%</td><td>0.05</td><td>2.67</td><td>97.3%</td></tr><tr><td>7</td><td>0.05</td><td>3.05</td><td>97.0%</td><td>0.03</td><td>2.72</td><td>98.4%</td><td>0.03</td><td>2.70</td><td>98.4%</td></tr><tr><td>8</td><td>0.03</td><td>3.08</td><td>98.1%</td><td>0.02</td><td>2.73</td><td>99.0%</td><td>0.02</td><td>2.72</td><td>99.0%</td></tr><tr><td>9</td><td>0.02</td><td>3.11</td><td>98.8%</td><td>0.01</td><td>2.74</td><td>99.4%</td><td>0.01</td><td>2.73</td><td>99.4%</td></tr><tr><td>10</td><td>0.01</td><td>3.12</td><td>99.2%</td><td>0.01</td><td>2.75</td><td>99.7%</td><td>0.01</td><td>2.74</td><td>99.7%</td></tr><tr><td>∞</td><td>0.00</td><td>3.14</td><td>100.0%</td><td>0.00</td><td>2.76</td><td>100.0%</td><td>0.00</td><td>2.75</td><td>100.0%</td></tr></tbody></table>

Source: Desnos et al. (2023, Table 15, page 52).

<figure><img src="image_419.png" /><figcaption>Figure 8.138: Multiplication coefficient $m_{(0-1)}$ and $m_{(0-\infty)}$ (global analysis)</figcaption></figure>

Source: Desnos et al. (2023, Figure 25, page 53).
In the case of Trucost, we can only compute $m_{(0-1)}$ and $m_{(0-\infty)}$. Results are shown in Figure 8.138. We notice that the multiplication coefficients obtained with Trucost are smaller than those computed with input-output models (1.45–1.54 versus 1.7 for the first tier). Nevertheless, the multiplication coefficient is very high when we integrate Scope 3 emissions since we obtain a value of 4.78 in 2019 and 6.75 in 2021.

Desnos et al. (2023) also suggested a country analysis<fnref n="141" />. They observed some large differences from one country to another. For instance, Figure 8.139 shows the multiplication coefficient $m_{(0-\infty)}$ of the different countries. The lowest value is obtained for the USA ($m_{(0-\infty)}$ = 2.19), while the highest factor is observed for Switzerland ($m_{(0-\infty)}$ = 7.21).

<figure><img src="image_420.png" /><figcaption>Figure 8.139: Multiplying coefficient $m_{(0-\infty)}$ (country analysis, WIOD 2014)</figcaption></figure>

Source: Desnos et al. (2023, Figure 26, page 54).

Desnos et al. (2023) applied the previous framework to estimate the upstream of the MSCI World index. They first estimated the total carbon intensity of all issuers in the portfolio:

$$ \mathcal{CI}_{\text{total},i} = \mathcal{CI}_{1,i}^{\text{reported}} + \mathcal{CI}_{\text{indirect},i}^{\text{estimated}} $$

where $\mathcal{CI}_{1,i}^{\text{reported}}$ is the Scope 1 carbon intensity reported by the issuer $i$ and $\mathcal{CI}_{\text{indirect},i}^{\text{estimated}}$ is the estimated indirect carbon intensity. In the case of Trucost, they used the values estimated by the data provider. For the input-output databases, they used the formula $\mathcal{CI}_{\text{indirect}}^{\text{estimated}} = ((I_n - A^\top)^{-1} - I_n) \mathcal{CI}_{\text{direct}}$ and considered the row corresponding to the sector and the country of the issuer $i$. They then aggregated the carbon intensity of the MSCI World Index at the GICS level 1 sectors. The results are shown in Table 8.53 and Figure 8.140. The direct plus indirect intensity of

<footnote n="141">¹⁴¹ In this case, $\omega_i = 1$ if $i$ belongs to the country and $\omega_i = 0$ otherwise.</footnote>
the MSCI World Index is equal to 299 tCO₂e/$ mn with Exiobase 2022, 281 tCO₂e/$ mn with Tru-cost 2021 and 278 tCO₂e/$ mn with WIOD 2014. The difference between the lowest and the highest value is then equal to 7.5%, which is a low figure. If we look at the GICS sectors, the differences are more significant, especially for consumer staples, energy and materials. For instance, the carbon intensity of the energy sector is equal to 757 tCO₂e/$ mn with WIOD 2014 and 1 373 tCO₂e/$ mn with Exiobase 2022. They also calculated the contribution of each sector to the carbon intensity of the MSCI World Index:

$$c_j(w) = \frac{\sum_{i \in S_{sector_j}} w_i \cdot C I_{total,i}}{\sum_i w_i \cdot C I_{total,i}}$$

where *w* is the vector of weights in the MSCI World Index and $c_j(w)$ is the contribution of the *j*th sector. In Table 8.54, we see some significant differences. This is true for the previously mentioned sectors (consumer staples, energy and materials), but also for consumer discretionary, health care and information technology.

<table><caption>Table 8.53: Direct + indirect carbon intensities of GICS sectors (MSCI World index, May 2023)</caption><thead><tr><th>Sector</th><th>Exiobase 2022</th><th>Trucost 2021</th><th>WIOD 2014</th></tr></thead><tbody><tr><td>Communication Services</td><td>66</td><td>78</td><td>102</td></tr><tr><td>Consumer Discretionary</td><td>168</td><td>209</td><td>219</td></tr><tr><td>Consumer Staples</td><td>437</td><td>387</td><td>277</td></tr><tr><td>Energy</td><td>1 373</td><td>796</td><td>757</td></tr><tr><td>Financials</td><td>83</td><td>55</td><td>83</td></tr><tr><td>Health Care</td><td>108</td><td>120</td><td>167</td></tr><tr><td>Industrials</td><td>276</td><td>277</td><td>307</td></tr><tr><td>Information Technology</td><td>110</td><td>138</td><td>131</td></tr><tr><td>Materials</td><td>791</td><td>973</td><td>747</td></tr><tr><td>Real Estate</td><td>128</td><td>134</td><td>138</td></tr><tr><td>Utilities</td><td>1 872</td><td>1 833</td><td>1 889</td></tr><tr><td>MSCI World</td><td>299</td><td>281</td><td>278</td></tr></tbody></table>

Source: Desnos et al. (2023, Table 57, page 178).

<table><caption>Table 8.54: Breakdown of the portfolio intensity by GICS sector (MSCI World Index, May 2023)</caption><thead><tr><th>Sector</th><th>Exiobase 2022</th><th>Trucost 2021</th><th>WIOD 2014</th></tr></thead><tbody><tr><td>Communication Services</td><td>1.5%</td><td>1.9%</td><td>2.5%</td></tr><tr><td>Consumer Discretionary</td><td>5.9%</td><td>7.8%</td><td>8.4%</td></tr><tr><td>Consumer Staples</td><td>11.6%</td><td>10.9%</td><td>7.9%</td></tr><tr><td>Energy</td><td>22.9%</td><td>14.1%</td><td>13.6%</td></tr><tr><td>Financials</td><td>4.2%</td><td>2.9%</td><td>4.5%</td></tr><tr><td>Health Care</td><td>4.8%</td><td>5.7%</td><td>8.0%</td></tr><tr><td>Industrials</td><td>10.2%</td><td>10.8%</td><td>12.1%</td></tr><tr><td>Information Technology</td><td>7.5%</td><td>10.0%</td><td>9.6%</td></tr><tr><td>Materials</td><td>11.7%</td><td>15.3%</td><td>11.9%</td></tr><tr><td>Real Estate</td><td>1.1%</td><td>1.2%</td><td>1.2%</td></tr><tr><td>Utilities</td><td>18.6%</td><td>19.4%</td><td>20.2%</td></tr></tbody></table>

Source: Desnos et al. (2023, Table 21, page 62).
<figure><img src="image_421.png" /><figcaption>Figure 8.140: Total carbon intensity $CI_{total}$ by GICS sector (MSCI World Index, May 2023)</figcaption></figure>

Source: Desnos et al. (2023, Figure 33, page 61).

### 8.4.4 Imported and exported carbon emissions

On page 819, we have analyzed the difference in carbon footprint when we use a production-based or consumption-based inventory. We can extend this framework by considering multiple regions. In this case, we can adjust a country's carbon footprint for imports and exports. For instance, it may be unfair to attribute the CO₂ emissions of a good to a country if that country exports 100% of that good to the rest of the world. The concept of imported and exported carbon emissions is then central when we would like to measuring a country's carbon footprint, which reflects the consumption and lifestyle choices of its citizens (Turner et al., 2007; Wiedmann et al., 2007). The groundbreaking research paper by Peters and Hertwich (2008) has been followed by a large number of studies<fnref n="142" />. They all conclude that there is a large bias when focusing on territorial emissions due to production.

#### Calculation of balanced emissions

Let's learn how to calculate the CO₂ embodied in international trade by using the input-output framework. We have:

* $y_{j,r}$ is the final demand of the $j$th sector and the $r$th region.

* $y = (y_1, \dots, y_n)$ is the vector of final demand where $y_j = \sum_{r=1}^{p} y_{j,r}$ is the final demand of the $j$th sector.

<footnote n="142">¹⁴²For example, we can cite the works of Davis and Caldeira (2010), Peters et al. (2011), Yamano and Guilhoto (2020), Lamb et al. (2021) and Friedlingstein et al. (2022).</footnote>
• $C^{(x)} = (C_{g,j}^{(x)})$ is the pollution output matrix where $C_{g,j}^{(x)}$ is the total amount of the $g^{th}$ pollutant generated by the output of the $j^{th}$ sector.

• $D^{(y)} = C^{(x)} \text{diag}(x)^{-1} = (D_{g,j}^{(y)})$ is the matrix of direct impact coefficients where $D_{g,j}^{(y)} = c_{g,j}^{(y)}/x_j$ is the amount of the $g^{th}$ pollutant generated by 1$ of the output of the $j^{th}$ sector.

• $\varpi = (\varpi_1, \dots, \varpi_m)$ is the vector of pollution level. We have:

$$ \varpi = D^{(x)}x = D^{(x)}(I_n - A)^{-1}y = D^{(y)}y $$

where $D^{(y)} = D^{(x)}(I_n - A)^{-1}$ is the pollutant multiplier matrix with respect to the final demand $y$.

In the production-based approach, the total contribution of the $r^{th}$ to the $g^{th}$ pollutant is equal to:

$$ C_{g}^{(x,r)} = \sum_{j \in r} C_{g,j}^{(x)} \qquad (8.42) $$

In the consumption-based approach, we have the following identity:

$$ \varpi_g = (D^{(y)}y)_g = \sum_{j=1}^{n} D_{g,j}^{(y)} y_j = \sum_{j=1}^{n} D_{g,j}^{(y)} \sum_{r=1}^{p} y_{j,r} = \sum_{r=1}^{p} \sum_{j=1}^{n} D_{g,j}^{(y)} y_{j,r} $$

we deduce that the total contribution of the $r^{th}$ region to the $g^{th}$ pollutant is equal to:

$$ \begin{aligned} C_{g}^{(y,r)} &= \sum_{j=1}^{n} \frac{\partial \varpi_{g}}{\partial y_{j,r}} y_{j,r} \\ &= \sum_{j=1}^{n} D_{g,j}^{(y)} y_{j,r} \end{aligned} \qquad (8.43) $$

**Example 32** We consider two sectors ($S_1$ and $S_2$) and three regions ($R_1$, $R_2$ and $R_3$), whose input-output table is given below:

<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="6">Z</th><th colspan="3">y</th><th rowspan="2">x</th></tr><tr><th colspan="2">R<sub>1</sub></th><th colspan="2">R<sub>2</sub></th><th colspan="2">R<sub>3</sub></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th></tr><tr><th></th><th></th><th>S<sub>1</sub></th><th>S<sub>2</sub></th><th>S<sub>1</sub></th><th>S<sub>2</sub></th><th>S<sub>1</sub></th><th>S<sub>2</sub></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><th rowspan="2">R<sub>1</sub></th><th>S<sub>1</sub></th><td>100</td><td>300</td><td>10</td><td>10</td><td>20</td><td>0</td><td>500</td><td>200</td><td>25</td><td>1165</td></tr><tr><th>S<sub>2</sub></th><td>250</td><td>150</td><td>20</td><td>0</td><td>10</td><td>0</td><td>800</td><td>100</td><td>17</td><td>1347</td></tr><tr><th rowspan="2">R<sub>2</sub></th><th>S<sub>1</sub></th><td>10</td><td>10</td><td>110</td><td>310</td><td>0</td><td>0</td><td>20</td><td>200</td><td>15</td><td>675</td></tr><tr><th>S<sub>2</sub></th><td>20</td><td>20</td><td>80</td><td>25</td><td>15</td><td>20</td><td>0</td><td>200</td><td>5</td><td>385</td></tr><tr><th rowspan="2">R<sub>3</sub></th><th>S<sub>1</sub></th><td>10</td><td>5</td><td>8</td><td>3</td><td>40</td><td>7</td><td>5</td><td>25</td><td>50</td><td>153</td></tr><tr><th>S<sub>2</sub></th><td>5</td><td>2</td><td>8</td><td>8</td><td>12</td><td>35</td><td>3</td><td>50</td><td>50</td><td>173</td></tr><tr><th colspan="2">V</th><td>770</td><td>860</td><td>439</td><td>29</td><td>56</td><td>111</td><td></td><td></td><td></td><td></td></tr><tr><th colspan="2">x</th><td>1165</td><td>1347</td><td>675</td><td>385</td><td>153</td><td>173</td><td></td><td></td><td></td><td></td></tr><tr><th rowspan="2">GHG</th><th>CO<sub>2</sub></th><td>50 000</td><td>20 000</td><td>10 000</td><td>10 000</td><td>5 000</td><td>5 000</td><td>100 000</td><td></td><td></td><td></td></tr><tr><th>CH<sub>4</sub></th><td>5 000</td><td>3 000</td><td>0</td><td>0</td><td>1 000</td><td>1 000</td><td>10 000</td><td></td><td></td><td></td></tr></tbody></table>

$Z$, $y$, $x$ and $V$ are expressed in $mn$, while the GHG are calculated in tonne. Final demand is split between the three regions. For instance, $x_1$ is equal to $1165 mn$, its intermediary demand is equal to $100+300+10+10+20+0$ or $440 mn$ and its final demand is equal to $725 mn$ with the following breakdown: 500 for the region $R_1$, 200 for the region $R_2$ and 25 for the region $R_3$.
If we take a production-based approach, $x_1$ and $x_2$ are in region $\mathcal{R}_1$, which means that this region is responsible for 70 000 tCO$_2$ and 8 000 tCH$_4$. Finally, we obtain the following distribution among the three regions:

<table><thead><tr><th rowspan="2"></th><th colspan="3">Absolute</th><th colspan="3">Relative</th></tr><tr><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th></tr></thead><tbody><tr><th>CO<sub>2</sub></th><td>70 000</td><td>20 000</td><td>10 000</td><td>70%</td><td>20%</td><td>10%</td></tr><tr><th>CH<sub>4</sub></th><td>8 000</td><td>0</td><td>2 000</td><td>80%</td><td>0%</td><td>20%</td></tr></tbody></table>

We have:

$$A = Z \operatorname{diag}(x)^{-1} = \begin{pmatrix} 0.0858 & 0.2227 & 0.0148 & 0.0260 & 0.1307 & 0.0000 \\ 0.2146 & 0.1114 & 0.0296 & 0.0000 & 0.0654 & 0.0000 \\ 0.0086 & 0.0074 & 0.1630 & 0.8052 & 0.0000 & 0.0000 \\ 0.0172 & 0.0148 & 0.1185 & 0.0649 & 0.0980 & 0.1156 \\ 0.0086 & 0.0037 & 0.0119 & 0.0078 & 0.2614 & 0.0405 \\ 0.0043 & 0.0015 & 0.0119 & 0.0208 & 0.0784 & 0.2023 \end{pmatrix}$$

and:

$$\mathcal{L} = (I_6 - A)^{-1} = \begin{pmatrix} 1.1674 & 0.2953 & 0.0454 & 0.0741 & 0.2450 & 0.0232 \\ 0.2848 & 1.1989 & 0.0586 & 0.0601 & 0.1663 & 0.0171 \\ 0.0482 & 0.0445 & 1.3705 & 1.1871 & 0.1893 & 0.1816 \\ 0.0351 & 0.0320 & 0.1817 & 1.2327 & 0.1927 & 0.1884 \\ 0.0167 & 0.0109 & 0.0263 & 0.0362 & 1.3706 & 0.0748 \\ 0.0101 & 0.0064 & 0.0280 & 0.0538 & 0.1442 & 1.2687 \end{pmatrix}$$

The direct impact matrix is equal to the GHG emissions divided by the output:

$$D^{(x)} = \begin{pmatrix} 50000/1165 & 20000/1347 & 10000/675 & 10000/385 & 5000/153 & 5000/173 \\ 5000/1165 & 3000/1347 & 0/675 & 0/385 & 1000/153 & 1000/173 \end{pmatrix} \\ = \begin{pmatrix} 42.9185 & 14.8478 & 14.8148 & 25.9740 & 32.6797 & 28.9017 \\ 4.2918 & 2.2272 & 0.0000 & 0.0000 & 6.5359 & 5.7803 \end{pmatrix}$$

The unit of $D^{(x)}$ is expressed in tonnes of the gas per million dollars. The carbon intensity of the sector $S_1$ in the region $\mathcal{R}_1$ is equal to 42.9185 tCO$_2$/S mn, while the methane intensity is equal to 4.2918 tCH$_4$/S mn. We have:

$$D^{(y)} = D^{(x)} \mathcal{L} = \begin{pmatrix} 56.7948 & 32.5033 & 29.5091 & 56.4128 & 69.7564 & 47.9460 \\ 5.8121 & 4.0453 & 0.6591 & 0.9993 & 11.2142 & 7.9600 \end{pmatrix}$$

Using Equation (8.43), we finally obtain the following decomposition:

<table><thead><tr><th rowspan="2"></th><th colspan="3">Absolute</th><th colspan="3">Relative</th></tr><tr><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th></tr></thead><tbody><tr><th>CO<sub>2</sub></th><td>55 483</td><td>35 935</td><td>8 582</td><td>55.5%</td><td>35.9%</td><td>8.6%</td></tr><tr><th>CH<sub>4</sub></th><td>6 235</td><td>2 577</td><td>1 188</td><td>62.4%</td><td>25.8%</td><td>11.9%</td></tr></tbody></table>

In the consumption-based approach, the region $\mathcal{R}_1$ is responsible for 55.5% of carbon emissions, which is lower than the 70% figure obtained in the production-based approach. The second region sees a high increase in carbon emissions of 79.5%.

Let us now look at how the concept of imported and exported emissions is defined. In Table 8.55, we report the allocation of carbon emissions to intermediate and final uses. For intermediate consumption, the allocation to $Z_{i,j}$ is equal to:

$$\mathcal{C}\mathcal{E}_i(Z_{i,j}) = \frac{Z_{i,j}}{x_i} \cdot \mathcal{C}\mathcal{E}_i$$
while we have for the final consumption:

$$ \mathcal{C}\mathcal{E}_i(y_{i,r}) = \frac{y_{i,r}}{x_i} \cdot \mathcal{C}\mathcal{E}_i $$

By construction, we verify the equality:

$$ \sum_{j=1}^{n} \mathcal{C}\mathcal{E}_i(Z_{i,j}) + \sum_{r=1}^{p} \mathcal{C}\mathcal{E}_i(y_{i,r}) = \left( \sum_{j=1}^{n} \frac{Z_{i,j}}{x_i} + \sum_{r=1}^{p} \frac{y_{i,r}}{x_i} \right) \mathcal{C}\mathcal{E}_i = \mathcal{C}\mathcal{E}_i $$

<table><caption>Table 8.55: Allocation of carbon emissions to intermediate and final uses (production-based approach)</caption><thead><tr><th rowspan="2"></th><th colspan="6">Z</th><th colspan="3">y</th><th rowspan="2">Total</th></tr><tr><th>R<sub>1</sub></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>R<sub>3</sub></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th></tr></thead><tbody><tr><th>(S<sub>1</sub>, R<sub>1</sub>)</th><td>4 292</td><td>12 876</td><td>429</td><td>429</td><td>858</td><td>0</td><td>21 459</td><td>8 584</td><td>1 073</td><td>50 000</td></tr><tr><th>(S<sub>2</sub>, R<sub>1</sub>)</th><td>3 712</td><td>2 227</td><td>297</td><td>0</td><td>148</td><td>0</td><td>11 878</td><td>1 485</td><td>252</td><td>20 000</td></tr><tr><th>(S<sub>1</sub>, R<sub>2</sub>)</th><td>148</td><td>148</td><td>1 630</td><td>4 593</td><td>0</td><td>0</td><td>296</td><td>2 963</td><td>222</td><td>10 000</td></tr><tr><th>(S<sub>2</sub>, R<sub>2</sub>)</th><td>519</td><td>519</td><td>2 078</td><td>649</td><td>390</td><td>519</td><td>0</td><td>5 195</td><td>130</td><td>10 000</td></tr><tr><th>(S<sub>1</sub>, R<sub>3</sub>)</th><td>327</td><td>163</td><td>261</td><td>98</td><td>1 307</td><td>229</td><td>163</td><td>817</td><td>1 634</td><td>5 000</td></tr><tr><th>(S<sub>2</sub>, R<sub>3</sub>)</th><td>145</td><td>58</td><td>231</td><td>231</td><td>347</td><td>1 012</td><td>87</td><td>1 445</td><td>1 445</td><td>5 000</td></tr></tbody><tfoot><tr><th>Total</th><td>9 143</td><td>15 992</td><td>4 926</td><td>6 000</td><td>3 050</td><td>1 760</td><td>33 884</td><td>20 488</td><td>4 757</td><td>100 000</td></tr></tfoot></table>

From Table 8.55, we can aggregate the columns to obtain the regional carbon allocation (Table 8.56). For instance, the carbon allocation to sector $S_1$ and region $R_1$ is equal to:

$$ 4 292 + 12 876 + 21 459 = 38 627 \text{ tCO}_2 $$

<table><caption>Table 8.56: Carbon allocation by region (production-based approach, column aggregation)</caption><thead><tr><th></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>Total</th></tr></thead><tbody><tr><th>(S<sub>1</sub>, R<sub>1</sub>)</th><td>38 627</td><td>9 442</td><td>1 931</td><td>50 000</td></tr><tr><th>(S<sub>2</sub>, R<sub>1</sub>)</th><td>17 817</td><td>1 782</td><td>401</td><td>20 000</td></tr><tr><th>(S<sub>1</sub>, R<sub>2</sub>)</th><td>593</td><td>9 185</td><td>222</td><td>10 000</td></tr><tr><th>(S<sub>2</sub>, R<sub>2</sub>)</th><td>1 039</td><td>7 922</td><td>1 039</td><td>10 000</td></tr><tr><th>(S<sub>1</sub>, R<sub>3</sub>)</th><td>654</td><td>1 176</td><td>3 170</td><td>5 000</td></tr><tr><th>(S<sub>2</sub>, R<sub>3</sub>)</th><td>289</td><td>1 908</td><td>2 803</td><td>5 000</td></tr></tbody><tfoot><tr><th>Total</th><td>59 018</td><td>31 415</td><td>9 567</td><td>100 000</td></tr></tfoot></table>

<table><caption>Table 8.57: Carbon allocation by region (production-based approach, row aggregation)</caption><thead><tr><th></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>Total</th></tr></thead><tbody><tr><th>R<sub>1</sub></th><td>56 444</td><td>11 224</td><td>2 332</td><td>70 000</td></tr><tr><th>R<sub>2</sub></th><td>1 632</td><td>17 107</td><td>1 261</td><td>20 000</td></tr><tr><th>R<sub>3</sub></th><td>943</td><td>3 084</td><td>5 973</td><td>10 000</td></tr></tbody><tfoot><tr><th>Total</th><td>59 018</td><td>31 415</td><td>9 567</td><td>100 000</td></tr></tfoot></table>

By aggregating the rows, we finally obtain the allocation given in Table 8.57. For example, the carbon allocation of goods produced in region $R_1$ and consumed in region $R_1$ is equal to:

$$ 3 8627 + 17 817 = 56 444 \text{ tCO}_2 $$
The carbon allocation of goods produced in region $R_1$ and consumed in region $R_2$ is equal to:

$$9442 + 1782 = 11224 \text{ tCO}_2$$

The row shows the allocation split of the emissions produced in a given region. The column shows the allocation split of the emissions consumed in a given region. For example, among the 70 000 tCO₂ emitted in region $R_1$, 11 224 and 2 332 are exported in regions $R_2$ and $R_3$. Therefore, 56 444 tCO₂ remain in region $R_1$. Since $R_1$ also imports 1 632 tCO₂ from region $R_2$ and 943 tCO₂ from region $R_3$, the total carbon emissions allocated to region $R_1$ is equal to 59 018 tCO₂. A summary of the allocation process is given below:

<table><thead><tr><th></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>Total</th></tr></thead><tbody><tr><td>Produced emissions</td><td>70 000</td><td>20 000</td><td>10 000</td><td>100 000</td></tr><tr><td>– Exported emissions</td><td>13 556</td><td>2 893</td><td>4 027</td><td>20 475</td></tr><tr><td>= Intermediate emissions</td><td>56 444</td><td>17 107</td><td>5 973</td><td>79 525</td></tr><tr><td>+ Imported emissions</td><td>2 574</td><td>14 308</td><td>3 593</td><td>20 475</td></tr><tr><td>= Regional emissions</td><td>59 018</td><td>31 415</td><td>9 567</td><td>100 000</td></tr></tbody></table>

By construction, exported emissions are equal to imported emissions at the aggregate level, but not at the regional level. In the case of region $R_1$, we have the following results:

$$\underbrace{70\,000}_{\text{produced}} - \underbrace{13\,556}_{\text{exported}} + \underbrace{2\,574}_{\text{imported}} = \underbrace{59\,018}_{\text{allocated}} \neq \underbrace{55\,483}_{\text{consumed}}$$

Note that we do not get the previous figure of 55 483 tCO₂ emissions. The reason for this is that the above analysis is based on the tier-one decomposition and does not include the full value chain relationships.

<table><caption>Table 8.58: Carbon allocation by region (consumption-based approach, column aggregation)</caption><thead><tr><th></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>Total</th><th>Produced emissions</th><th>EEIO correction</th></tr></thead><tbody><tr><td>(S<sub>1</sub>, R<sub>1</sub>)</td><td>28 397</td><td>11 359</td><td>1 420</td><td>41 176</td><td>50 000</td><td>-8 824</td></tr><tr><td>(S<sub>2</sub>, R<sub>1</sub>)</td><td>26 003</td><td>3 250</td><td>553</td><td>29 806</td><td>20 000</td><td>9 806</td></tr><tr><td>(S<sub>1</sub>, R<sub>2</sub>)</td><td>590</td><td>5 902</td><td>443</td><td>6 935</td><td>10 000</td><td>-3 065</td></tr><tr><td>(S<sub>2</sub>, R<sub>2</sub>)</td><td>0</td><td>11 283</td><td>282</td><td>11 565</td><td>10 000</td><td>1 565</td></tr><tr><td>(S<sub>1</sub>, R<sub>3</sub>)</td><td>349</td><td>1 744</td><td>3 488</td><td>5 581</td><td>5 000</td><td>581</td></tr><tr><td>(S<sub>2</sub>, R<sub>3</sub>)</td><td>144</td><td>2 397</td><td>2 397</td><td>4 938</td><td>5 000</td><td>-62</td></tr><tr><td>Total</td><td>55 483</td><td>35 935</td><td>8 582</td><td>100 000</td><td>100 000</td><td>0</td></tr></tbody></table>

<table><caption>Table 8.59: Carbon allocation by region (consumption-based approach, row aggregation)</caption><thead><tr><th></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>Total</th></tr></thead><tbody><tr><td>R<sub>1</sub></td><td>54 400</td><td>14 609</td><td>1 972</td><td>70 982</td></tr><tr><td>R<sub>2</sub></td><td>590</td><td>17 184</td><td>725</td><td>18 499</td></tr><tr><td>R<sub>3</sub></td><td>493</td><td>4 141</td><td>5 885</td><td>10 519</td></tr><tr><td>Total</td><td>55 483</td><td>35 935</td><td>8 582</td><td>100 000</td></tr></tbody></table>

In order to take into account the full value chain, we proceed as previously by considering the consumption-based emissions $D_{g,j}^{(y)}y_{j,r}$ instead of the production-based emissions $C_{g,j}^{(x)}$. We obtain
the results shown in Table 8.58. We can then compute the EEIO correction $cr_{g,j}^{(y)}$, which is defined by the following equation:

$$ \sum_{r=1}^{p} D_{g,j}^{(y)} y_{j,r} = C_{g,j}^{(x)} + cr_{g,j}^{(y)} $$

For example, it is equal to -8 824 tCO₂ for ($S_1$, $R_1$). We then perform the row aggregation (Table 8.59), and we finally obtain the following allocation results:

<table><thead><tr><th></th><th>R<sub>1</sub></th><th>R<sub>2</sub></th><th>R<sub>3</sub></th><th>Total</th></tr></thead><tbody><tr><td>Produced emissions</td><td>70 000</td><td>20 000</td><td>10 000</td><td>100 000</td></tr><tr><td>+ EEIO correction</td><td>982</td><td>-1 501</td><td>519</td><td>0</td></tr><tr><td>= Consumed emissions</td><td>70 982</td><td>18 499</td><td>10 519</td><td>100 000</td></tr><tr><td>- Exported emissions</td><td>16 582</td><td>1 315</td><td>4 634</td><td>22 530</td></tr><tr><td>= Intermediate emissions</td><td>54 400</td><td>17 184</td><td>5 885</td><td>77 470</td></tr><tr><td>+ Imported emissions</td><td>1 083</td><td>18 751</td><td>2 697</td><td>22 530</td></tr><tr><td>= Regional emissions</td><td>55 483</td><td>35 935</td><td>8 582</td><td>100 000</td></tr><tr><td>Net imported emissions</td><td>-14 517</td><td>15 935</td><td>-1 418</td><td>0</td></tr></tbody></table>

We can define the concept of net imported emissions, which takes into account the EEIO correction, imported and exported emissions. It is the difference between the emissions due to consumption and the emissions due to production. In our example, $R_1$ and $R_3$ are net exporting regions, while $R_2$ is a net importing region.

**Remark 93** The previous formulas can be easily translated using the traditional notations. Let $CI_1 = (CI_{1,1}, \dots, CI_{n,1})$ be the vector of carbon intensities evaluated in CO₂e, where $CI_{j,1}$ measures the direct emission intensity of sector j. The vector of consumption-based carbon intensities is equal to:

$$ CI^{(y)} = \tilde{L} CI_{1} := CI_{\text{total}} \qquad (8.44) $$

Let $y = (y_{j,r})$ be the $n \times p$ matrix, where $y_{j,r}$ is the final demand of the $j^{th}$ sector and the $r^{th}$ region. We have:

$$ CE^{(y)} = CI^{(y)} \odot y \qquad (8.45) $$

$CE^{(y)}$ is the $n \times p$ matrix of carbon emissions. The consumption-based carbon emissions of the $r^{th}$ region is then equal to:

$$ CE^{(y,r)} = 1_n^T (CE^{(y)} e_r) \qquad (8.46) $$

while the imported and exported carbon emissions of the $r^{th}$ region are:

$$ CE_{\text{imported}}^{(y,r)} = \sum_{j \notin r} (CE^{(y)} e_r)_j \qquad (8.47) $$

and:

$$ CE_{\text{exported}}^{(y,r)} = \sum_{j \in r} \sum_{k \neq r} (CE^{(y)} e_r)_j \qquad (8.48) $$

Equations (8.44–8.48) are the basic formulas for calculating imported and exported carbon emissions. We can also easily define consumed, intermediate and net imported emissions and the EEIO correction at both the sectoral and regional levels.
Stylized facts

Figure 8.141 shows a global picture of carbon emissions embedded in trade, which can be found at https://ourworldindata.org/consumption-based-co2. The data are taken from the famous *Global Carbon Budget* (2022) report (Friedlingstein et al., 2022). The exported (blue color) or imported (red color) emissions are normalized to domestic production emissions. For example, a value of 100% indicates a country whose imported emissions are equal to its domestic production emissions. We notice that the European countries and the United States are colored red, which means that they are net importers of CO2. On the contrary, the BRICS countries<fnref n="143" /> are net exporters of CO2, as are Australia, Canada and the Middle East region.

<figure><img src="image_422.png" /><figcaption>Figure 8.141: CO₂ emissions embedded in trade, 2020</figcaption></figure>

This is measured as emissions exported or imported as a percentage of domestic production emissions. Positive values (red) represent net importers of CO2. Negative values (blue) represent net exporters of CO2.

Source: https://ourworldindata.org/consumption-based-co2.

The OECD maintains a database of imported and exported CO2 at www.oecd.org/industry/ind/carbondioxideemissionembodiedininternationaltrade.htm. Calculations are detailed in Yamano and Guilhoto (2020). We use an updated version of these data, which can be obtained from the OECD statistics website<fnref n="144" />. In our case, we are interested in three variables:

* FD_CO2: CO2 emissions embodied in domestic final demand, by source country and industry;

* PROD_CO2: CO2 emissions based on production;

<footnote n="143">¹⁴³Brazil, Russia, India, China and South Africa.</footnote>

<footnote n="144">¹⁴⁴The exact link is https://stats.oecd.org/Index.aspx?DataSetCode=IO_GHG_2021.</footnote>
• BALCO2 FD: CO2 embodied in final demand, balance.

The first variable is the consumption-based emissions, the second variable is the production-based emissions, while the third variable is the difference between the previous two: BALCO2 FD = PROD CO2 – FD CO2. Using our notations, the GHG balance is equal to:

$$ \mathcal{C}\mathcal{E}^{(x-y,r)} = \mathcal{C}\mathcal{E}^{(x,r)} - \mathcal{C}\mathcal{E}^{(y,r)} $$

We can choose a specific industry or all activities (DTOTAL) and also the partner for the trade analysis. If the partner is WLD, we get the emissions of each country/region with respect to the world. If the partner is CHN, we obtain the emissions of each country/region with respect to China. In Table 8.60, we report the top importing and exporting countries. In 2018, the largest importer of carbon emissions is the United States, followed by Japan and Germany, while the largest exporter of carbon emissions is China, followed by Russia and South Africa. In the first case the GHG balance is negative, in the second case it is positive. In a sense, these rankings show a contrast between developed and emerging economies.

<table><caption>Table 8.60: Top importing and exporting countries by carbon emissions (in MtCO2e, 2018)</caption><thead><tr><th colspan="4">Top importers</th><th colspan="4">Top exporters</th></tr><tr><th>Rank</th><th>ISO</th><th>Country</th><th>Balance</th><th>Rank</th><th>ISO</th><th>Country</th><th>Balance</th></tr></thead><tbody><tr><td>1</td><td>USA</td><td>United States</td><td>-752.10</td><td>1</td><td>CHN</td><td>China</td><td>895.45</td></tr><tr><td>2</td><td>JPN</td><td>Japan</td><td>-160.62</td><td>2</td><td>RUS</td><td>Russian Federation</td><td>343.48</td></tr><tr><td>3</td><td>DEU</td><td>Germany</td><td>-128.73</td><td>3</td><td>ZAF</td><td>South Africa</td><td>122.47</td></tr><tr><td>4</td><td>GBR</td><td>United Kingdom</td><td>-123.77</td><td>4</td><td>IND</td><td>India</td><td>106.10</td></tr><tr><td>5</td><td>FRA</td><td>France</td><td>-111.65</td><td>5</td><td>TWN</td><td>Chinese Taipei</td><td>77.03</td></tr><tr><td>6</td><td>ITA</td><td>Italy</td><td>-80.09</td><td>6</td><td>SGP</td><td>Singapore</td><td>62.19</td></tr><tr><td>7</td><td>HKG</td><td>Hong Kong, China</td><td>-70.14</td><td>7</td><td>KOR</td><td>Korea</td><td>54.35</td></tr><tr><td>8</td><td>CHE</td><td>Switzerland</td><td>-44.53</td><td>8</td><td>CAN</td><td>Canada</td><td>53.12</td></tr><tr><td>9</td><td>PHL</td><td>Philippines</td><td>-40.49</td><td>9</td><td>VNM</td><td>Viet Nam</td><td>52.31</td></tr><tr><td>10</td><td>SWE</td><td>Sweden</td><td>-29.67</td><td>10</td><td>MYS</td><td>Malaysia</td><td>46.52</td></tr></tbody></table>

Source: Yamano and Guilhoto (2020), https://stats.oecd.org & Author's calculations.

These rankings also show a divide between OECD and non-OECD countries. In Figure 8.142, we reproduce Figure 4-1 on page 24 of the OECD research report written by Yamano and Guilhoto (2020). The comparison between aggregate OECD and aggregate non-OECD production- and consumption-based emissions shows two different dynamics. For OECD countries, the long-term trend between 1995 and 2018 is quite stable. On the contrary, there is a general increase in emissions from non-OECD economies over the same period.

“These increases are in great part linked with the need of these countries to sustain their own development and to improve the quality of life of their population. A consequence being that many of these countries are important net exporters of CO2 emissions as they develop a strong manufacturing base to meet the consumption needs of more developed nations. Despite increasing industrialisation, emissions per capita in non-OECD economies are still low compared to OECD countries” (Yamano and Guilhoto, 2020, page 23).
<figure><img src="image_423.png" /><figcaption>Figure 8.142: Total production- and consumption-based CO₂ emitted by OECD and non-OECD countries (in GtCO₂e)</figcaption></figure>

Source: Yamano and Guilhoto (2020), https://stats.oecd.org & Author's calculations.

<figure><img src="image_424.png" /><figcaption>Figure 8.143: Decomposition of OECD imported emissions (in GtCO₂e)</figcaption></figure>

Source: Yamano and Guilhoto (2020), https://stats.oecd.org & Author's calculations.
<figure><img src="image_425.png" /><figcaption>Figure 8.144: Net exported emissions (in MtCO₂e)</figcaption></figure>

Source: Yamano and Guilhoto (2020), https://stats.oecd.org & Author's calculations.

<figure><img src="image_426.png" /><figcaption>Figure 8.145: Net exported emissions (in MtCO₂e)</figcaption></figure>

Source: Yamano and Guilhoto (2020), https://stats.oecd.org & Author's calculations.
Figures 8.144 and 8.145 show the evolution of net exported emissions for some countries. First, we see that they are highly time-varying. One reason for this is that the calculation is sensitive to the Leontief matrix of input-output analysis. Second, we observe different patterns across countries. For example, France and Germany are at similar levels in 2018, but this was not the case in 1995. Canada was a net exporter before 2005, a net importer between 2005 and 2015, and a net exporter again after 2015.

**Remark 94** The calculation of imported and exported emissions depends on many factors, especially when using input-output analysis. Depending on the data source, we may find different results. For instance, Australia is a net exporter according to Figure 8.141 and a net importer according to Figure 8.144. According to Commissariat Général au Développement Durable (2020), France's carbon footprint is made up of 45% domestic emissions and 55% imported emissions. The estimated value of imported emissions is then more than 350 MtCO₂e. According to the OECD calculation, this figure is close to 110 MtCO₂e. These two examples illustrate the large uncertainty in the calculation of GHG emissions embodied in trade.

### 8.4.5 Taxation, pass-through and price dynamics

To study the impact of taxation on production costs, we need to diffuse the carbon tax in the input-output economic model to account for the cascading effects through the value chain. The diffusion of the carbon tax depends on the assumption of the reaction function of suppliers and pass-through mechanisms.

#### Value added approach

By construction, a carbon tax affects the income of producers, who may react in different ways. We first consider a flexible price model and assume that they want to maintain their level of value added.

**Remark 95** In the following, $p^-$ is the price vector before the carbon tax, while $p$ is the price vector including the tax effect.

**Impact on production prices** The absolute amount of the carbon tax for sector j is equal to:

$$T_{\text{direct},j} = \tau_j \mathcal{C} \mathcal{E}_{1,j}$$

where $\tau_j$ is the nominal carbon tax expressed in $/tCO₂e and $\mathcal{C}\mathcal{E}_{1,j}$ is the Scope 1 emissions of the sector. We deduce that the carbon tax rate is equal to:

$$t_{\text{direct},j} = \frac{T_{\text{direct},j}}{x_j} = \frac{\tau_j \mathcal{C}\mathcal{E}_{1,j}}{x_j} = \tau_j \mathcal{C}\mathcal{I}_{1,j}$$

Note that $t_{\text{direct},j}$ has no unit and is equal to the product of the tax and the Scope 1 carbon intensity. The input-output model implies that:

$$p_j x_j = \sum_{i=1}^{n} Z_{i,j} p_i + \sum_{k=1}^{m} V_{k,j} \psi_k + T_{\text{direct},j}$$
We deduce that:

$$p_j = \sum_{i=1}^{n} A_{i,j} p_i + \sum_{k=1}^{m} B_{k,j} \psi_k + t_{\text{direct},j} = \sum_{i=1}^{n} A_{i,j} p_i + v_j + t_{\text{direct},j}$$

It follows that:

$$p = (I_n - A^{\top})^{-1} (v + t_{\text{direct}})$$

where $t_{\text{direct}} = (t_{\text{direct},1}, \dots, t_{\text{direct},n})$ is the vector of direct tax rates. We recover the cost-push price model, where the vector $v$ of value added ratios is replaced by $v + t_{\text{direct}}$. It follows that the vector of price changes due to the carbon tax is equal to:

$$\Delta p = (I_n - A^{\top})^{-1} t_{\text{direct}} \qquad (8.49)$$

This result is obvious since Equation (8.30) implies that $\Delta p = (I_n - A^{\top})^{-1} \Delta v$ and $\Delta v$ corresponds to the vector $t_{\text{direct}}$ of direct tax rates.

**Impact on the price index** The definition of a price index is:

$$\mathcal{PI} = \sum_{i=1}^{n} \alpha_i p_i = \alpha^{\top} p$$

where $\alpha = (\alpha_1, \dots, \alpha_n)$ are the weights of the items in the basket. We deduce that the inflation rate is:

$$\pi = \frac{\Delta \mathcal{PI}}{\mathcal{PI}^{-}} = \frac{\mathcal{PI} - \mathcal{PI}^{-}}{\mathcal{PI}^{-}} = \frac{\alpha^{\top} (I_n - A^{\top})^{-1} t_{\text{direct}}}{\alpha^{\top} (I_n - A^{\top})^{-1} v}$$

We can simplify this formula because $p^{-} = (I_{n} - A^{\top})^{-1} v = \mathbf{1}_{n}$ and $\mathbf{1}_{n}^{\top} \alpha = 1$. Finally, we have:

$$\pi = \alpha^{\top} (I_n - A^{\top})^{-1} t_{\text{direct}} \qquad (8.50)$$

**Computation of the total tax amount** The total tax cost is equal to:

$$T_{\text{total}} = x \odot \Delta p = x \odot (I_n - A^{\top})^{-1} t_{\text{direct}} \qquad (8.51)$$

while the direct tax cost is $T_{\text{direct}} = x \odot t_{\text{direct}}$. Since we have $x \succeq \mathbf{0}_n$ and $(I_n - A^{\top})^{-1} \succeq I_n$ and using Hadarmard properties$^{145}$, then we conclude that the total tax cost is greater than the direct tax cost for all the sectors:

$$T_{\text{total},j} \geq T_{\text{direct},j}$$

Since the total cost to the economy is equal to $\mathcal{C}_{\text{total}} = \sum_{j=1}^{n} T_{\text{total},j} = x^{\top} (I_n - A^{\top})^{-1} t_{\text{direct}}$, the tax incidence is then equal to:

$$\mathcal{TI} = \frac{\mathcal{C}_{\text{total}}}{\mathbf{1}_{n}^{\top} x} = \frac{x^{\top} (I_n - A^{\top})^{-1} t_{\text{direct}}}{\mathbf{1}_{n}^{\top} x}$$

<footnote>$^{145}$Let A, B and C be three nonnegative matrices. If $B \preceq C$, then $A \odot B \preceq A \odot C$.</footnote>
**Common mistakes in calculating total tax costs** In some research papers, we can find two formulas that seem to be intuitive:

$$T'_{\text{total}} = (I_n - A^{\top})^{-1} T_{\text{direct}}$$

and:

$$T''_{\text{total}} = \tau \odot C\mathcal{E}_{\text{total}}$$

The two previous equations are generally wrong because the Hadamard and matrix products are not associative: $A \odot (BC) \neq (A \odot B) C$.

**Mathematical properties** Let us denote by $f(\tau)$ the function $f$ that depends on the vector $\tau = (\tau_1, \dots, \tau_n)$ of carbon taxes. Let $\lambda \ge 0$ be a positive scalar. The functions $\Delta p, \pi, T_{\text{total}}, C_{\text{total}}$ and $TI$ are homogeneous$^{146}$ and additive$^{147}$. For instance, we have:

$$\Delta p(\lambda \tau) = (I_n - A^{\top})^{-1} t_{\text{direct}}(\lambda \tau) = \lambda (I_n - A^{\top})^{-1} t_{\text{direct}}(\tau) = \lambda \Delta p(\tau)$$

If the tax is uniform $\tau = \tau \mathbf{1}_n$, the vector of total tax amount is the product of the tax by the total emissions:

$$T_{\text{total}}(\tau \mathbf{1}_n) = \tau C\mathcal{E}_{\text{total}}$$

The tax incidence for a given sector is then proportional to the direct plus indirect carbon emissions of the sector. At the global level, the tax incidence is equal to the carbon tax multiplied by the total carbon intensity of the world:

$$TI(\tau \mathbf{1}_n) = \frac{\mathbf{1}_n^{\top} \tau C\mathcal{E}_{\text{total}}}{\mathbf{1}_n^{\top} x} = \tau C\mathcal{I}_{\text{total}}$$

**Illustration** Consider a variant of Example 29. Table 8.61 gives the values of $Z_{i,j}, y_j, x_j$ and $V_{1,j}$ in $\$ mn$. The carbon emissions are expressed in ktCO$_2$e, while the carbon intensities are expressed in tCO$_2$e/$\$ mn. We have:

$$A = Z \operatorname{diag}^{-1}(x) = \begin{pmatrix} 0.10 & 0.20 & 0.20 & 0.10 \\ 0.10 & 0.10 & 0.20 & 0.05 \\ 0.05 & 0.20 & 0.30 & 0.10 \\ 0.02 & 0.05 & 0.10 & 0.35 \end{pmatrix}$$

and:

$$\tilde{\mathcal{L}} = (I_4 - A^{\top})^{-1} = \begin{pmatrix} 1.1881 & 0.1678 & 0.1430 & 0.0715 \\ 0.3894 & 1.2552 & 0.4110 & 0.1718 \\ 0.4919 & 0.4336 & 1.6303 & 0.2993 \\ 0.2884 & 0.1891 & 0.3044 & 1.6087 \end{pmatrix}$$

Then, we calculate the vector $v$ of value added ratios:

$$v = \begin{pmatrix} 3650/5000 \\ 1800/4000 \\ 1600/8000 \\ 5000/12500 \end{pmatrix} = \begin{pmatrix} 0.73 \\ 0.45 \\ 0.20 \\ 0.40 \end{pmatrix}$$

<footnote>$^{146}$This means that $f(\lambda\tau) = \lambda f(\tau)$.</footnote>

<footnote>$^{147}$We have $f(\tau + \tau') = f(\tau) + f(\tau')$.</footnote>
<table><caption>Table 8.61: Environmentally extended monetary input-output table</caption><thead><tr><th>Sector</th><th colspan="4">Z</th><th>y</th><th>x</th><th>CE<sub>1</sub></th><th>CI<sub>1</sub></th></tr></thead><tbody><tr><td>Energy</td><td>500</td><td>800</td><td>1 600</td><td>1 250</td><td>850</td><td>5 000</td><td>500</td><td>100</td></tr><tr><td>Materials</td><td>500</td><td>400</td><td>1 600</td><td>625</td><td>875</td><td>4 000</td><td>200</td><td>50</td></tr><tr><td>Industrials</td><td>250</td><td>800</td><td>2 400</td><td>1 250</td><td>3 300</td><td>8 000</td><td>200</td><td>25</td></tr><tr><td>Services</td><td>100</td><td>200</td><td>800</td><td>4 375</td><td>7 025</td><td>12 500</td><td>125</td><td>10</td></tr><tr><td>Value added</td><td>3 650</td><td>1 800</td><td>1 600</td><td>5 000</td><td></td><td></td><td></td><td></td></tr><tr><td>Income</td><td>5 000</td><td>4 000</td><td>8 000</td><td>12 500</td><td></td><td></td><td></td><td></td></tr></tbody></table>

We check that $p^{-} = \tilde{\mathcal{L}}v = \mathbf{1}_{4}$. By construction, all the prices are standardized and equal to one in a monetary input-output model. We now introduce a differentiated carbon tax: $\tau_{1} = \$200/\text{tCO}_{2}\text{e}$ and $\tau_{2} = \tau_{3} = \tau_{4} = \$100/\text{tCO}_{2}\text{e}$. The direct tax costs are 100, 20, 20 and 12.5 million dollars for Energy, Materials, Industrials and Services respectively. We deduce that the vector of carbon tax rates is $t_{\text{direct}} = (2.00\%, 0.50\%, 0.25\%, 0.10\%)$. It follows that:

$$p = (I_n - A^{\top})^{-1} (v + t_{\text{direct}}) = \begin{pmatrix} 1.0250 \\ 1.0153 \\ 1.0164 \\ 1.0091 \end{pmatrix}$$

If we assume that the basket of goods and services is $\alpha = (10\%, 20\%, 30\%, 40\%)$, the price index $\mathcal{PI}$ is 1.0141 while the inflation rate $\pi$ is 1.410%. Finally, we calculate the total tax cost and obtain the results shown in Table 8.62. The direct tax cost is multiplied by a factor of 2.8 when we consider the diffusion of the carbon tax. We check that $T_{\text{total}} \neq T'_{\text{total}} \neq T''_{\text{total}}$. Services is the most affected sector, followed by Industrials, Materials and Energy with impact ratios $T_{\text{total}}/T_{\text{direct}}$ of 9.1, 6.6, 3.1 and 1.3 respectively. In Table 8.63, we consider a uniform tax of $\$100/\text{tCO}_{2}\text{e}$. We check that $T_{\text{total}} = T''_{\text{total}}$ but $T_{\text{total}} \neq T'_{\text{total}}$.

<table><caption>Table 8.62: Total carbon costs (in $ mn) (differentiated tax)</caption><thead><tr><th>Sector</th><th><i>T</i><sub>direct</sub></th><th><i>T</i><sub>total</sub></th><th><i>T'</i><sub>total</sub></th><th><i>T''</i><sub>total</sub></th><th><i>CE</i><sub>direct</sub></th><th><i>CE</i><sub>total</sub></th></tr></thead><tbody><tr><td>Energy</td><td>100.00</td><td>125.15</td><td>125.92</td><td>131.49</td><td>500.00</td><td>657.44</td></tr><tr><td>Materials</td><td>20.00</td><td>61.05</td><td>74.41</td><td>45.48</td><td>200.00</td><td>454.76</td></tr><tr><td>Industrials</td><td>20.00</td><td>131.05</td><td>94.21</td><td>91.70</td><td>200.00</td><td>916.97</td></tr><tr><td>Services</td><td>12.50</td><td>113.54</td><td>58.82</td><td>77.49</td><td>125.00</td><td>774.92</td></tr><tr><td>Sum</td><td>152.50</td><td>430.79</td><td>353.36</td><td>346.15</td><td>1 025.00</td><td>2 804.10</td></tr></tbody></table>

<table><caption>Table 8.63: Total carbon costs (in $ mn) (uniform taxation)</caption><thead><tr><th>Sector</th><th><i>T</i><sub>direct</sub></th><th><i>T</i><sub>total</sub></th><th><i>T'</i><sub>total</sub></th><th><i>T''</i><sub>total</sub></th><th><i>CE</i><sub>direct</sub></th><th><i>CE</i><sub>total</sub></th></tr></thead><tbody><tr><td>Energy</td><td>50.00</td><td>65.74</td><td>66.51</td><td>65.74</td><td>500.00</td><td>657.44</td></tr><tr><td>Materials</td><td>20.00</td><td>45.48</td><td>54.94</td><td>45.48</td><td>200.00</td><td>454.76</td></tr><tr><td>Industrials</td><td>20.00</td><td>91.70</td><td>69.62</td><td>91.70</td><td>200.00</td><td>916.97</td></tr><tr><td>Services</td><td>12.50</td><td>77.49</td><td>44.40</td><td>77.49</td><td>125.00</td><td>774.92</td></tr><tr><td>Sum</td><td>102.50</td><td>280.41</td><td>235.47</td><td>280.41</td><td>1 025.00</td><td>2 804.10</td></tr></tbody></table>
Mark-up pricing approach

**Theoretical framework** We consider a second approach proposed by Gemechu et al. (2014) and Mardones and Mena (2020). Mark-up pricing refers to a business strategy in which the suppliers determine the selling price by adding a fixed percentage to the cost of production. Let $p_j^-$ be the price before the introduction of the carbon tax. We define $\xi_j$ as the price factor induced by the carbon tax: $t_{direct,j} = \xi_j p_j^-$. It follows that $p_j^- = \sum_{i=1}^n A_{i,j}p_i^- + v_j$ and$^{148}$:

$$ 
\begin{aligned}
p_j &= \left( \sum_{i=1}^{n} A_{i,j}p_i + v_j \right) + t_{\text{direct},j} \\
&= \left( \sum_{i=1}^{n} A_{i,j}p_i + v_j \right) + \xi_j p_j^- \\
&= (1 + \xi_j) \left( \sum_{i=1}^{n} A_{i,j}p_i + v_j \right)
\end{aligned}
$$

We deduce that:

$$ \frac{p_j}{1 + \xi_j} = \sum_{i=1}^{n} A_{i,j}p_i + v_j $$

and:

$$ p_j \left( 1 - \frac{\xi_j}{1 + \xi_j} \right) = \sum_{i=1}^{n} A_{i,j}p_i + v_j $$

It follows that:

$$ 
\begin{aligned}
p_j &= \sum_{i=1}^{n} A_{i,j}p_i + \frac{\xi_j}{1 + \xi_j}p_j + v_j \\
&= \sum_{i=1}^{n} A_{i,j}p_i + p_j \left( 1 - \frac{1}{1 + \xi_j} \right) + v_j
\end{aligned}
$$

In a matrix form, we have:

$$ p = A^{\top} p + (I_n - D_{\xi}) p + v $$

where:

$$ D_{\xi} = \operatorname{diag} \left( \frac{1}{1 + \xi_1}, \dots, \frac{1}{1 + \xi_n} \right) $$

Finally, we obtain:

$$ p = (I_n - A_{\xi}^{\top})^{-1} v $$

where $A_{\xi} = A + I_n - D_{\xi}$. Another expression is:

$$ p = \tilde{\mathcal{L}}_m v = (D_{\xi} - A^{\top})^{-1} v \qquad (8.52) $$

where $\tilde{\mathcal{L}}_m = (D_{\xi} - A^{\top})^{-1}$ is the mark-up inverse matrix. The vector of price variations is then:

$$ \Delta p = (\tilde{\mathcal{L}}_m - \tilde{\mathcal{L}}) v \qquad (8.53) $$

<footnote n="148">148 We assume that $p_j \approx p_j^-$.</footnote>
The expression of the price index is $PI = \alpha^\top (D_\xi - A^\top)^{-1} v$ whereas the inflation rate is equal to $\pi = \alpha^\top (\tilde{\mathcal{L}}_m - \tilde{\mathcal{L}}) v$. From Equation (8.53), we also deduce the total tax cost:

$$T_{\text{total}} = x \odot (\tilde{\mathcal{L}}_m - \tilde{\mathcal{L}}) v \qquad (8.54)$$

Note that the mark-up approach implies replacing the identity matrix $I_n$ with the diagonal matrix $D_\xi$ in the cost-push price model. Since we have $D_\xi \preceq I_n$, we deduce that $D_\xi^{-1} \succeq I_n$. Desnos et al. (2023) also showed that $\tilde{\mathcal{L}}_m \succeq \tilde{\mathcal{L}}$.

**Illustration** Considering the previous example, we have:

$$\tilde{\mathcal{L}}_m = (D_\xi - A^\top)^{-1} = \begin{pmatrix} 1.2170 & 0.1730 & 0.1474 & 0.0735 \\ 0.4017 & 1.2650 & 0.4165 & 0.1740 \\ 0.5067 & 0.4398 & 1.6394 & 0.3021 \\ 0.2965 & 0.1919 & 0.3074 & 1.6121 \end{pmatrix}$$

In the case of differentiated taxation, we obtain:

$$p = \tilde{\mathcal{L}}_m v = \begin{pmatrix} 1.0252 \\ 1.0154 \\ 1.0165 \\ 1.0091 \end{pmatrix}$$

The inflation rate $\pi$ is equal to 1.421% and the total carbon costs (in $ mn) are 125.82, 61.53, 132.10 and 114.33. The global cost is then $433.78 mn compared to $430.79 mn in the value added approach.

## Pass-through integration

**Definition** According to RBB Economics (2014), “cost pass-through describes what happens when a business changes the price of the production or services it sells following a change in the cost of producing them.” Therefore, a pass-through rate is closely related to the supply and demand elasticity. This concept of price adjustment is extremely common in many fields of economics: exchange rates, imperfect competition and Cournot-Bertrand equilibria, product taxation and retail prices, inflation regimes, etc. In other words, pass-through is the ability of a sector or a company to pass costs through its supply chain. In general, this parameter ranges from 0%, where the entire amount is supported by the agent, to 100%, where the entire amount is passed on to customers. As this parameter depends on several factors, such as supply and demand elasticity, international trade exposure, market concentration, product homogeneity, etc., its estimation is not easy, which implies a large uncertainty about the tax incidence in a transition risk framework.

Pass-through strongly depends on the market structure and the supply-demand equilibrium. In Figure 8.146, we show different demand curves whose slope depends on the consumer response to different price levels. If the slope of the curve is steep, it suggests that an increase in price would lead to a marginal decrease in sales. This scenario represents inelastic demand, where consumer demand is relatively unchanged when the price moves up or down. Conversely, if the demand curve is flatter, an increase in price will result in a significant reduction in the quantity demanded. This situation represents elastic demand, where consumers are highly responsive to price changes. If the demand curve is linear, there is no curvature, which means that the rate of decline in demand remains
<figure><img src="image_427.png" /><figcaption>Figure 8.146: Demand curvature</figcaption></figure>

Source: RBB Economics (2014, Figure 2, page 16).

constant as the price increases (top/left panel in Figure 8.146). In situations where demand falls more sharply as the price rises, this type of demand is classified as concave to the origin (top/right panel in Figure 8.146). As prices rise in this scenario, the demand curve becomes increasingly flatter, indicating increased price sensitivity or greater elasticity. In this scenario, firms should absorb part of the cost, implying a relatively low pass-through rate. Finally, if the rate of decline in demand slows with each price increase, this type of demand curve is said to be convex to the origin. In this case, as prices escalate, the remaining demand becomes less sensitive to these price fluctuations (bottom/left panel of the figure). Firms can then pass on the costs and set a relatively high pass-through rate.

From an economic point of view, the specification of the pass-through depends on several factors. In the case of competition, the general formula for the pass-through rate $\phi$ is:

$$ \phi = \frac{\frac{\text{d}p}{\text{d}\tau}}{\frac{\text{d}p}{\text{d}\tau} = \frac{\text{price sensitivity of supply}}{\text{price sensitivity of supply} - \text{price sensitivity of demand}}} $$

We deduce that $\phi \in [0, 100\%]$. In a monopolistic situation, the previous formula becomes:

$$ \phi = \frac{1}{2 + \text{elasticity of the slope of inverse demand}} $$
Since the slope elasticity of inverse demand is negative, $\phi \ge 50\%$. We get similar results in oligopolistic situations. In monopolistic and oligopolistic situations, it can also be greater than 100% if demand is highly convex. In Table 8.64, we report some estimates of path-through rates.

<table><caption>Table 8.64: Pass-through rates (in %) for intensive sectors</caption><thead><tr><th>Sector</th><th>Rate</th></tr></thead><tbody><tr><td>Electricity, gas and steam</td><td>100%</td></tr><tr><td>Petroleum refining</td><td>100%</td></tr><tr><td>Base metals</td><td>78%</td></tr><tr><td>Mining</td><td>78%</td></tr><tr><td>Waste/wastewater</td><td>78%</td></tr><tr><td>Land transport</td><td>78%</td></tr><tr><td>Fishery</td><td>75%</td></tr><tr><td>Non-metallic minerals</td><td>60%</td></tr><tr><td>Agriculture</td><td>50%</td></tr><tr><td>Chemicals</td><td>40%</td></tr><tr><td>Maritime transport</td><td>30%</td></tr><tr><td>Aviation</td><td>30%</td></tr><tr><td>Paper</td><td>10%</td></tr></tbody></table>

Source: Sautel et al. (2022, page 35).

**Analytical formula** We focus on the value added model, which is the most widely used approach in the academic literature. It is also the simplest model to introduce the pass-through mechanism. We have:

$$ \Delta p = \tilde{\mathcal{L}} \Delta v = \sum_{k=0}^{\infty} (A^{\top})^{k} \Delta v = \sum_{k=0}^{\infty} \Delta p_{(k)} $$

where $\Delta p_{(k)} = (A^{\top})^{k} \Delta v$ is the price impact at the $k^{th}$ tier. In fact, $\Delta p_{(k)}$ satisfies the following recurrence relation:

$$ \left\{ \begin{array}{l} \Delta p_{(k)} = A^{\top} \Delta p_{(k-1)} \\ \Delta p_{(0)} = \Delta v \end{array} \right. $$

If we consider the price $p_j$ of sector $j$, we have $\Delta p_{(0),j} = \Delta v_j$ and:

$$ \Delta p_{(k),j} = \sum_{i=1}^{n} A_{i,j} \Delta p_{(k-1),i} $$

This representation helps to better understand the cascading effect of the carbon tax. In the zeroth round, it induces an additional cost $\Delta v_j$, which is fully passed on to the price $p_j$ of the sector. The new price is then $p_j + \Delta p_{(0),j} = p_j + \Delta v_j$. In the first round, sector $j$ faces new additional costs due to the price increase of intermediate consumption. We have $\Delta p_{(1),j} = \sum_{i=1}^{n} A_{i,j} \Delta p_{(0),i} = \sum_{i=1}^{n} A_{i,j} \Delta v_i$. The iteration process continues and we have $\Delta p_{(2),j} = \sum_{i=1}^{n} A_{i,j} \Delta p_{(1),i} = \sum_{i=1}^{n} \sum_{k=1}^{n} A_{i,j} A_{k,i} \Delta v_k$ at the second round.

Now let us introduce the pass-through mechanism. By definition, we have $\Delta p_{(0),j} = \phi_j \Delta v_j$ where $\phi_j$ denotes the pass-through rate of sector $j$. In the first round, we have:

$$ \Delta p_{(1),j} = \sum_{i=1}^{n} A_{i,j} (\phi_i \Delta p_{(0),i}) = \sum_{i=1}^{n} A_{i,j} (\phi_i \Delta v_i) $$
More generally, the recurrence relation is:

$$ \Delta p_{(k),j} = \sum_{i=1}^{n} A_{i,j} \phi_i \Delta p_{(k-1),i} $$

Let $\phi = (\phi_1, \dots, \phi_n)$ and $\Phi = \text{diag}(\phi)$ be the pass-through vector and matrix. The recurrence matrix form is:

$$ \left\{ \begin{array}{l} \Delta p_{(k)} = A^{\top} \Phi \Delta p_{(k-1)} \\ \Delta p_{(0)} = \Phi \Delta v \end{array} \right. $$

We deduce that:

$$ \begin{align} \Delta p &= \sum_{k=0}^{\infty} (A^{\top} \Phi)^k \Phi \Delta v \nonumber \\ &= (I_n - A^{\top} \Phi)^{-1} \Phi \Delta v \nonumber \\ &= \tilde{\mathcal{L}}(\phi) \Delta v \tag{8.55} \end{align} $$

where $\tilde{\mathcal{L}}(\phi) = (I_n - A^{\top} \Phi)^{-1} \Phi$.

Since $A$ is a substochastic matrix and $\Phi$ is a positive diagonal matrix, we verify that $\phi' \succeq \phi \Rightarrow \tilde{\mathcal{L}}(\phi') \succeq \tilde{\mathcal{L}}(\phi)$. The lower bound is then reached when $\phi = \mathbf{0}_n$ while the upper bound is reached when $\phi = \mathbf{1}_n$.

**Application to the carbon tax** Applying the previous analysis to the carbon tax, we have $\Delta v = t_{\text{direct}}$. In this case, the concept of total tax cost must be redefined because part of the cost is paid by producers and part by consumers. By consumers, we must understand the downstream of the value chain. We have:

$$ \begin{align*} T_{\text{producer}} &= x \odot (I_n - \Phi) t_{\text{direct}} \\ &= x \odot (\mathbf{1}_n - \phi) \odot t_{\text{direct}} \\ &= (\mathbf{1}_n - \phi) \odot T_{\text{direct}} \end{align*} $$

and:

$$ T_{\text{consumer}} = T_{\text{downstream}} = x \odot \tilde{\mathcal{L}}(\phi) t_{\text{direct}} $$

We deduce that:

$$ \begin{align*} T_{\text{total}} &= T_{\text{producer}} + T_{\text{consumer}} \\ &= x \odot (I_n - \Phi + \tilde{\mathcal{L}}(\phi)) t_{\text{direct}} \end{align*} $$

If $\phi_j = 100\%$, we have $\tilde{\mathcal{L}}(\mathbf{1}_n) = \tilde{\mathcal{L}}$ and $\Delta p = \tilde{\mathcal{L}} t_{\text{direct}}$. This is the original approach. If $\phi_j = 0\%$, we have $\tilde{\mathcal{L}}(\mathbf{0}_n) = \mathbf{0}_{n,n}$, $\Delta p = \mathbf{0}_n$, $T_{\text{producer}} = T_{\text{direct}}$ but $T_{\text{consumer}} = \mathbf{0}_n$. The costs passed on to consumers (or the downstream of the value chain) are zero because the direct costs are initially absorbed by the producers.

**Remark 96** The functions $\Delta p$, $\pi$, $T_{\text{total}}$, $C_{\text{total}}$ and $TI$ remain homogeneous and additive with respect to $\tau$. We can also show that:

$$ \phi' \succeq \phi \Rightarrow T_{\text{total}}(\tau, \phi') \succeq T_{\text{total}}(\tau, \phi) $$

The effects of the tax is maximum when $\phi = \mathbf{1}_n$ and minimum when $\phi = \mathbf{0}_n$. If we consider a uniform pass-through, the total cost of the carbon tax is an increasing function of the pass-through rate.
<figure><img src="image_428.png" /><figcaption>Figure 8.148: Producer and consumer cost contributions ($\phi_2 = \phi_3 = \phi_4 = 0\%$)</figcaption></figure>

<figure><img src="image_429.png" /></figure>
**Illustration** Continuing our example of differential taxation, let's assume that pass-through rates are uniform ($\phi_1 = \phi_2 = \phi_3 = \phi_4$). The evolution of the total cost is shown in Figure 8.147. When $\phi_j = 0\%$, $T_{total}$ is equal to $152.50 mn$ and is the lower bound. The upper bound is reached when $\phi_j = 100\%$ and we get $T_{total} = $430.79 mn. We have also shown the contribution of each sector by distinguishing between direct and indirect costs. Figure 8.148 corresponds to the case where the Energy sector passes on the direct costs to the other sectors.

## Empirical results

There are many studies of carbon taxation and input-output models (Köppl and Schratzenstaller, 2023). In the following, we focus on two studies that have extensively examined the impact of pass-through rates on the economic cost of a carbon tax and its impact on inflation. We also produce new results by implementing a carbon tax of $100/tCO₂e and differentiated pass-through rates.

**The study of Desnos et al. (2023)** The authors analyze the impact of a uniform tax $\tau$ for all countries and a uniform pass-through rate $\phi$ for all sectors. The direct cost $C_{direct} = \sum_{j=1}^{n} T_{direct}$ and the total cost $C_{total} = \sum_{j=1}^{n} T_{total}$ are shown in Figure 8.149. If the carbon tax is set at $100/tCO₂e, the direct cost is $4.8 tn, while the total cost is $6.1 tn if $\phi$ = 50% and $13.3 tn if $\phi$ = 100%. These correspond to 2.8%, 3.6% and 7.8% of the world GDP respectively. If we apply a carbon tax of $500/tCO₂e, these costs become $24.2, $30.4 and $66.4 tn respectively. They show that the relationship between total costs and the pass-through parameter is cubic. They suggest the following approximation:

$$ \frac{C_{\text{total}}(\tau, \phi \mathbf{1}_n)}{C_{\text{direct}}(\tau, \phi \mathbf{1}_n)} \approx 1 + m_{(1-\infty)} \phi^3 $$

Therefore, a small error in pass-through rate estimate can lead to a large error in the cost estimate.

Desnos et al. (2023) then analyze the impact of the carbon tax on the inflation. To do this, they define two price indexes: the producer price index (PPI), where the basket weights are proportional to the output ($\alpha_j \propto x_j$) and the consumer price index (CPI), where the basket weights are proportional to the final demand ($\alpha_j \propto y_j$). Results are shown in Figure 8.150. Again, the inflation rate depends on the pass-through rate. For a carbon tax of $500/tCO₂e and a pass-through rate of 100%, the PPI inflation rate is close to 40%, while the CPI inflation rate reaches 30%. These global figures are the result of a large discrepancy between country inflation rates. Figure 8.151 shows the world map of the country inflation rates for a uniform tax of $100/tCO₂e. There are three factors (basket composition, value chain impact and direct carbon emissions of the country) that explain the dispersion of the inflation rates:

$$ \pi = \underbrace{\alpha^\top}_{\text{Basket}} \cdot \underbrace{\hat{\mathcal{L}}(\phi)}_{\text{Value chain}} \cdot \underbrace{t_{\text{direct}}}_{\text{Scope 1}} $$

Direct costs are the main contributor, followed by the impact of the downstream diffusion of the carbon tax. For example, Europe's low inflation rate is explained by its low direct emissions, but Europe is heavily penalized by its value chain. China is affected by both factors, while Russia's high inflation is mainly due to its direct emissions, as its value chain impact is among the lowest in the world.

Desnos et al. (2023) analyze a regional taxation scenario in which a carbon tax is imposed uniquely within a specific region of the world. This situation is likely to arise due to the lack of uniformity in carbon pricing. They first consider a uniform taxation across EU Member States, which is certainly the most likely scenario. In this scenario, a $500/tCO₂e carbon tax with a 100% pass-through would result in a global cost of $4.5 tn, of which $4 tn would be borne by EU countries
<figure><img src="image_430.png" /><figcaption>Figure 8.149: World economic cost in $ tn (global analysis, uniform tax, Exiobase 2022)</figcaption></figure>

Source: Desnos et al. (2023, Figure 41, page 82).

<figure><img src="image_431.png" /><figcaption>Figure 8.150: World inflation rate in % (global analysis, uniform tax, Exiobase 2022)</figcaption></figure>

Source: Desnos et al. (2023, Figure 44, page 84).
<figure><img src="image_432.png" /><figcaption>Figure 8.151: Production inflation rate in % (global analysis, uniform tax, $\tau$ = $100/tCO_2e$, $\phi$ = 100%, Exiobase 2022)</figcaption></figure>

Source: Desnos et al. (2023, Figure 45, page 85).

and a $0.5 tn by non-EU countries. If EU sectors absorb their increased costs by passing only 50% through the value chain, non-EU countries are less affected by carbon tax diffusion, and their costs fall from $521 bn to $54 bn. Moreover, the costs relative to GDP for EU sectors fall as they absorb the carbon tax, from 14% if they pass on the carbon tax in full to 8% if direct emitters bear half of the cost of the carbon tax. Among the EU countries, Germany is the most affected, followed by Poland and Italy (Figure 8.152).

<table><caption>Table 8.65: Domestic and foreign impacts (in $ bn) of a regional tax (uniform taxation, $\phi$ = 100%, Exiobase 2022)</caption><thead><tr><th rowspan="2">Carbon tax</th><th colspan="3">Domestic impact</th><th colspan="3">Foreign impact</th></tr><tr><th>EU</th><th>USA</th><th>China</th><th>EU</th><th>USA</th><th>China</th></tr></thead><tbody><tr><td>$100/tCO<sub>2</sub>e</td><td>792</td><td>886</td><td>4710</td><td>104</td><td>118</td><td>257</td></tr><tr><td>$250/tCO<sub>2</sub>e</td><td>1979</td><td>2215</td><td>11774</td><td>261</td><td>296</td><td>643</td></tr><tr><td>$500/tCO<sub>2</sub>e</td><td>3959</td><td>4430</td><td>23549</td><td>521</td><td>592</td><td>1287</td></tr></tbody></table>

Source: Desnos et al. (2023, Table 30, page 91).

Table 8.65 shows the global foreign impact of a tax in the EU, US and China for three values of the carbon price. China always has the highest external impact, with a cost of $1287 bn for a $500/tCO₂e carbon price, while the EU's external impact is only $521 bn. To better understand the winners and the losers, we report the fifteen largest countries affected by a carbon tax in Table 8.66. In this scenario, the tax is set at $100/tCO₂e and the pass-through parameter is set to 100%. When the tax is applied in the European Union, the rest-of-the-world region is most affected, accounting for 25.25% of the total costs supported by foreign countries. This is followed by China (23.62%).
<figure><img src="image_433.png" /><figcaption>Figure 8.152: Cost breakdown (EU, uniform tax, $\phi$ = 50%, Exiobase 2022)</figcaption></figure>

Source: Desnos et al. (2023, Figure 49, page 88).

<table><caption>Table 8.66: Fifteen most affected foreign countries (uniform tax, $\tau$ = $100/tCO_2e$, $\phi$ = 100%, Exiobase 2022)</caption><thead><tr><th>Rank</th><th>EU tax</th><th>US tax</th><th>Chinese tax</th></tr></thead><tbody><tr><td>1</td><td>ROW 25.25%</td><td>CHN 24.74%</td><td>ROW 36.89%</td></tr><tr><td>2</td><td>CHN 23.62%</td><td>ROW 18.60%</td><td>USA 12.95%</td></tr><tr><td>3</td><td>USA 11.45%</td><td>CAN 9.35%</td><td>KOR 8.87%</td></tr><tr><td>4</td><td>GBR 8.77%</td><td>MEX 8.51%</td><td>IND 6.91%</td></tr><tr><td>5</td><td>CHE 4.32%</td><td>KOR 6.89%</td><td>JPN 6.44%</td></tr><tr><td>6</td><td>KOR 4.05%</td><td>JPN 5.05%</td><td>DEU 3.61%</td></tr><tr><td>7</td><td>IND 3.67%</td><td>IND 4.28%</td><td>MEX 2.19%</td></tr><tr><td>8</td><td>JPN 3.31%</td><td>DEU 2.80%</td><td>FRA 1.88%</td></tr><tr><td>9</td><td>TUR 2.62%</td><td>BRA 2.51%</td><td>GBR 1.83%</td></tr><tr><td>10</td><td>TWN 2.08%</td><td>GBR 2.34%</td><td>BRA 1.75%</td></tr><tr><td>11</td><td>CAN 2.06%</td><td>FRÁ 1.63%</td><td>IDN 1.74%</td></tr><tr><td>12</td><td>RUS 1.96%</td><td>TWN 1.59%</td><td>CAN 1.62%</td></tr><tr><td>13</td><td>BRA 1.90%</td><td>IRL 1.47%</td><td>ITA 1.59%</td></tr><tr><td>14</td><td>MEX 1.70%</td><td>ITA 1.43%</td><td>AUS 1.31%</td></tr><tr><td>15</td><td>NOR 1.46%</td><td>NLD 1.23%</td><td>TUR 1.11%</td></tr></tbody></table>

Source: Desnos et al. (2023, Table 31, page 91).
the United States (11.45%) and the United Kingdom (8.77%). This would also be the case with a Chinese tax, but it now represents more than 36% of the foreign costs. This is followed by the United States (12.95%), South Korea (8.87%) and India (6.91%). In the case of a US tax, China would be the most affected country with 24.74% of the total impact, followed by the rest-of-the-world region (18.60%), Canada (9.35%) and Mexico (8.51%). It is important to note that the US has strong trade relations with China, but also with other countries in the Americas (Canada, Mexico, Brazil). These results highlight the trade links between countries, and consequently, the potential exposure to a carbon tax. For example, if we focus on Turkey, it is highly linked to the EU, as it would be the 9th country most affected by an EU carbon tax. It would also be affected by a Chinese tax, as it would be the 15th most affected country. The impact would be smaller in the case of a US carbon tax. Similarly, we see the importance of Germany in the Chinese and US supply chain.

**The study of Sautel et al. (2022)** The analysis carried out by Sautel et al. (2022) uses the previous input-output framework. It focuses on France and considers a carbon tax of €250 per tonne of CO₂. The estimates are based on the WIOD database and default values of pass-through rates (see Table 8.64 on page 853). The conclusions are as follows:

> "The total additional cost of introducing a price of €250 per tonne of CO₂ to be paid by French emitting installations is €57.6 billion, or about 2.5 points of GDP. Of this total, €7 billion corresponds to purchases by foreign operators and investments by French and foreign operators. [...] Of this €50.3 billion, French companies would ultimately bear 57% of the additional costs, or about €28.7 billion. The rest would be passed on to final demand, i.e., 21.6 billion euros." (Sautel et al., 2022, page 39).

**The study of Roncalli and Semet (2024)** It is common to assume that the pass-through rate follows a beta distribution, as it is a parameter between 0 and 1:

$$ \phi \sim B(\alpha, \beta) $$

Following Sautel et al. (2022), Desnos et al. (2023) considered four types of sectors with respect to price-demand elasticity (highly-elastic, high-elastic, medium-elastic and low-elastic), and used the expert-opinion values of the parameters α and β given in Table 8.67. We give the mean, the standard deviation and the 95% range, and show the associated probability density functions in Figure 8.153. The first type is right-skewed, while the fourth type is left-skewed. The second and third types are more symmetric. Moreover, these four distribution functions are ordered since they verify the first-order stochastic dominance principle (Figure 8.154).

<table><caption>Table 8.67: Probabilistic characterization of the four pass-through types</caption><thead><tr><th></th><th>Statistic</th><th>Highly-elastic</th><th>High-elastic</th><th>Medium-elastic</th><th>Low-elastic</th></tr></thead><tbody><tr><td rowspan="2">Parameters</td><td>α</td><td>3.0</td><td>4.0</td><td>14.0</td><td>12.0</td></tr><tr><td>β</td><td>12.0</td><td>6.0</td><td>6.0</td><td>0.6</td></tr><tr><td rowspan="2">Moments</td><td>μ<sub>φ</sub></td><td>20%</td><td>40%</td><td>70%</td><td>95%</td></tr><tr><td>σ<sub>φ</sub></td><td>10%</td><td>15%</td><td>10%</td><td>6%</td></tr><tr><td rowspan="2">Range</td><td>Q<sub>φ</sub> (2.5%)</td><td>5%</td><td>14%</td><td>49%</td><td>79%</td></tr><tr><td>Q<sub>φ</sub> (97.5%)</td><td>43%</td><td>70%</td><td>87%</td><td>100%</td></tr></tbody></table>

Desnos et al. (2023) defined a mapping between the WIOD sectors and the four types of pass-through mechanisms. They used this classification to build a Monte Carlo Value-at-Risk engine<fnref n="149" />.

<footnote n="149">¹⁴⁹See Section 13.3 on page 1131.</footnote>
<figure><img src="image_434.png" /><figcaption>Figure 8.154: Probability density function of pass-through rates</figcaption></figure>

<figure><img src="image_435.png" /></figure>
Roncalli and Semet (2024) use their classification to measure the impact of a $100/tCO₂e carbon tax. They assume that the carbon tax is applied to a given region and compute the total cost vector as follows:

$$T_{\text{total}} = x \odot (I_n - \text{diag}(\phi) + \tilde{\mathcal{L}}(\phi)) t_{\text{direct}}$$

They assume that the pass-through rate of the sector is constant and equal to the mean of the corresponding beta distribution: $\phi = \mu_{\phi}$. They decompose the total costs into what is paid by the producer and what is paid by the downstream value chain, including the final consumer:

$$\begin{cases} T_{\text{producer}} = x \odot (\mathbf{1}_n - \phi) \odot t_{\text{direct}} \\ T_{\text{downstream}} = x \odot \tilde{\mathcal{L}}(\phi) t_{\text{direct}} \end{cases}$$

They also consider a second decomposition between the direct costs of the carbon tax and the indirect costs due to the pass-through mechanism:

$$\begin{cases} T_{\text{direct}} = x \odot t_{\text{direct}} \\ T_{\text{indirect}} = T_{\text{total}} - T_{\text{direct}} = x \odot (\tilde{\mathcal{L}}(\phi) - \text{diag}(\phi)) t_{\text{direct}} \end{cases}$$

By definition, government revenue is equal to the direct cost of the carbon tax:

$$R_{\text{government}} = T_{\text{direct}} = x \odot t_{\text{direct}}$$

All of the previous measures can be aggregated at a global level or at a country/regional level<fnref n="150" />. Inflation rates are calculated using the following formula:

$$\pi = \alpha^{\top} \Delta p = \alpha^{\top} \tilde{\mathcal{L}}(\phi) t_{\text{direct}}$$

They distinguish between PPI and CPI inflation. In the first case, the basket weights $\alpha$ are defined with respect to the output vector, while in the second case they use the final demand vector: $\alpha_{\text{ppi}} = x / (\mathbf{1}_n^{\top} x)$ and $\alpha_{\text{cpi}} = y / (\mathbf{1}_n^{\top} y)$. If the analysis is conducted at the country/regional level, the weights are calculated using the country's output and final demand vectors.

In Table 8.68, we report the results of Roncalli and Semet (2024) when considering a uniform global carbon tax of $100/tCO₂e. The cost and revenue figures are expressed as a percentage of GDP<fnref n="151" />. At the global level, the implementation of a uniform carbon tax affects GDP by 5.01% and generates revenue of 2.82%. The net cost would then be 2.18% of world GDP. While the government revenue is exactly the direct cost of the carbon tax, the net cost is the indirect cost of the carbon tax due to the pass-through mechanism. If we split the total cost between what is paid by the producer and what is paid by the downstream supply chain, including the final consumer, we find that of the 5.01% of the total cost, only 0.93% is paid by the producers, while 4.08% is paid by the value chain and consumers. This means that only 20% of the carbon tax is effectively borne by the producers themselves. Analysis at the regional level highlights economic disparities between countries. Overall, four countries have net costs above 3%. India is the most affected country with a net cost of 4.55% of GDP, followed by Russia (4.24%), China (4.03%) and Bulgaria (3.12%).

The previous results are not realistic because it is unlikely that common measures such as an international carbon pricing system will see the light of day. In this context, Roncalli and Semet (2024) propose to analyze the economic impact of the implementation of individual climate policies, in particular the implementation of an individual carbon tax in Europe, the United States and

<footnote n="150">¹⁵⁰Let S be a set of sectors. The aggregated costs and revenues are equal to $C = \sum_{j \in S} T_j$ and $R = \sum_{j \in S} R_j$.</footnote>

<footnote n="151">¹⁵¹This means that we normalize these figures by the total output $\mathbf{1}_n^{\top} x$.</footnote>
<table><caption>Table 8.68: Economic impact of a global carbon tax ($100/tCO₂e, Exiobase 2022)</caption><thead><tr><th rowspan="2">Region</th><th rowspan="2">C<sub>total</sub></th><th rowspan="2">C<sub>direct</sub></th><th rowspan="2">C<sub>indirect</sub></th><th colspan="2">Cost</th><th rowspan="2">C<sub>net</sub></th><th rowspan="2">R<sub>government</sub></th></tr><tr><th>C<sub>producer</sub></th><th>C<sub>downstream</sub></th></tr></thead><tbody><tr><td>World</td><td>5.01%</td><td>2.82%</td><td>2.18%</td><td>0.93%</td><td>4.08%</td><td>2.18%</td><td>2.82%</td></tr><tr><td>AUS</td><td>4.63%</td><td>2.93%</td><td>1.70%</td><td>0.81%</td><td>3.82%</td><td>1.70%</td><td>2.93%</td></tr><tr><td>AUT</td><td>2.08%</td><td>0.91%</td><td>1.17%</td><td>0.30%</td><td>1.77%</td><td>1.17%</td><td>0.91%</td></tr><tr><td>BEL</td><td>1.77%</td><td>0.94%</td><td>0.82%</td><td>0.44%</td><td>1.33%</td><td>0.82%</td><td>0.94%</td></tr><tr><td>BGR</td><td>7.07%</td><td>3.94%</td><td>3.12%</td><td>0.89%</td><td>6.18%</td><td>3.12%</td><td>3.94%</td></tr><tr><td>BRA</td><td>5.22%</td><td>3.78%</td><td>1.44%</td><td>2.01%</td><td>3.21%</td><td>1.44%</td><td>3.78%</td></tr><tr><td>CAN</td><td>3.74%</td><td>2.25%</td><td>1.49%</td><td>0.57%</td><td>3.17%</td><td>1.49%</td><td>2.25%</td></tr><tr><td>CHE</td><td>0.75%</td><td>0.30%</td><td>0.45%</td><td>0.16%</td><td>0.59%</td><td>0.45%</td><td>0.30%</td></tr><tr><td>CHN</td><td>7.47%</td><td>3.44%</td><td>4.03%</td><td>1.21%</td><td>6.26%</td><td>4.03%</td><td>3.44%</td></tr><tr><td>CYP</td><td>5.05%</td><td>3.94%</td><td>1.11%</td><td>2.49%</td><td>2.56%</td><td>1.11%</td><td>3.94%</td></tr><tr><td>CZE</td><td>4.47%</td><td>2.13%</td><td>2.34%</td><td>0.44%</td><td>4.03%</td><td>2.34%</td><td>2.13%</td></tr><tr><td>DEU</td><td>1.99%</td><td>1.10%</td><td>0.89%</td><td>0.35%</td><td>1.64%</td><td>0.89%</td><td>1.10%</td></tr><tr><td>DNK</td><td>1.47%</td><td>0.98%</td><td>0.49%</td><td>0.54%</td><td>0.93%</td><td>0.49%</td><td>0.98%</td></tr><tr><td>ESP</td><td>2.25%</td><td>1.15%</td><td>1.11%</td><td>0.41%</td><td>1.84%</td><td>1.11%</td><td>1.15%</td></tr><tr><td>FIN</td><td>2.80%</td><td>1.36%</td><td>1.44%</td><td>0.36%</td><td>2.44%</td><td>1.44%</td><td>1.36%</td></tr><tr><td>FRA</td><td>1.39%</td><td>0.79%</td><td>0.60%</td><td>0.35%</td><td>1.04%</td><td>0.60%</td><td>0.79%</td></tr><tr><td>GBR</td><td>1.53%</td><td>0.88%</td><td>0.65%</td><td>0.33%</td><td>1.20%</td><td>0.65%</td><td>0.88%</td></tr><tr><td>GRC</td><td>6.39%</td><td>4.61%</td><td>1.78%</td><td>2.52%</td><td>3.87%</td><td>1.78%</td><td>4.61%</td></tr><tr><td>HRV</td><td>3.57%</td><td>2.18%</td><td>1.38%</td><td>0.89%</td><td>2.67%</td><td>1.38%</td><td>2.18%</td></tr><tr><td>HUN</td><td>3.41%</td><td>1.83%</td><td>1.58%</td><td>0.61%</td><td>2.80%</td><td>1.58%</td><td>1.83%</td></tr><tr><td>IDN</td><td>7.85%</td><td>5.53%</td><td>2.31%</td><td>2.08%</td><td>5.77%</td><td>2.31%</td><td>5.53%</td></tr><tr><td>IND</td><td>11.38%</td><td>6.83%</td><td>4.55%</td><td>2.28%</td><td>9.11%</td><td>4.55%</td><td>6.83%</td></tr><tr><td>IRL</td><td>1.47%</td><td>0.95%</td><td>0.52%</td><td>0.57%</td><td>0.89%</td><td>0.52%</td><td>0.95%</td></tr><tr><td>ITA</td><td>2.22%</td><td>0.93%</td><td>1.29%</td><td>0.28%</td><td>1.93%</td><td>1.29%</td><td>0.93%</td></tr><tr><td>JPN</td><td>2.85%</td><td>1.38%</td><td>1.47%</td><td>0.32%</td><td>2.53%</td><td>1.47%</td><td>1.38%</td></tr><tr><td>KOR</td><td>4.23%</td><td>1.61%</td><td>2.61%</td><td>0.38%</td><td>3.85%</td><td>2.61%</td><td>1.61%</td></tr><tr><td>LTU</td><td>4.06%</td><td>2.41%</td><td>1.65%</td><td>1.00%</td><td>3.06%</td><td>1.65%</td><td>2.41%</td></tr><tr><td>LUX</td><td>1.15%</td><td>0.51%</td><td>0.64%</td><td>0.35%</td><td>0.80%</td><td>0.64%</td><td>0.51%</td></tr><tr><td>LVA</td><td>3.43%</td><td>2.15%</td><td>1.28%</td><td>1.07%</td><td>2.36%</td><td>1.28%</td><td>2.15%</td></tr><tr><td>MEX</td><td>5.59%</td><td>3.60%</td><td>1.99%</td><td>1.02%</td><td>4.57%</td><td>1.99%</td><td>3.60%</td></tr><tr><td>MLT</td><td>1.82%</td><td>0.64%</td><td>1.18%</td><td>0.17%</td><td>1.65%</td><td>1.18%</td><td>0.64%</td></tr><tr><td>NLD</td><td>2.25%</td><td>1.14%</td><td>1.12%</td><td>0.51%</td><td>1.74%</td><td>1.12%</td><td>1.14%</td></tr><tr><td>NOR</td><td>1.81%</td><td>1.31%</td><td>0.51%</td><td>0.58%</td><td>1.23%</td><td>0.51%</td><td>1.31%</td></tr><tr><td>POL</td><td>5.84%</td><td>3.44%</td><td>2.40%</td><td>0.98%</td><td>4.86%</td><td>2.40%</td><td>3.44%</td></tr><tr><td>PRT</td><td>3.77%</td><td>2.13%</td><td>1.64%</td><td>0.70%</td><td>3.07%</td><td>1.64%</td><td>2.13%</td></tr><tr><td>ROU</td><td>4.10%</td><td>2.19%</td><td>1.91%</td><td>0.69%</td><td>3.42%</td><td>1.91%</td><td>2.19%</td></tr><tr><td>RUS</td><td>12.79%</td><td>8.55%</td><td>4.24%</td><td>1.44%</td><td>11.34%</td><td>4.24%</td><td>8.55%</td></tr><tr><td>SVK</td><td>3.29%</td><td>1.62%</td><td>1.66%</td><td>0.42%</td><td>2.87%</td><td>1.66%</td><td>1.62%</td></tr><tr><td>SVN</td><td>2.79%</td><td>1.51%</td><td>1.28%</td><td>0.47%</td><td>2.32%</td><td>1.28%</td><td>1.51%</td></tr><tr><td>SWE</td><td>1.21%</td><td>0.59%</td><td>0.62%</td><td>0.21%</td><td>1.00%</td><td>0.62%</td><td>0.59%</td></tr><tr><td>TUR</td><td>5.78%</td><td>3.73%</td><td>2.05%</td><td>1.39%</td><td>4.39%</td><td>2.05%</td><td>3.73%</td></tr><tr><td>TWN</td><td>5.16%</td><td>2.21%</td><td>2.95%</td><td>0.75%</td><td>4.41%</td><td>2.95%</td><td>2.21%</td></tr><tr><td>USA</td><td>2.17%</td><td>1.40%</td><td>0.78%</td><td>0.34%</td><td>1.83%</td><td>0.78%</td><td>1.40%</td></tr><tr><td>ROW</td><td>7.55%</td><td>5.14%</td><td>2.40%</td><td>1.87%</td><td>5.68%</td><td>2.40%</td><td>5.14%</td></tr></tbody></table>

Source: Roncalli and Semet (2024, Table 1, page 16).
<table><caption>Table 8.69: Economic impact of a EU carbon tax ($100/tCO₂e, Exiobase 2022)</caption><thead><tr><th rowspan="2">Region</th><th colspan="6">Cost</th><th>Revenue</th></tr><tr><th>C<sub>total</sub></th><th>C<sub>direct</sub></th><th>C<sub>indirect</sub></th><th>C<sub>producer</sub></th><th>C<sub>downstream</sub></th><th>C<sub>net</sub></th><th>R<sub>government</sub></th></tr></thead><tbody><tr><td>World</td><td>0.36%</td><td>0.22%</td><td>0.14%</td><td>0.07%</td><td>0.28%</td><td>0.14%</td><td>0.22%</td></tr><tr><td>BGR</td><td>6.30%</td><td>3.94%</td><td>2.35%</td><td>0.89%</td><td>5.41%</td><td>2.35%</td><td>3.94%</td></tr><tr><td>GRC</td><td>5.64%</td><td>4.61%</td><td>1.03%</td><td>2.52%</td><td>3.12%</td><td>1.03%</td><td>4.61%</td></tr><tr><td>POL</td><td>5.21%</td><td>3.44%</td><td>1.77%</td><td>0.98%</td><td>4.24%</td><td>1.77%</td><td>3.44%</td></tr><tr><td>CYP</td><td>4.86%</td><td>3.94%</td><td>0.92%</td><td>2.49%</td><td>2.37%</td><td>0.92%</td><td>3.94%</td></tr><tr><td>CZE</td><td>3.90%</td><td>2.13%</td><td>1.76%</td><td>0.44%</td><td>3.46%</td><td>1.76%</td><td>2.13%</td></tr><tr><td>ROU</td><td>3.60%</td><td>2.19%</td><td>1.41%</td><td>0.69%</td><td>2.91%</td><td>1.41%</td><td>2.19%</td></tr><tr><td>PRT</td><td>3.28%</td><td>2.13%</td><td>1.15%</td><td>0.70%</td><td>2.58%</td><td>1.15%</td><td>2.13%</td></tr><tr><td>LTU</td><td>3.22%</td><td>2.41%</td><td>0.82%</td><td>1.00%</td><td>2.22%</td><td>0.82%</td><td>2.41%</td></tr><tr><td>LVA</td><td>3.11%</td><td>2.15%</td><td>0.96%</td><td>1.07%</td><td>2.05%</td><td>0.96%</td><td>2.15%</td></tr><tr><td>HRV</td><td>2.88%</td><td>2.18%</td><td>0.70%</td><td>0.89%</td><td>1.99%</td><td>0.70%</td><td>2.18%</td></tr><tr><td>SVK</td><td>2.72%</td><td>1.62%</td><td>1.09%</td><td>0.42%</td><td>2.30%</td><td>1.09%</td><td>1.62%</td></tr><tr><td>HUN</td><td>2.70%</td><td>1.83%</td><td>0.87%</td><td>0.61%</td><td>2.08%</td><td>0.87%</td><td>1.83%</td></tr><tr><td>SVN</td><td>2.38%</td><td>1.51%</td><td>0.87%</td><td>0.47%</td><td>1.91%</td><td>0.87%</td><td>1.51%</td></tr><tr><td>FIN</td><td>2.27%</td><td>1.36%</td><td>0.91%</td><td>0.36%</td><td>1.91%</td><td>0.91%</td><td>1.36%</td></tr><tr><td>ESP</td><td>1.82%</td><td>1.15%</td><td>0.68%</td><td>0.41%</td><td>1.41%</td><td>0.68%</td><td>1.15%</td></tr></tbody></table>

Source: Roncalli and Semet (2024, Table 2, page 18).

<table><caption>Table 8.70: Economic impact of a US carbon tax ($100/tCO₂e, Exiobase 2022)</caption><thead><tr><th rowspan="2">Region</th><th colspan="6">Cost</th><th>Revenue</th></tr><tr><th>C<sub>total</sub></th><th>C<sub>direct</sub></th><th>C<sub>indirect</sub></th><th>C<sub>producer</sub></th><th>C<sub>downstream</sub></th><th>C<sub>net</sub></th><th>R<sub>government</sub></th></tr></thead><tbody><tr><td>World</td><td>0.44%</td><td>0.29%</td><td>0.14%</td><td>0.07%</td><td>0.37%</td><td>0.14%</td><td>0.29%</td></tr><tr><td>USA</td><td>1.96%</td><td>1.40%</td><td>0.57%</td><td>0.34%</td><td>1.62%</td><td>0.57%</td><td>1.40%</td></tr><tr><td>CAN</td><td>0.18%</td><td>0.00%</td><td>0.18%</td><td>0.00%</td><td>0.18%</td><td>0.18%</td><td>0.00%</td></tr><tr><td>MEX</td><td>0.18%</td><td>0.00%</td><td>0.18%</td><td>0.00%</td><td>0.18%</td><td>0.18%</td><td>0.00%</td></tr><tr><td>KOR</td><td>0.07%</td><td>0.00%</td><td>0.07%</td><td>0.00%</td><td>0.07%</td><td>0.07%</td><td>0.00%</td></tr><tr><td>IRL</td><td>0.06%</td><td>0.00%</td><td>0.06%</td><td>0.00%</td><td>0.06%</td><td>0.06%</td><td>0.00%</td></tr><tr><td>BRA</td><td>0.05%</td><td>0.00%</td><td>0.05%</td><td>0.00%</td><td>0.05%</td><td>0.05%</td><td>0.00%</td></tr><tr><td>TWN</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.04%</td><td>0.00%</td></tr><tr><td>ROW</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.04%</td><td>0.00%</td></tr><tr><td>IND</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.04%</td><td>0.00%</td></tr><tr><td>NLD</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.03%</td><td>0.00%</td></tr><tr><td>GBR</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.03%</td><td>0.00%</td></tr><tr><td>NOR</td><td>0.02%</td><td>0.00%</td><td>0.02%</td><td>0.00%</td><td>0.02%</td><td>0.02%</td><td>0.00%</td></tr><tr><td>BEL</td><td>0.02%</td><td>0.00%</td><td>0.02%</td><td>0.00%</td><td>0.02%</td><td>0.02%</td><td>0.00%</td></tr><tr><td>JPN</td><td>0.02%</td><td>0.00%</td><td>0.02%</td><td>0.00%</td><td>0.02%</td><td>0.02%</td><td>0.00%</td></tr><tr><td>CHN</td><td>0.02%</td><td>0.00%</td><td>0.02%</td><td>0.00%</td><td>0.02%</td><td>0.02%</td><td>0.00%</td></tr></tbody></table>

Source: Roncalli and Semet (2024, Table 3, page 18).
<table><caption>Table 8.71: Economic impact of a carbon tax in China ($100/tCO₂e, Exiobase 2022)</caption><thead><tr><th rowspan="2">Region</th><th colspan="6">Cost</th><th rowspan="2">Revenue<br>R<sub>government</sub></th></tr><tr><th>C<sub>total</sub></th><th>C<sub>direct</sub></th><th>C<sub>indirect</sub></th><th>C<sub>producer</sub></th><th>C<sub>downstream</sub></th><th>C<sub>net</sub></th></tr></thead><tbody><tr><td>World</td><td>1.66%</td><td>0.81%</td><td>0.85%</td><td>0.29%</td><td>1.38%</td><td>0.85%</td><td>0.81%</td></tr><tr><td>CHN</td><td>6.89%</td><td>3.44%</td><td>3.45%</td><td>1.21%</td><td>5.68%</td><td>3.45%</td><td>3.44%</td></tr><tr><td>ROW</td><td>0.13%</td><td>0.00%</td><td>0.13%</td><td>0.00%</td><td>0.13%</td><td>0.13%</td><td>0.00%</td></tr><tr><td>KOR</td><td>0.12%</td><td>0.00%</td><td>0.12%</td><td>0.00%</td><td>0.12%</td><td>0.12%</td><td>0.00%</td></tr><tr><td>MEX</td><td>0.06%</td><td>0.00%</td><td>0.06%</td><td>0.00%</td><td>0.06%</td><td>0.06%</td><td>0.00%</td></tr><tr><td>IND</td><td>0.05%</td><td>0.00%</td><td>0.05%</td><td>0.00%</td><td>0.05%</td><td>0.05%</td><td>0.00%</td></tr><tr><td>IDN</td><td>0.05%</td><td>0.00%</td><td>0.05%</td><td>0.00%</td><td>0.05%</td><td>0.05%</td><td>0.00%</td></tr><tr><td>JPN</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.04%</td><td>0.00%</td></tr><tr><td>POL</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.04%</td><td>0.00%</td></tr><tr><td>CZE</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.04%</td><td>0.00%</td></tr><tr><td>HUN</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.04%</td><td>0.00%</td></tr><tr><td>TUR</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.00%</td><td>0.04%</td><td>0.04%</td><td>0.00%</td></tr><tr><td>CAN</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.03%</td><td>0.00%</td></tr><tr><td>BEL</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.03%</td><td>0.00%</td></tr><tr><td>SVK</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.03%</td><td>0.00%</td></tr><tr><td>AUS</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.00%</td><td>0.03%</td><td>0.03%</td><td>0.00%</td></tr></tbody></table>

Source: Roncalli and Semet (2024, Table 4, page 19).

China. In Table 8.69, we observe that the implementation of a carbon tax within the EU member states will primarily affect the European economy since more than 95% of the total costs will fall on the countries of the European Union. In Tables 8.70 and 8.71, we report estimates for a US and a Chinese carbon tax, respectively. As in the case of the EU tax, the economic incidence of the carbon tax is almost entirely borne by the domestic economy. For the US, the total cost is less than 2% of GDP, while for China this figure reaches 6.89%. The two main trading partners of the United States, Canada and Mexico, will be indirectly affected by the tax by up to $5.5 bn and $3.7 bn, respectively, which is less than 0.2% of their respective GDP. Despite the limited cascading effect of the domestic carbon tax, the estimates illustrate the distortionary effects of an isolated climate policy. Finally, comparing the macroeconomic effects on a global scale, we see that the impact of an American or Chinese tax on world GDP is larger than that of a European tax. A Chinese carbon tax could induce a significant cost of 1.66% on the global economy, which is more than four times the total impact of a European tax.

In Table 8.72 and 8.73, we present the PPI and CPI values for the fifteen most affected countries, categorized according to different tax structures. With a uniform global tax of $100/tCO₂e, cost-push inflation reaches 4.08% worldwide, while escalating producer prices lead to a 3.53% increase in consumer prices worldwide. Focusing on the regional tax, inflation is mainly supported by the region that implements the carbon tax. This is especially true for the European and Chinese tax.

In summary, the introduction of a regional carbon tax creates a competitive distortion between the region introducing the tax and the rest of the world. Moreover, the introduction of a carbon tax leads to inflation. This is because path-through mechanisms amplify the price dynamics of the costs passed on by producers. The real benefits of a carbon tax are then challenging and are discussed in detail in Chapter 10.
<table><caption>Table 8.72: Producer price index ($100/tCO₂e, Exiobase 2022)</caption><thead><tr><th rowspan="2">Rank</th><th colspan="2">Global tax</th><th colspan="2">EU tax</th><th colspan="2">US tax</th><th colspan="2">China tax</th></tr><tr><th>World</th><th>4.08%</th><th>World</th><th>0.28%</th><th>World</th><th>0.37%</th><th>World</th><th>1.38%</th></tr></thead><tbody><tr><td>1</td><td>RUS</td><td>11.34%</td><td>BGR</td><td>5.41%</td><td>USA</td><td>1.62%</td><td>CHN</td><td>5.68%</td></tr><tr><td>2</td><td>IND</td><td>9.11%</td><td>POL</td><td>4.24%</td><td>CAN</td><td>0.18%</td><td>ROW</td><td>0.13%</td></tr><tr><td>3</td><td>CHN</td><td>6.26%</td><td>CZE</td><td>3.46%</td><td>MEX</td><td>0.18%</td><td>KOR</td><td>0.12%</td></tr><tr><td>4</td><td>BGR</td><td>6.18%</td><td>GRC</td><td>3.12%</td><td>KOR</td><td>0.07%</td><td>MEX</td><td>0.06%</td></tr><tr><td>5</td><td>IDN</td><td>5.77%</td><td>ROU</td><td>2.91%</td><td>IRL</td><td>0.06%</td><td>IND</td><td>0.05%</td></tr><tr><td>6</td><td>ROW</td><td>5.68%</td><td>PRT</td><td>2.58%</td><td>BRÁ</td><td>0.05%</td><td>IDN</td><td>0.05%</td></tr><tr><td>7</td><td>POL</td><td>4.86%</td><td>CYP</td><td>2.37%</td><td>TWN</td><td>0.04%</td><td>JPN</td><td>0.04%</td></tr><tr><td>8</td><td>MEX</td><td>4.57%</td><td>SVK</td><td>2.30%</td><td>ROW</td><td>0.04%</td><td>POL</td><td>0.04%</td></tr><tr><td>9</td><td>TWN</td><td>4.41%</td><td>LTU</td><td>2.22%</td><td>IND</td><td>0.04%</td><td>CZE</td><td>0.04%</td></tr><tr><td>10</td><td>TUR</td><td>4.39%</td><td>HUN</td><td>2.08%</td><td>NLD</td><td>0.03%</td><td>HUN</td><td>0.04%</td></tr><tr><td>11</td><td>ČŽE</td><td>4.03%</td><td>LVÁ</td><td>2.05%</td><td>GBR</td><td>0.03%</td><td>TUR</td><td>0.04%</td></tr><tr><td>12</td><td>GRC</td><td>3.87%</td><td>HRV</td><td>1.99%</td><td>NOR</td><td>0.02%</td><td>CAN</td><td>0.03%</td></tr><tr><td>13</td><td>KOR</td><td>3.85%</td><td>SVN</td><td>1.91%</td><td>BEL</td><td>0.02%</td><td>BEL</td><td>0.03%</td></tr><tr><td>14</td><td>AUS</td><td>3.82%</td><td>FIN</td><td>1.91%</td><td>JPN</td><td>0.02%</td><td>SVK</td><td>0.03%</td></tr><tr><td>15</td><td>ROU</td><td>3.42%</td><td>AUT</td><td>1.50%</td><td>CHN</td><td>0.02%</td><td>AUS</td><td>0.03%</td></tr></tbody></table>

Source: Roncalli and Semet (2024, Table 5, page 20).

<table><caption>Table 8.73: Consumer price index ($100/tCO₂e, Exiobase 2022)</caption><thead><tr><th rowspan="2">Rank</th><th colspan="2">Global tax</th><th colspan="2">EU tax</th><th colspan="2">US tax</th><th colspan="2">China tax</th></tr><tr><th>World</th><th>3.53%</th><th>World</th><th>0.48%</th><th>World</th><th>0.27%</th><th>World</th><th>1.15%</th></tr></thead><tbody><tr><td>1</td><td>IDN</td><td>6.75%</td><td>FRÁ</td><td>5.95%</td><td>USA</td><td>1.06%</td><td>CHN</td><td>5.88%</td></tr><tr><td>2</td><td>CHN</td><td>6.35%</td><td>CZE</td><td>4.07%</td><td>MEX</td><td>0.16%</td><td>ROW</td><td>0.16%</td></tr><tr><td>3</td><td>FRA</td><td>6.29%</td><td>HRV</td><td>3.83%</td><td>CAN</td><td>0.16%</td><td>KOR</td><td>0.08%</td></tr><tr><td>4</td><td>IND</td><td>5.98%</td><td>GRC</td><td>3.59%</td><td>IRL</td><td>0.05%</td><td>AUS</td><td>0.07%</td></tr><tr><td>5</td><td>RUS</td><td>5.72%</td><td>POL</td><td>3.49%</td><td>BRA</td><td>0.04%</td><td>IND</td><td>0.07%</td></tr><tr><td>6</td><td>ČŽE</td><td>4.63%</td><td>ČYP</td><td>3.32%</td><td>GBR</td><td>0.04%</td><td>ČAN</td><td>0.07%</td></tr><tr><td>7</td><td>HRV</td><td>4.42%</td><td>BGR</td><td>3.16%</td><td>ROW</td><td>0.04%</td><td>MEX</td><td>0.06%</td></tr><tr><td>8</td><td>GRC</td><td>4.35%</td><td>SVK</td><td>2.80%</td><td>KOR</td><td>0.03%</td><td>TUR</td><td>0.05%</td></tr><tr><td>9</td><td>POL</td><td>4.14%</td><td>MLT</td><td>2.69%</td><td>IND</td><td>0.03%</td><td>IDN</td><td>0.04%</td></tr><tr><td>10</td><td>BGR</td><td>3.89%</td><td>PRT</td><td>2.58%</td><td>NOR</td><td>0.03%</td><td>BRA</td><td>0.04%</td></tr><tr><td>11</td><td>ROW</td><td>3.82%</td><td>LUX</td><td>2.30%</td><td>NLD</td><td>0.03%</td><td>JPN</td><td>0.04%</td></tr><tr><td>12</td><td>TWN</td><td>3.73%</td><td>HUN</td><td>2.20%</td><td>LUX</td><td>0.03%</td><td>BEL</td><td>0.04%</td></tr><tr><td>13</td><td>CYP</td><td>3.57%</td><td>LTU</td><td>2.11%</td><td>TWN</td><td>0.02%</td><td>RUS</td><td>0.04%</td></tr><tr><td>14</td><td>MLT</td><td>3.38%</td><td>NLD</td><td>2.11%</td><td>BEL</td><td>0.02%</td><td>GRC</td><td>0.04%</td></tr><tr><td>15</td><td>SVK</td><td>3.36%</td><td>SVN</td><td>1.90%</td><td>TUR</td><td>0.02%</td><td>POL</td><td>0.04%</td></tr></tbody></table>

Source: Roncalli and Semet (2024, Table 6, page 21).
8.5 Exercises
<figure><img src="image_436.png" /></figure>
# Chapter 9

## Climate Risk Measures

In this chapter, we list the various metrics that are helpful in assessing climate risk. We focus on traditional metrics related to the concept of carbon footprint. This concept seems easy to understand, but difficult to define precisely. In their seminal book on the ecological footprint, Wackernagel and Rees (1996) stated that “*the carbon footprint stands for a certain amount of gaseous emissions that are relevant to climate change and associated with human production or consumption activities.*” Wiedmann and Minx (2008, pages 2-4) listed several definitions found in the scientific literature and the grey literature (NGOs, consultants, data providers, etc.) between 1960 and 2007. They also proposed a definition of their own:

> “*The carbon footprint is a measure of the exclusive total amount of carbon dioxide emissions that is directly and indirectly caused by an activity or is accumulated over the life stages of a product.*” (Wiedmann and Minx, 2008, page 5).

Note that this definition includes only carbon dioxide emissions. The authors excluded methane, because they felt that it should be reserved for another measure, that could be called climate footprint. Furthermore, they pointed out that the carbon footprint should be measured physically in a unit of mass (e.g., kg) and not converted into a land area unit (e.g., ha). This digression shows that the concepts of climate risk measurement are very recent<fnref n="1" /> and not necessarily clear and stable. In the first section, we will then discuss how to measure carbon emissions. In particular, we will distinguish the three scopes, define the carbon dioxide equivalent (CO₂e) unit based on global warming potential (GWP) values, and present some examples of carbon footprint calculations. We will examine some specific sectors, such as transportation and agriculture, and the two main approaches: activity-based and energy-based methods. In the second section, we will analyze the concept of carbon intensity, which is a normalization of the carbon emissions measure in order to compare countries, companies or portfolios. In the case of physical intensity ratios, total emissions are divided by physical quantities. For instance, it is common to measure the carbon footprint of transportation in terms of kgCO₂e per kilometer traveled, while the carbon footprint of a country is typically measured in terms of tCO₂e per capita. The second way to calculate carbon intensity is to use monetary quantities in the denominator. In the previous example, we can divide the total carbon emissions by the revenue generated by the transportation activity or the GDP of the country. Carbon intensity based on monetary units plays an important role in finance and especially in portfolio management. While carbon emissions and intensity are two static measures, in the third section we will develop some dynamic risk measures based on carbon budget and trend. Finally, the last section is dedicated to green intensity measures, whose objective is to assess the greenness of products, countries, companies or assets.

<footnote n="1">¹The publication of this much cited research paper was only fifteen years ago.</footnote>
## 9.1 Carbon emissions

Carbon footprint is a generic term used to define the total greenhouse gas (GHG) emissions caused by a given system, activity, company, country, or region. Greenhouse gases are made up of water vapor (H₂O), carbon dioxide (CO₂), methane (CH₄), nitrous oxide (N₂O), Ozone (O₃), etc. They absorb and emit radiation energy, causing the greenhouse effect. We remind that the greenhouse effect was a crucial factor for the development of human life on Earth. Indeed, without the greenhouse effect, the average temperature of Earth's surface would be about -18°C. With the greenhouse effect, the current temperature of Earth's surface is about +15°C. Nevertheless, the increasing concentration of some GHGs is an issue because it is a factor in global warming. It mainly concerns carbon dioxide, and to a lesser extent, methane and nitrous oxide.

### 9.1.1 Global warming potential

Carbon footprint is generally measured in carbon dioxide equivalent (CO₂e), which is a term for describing different GHGs in a common unit. In this framework, a quantity of GHG is expressed as CO₂e by multiplying the GHG amount by its global warming potential (GWP):

$$ \text{equivalent mass of CO}_2 = \text{mass of the gas} \times \text{gwp of the gas} $$

where the GWP of a gas is the amount of CO₂ that would warm the earth equally. Since the mass of the gas is expressed in kilogram and the GWP has no unit, the mass of CO₂ equivalent is also expressed in kilogram. For instance, the IPCC's 5th assessment report has used the following rules (IPCC, 2014a): 1 kg of methane corresponds to 28 kg of CO₂ and 1 kg of nitrous oxide corresponds to 265 kg of CO₂. The definition of a common unit allows two companies to be compared properly. To compute the carbon footprint of a system that is made up of several gases, we apply the weighted sum formula:

$$ m = \sum_{i=1}^{n} m_i \cdot \text{gwp}_i $$

where m is the mass of CO₂ equivalent, mᵢ and gwpᵢ are the mass and the global warming potential of the i-th gas, and n is the number of gases. m and mᵢ have the same mass unit (eg., kilogram or kg, tonne or t, kilotonne or kt, megatonne or Mt, gigatonne or Gt). However, m measures a mass of CO₂ equivalent. Therefore, it better to use the following units: kgCO₂e, tCO₂e, ktCO₂e, MtCO₂e and GtCO₂e.

**Example 33** We consider a company A that emits 3017 tonnes of CO₂, 10 tonnes of CH₄ and 1.8 tonnes of N₂O. For the company B, the GHG emissions are respectively equal to 2302 tonnes of CO₂, 32 tonnes of CH₄ and 3.0 tonnes of N₂O.

The mass of CO₂ equivalent for companies A and B is equal to:

$$ m_A = 3017 \times 1 + 10 \times 28 + 1.8 \times 265 = 3774 \text{ tCO}_2\text{e} $$

and:

$$ m_B = 2302 \times 1 + 32 \times 28 + 3.0 \times 265 = 3993 \text{ tCO}_2\text{e} $$

We notice that company B emits more carbon emissions than company A when they are measured in CO₂ equivalent. We can also compute the mass contribution of each gas:

$$ c_i = \frac{m_i \cdot \text{gwp}_i}{m} $$
The mass decomposition is reported below:

<table><thead><tr><th>Company</th><th>Mass</th><th>Absolute contribution</th><th>Relative contribution</th></tr></thead><tbody><tr><td>A</td><td>3 774</td><td>3 017 280</td><td>79.94%</td></tr><tr><td>B</td><td>3 993</td><td>2 302 896</td><td>57.65%</td></tr></tbody></table>

The contribution of the carbon dioxide gas is equal to 79.94% for company A and 57.65% for company B. Concerning methane gas, its contribution is respectively equal to 7.42% and 22.44%.

Box 9.1: Estimation of the global warming potential

According to IPCC (2007), GWP is defined as “the cumulative radiative forcing, both direct and indirect effects, over a specified time horizon resulting from the emission of a unit mass of gas related to some reference gas.” Since each gas differs in their capacity to absorb the energy (radiative efficiency) and how long it stays in the atmosphere (lifetime), its impact on global warming depends on these two factors. GWP is then a synthetic measure that combines radiative efficiency and lifetime.

The mathematical definition of the global warming potential is:

$$gwp_i(t) = \frac{Agwp_i(t)}{Agwp_0(t)} = \frac{\int_{0}^{t} RF_i(s) \, ds}{\int_{0}^{t} RF_0(s) \, ds} = \frac{\int_{0}^{t} A_i(s) S_i(s) \, ds}{\int_{0}^{t} A_0(s) S_0(s) \, ds} \quad (9.1)$$

where $A_i(t)$ is the radiative efficiency value of gas i (or the radiative forcing increase per unit mass increase of gas i in the atmosphere), $S_i(t)$ is the decay function (or the fraction of gas i remaining in the atmosphere after t years following an incremental pulse of the gas) and $i = 0$ is the reference gas (e.g, CO$_2$). The radiative forcing $RF_i(t) = A_i(t) S_i(t)$ is the product of the radiative efficiency and the decay function, whereas the absolute global warming potential $Agwp_i(t)$ is the cumulative radiative forcing of the gas i between 0 and t. GWP is then the ratio between the cumulative radiative forcing of the gas and this of the reference gas. We also notice that it depends on the time horizon t.

It is generally accepted to describe the decay function (or impulse response function) by exponential functions (Joos et al., 2013):

$$S_i(t) = \sum_{j=1}^{m} a_{i,j} e^{-\lambda_{i,j}t} \quad (9.2)$$

where $\sum_{j=1}^{m} a_{i,j} = 1$. Once we have defined the radiative efficiency function $A_i(t)$ and the set of parameters $\{(a_{i,j}, \lambda_{i,j}), j = 1, \dots, m\}$ of the impulse function, we compute Equation (9.1) using numerical integration. In the case where $A_i(t)$ and $A_0(t)$ are constant, we obtain:

$$gwp_i(t) = \frac{A_i \sum_{j=1}^{m} a_{i,j} \lambda_{i,j}^{-1} (1 - e^{-\lambda_{i,j}t})}{A_0 \sum_{j=1}^{m} a_{0,j} \lambda_{0,j}^{-1} (1 - e^{-\lambda_{0,j}t})}$$

In Box 9.1, we explain how GWP is computed. We notice that the global warming potential value depends on the time horizon. For instance, the relative warming impact of one molecule of a greenhouse gas is not the same at 20 years than 100 years. We also notice that the estimation of GWP implies to make some assumptions about the impulse response (or decay) function and the radiative efficiency value. Let us see how the value for methane has been obtained. IPCC (2013)
assumed that the radiative intensity is constant and used the following values<fnref n="2" />: $A_{CO_2} = 1.76 \times 10^{-18}$ and $A_{CH_4} = 2.11 \times 10^{-16}$. The impulse response functions were estimated by least squares and they found the following approximated curve:

$$S_{CO_2}(t) = 0.2173 + 0.2240 \cdot \exp\left(-\frac{t}{394.4}\right) + 0.2824 \cdot \exp\left(-\frac{t}{36.54}\right) + 0.2763 \cdot \exp\left(-\frac{t}{4.304}\right)$$

and:

$$S_{CH_4}(t) = \exp\left(-\frac{t}{12.4}\right)$$

These two decay functions are reported in Figure 9.1. We can interpret them as survival functions<fnref n="3" />, meaning that the density function can be computed as $f_i(t) = -\partial_t S_i(t)$.

<figure><img src="image_437.png" /><figcaption>Figure 9.1: Fraction of gas remaining in the atmosphere</figcaption></figure>

Source: Kleinberg (2020) & Author's calculations.

In the case of the exponential distribution $\mathcal{E}(\lambda)$, we have $S_i(t) = e^{-\lambda t}$ and $f_i(t) = \lambda e^{-\lambda t}$ where $\lambda$ is the rate parameter. Let $\tau_i$ be random time that the gas remains in the atmosphere. We have $\mathbb{E}[\tau_i] = 1/\lambda$ for the exponential random time. The survival function of the CH$_4$ gas is exponential with a mean time equal to 12.4 years ($\lambda = 1/12.4$). In the case of the general formula (9.2), the probability density function is equal to:

$$f_i(t) = -\partial_t S_i(t) = \sum_{j=1}^{m} a_{i,j} \lambda_{i,j} e^{-\lambda_{i,j} t}$$

<footnote n="2">²The unit is W · m⁻²g⁻¹.</footnote>

<footnote n="3">³This is why we use the notation $S(t)$.</footnote>
<figure><img src="image_438.png" /><figcaption>Figure 9.2: Probability density function of the random time</figcaption></figure>

Source: Kleinberg (2020) & Author's calculations.

and the mean time $\mathcal{T}_i$ is given by:

$$ \begin{aligned} \mathcal{T}_i &:= \mathbb{E}[\tau_i] = \int_0^\infty s f_i(s) \, ds \\ &= \sum_{j=1}^m a_{i,j} \int_0^\infty \lambda_{i,j} s e^{-\lambda_{i,j}s} \, ds \\ &= \sum_{j=1}^m \frac{a_{i,j}}{\lambda_{i,j}} \end{aligned} $$

Another way to find this result is to notice that $f_i(t)$ is an exponential mixture distribution where $m$ is the number of mixture components, $\mathcal{E}(\lambda_{i,j})$ is the probability distribution associated with the $j$th component and $a_{i,j}$ is the mixture weight of the $j$th component. Therefore, we deduce that the mean time is equal to the weighted average of the mean times of the mixture components:

$$ \mathcal{T}_i = \mathbb{E}[\tau_i] = \sum_{j=1}^{m} a_{i,j} \mathbb{E}[\tau_{i,j}] = \sum_{j=1}^{m} a_{i,j} \mathcal{T}_{i,j} $$

For the CO₂ gas, the exponential mixture distribution is defined by the following parameters:

<table><thead><tr><th>j</th><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tbody><tr><td>a<sub>i,j</sub></td><td>0.2173</td><td>0.2240</td><td>0.2824</td><td>0.2763</td></tr><tr><td>λ<sub>i,j</sub> (×10<sup>3</sup>)</td><td>0.00</td><td>2.535</td><td>27.367</td><td>232.342</td></tr><tr><td>T<sub>i,j</sub> (in years)</td><td>∞</td><td>394.4</td><td>36.54</td><td>4.304</td></tr></tbody></table>

We can now explain why the carbon dioxide stays longer in the atmosphere than the methane. When we compare the density functions of the two gases (Figure 9.2), we observe that the disappearance
probabilities are located before 50 years, since the probability to stay in the atmosphere after 50 years is less than 2%. For the CO₂ gas, we have roughly 50% that the molecule disappears and 50% that the molecule stays. In fact, we notice that one mixture component corresponds to a permanent state ($\lambda_{i,1} = 0$) with a weight of 21.73%. This explains that the CO₂ molecule can stay in the atmosphere, and we have $S_{CO_2}(\infty) = 21.73\%$.

We compute $Agwp_{CO_2}(t)$ and $Agwp_{CH_4}(t)$ and report their values in Figure 9.3. Even if the methane has a much shorter atmospheric lifetime than the carbon dioxide, it absorbs much more energy — because of the value of $A_{CH_4}$ compared to the value of $A_{CO_2}$. Nevertheless, the absolute global warming potential is unbounded for COtwo because we have $Agwp_{CO_2}(\infty) = \infty$. For the methane, it reaches an upper bound, which corresponds to the ratio $A_{CH_4} \times T_{CH_4} \propto 2.11 \times 12.4 = 26.164$. Therefore, $gwp_{CH_4}(t)$ is a decreasing function with respect to the time horizon (Figure 9.4). The instantaneous global warming potential of the methane is equal to:

$$gwp_{CH_4}(0) = \frac{A_{CH_4}}{A_{CO_2}} = \frac{2.11 \times 10^{-16}}{1.76 \times 10^{-18}} \approx 119.9$$

After 100 years, we obtain $gwp_{CH_4}(100) = 28.3853$, which is the value calculated by IPCC (2013, 2014a). Because of the persistant regime of the carbon dioxyde, we have $gwp_{CH_4}(\infty) = 0$. In fact, the global warming potential of CO₂ becomes greater than this of CH₄ when $t \ge 6382$ years.

The previous analysis shows that the estimation of the global warming potential involves significant scientific uncertainty. First, the choice of a 100-year time horizon is arbitrary, and any other choice will change the GWP value. Second, the survival function $S_i(t)$ is estimated and based on empirical experiments. Third, we have assumed that the radiative efficiency $A_i(t)$ is constant and equal to the initial value $A_i(0)$. In this context, we can consider that $gwp_i(t)$ is stochastic or cannot be observed without any error. The GHG protocol considers the six gases listed in the Kyoto Protocol: carbon dioxide, methane, nitrous oxide, sulphur hexafluoride, hydrofluorocarbons, and perfluorocarbons. Table 9.1 gives their GWP values according to the different IPCC reports. We notice that they have continuously changed.

<table><caption>Table 9.1: GWP values for 100-year time horizon</caption><thead><tr><th>Name</th><th>Formula</th><th>AR2</th><th>AR4</th><th>AR5</th><th>AR6</th></tr></thead><tbody><tr><td>Carbon dioxide</td><td>CO<sub>2</sub></td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Methane</td><td>CH<sub>4</sub></td><td>21</td><td>25</td><td>28</td><td>27.9</td></tr><tr><td>Nitrous oxide</td><td>N<sub>2</sub>O</td><td>310</td><td>298</td><td>265</td><td>273</td></tr><tr><td>Sulphur hexafluoride</td><td>SF<sub>6</sub></td><td>23 900</td><td>22 800</td><td>23 500</td><td>25 200</td></tr><tr><td rowspan="3">Hydrofluorocarbons<br>(HFC)</td><td>CHF<sub>3</sub></td><td>11 700</td><td>14 800</td><td>12 400</td><td>14 600</td></tr><tr><td>CH<sub>2</sub>F<sub>2</sub></td><td>650</td><td>675</td><td>677</td><td>771</td></tr><tr><td>Etc.</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">Perfluorocarbons<br>(PFC)</td><td>CF<sub>4</sub></td><td>6 500</td><td>7 390</td><td>6 630</td><td>7 380</td></tr><tr><td>C<sub>2</sub>F<sub>6</sub></td><td>9 200</td><td>12 200</td><td>11 100</td><td>12 400</td></tr><tr><td>Etc.</td><td></td><td></td><td></td><td></td></tr></tbody></table>

**Remark 97** The GWP has been subjected to many criticisms because it does not directly measure the impact on the temperature and it is a single-pulse emission metric (Kleinberg, 2020). The alternative metric is the global temperature potential (GTP) proposed by Shine et al. (2005). While the GWP is a measure of the energy absorbed over a given time period, the GTP is a measure of the temperature change at the end of that time period. The adoption of the GTP is however extremely rare, because its calculation is complicated and it appeared after the Kyoto Protocol.
<figure><img src="image_439.png" /><figcaption>Figure 9.3: Absolute global warming potential</figcaption></figure>

Source: Kleinberg (2020) & Author's calculations.

<figure><img src="image_440.png" /><figcaption>Figure 9.4: Global warming potential for methane</figcaption></figure>

Source: Kleinberg (2020) & Author's calculations.
### 9.1.2 Consolidation accounting at the company level

Greenhouse gas accounting requires to define rules about what is reported and what is not reported. For instance, we have seen that the GHG protocol does not consider all greenhouse gases. In a similar way, the consolidation of GHG emissions (also named organizational boundary) follows specific principles, which are similar to those we can find in financial consolidation accounting. For corporate reporting, the GHG protocol distinguishes two approaches:

1. Equity share approach

2. Control approach

Under the first approach, a parent company must report carbon emissions of a subsidiary company according to its share of equity (or ownership ratio). This is the simplest accounting method. For example, if company A owns 25% of company B, company A have to take into account 25% of the company B's GHG emissions. Under the second approach, it can use either the financial control method or the operational control method. The company financially controls an operation if it bears the majority risks and rewards of this operation<fnref n="4" />, whereas it has operational control if it has the full authority to implement the operation. The control approach is based on the all-or-none principle: Company A includes 100% of the company B's GHG emissions if A controls B, otherwise it includes 0%. In Table 9.2, we report the three accounting principles. By definition, the company has financial (and operational) control on group companies or subsidiaries. This explains that 100% of GHG emissions are consolidated. Associated and affiliated companies differ from the previous categories, because the company do not have the financial control. In this case, no GHG emissions are consolidated under the financial control method. Nevertheless, the company may have operational control, which explains that 100% of GHG emissions may be consolidated<fnref n="5" />. For joint ventures and partnerships, we apply the equity share principle for the financial control method, and we use the same rule than for associated and affiliated companies when we consider the financial control approach. Since the category fixed assets correspond to investments, the company receives dividends but has no control and GHG emissions are not consolidated. The same case applies to franchises because they are separate legal entities and franchisers have no equity rights or control. In the opposite situation, the franchise is considered as a subsidiary.

<table><caption>Table 9.2: Percent of reported GHG emissions under each consolidation method</caption><thead><tr><th rowspan="2">Accounting categories</th><th colspan="3">GHG accounting based on</th></tr><tr><th>equity share</th><th>financial control</th><th>operational control</th></tr></thead><tbody><tr><td>Wholly owned asset</td><td>100%</td><td>100%</td><td>100%</td></tr><tr><td>Group companies/subsidiaries</td><td>Ownership ratio</td><td>100%</td><td>100%</td></tr><tr><td>Associated/affiliated companies</td><td>Ownership ratio</td><td>0%</td><td>0%/100%</td></tr><tr><td>Joint ventures/partnerships</td><td>Ownership ratio</td><td>Ownership ratio</td><td>0%/100%</td></tr><tr><td>Fixed asset investments</td><td>0%</td><td>0%</td><td>0%</td></tr><tr><td>Franchises</td><td>0%</td><td>0%</td><td>0%</td></tr><tr><td></td><td>Ownership ratio</td><td>100%</td><td>100%</td></tr></tbody></table>

Source: GHG Protocol (2004, Table 1, page 19).

**Remark 98** An illustration of the differences between the equity share and control approaches is given in GHG Protocol (2004) on pages 22 and 23. This concerns the Holland Industries Group.

<footnote n="4">⁴For instance when the company has more than 50% voting rights.</footnote>

<footnote n="5">⁵If the company has no operational control, GHG emissions are not consolidated.</footnote>
**Example 34** We report the organizational structure of company A in Figure 9.5. This industrial group has several subsidiaries, partnerships and joint ventures. For instance, Companies B and C are integrated to company A, implying that the latter has financial and operational control on them. Company D is in the same situation even if company A has not the majority of capital. Indeed, company A treats company D as a subsidiary in its financial accounts, because the remaining capital is diluted and they control the management. The participation in company E is an investment. Company C has two joint ventures. For company F, it owns 75% of the capital, while company G is held in equal proportion by company A and another partner. Finally, company H is affiliated to company D, which has no financial or operational control.

<figure><img src="image_441.png" /><figcaption>Figure 9.5: Defining the organizational boundary of company A</figcaption></figure>

For each company, the brown number corresponds to the carbon emissions in tCO₂e. The three figures at the right or left of the node corresponds respectively to the equity share, the financial control and the operational control.

Computing the carbon emissions reported by company A will depend on the accounting method. In the case of the equity share approach, we exclude the investment in company H and obtain:

$$ \begin{aligned} \mathcal{CE}_A &= 827 + 100\% \times 135 + 90\% \times 261 + 45\% \times 220 + 0\% \times 1385 + \\ & 90\% \times 75\% \times 63 + 90\% \times 50\% \times 179 + 45\% \times 33\% \times 37 \\ &= 1424.4 \text{ tCO}_2\text{e} \end{aligned} $$

If we use the financial control approach, the reported carbon emissions become:

$$ \begin{aligned} \mathcal{CE}_A &= 827 + 100\% \times 135 + 100\% \times 261 + 100\% \times 220 + 0\% \times 1385 + \\ & 100\% \times 100\% \times 63 + 100\% \times 50\% \times 179 + 100\% \times 0\% \times 37 \\ &= 1595.50 \text{ tCO}_2\text{e} \end{aligned} $$

With the operational control approach, they are slightly different from above because of the treatment of company G:

$$ \begin{aligned} \mathcal{CE}_A &= 827 + 100\% \times 135 + 100\% \times 261 + 100\% \times 220 + 0\% \times 1385 + \\ & 100\% \times 100\% \times 63 + 100\% \times 0\% \times 179 + 100\% \times 0\% \times 37 \\ &= 1506.00 \text{ tCO}_2\text{e} \end{aligned} $$
Generally, an equity share of 50% or more induces a financial/operational control, which implies that 100% of carbon emissions are consolidated and reported by the parent company. By contrast, the consolidation factor may be equal to 0% if the parent company has no control, even in the case it has a significant equity share (e.g., between 30% and 40%).

### 9.1.3 Scope 1, 2 and 3 emissions

The GHG Protocol corporate standard classifies a company's greenhouse gas emissions in three scopes<fnref n="6" />:

* Scope 1 denotes direct GHG emissions occurring from sources that are owned and controlled by the issuer.

* Scope 2 corresponds to the indirect GHG emissions from the consumption of purchased electricity, heat or steam.

* Scope 3 are other indirect emissions (not included in scope 2) of the entire value chain. They can be divided into two main categories<fnref n="7" />:

- Upstream scope 3 emissions are defined as indirect carbon emissions related to purchased goods and services.

- Downstream scope 3 emissions are defined as indirect carbon emissions related to sold goods and services.

Scope 1 emissions are also called direct emissions, whereas indirect emissions encompass both scope 2 and 3 GHG emissions. Unlike scope 1 and 2, scope 3 is an optional reporting category.

**Remark 99** The GHG protocol defines “the operational boundary as the scope of direct and indirect emissions for operations that fall within a company's established organizational boundary. The operational boundary (scope 1, scope 2, scope 3) is decided at the corporate level after setting the organizational boundary. The selected operational boundary is then uniformly applied to identify and categorize direct and indirect emissions at each operational level. The established organizational and operational boundaries together constitute a company's inventory boundary.”

Before explaining in fine detail the different scopes and their computation, we report six examples of carbon footprint reporting in Table 9.3. We consider the CDP database, since most of companies reporting to CDP use the GHG protocol framework. The CDP reporting framework is based on a questionnaire (see Box 9.2 on page 880). We first notice all the figures are not calculated. This is normal since scope 3 is not mandatory in the GHG protocol framework. When the sub-category is empty, we don't know whether it is equal to zero or a missing value, meaning that the company has not the capacity or implemented the method to compute it. In the case of Amazon, the sub-category end-of-life treatment of sold products is equal to zero and not an empty case. A second remark concerns the definition of the scope 2, because there are two approaches: location-based and market-based. How to read this table? If we consider the first company Amazon, its scope 1 emissions are equal to 9.62 MtCO₂e. For the scope 2 emissions, they are equal to 9.02 MtCO₂e when they are calculated with the location-based method or 5.27 MtCO₂e when they are calculated

<footnote n="6">⁶The latest version of corporate accounting and reporting standard can be found at www.ghgprotocol.org/corporate-standard.</footnote>

<footnote n="7">⁷The upstream value chain includes all activities related to the suppliers whereas the downstream value chain refers to post-manufacturing activities.</footnote>
<table><caption>Table 9.3: Examples of CDP reporting (carbon emissions in tCO₂e, reporting year 2020)</caption><thead><tr><th>Scope</th><th>Category</th><th>Sub-category</th><th>Amazon</th><th>Danone</th><th>ENEL</th><th>Pfizer</th><th>Netflix</th><th>Walmart</th></tr></thead><tbody><tr><td>1</td><td></td><td></td><td>9 623 138</td><td>668 354</td><td>45 255 000</td><td>654 460</td><td>30 883</td><td>7 236 499</td></tr><tr><td rowspan="2">2</td><td rowspan="2">Location-based (2a)<br>Market-based (2b)</td><td>Purchased goods and services</td><td>16 683 423</td><td>19 920 918</td><td></td><td>2 526 537</td><td>765 208</td><td>130 200 000</td></tr><tr><td>Capital goods</td><td>13 202 065</td><td></td><td></td><td>191 894</td><td>116 366</td><td>645 328</td></tr><tr><td rowspan="7">3</td><td rowspan="7">Upstream</td><td>Fuel and energy related activities</td><td>1 248 847</td><td>283 764</td><td>1 061 268</td><td>203 093</td><td>12 287</td><td>3 327 874</td></tr><tr><td>Upstream transportation and distribution</td><td>8 563 695</td><td>321 558</td><td>112 358</td><td>723 558</td><td>64 693</td><td>342 577</td></tr><tr><td>Waste generated in operations</td><td>16 628</td><td>152 789</td><td></td><td>14 940</td><td></td><td>869 927</td></tr><tr><td>Business travel</td><td>313 043</td><td></td><td></td><td>35 128</td><td>41 439</td><td>37 439</td></tr><tr><td>Employee commuting</td><td>306 033</td><td></td><td></td><td>48 414</td><td>19 116</td><td>3 500 000</td></tr><tr><td>Upstream leased assets</td><td>1 223 903</td><td></td><td></td><td>30 522</td><td>131</td><td></td></tr><tr><td>Downstream transportation and distribution</td><td>2 785 676</td><td>1 627 090</td><td></td><td>7 295</td><td></td><td>5 099</td></tr><tr><td rowspan="5">Downstream</td><td rowspan="5">End-of-life treatment of sold products</td><td>Processing of sold products</td><td>1 426 543</td><td>1 885 548</td><td>46 524 860</td><td></td><td>952</td><td>32 211 000</td></tr><tr><td>Use of sold products</td><td>0</td><td>782 649</td><td></td><td></td><td></td><td>130</td></tr><tr><td>Downstream leased assets</td><td></td><td></td><td></td><td></td><td>349</td><td>130 000</td></tr><tr><td>Franchises</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Investments</td><td></td><td></td><td></td><td>36 839</td><td></td><td></td></tr><tr><td rowspan="10">Total</td><td rowspan="10">Scope 1 + 2a<br>Scope 1 + 2b<br>Scope 3 upstream<br>Scope 3 downstream<br>Scope 3<br>Scope 1 + 2a + 3<br>Scope 1 + 2b + 3</td><td></td><td>18 642 924</td><td>1 533 064</td><td>50 245 685</td><td>1 206 037</td><td>59 468</td><td>18 268 299</td></tr><tr><td></td><td>14 888 227</td><td>1 147 564</td><td>53 110 954</td><td>1 196 981</td><td>31 024</td><td>16 426 836</td></tr><tr><td></td><td>41 557 637</td><td>20 679 029</td><td>1 176 787</td><td>3 774 086</td><td>1 019 240</td><td>138 923 145</td></tr><tr><td></td><td>4 212 219</td><td>4 295 287</td><td>46 524 860</td><td>44 134</td><td>1 301</td><td>32 346 229</td></tr><tr><td></td><td>45 769 856</td><td>24 974 316</td><td>47 701 647</td><td>3 818 220</td><td>1 020 541</td><td>171 269 374</td></tr><tr><td></td><td>64 412 780</td><td>26 507 380</td><td>97 947 332</td><td>5 024 257</td><td>1 080 009</td><td>189 537 673</td></tr><tr><td></td><td>60 658 083</td><td>26 121 880</td><td>100 812 601</td><td>5 015 201</td><td>1 051 565</td><td>187 696 210</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

Source: CDP database as of 01/07/2022 & Author's computation.
with the market-based method. The fifteen sub-categories of scope 3 emissions are not aggregated, and each item is filled separately. The last of part of the table (from Scope 1 + 2a to Scope 1 + 2a + 3 is not included in the CDP questionnaire. These figures are calculated by summing the different items.

Box 9.2: CDP questionnaire for corporates

In order to report to CDP, companies must fill the CDP questionnaire on climate change, which is available at www.cdp.net/en/guidance/guidance-for-companies in HTML, Word and PDF formats. The full questionnaire has 129 pages and 16 sections. Emissions methodology corresponds to section C5, while emissions data are reported in section C6 with the following breakdown: scope 1 emissions (§C6.1), scope 2 emissions (§C6.3), scope 3 emissions (§C6.5) and emissions intensities (§C6.10).

<figure><img src="image_442.png" /></figure>

Scope 1 emissions

According to GHG Protocol (2004, page 40)GHG-Protocol, once the inventory boundary has been established, “companies generally calculate GHG emissions using the following steps: (1) Identify GHG emissions sources; (2) Select a GHG emissions calculation approach; (3) Collect activity data and choose emission factors; (4) Apply calculation tools and (5) Roll-up GHG emissions data to corporate level. The identification step helps to categorize the GHG emission sources.” The identification step consists in categorizing the GHG emissions in the four main source categories:

1. Stationary combustion: combustion of fuels in stationary equipment (e.g., boilers, turbines, heaters, incinerators);

2. Mobile combustion: combustion of fuels in transportation devices (e.g., automobiles, trucks, trains, airplanes, boats);

3. Process emissions: emissions from physical or chemical processes (e.g., cement manufacturing, petrochemical processing, aluminum smelting);
4. Fugitive emissions: intentional and unintentional releases as well as fugitive emissions (e.g., such as equipment leaks, coal piles, wastewater treatment, cooling towers).

Concretely, the company lists all the activities that result in a GHG emission, and allocates them to the three scopes. Then, we apply an emission factor to each activity and each gas:

$$E_{g,h} = A_h \cdot \mathcal{E}\mathcal{F}_{g,h}$$

where $A_h$ is the $h^{th}$ activity rate (also called activity data) and $\mathcal{E}\mathcal{F}_{g,h}$ is the emission factor for the $h^{th}$ activity and the $g^{th}$ gas. $A_h$ can be measured in volume, weight, distance, duration, surface, frequency, etc. Since $E_{g,h}$ is expressed in tonne, $\mathcal{E}\mathcal{F}_{g,h}$ is measured in tonne per activity unit. For instance, if $A_h$ is measured in hectare, $EF_{g,h}$ is measured in tonne per hectare. In fact, the emission factor is a coefficient that attempts to quantify how much of a greenhouse gas is released into the atmosphere by an activity that releases that gas$^8$. For each gas, we calculate the total emissions:

$$E_g = \sum_{h=1}^{n_A} E_{g,h} = \sum_{h=1}^{n_A} A_h \cdot \mathcal{E}\mathcal{F}_{g,h}$$

where $n_A$ is the number of activities. Finally, we estimate the carbon emissions by applying the right GWP and summing up all the gases:

$$\mathcal{C}\mathcal{E} = \sum_{g=1}^{n_G} \text{gwp}_g \cdot E_g$$

where $n_G$ is the number of gases$^9$. Therefore, the compact formula is:

$$\mathcal{C}\mathcal{E} = \sum_{g=1}^{n_G} \text{gwp}_g \cdot \left( \sum_{h=1}^{n_A} A_h \cdot \mathcal{E}\mathcal{F}_{g,h} \right)$$

The carbon footprint of the company can be split into activities:

$$\mathcal{C}\mathcal{E} = \sum_{h=1}^{n_A} A_h \left( \sum_{g=1}^{n_G} \text{gwp}_g \cdot \mathcal{E}\mathcal{F}_{g,h} \right) = \sum_{h=1}^{n_A} A_h \cdot \mathcal{E}\mathcal{F}_h = \sum_{h=1}^{n_A} \mathcal{C}\mathcal{E}_h$$

where $\mathcal{E}\mathcal{F}_h$ and $\mathcal{C}\mathcal{E}_h$ are the global emission factor and carbon emissions related to the $h^{th}$ activity. We can also aggregate several activities:

$$\mathcal{C}\mathcal{E}_A = \sum_{h \in A} \mathcal{C}\mathcal{E}_h = \sum_{h \in A} A_h \cdot \mathcal{E}\mathcal{F}_h$$

where $A$ is the set of activities. It may happen that some emission factors are defined without a reference to a specific gas (e.g., CO$_2$ or CH$_4$). In this case, the emission factor is a synthetic measure which already take into account the GWP of the gases:

$$\mathcal{E}\mathcal{F}_h = \sum_{g=1}^{n_G} \text{gwp}_g \cdot \mathcal{E}\mathcal{F}_{g,h}$$

<footnote>$^8$For example, how many kg of GHG are emitted by 1 kWh of natural gas?</footnote>

<footnote>$^9$n$_G$ is equal to six in the GHG Protocol.</footnote>
The expression of the carbon footprint becomes:

$$ \mathcal{CE} = \sum_{h \in A_1} A_h \left( \sum_{g=1}^{n_G} \text{gwp}_g \cdot \mathcal{EF}_{g,h} \right) + \sum_{h \in A_2} A_h \cdot \mathcal{EF}_h $$

where $A_1$ and $A_2$ are the sets of activities without and with synthetic emission factors.

The choice of data inputs is codified by IPCC (2006, 2019):

* Tier 1 methods use global default emission factors;

* Tier 2 methods use country-level or region-specific emission factors;

* Tier 3 methods use directly monitored or site-specific emission factors.

We can find emission factors in several sources: IPCC Emission Factor Database (Box 9.3), National Inventory Reports<fnref n="10" /> (NIRs), country emission factor databases<fnref n="11" />, international agencies or academic publications. In the US, the emission factors are calculated by the Environmental Protection Agency (US EPA). In the UK, this is the National Atmospheric Emissions Inventory (NAEI) agency, which is in charge to define the emission factors. In France, the database is managed by ADEME (Agence de l'Environnement et de la Maîtrise de l'Energie) and contains about 5 300 validated emission factors. Generally, we can download these emission factors in an Excel or PDF file.

Let us see an example. We consider the GHG inventory document<fnref n="12" /> published by Enel. The scope 1 is based on the following activities: (1) combustion of fossil fuels in electricity generation activities; (2) combustion of fossil fuels in generators used for electricity generation and distribution activities; (3) combustion of fossil fuels in vehicles under the Company's control; (4) combustion of fuels for heating offices and canteens; (5) CH₄ leakage in gas-fired thermoelectric power plants; (6) SF₆ losses in electricity generation and distribution activities; (7) HFCs gas losses from cooling systems; (8) NF₃ losses from the production of solar panels; (9) Transportation of fuel (LNG and coal) on vessels under own operational control and (10) CH₄ emissions from the decomposition of organic matter in hydroelectric basins. For the calculations, they use the parameter values of IPCC (2006) for emission factors and the GWP figures of IPCC (2014a). They obtained the following results in 2021 expressed in ktCO₂e:

<table><thead><tr><th></th><th>CO<sub>2</sub></th><th>CH<sub>4</sub></th><th>N<sub>2</sub>O</th><th>NF<sub>3</sub></th><th>SF<sub>6</sub></th><th>HFCs</th><th>Total</th></tr></thead><tbody><tr><td>Electricity power generation</td><td>50 643.54</td><td>385.25</td><td>98.14</td><td>0.014</td><td>31.15</td><td>10.22</td><td>51 168.32</td></tr><tr><td>Electricity distribution</td><td>208.33</td><td>0.24</td><td>0.45</td><td></td><td>111.62</td><td></td><td>320.64</td></tr><tr><td>Real estate</td><td>79.87</td><td>0.22</td><td>1.24</td><td></td><td></td><td></td><td>81.30</td></tr><tr><td>Total</td><td>50 931.72</td><td>385.71</td><td>99.83</td><td>0.014</td><td>142.77</td><td>10.22</td><td>51 750.26</td></tr></tbody></table>

The scope 1 emissions of Enel is then equal to 51.75 MtCO₂e. The contribution of CO₂ is the most important since it represents 98.4% of the total emissions, implying that the other gases have a small impact. In terms of activities, GHG emissions are mainly located in the electricity power generation. Buildings has a contribution of 0.2%.

<footnote n="10">¹⁰The NIR reports can be found at the UNFCCC website: https://unfccc.int/ghg-inventories-annex-i-parties/2021.</footnote>

<footnote n="11">¹¹Here are some websites: www.epa.gov/climateleadership/ghg-emission-factors-hub (US), https://naei.beis.gov.uk/data/ef-all (UK), https://bilans-ges.ademe.fr (France), www.dcceew.gov.au/climate-change/publications/national-greenhouse-accounts-factors-2021 (Australia), www.isprambiente.gov.it (Italy), https://publications.gc.ca/site/eng/9.911206/publication.html (Canada).</footnote>

<footnote n="12">¹²Enel (2022). Quantification and Reporting of Greenhouse Gas Emissions in Accordance with the Corporate GHG Protocol. 12th April 2022, www.enel.com/investors/sustainability.</footnote>
Box 9.3: IPCC emission factor database (EFDB)

The IPCC emission factor database<sup>*a*</sup> (EFDB) is a database on various parameters to be used in calculation of anthropogenic emissions by sources and removals by sinks of greenhouse gases. It contains the IPCC default data<sup>*b*</sup>, and data from peer-reviewed journals and other publications including national inventory reports (NIRs). The database includes emission factors for five categories:

1. Energy (fuel combustion activities, fugitive emissions from fuels, carbon dioxide transport and storage);

2. Industrial processes and product use (mineral industry, chemical industry, metal industry, non-energy products from fuels and solvent use, electronics industry, product uses as substitutes for ozone depleting substances, other product manufacture and use, other);

3. Agriculture, forestry, and other land use (livestock, land, aggregate sources and non-CO2 emissions sources on land, other);

4. Waste (solid waste disposal, biological treatment of solid waste, incineration and open burning of waste, wastewater treatment and discharge, other)

5. Other (Indirect N₂O emissions from the atmospheric deposition of nitrogen in NOx and NH₃, other)

Some figures of the library are given in Table 9.4. We notice that they may depend on the region and technical criteria. The unit is also important. For instance, if the emission factors are expressed in tCarbon/TeraJoule (tonne of carbon per terajoule energy), we multiply the emission factor *EF* (in tC/TJ) by the energy consumption *C* (in TJ) to obtain the carbon content (in tonnes of carbon). Since one tonne of CO₂ contains 0.2727 tonne of carbon, we then deduce that the CO₂ emissions are equal to $\frac{C \cdot EF}{0.2727}$ tCO₂e. In this case, the activity data corresponds to the energy consumption.

<footnote>*a*The website is www.ipcc-nggip.iges.or.jp/EFDB.</footnote>

<footnote><sup>*b*</sup>Revised 1996 IPCC Guidelines, IPCC Good Practice Guidance and Uncertainty Management in National Greenhouse Gas Inventories, IPCC Good Practice Guidance for Land Use, Land-Use Change and Forestry, 2006 IPCC Guidelines for National Greenhouse Gas Inventories and 2013 Supplement to the 2006 IPCC Guidelines for National Greenhouse Gas Inventories: Wetlands.</footnote>
<table><caption>Table 9.4: Examples of emission factors (EFDB, IPCC)</caption><thead><tr><th>Category</th><th>Description</th><th>Gas</th><th>Region</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td rowspan="2">Iron and steel production</td><td>Integrated facility</td><td>CO<sub>2</sub></td><td>Canada</td><td>1.6</td><td>t/tonne</td></tr><tr><td>Electrode consumption from steel produced in electric arc furnaces</td><td>CO<sub>2</sub></td><td>Global</td><td>5.0</td><td>kg/tonne</td></tr><tr><td rowspan="2">Manufacture of solid fuels</td><td>Steel processing (rolling mills)</td><td>N<sub>2</sub>O</td><td>Global</td><td>40</td><td>g/tonne</td></tr><tr><td>Metallurgical coke production</td><td>CO<sub>2</sub></td><td>Global</td><td>0.56</td><td>t/tonne</td></tr><tr><td rowspan="3">Fuel combustion activities</td><td>Crude oil</td><td>CH<sub>4</sub></td><td>Global</td><td>0.1</td><td>g/tonne</td></tr><tr><td>Natural gas</td><td>CO<sub>2</sub></td><td>Global</td><td>20</td><td>t/Carbon/TeraJoule</td></tr><tr><td>Ethane</td><td>CO<sub>2</sub></td><td>Global</td><td>15.3</td><td>t/Carbon/TeraJoule</td></tr><tr><td rowspan="2">Integrated circuit or semiconductor</td><td>Semiconductor manufacturing (silicon)</td><td>CF<sub>4</sub></td><td>Global</td><td>16.8</td><td>t/Carbon/TeraJoule</td></tr><tr><td></td><td></td><td></td><td>0.9</td><td>kg/m<sup>2</sup></td></tr><tr><td rowspan="2">Cement production</td><td>Cement production</td><td>CO<sub>2</sub></td><td>Global</td><td>0.4985</td><td>t/tonne</td></tr><tr><td>Enteric fermentation</td><td>CH<sub>4</sub></td><td>Global</td><td>18</td><td>kg/head/year</td></tr><tr><td rowspan="2">Horses</td><td>Manure management (annual average temperature is less than 15oC)</td><td>CH<sub>4</sub></td><td>Developed countries</td><td>1.4</td><td>kg/head/year</td></tr><tr><td>Manure management (annual average temperature is between 15oC and 25oC)</td><td>CH<sub>4</sub></td><td>Developed countries</td><td>2.1</td><td>kg/head/year</td></tr><tr><td rowspan="2">Buffalo</td><td>Enteric fermentation</td><td>CH<sub>4</sub></td><td>Global</td><td>55</td><td>kg/head/year</td></tr><tr><td>Manure management (annual average temperature is less than 15oC)</td><td>CH<sub>4</sub></td><td>Developed countries</td><td>0.078</td><td>kg/head/year</td></tr><tr><td rowspan="4">Poultry</td><td>Manure management (annual average temperature is between 15oC and 25oC)</td><td>CH<sub>4</sub></td><td>Developed countries</td><td>0.117</td><td>kg/head/year</td></tr><tr><td>Manure management (annual average temperature is greater than 25oC)</td><td>CH<sub>4</sub></td><td>Developed countries</td><td>0.157</td><td>kg/head/year</td></tr><tr><td>Manure management (annual average temperature is greater than 25oC)</td><td>CH<sub>4</sub></td><td>Developing countries</td><td>0.023</td><td>kg/head/year</td></tr></tbody></table>

Source: EFDB, www.ipcc-nggip.iges.or.jp/EFDB.
## Scope 2 emissions

Scope 2 is “an indirect emission category that includes GHG emissions from the purchased or acquired electricity, steam, heat, or cooling consumed” (GHG Protocol, 2015, page 34). There are then four forms of energy that are tracked in scope 2:

* **Electricity**

People use electricity for operating machines, lighting, heating, cooling, electric vehicle charging, computers, electronics, public transportation systems, etc.

* **Steam**

Industries use steam for mechanical work, heating, propulsion, driven turbines in electric power plants, etc.

* **Heat**

Buildings use heat to control inside temperature and heat water, while the industrial sector uses heat for washing, cooking, sterilizing, drying, etc. Heat may be produced from electricity, solar heat processes or thermal combustion.

* **Cooling**

It is produced from electricity or through the processes of forced air, conduction, convection, etc.

Scope 2 includes indirect emissions from generation only. For instance, the distribution of energy within a grid is tracked in scope 3. In Figures 9.6–9.9, we report the different cases that are illustrated in the GHG Protocol: if the consumed electricity comes from owned/operated equipment, no scope 2 emissions are reported (Figure 9.6); if the consumed electricity comes from a direct line transfer or the grid<fnref n="13" />, the consumer of the energy reports the emissions in scope 2 (Figures 9.7 and (Figure 9.8)); if some consumed electricity comes from the owned/operated equipment, and some is purchased from the grid, the operator (company A) has both scope 1 emissions from energy generation, and scope 3 emissions from energy purchased on the grid (Figure 9.9).

<figure><img src="image_443.png" /><figcaption>Figure 9.7: Direct line energy transfer</figcaption></figure>

<figure><img src="image_444.png" /></figure>

Source: GHG Protocol (2015, Figures 5.1 and 5.2, pages 35-36).

<footnote n="13">¹³A grid is “a system of power transmission and distribution lines under the control of a coordinating entity or grid operator, which transfers electrical energy generated by power plants to energy users — also called a power grid.” (CDP, 2022, page 8).</footnote>
<figure><img src="image_445.png" /><figcaption>Figure 9.8: Electricity production on a grid</figcaption></figure>

Source: GHG Protocol (2015, Figure 5.4, page 38).

<figure><img src="image_446.png" /><figcaption>Figure 9.9: Facility consuming both energy generated on-site and purchased from the grid</figcaption></figure>

Source: GHG Protocol (2015, Figure 5.3, page 37).
Scope 2 emissions are calculated using activity data and emission factors<fnref n="14" /> expressed in MWh and tCO₂e/MWh:

$$CE = \sum_s A_s \cdot EF_s$$

where $A_s$ is the amount of purchased electricity for the energy generation source s and $EF_s$ is the emission factor of the source s. The source can be an electricity supplier<fnref n="15" />, a specific or country grid, a specific power station, etc.

**Remark 100** A Megawatt-hour is the common billing unit for electrical energy delivered to consumers. 1000 MWh is equivalent to 3.6 TeraJoule (TJ). The TJ unit is used by IPCC (2006) (see Table 9.4 on page 884). A third energy unit is also used to defined emissions factors in North America (Canada and the US) and the United Kingdom: the British thermal unit or Btu (1 Btu is equivalent to 1.0551 KJ or 0.2931 Wh).

**Example 35** We consider a company, whose electricity consumption is equal to 2000 MWh per year. The electricity comes from two sources: 60% from a direct line with an electricity supplier (source $S_1$) and 40% from the country grid (source $S_2$). The emission factors are respectively equal to 200 and 350 gCO₂e/kWh.

The electricity consumption from source $S_1$ is equal to $60\% \times 2000 = 1200$ MWh or 1200 000 kWh. We deduce that the carbon emissions from this source is:

$$CE(S_1) = (1.2 \times 10^6) \times 200 = 240 \times 10^6 \text{ gCO}_2\text{e} = 240 \text{ tCO}_2\text{e}$$

For the second source, we obtain:

$$CE(S_2) = (0.8 \times 10^6) \times 350 = 280 \times 10^6 \text{ gCO}_2\text{e} = 280 \text{ tCO}_2\text{e}$$

We deduce that the scope 2 carbon emissions of this company is equal to 520 tCO₂e.

Let us consider again the GHG inventory report of Enel (page 882). We remind that the scope 1 emissions of Enel is equal to 51 750 265 tCO₂e. In the same document, we learn that the ratio between scope 1 emissions and the total electricity production is equal to 227 gCO₂e/kWh (or 0.227 tCO₂e/MWh). We deduce that the 2021 electricity production of Enel is <fnref n="16" />:

$$A = \frac{51,750,265}{0.227} = 227,974,735 \text{ MWh} = 228 \text{ TWh}$$

Two main methods are available for accounting scope 2 emissions:

* Location-based method

In this approach, the company uses the average emission factor of the region or the country.

For instance, if the electricity consumption is located in France, the company can use the emission intensity of the French energy mix;

$$A = \frac{CE}{EF}$$

where CE is the scope 1 emissions of the utility company.

<footnote n="14">¹⁴This approach is also known as the emission rate approach.</footnote>

<footnote n="15">¹⁵The largest electricity companies are EDF, Enel, Engie, E.ON, Fortum, Marubeni, Siemens, State Grid Corporation of China, Tokyo Electric Power, and Uniper.</footnote>

<footnote n="16">¹⁶In this example, we inverse the equation in order to estimate the activity data of an electricity supplier:</footnote>
* Market-based method
This approach reflects the GHG emissions from the electricity that the company has chosen in the market. This means that the scope 2 carbon emissions will depend on the scope 1 carbon intensity of the electricity supplier.

Under the market-based method, an emission factor is associated to each electricity contract. To be relevant, contacts must meet some quality criteria and concern some specific instruments: energy attribute certificates <fnref n="17" /> (e.g., RECs, GOs), power purchase agreements with energy generators and green electricity products.

<figure><img src="image_447.png" /><figcaption>Figure 9.10: Emission factor in gCO₂e/kWh of electricity generation (European Union, 1990–1992)</figcaption></figure>

Source: European Environment Agency (2022), www.eea.europa.eu/data-and-maps & Author's calculations.

The location-based method depends on the emission factor of the regional, subnational or national grid. Its value highly depends on the energy mix and the grid infrastructure. For instance, we report the evolution of national emission factors of some European countries in Figure 9.10. We notice that they tend to decrease since thirty years and they differ from one country to another. On average, the emission factor is equal to 275 gCO₂e/kWh in the European Union in 2021. The two extreme countries are Sweden (9 gCO₂e/kWh) and Poland (750 gCO₂e/kWh). The reason is that most of Sweden's electricity supply comes from hydropower and nuclear, while Poland produces 83% of its electricity from fossil fuels (and 72% from coal). Emission factors for several region and countries in the world are given in Table 9.5. We notice the high heterogeneity of the figures. The continent with the lowest value is South America (204 gCO₂e/kWh) while Asia has the largest emission factor (539 gCO₂e/kWh). Two countries which are geographically close may have different

<footnote n="17">¹⁷REC (or renewable energy certificate) is an energy attribute certificate used in Australia, Canada, India and the US, while GO (guarantee of origin) is an energy attribute certificate used in Europe.</footnote>
emission factors. This is the case of France and Germany (58 vs. 354 gCO₂e/kWh), Canada and the US (128 vs. 380 gCO₂e/kWh), etc.

<table><caption>Table 9.5: Emission factor in gCO₂e/kWh of electricity generation in the world</caption><thead><tr><th>Region</th><th>EF</th><th>Country</th><th>EF</th><th>Country</th><th>EF</th><th>Country</th><th>EF</th></tr></thead><tbody><tr><td>Africa</td><td>484</td><td>Australia</td><td>531</td><td>Germany</td><td>354</td><td>Portugal</td><td>183</td></tr><tr><td>Asia</td><td>539</td><td>Canada</td><td>128</td><td>India</td><td>637</td><td>Russia</td><td>360</td></tr><tr><td>Europe</td><td>280</td><td>China</td><td>544</td><td>Iran</td><td>492</td><td>Spain</td><td>169</td></tr><tr><td>North America</td><td>352</td><td>Costa Rica</td><td>33</td><td>Italy</td><td>226</td><td>Switzerland</td><td>47</td></tr><tr><td>South America</td><td>204</td><td>Cuba</td><td>575</td><td>Japan</td><td>479</td><td>United Kingdom</td><td>270</td></tr><tr><td>World</td><td>442</td><td>France</td><td>58</td><td>Norway</td><td>26</td><td>United States</td><td>380</td></tr></tbody></table>

Source: https://ourworldindata.org/grapher/carbon-intensity-electricity

**Example 36** We consider a French bank, whose activities are mainly located in France and the Western Europe. Below, we report the energy consumption (in MWh) by country:

<table><tr><td>Belgium</td><td>125 807</td><td>France</td><td>1 132 261</td><td>Germany</td><td>71 890</td><td>Ireland</td><td>125 807</td></tr><tr><td>Italy</td><td>197 696</td><td>Luxembourg</td><td>33 069</td><td>Netherlands</td><td>18 152</td><td>Portugal</td><td>12 581</td></tr><tr><td>Spain</td><td>61 106</td><td>Switzerland</td><td>73 148</td><td>UK</td><td>124 010</td><td>World</td><td>37 742</td></tr></table>

If we consider a Tier 1 approach, we can estimate scope 2 emissions of the bank by computing the total activity data and multiplying by the global emission factor. Since we have twelve sources, we obtain:

$$A = \sum_{s=1}^{12} A_s = 125\,807 + 1\,132\,261 + \dots + 37\,742 = 2\,013\,269 \text{ MWh}$$

and:

$$ \begin{aligned} \mathcal{CE} &= A \cdot \mathcal{EF}_{\text{World}} \\ &= (2\,013\,269 \times 10^3) \times 442 \\ &= 889\,864\,898\,000 \text{ gCO}_2\text{e} \\ &= 889.86 \text{ ktCO}_2\text{e} \end{aligned} $$

Another Tier 1 approach is to consider the emission factor of the European Union, because the rest of the world represents less than 2% of the electricity consumption. Using $\mathcal{EF}_{EU} = 275$, we obtain $\mathcal{CE} = 553.65 \text{ ktCO}_2\text{e}$. The third approach uses a Tier 2 method by considering the emission factor of each country. In this case, we have to collect the data. We use figures in Table 9.5 and the following emission factors: Belgium (143); Ireland (402); Luxembourg (68) and Netherlands (331). It follows that:

$$ \begin{aligned} \mathcal{CE} &= \sum_{s=1}^{12} A_s \cdot \mathcal{EF}_s \\ &= (125\,807 \times 143 + 1\,132\,261 \times 58 + \dots + 124\,010 \times 270 + 37\,742 \times 442) \times \frac{10^3}{10^9} \\ &= 278.85 \text{ ktCO}_2\text{e} \end{aligned} $$

We notice that the estimated scope 2 emissions of this bank are sensitive to the chosen approach.
The market-based accounting approach requires to track the electricity of each supplier. For that, the company can use reliable tracking systems (North American REC, European Energy Certificate System GO, International REC standard, TIGR registry) and supplier-based contactual instruments. This Tier 1 approach is based on contract-specific emission factors. Nevertheless, when they are not available, the company can use supplier-based average and residual mix emission factors.

**Example 37** We consider a Norwegian company, whose current electricity consumption is equal to 1351 Mwh. 60% of the electricity comes from the Norwegian hydroelectricity and the GO system guarantees that this green electricity emits 1 gCO₂e/kWh.

If we assume that the remaining 40% of the electricity consumption comes from the Norwegian grid<fnref n="18" />, the market-based scope 2 emissions of this company are equal to:

$$ \begin{aligned} CE &= \frac{10^6 \times 60\% \times 1 + 10^6 \times 40\% \times 26}{10^6} \\ &= 11 \text{ ktCO}_2\text{e} \end{aligned} $$

The market-based approach may reduce the scope 2 emissions when the company purchases green electricity. For instance, the emission factors in France are the following: 6 for nuclear, 418 for natural gas, 730 for fuel oil and 1058 for coal. In Table 9.6, we have reported the life cycle emission factors for several technologies. Even if these figures depend on many parameters (vintage, country, etc.) and the ranges are relatively wide, we clearly observe an ordering. Wind, nuclear, hydro and solar electricity generates less GHG emissions than gas, fuel oil and coal.

<table><caption>Table 9.6: Emission factor in gCO₂e/KWh from electricity supply technologies (IPCC, 2014a; UN-ECE, 2022)</caption><thead><tr><th rowspan="2">Technology</th><th rowspan="2">Characteristic</th><th colspan="2">IPCC</th><th colspan="2">UNECE</th></tr><tr><th>Mean</th><th>Min-Max</th><th>Mean</th><th>Min-Max</th></tr></thead><tbody><tr><td rowspan="2">Wind</td><td>Onshore</td><td>11</td><td>7-56</td><td>12</td><td>8-16</td></tr><tr><td>Offshore</td><td>12</td><td>8-35</td><td>13</td><td>13-23</td></tr><tr><td>Nuclear</td><td></td><td>12</td><td>3-110</td><td>6</td><td></td></tr><tr><td>Hydro power</td><td></td><td>24</td><td>1-2200</td><td>11</td><td>6-147</td></tr><tr><td rowspan="3">Solar power</td><td>CSP</td><td>27</td><td>9-63</td><td>32</td><td>14-122</td></tr><tr><td>Rooftop (PV)</td><td>41</td><td>26-60</td><td>22</td><td>9-83</td></tr><tr><td>Utility/Ground (PV)</td><td>48</td><td>18-180</td><td>20</td><td>8-82</td></tr><tr><td>Geothermal</td><td></td><td>38</td><td>6-79</td><td></td><td></td></tr><tr><td>Biomass</td><td>Dedicated</td><td>230</td><td>130-420</td><td></td><td></td></tr><tr><td rowspan="2">Gas</td><td>CCUS</td><td>169</td><td>90-370</td><td>130</td><td>92-221</td></tr><tr><td>Combined cycle</td><td>490</td><td>410-650</td><td>430</td><td>403-513</td></tr><tr><td>Fuel oil</td><td></td><td></td><td>510-1170</td><td></td><td></td></tr><tr><td rowspan="2">Coal</td><td>CCUS</td><td>161</td><td>70-290</td><td>350</td><td>190-470</td></tr><tr><td>PC</td><td>820</td><td>740-650</td><td>1000</td><td>912-1095</td></tr></tbody></table>

CSP: concentrated solar power; PV: photovoltaic power; CCUS: carbon capture, use, and storage; PC: pulverized coal.

<footnote n="18">¹⁸The emission factor for Norway is 26 gCO₂e/kWh.</footnote>
Let us consider again the example of Enel (page 882). They obtained the following figures (expressed in ktCO₂e):

<table><thead><tr><th></th><th>Electricity purchased<br>from the grid</th><th>Losses on the<br>distribution grid</th><th>Total</th></tr></thead><tbody><tr><td>Location-based</td><td>1 336.67</td><td>2 966.52</td><td>4 303.18</td></tr><tr><td>Market-based</td><td>2 351.00</td><td>4 763.15</td><td>7 114.15</td></tr></tbody></table>

The first category derives from the generation of electricity purchased and consumed by Enel (electricity consumption taken from the network for civil use or for energy generation in thermoelectric and hydroelectric plants). The second category includes indirect emissions due to dissipated energy emissions from technical losses from Enel's distribution network and from the transmission system. We notice that this second category represented 67% and 69% of scope 2 emissions. Curiously, the market-based figure is greater than the location-based approach: 7.11 vs. 4.30 MtCO₂e.

We consider the CDP database and compare the location-based and market-based values for the year 2020. Statistics are reported in Table 9.7. Less than 1% of issuers have declared zero scope 2 carbon emissions with the location-based approach. This figure becomes 8.78% when we consider the market-based approach. 70% of issuers have greater location-based emissions than market-based emissions. About 10% have the same value, meaning that these issuers have certainly used the mix residual approach to compute the scope 2 emissions with the market-based approach. The mean variation ratio<fnref n="19" /> is equal to +26.59%. This result is explained by the frequency asymmetry, but also by the fact that the variation is higher for issuers that have greater location-based emissions than market-based emissions (+43.29% vs. -22.04%).

<table><caption>Table 9.7: Statistics of CDP scope 2 emissions (2020)</caption><thead><tr><th></th><th>Cε<sub>loc</sub> = 0</th><th>Cε<sub>loc</sub> = Cε<sub>mkt</sub> = 0</th><th>Cε<sub>mkt</sub> = 0</th></tr></thead><tbody><tr><td>Frequency</td><td>0.89%</td><td>0.39%</td><td>8.78%</td></tr><tr><td></td><td>Cε<sub>loc</sub> > Cε<sub>mkt</sub></td><td>Cε<sub>loc</sub> = Cε<sub>mkt</sub></td><td>Cε<sub>loc</sub> < Cε<sub>mkt</sub></td></tr><tr><td>Frequency</td><td>70.43%</td><td>9.48%</td><td>20.09%</td></tr><tr><td>Mean variation ratio</td><td>+43.89%</td><td>0.00%</td><td>-22.04%</td></tr></tbody></table>

Source: CDP database as of 01/07/2022 & Author's computation.

## Scope 3 emissions

Scope 3 emissions are all the indirect emissions in the company's value chain, apart from indirect emissions which are reported in scope 2. They are divided into fifteen categories of emissions: eight upstream categories and seven downstream categories (Table 9.8). We report below their description as it appears in GHG Protocol (2011, Table 5.4, pages 34-37):

1. Purchased goods and services (not included in categories 2-8)

Extraction, production, and transportation of goods and services purchased or acquired by the company;

2. Capital goods

Extraction, production, and transportation of capital goods purchased or acquired by the company;

<footnote n="19">¹⁹The variation ratio is equal to $\frac{CE_{loc} - CE_{mkt}}{\max(CE_{loc}, CE_{mkt})}$.</footnote>
<table><caption>Table 9.8: The scope 3 carbon emissions categories</caption><thead><tr><th>Upstream</th><th>Downstream</th></tr></thead><tbody><tr><td><ol><li>Purchased goods and services</li><li>Capital goods</li><li>Fuel and energy related activities</li><li>Upstream transportation and distribution</li><li>Waste generated in operations</li><li>Business travel</li><li>Employee commuting</li><li>Upstream leased assets</li><li>Other upstream</li></ol></td><td><ol><li>Downstream transportation and distribution</li><li>Processing of sold products</li><li>Use of sold products</li><li>End-of-life treatment of sold products</li><li>Downstream leased assets</li><li>Franchises</li><li>Investments</li><li>Other downstream</li></ol></td></tr></tbody></table>

3. Fuel- and energy-related activities (not included in scope 1 or 2)
Extraction, production, and transportation of fuels and energy purchased or acquired by the company;

4. Upstream transportation and distribution
Transportation and distribution of products purchased by the company between the company's tier 1 suppliers and its own operations; Transportation and distribution services purchased by the company, including inbound logistics, outbound logistics (e.g., sold products), and transportation and distribution between the company's own facilities;

5. Waste generated in operations
Disposal and treatment of waste generated in the company's operations;

6. Business travel
Transportation of employees for business-related activities;

7. Employee commuting
Transportation of employees between their homes and their work sites;

8. Upstream leased assets
Operation of assets leased by the company (lessee);

9. Downstream transportation and distribution
Transportation and distribution of products sold by the company between the company's operations and the end consumer (if not paid for by the company);

10. Processing of sold products
Processing of intermediate products sold by downstream companies (e.g., manufacturers);

11. Use of sold products
End use of goods and services sold by the company;
12. End-of-life treatment of sold products
Waste disposal and treatment of products sold by the company at the end of their life;

13. Downstream leased assets
Operation of assets owned by the company (lessor) and leased to other entities;

14. Franchises
Operation of franchises reported by franchisor;

15. Investments
Operation of investments (including equity and debt investments and project finance).

All these categories share the principle that there is no double counting of emissions between the scopes. For instance, the transport categories do not concern vehicles and facilities owned, controlled or operated by the company, because their GHG emissions are already reported in scope 1 and 2. This means that the transport of employees with a company's vehicle is reported in scope 1 and 2, but not in scope 3. On the contrary, the public transport of employees is reported in scope 3.

<table><caption>Table 9.9: Scope 3 emission factors for business travel and employee commuting (United States)</caption><thead><tr><th>Vehicle type</th><th>CO<sub>2</sub><br>(kg/unit)</th><th>CH<sub>4</sub><br>(g/unit)</th><th>N<sub>2</sub>O<br>(g/unit)</th><th>Unit</th></tr></thead><tbody><tr><td>Passenger car</td><td>0.332</td><td>0.0070</td><td>0.0070</td><td>vehicle-mile</td></tr><tr><td>Light-duty truck</td><td>0.454</td><td>0.0120</td><td>0.0090</td><td>vehicle-mile</td></tr><tr><td>Motorcycle</td><td>0.183</td><td>0.0700</td><td>0.0070</td><td>vehicle-mile</td></tr><tr><td>Intercity rail (northeast corridor)</td><td>0.058</td><td>0.0055</td><td>0.0007</td><td>passenger-mile</td></tr><tr><td>Intercity rail (other routes)</td><td>0.150</td><td>0.0117</td><td>0.0038</td><td>passenger-mile</td></tr><tr><td>Intercity rail (national average)</td><td>0.113</td><td>0.0092</td><td>0.0026</td><td>passenger-mile</td></tr><tr><td>Commuter rail</td><td>0.139</td><td>0.0112</td><td>0.0028</td><td>passenger-mile</td></tr><tr><td>Transit rail (subway, tram)</td><td>0.099</td><td>0.0084</td><td>0.0012</td><td>passenger-mile</td></tr><tr><td>Bus</td><td>0.056</td><td>0.0210</td><td>0.0009</td><td>passenger-mile</td></tr><tr><td>Air travel (short haul, &lt; 300 miles)</td><td>0.207</td><td>0.0064</td><td>0.0066</td><td>passenger-mile</td></tr><tr><td>Air travel (medium haul, 300-2300 miles)</td><td>0.129</td><td>0.0006</td><td>0.0041</td><td>passenger-mile</td></tr><tr><td>Air travel (long haul, &gt; 2300 miles)</td><td>0.163</td><td>0.0006</td><td>0.0052</td><td>passenger-mile</td></tr></tbody></table>

These factors are intended for use in the distance-based method defined in the scope 3 calculation guidance. If fuel data are available, then the fuel-based method should be used.

Source: US EPA (2020), Table 10, www.epa.gov, ghg-emission-factors-hub.xlsx.

The computation of scope 3 emissions requires specific emission factors. For example, Table 9.9 gives their values for business travel (category 6) and employee commuting (category 7) in the US. In the same document, we can find other scope 3 emissions factors (categories 4, 5, 9 and 12). Collecting data is not an easy task since there is no available comprehensive database at the global level. Nevertheless, we can find documented databases at the sector level. For instance, AGRIBALYSE provides references data on the environmental impacts of agricultural and food products through a database built according to the life cycle analysis (LCA) methodology<fnref n="20" />. Other databases can be found in the GHG Protocol website (https://ghgprotocol.org/life-cycle-databases). The

<footnote n="20">²⁰The web site is https://doc.agribalyse.fr.</footnote>
GHG protocol has also developed several calculation tools (cross-sector, country-specific, sector-specific and cities). With Quantis, they also provide scope 3 evaluator (S3E), which is a free web-based tool<fnref n="21" />.

Since it may be sometimes difficult to manipulate physical units, the organizations have also developed monetary emission factors, which are expressed in kgCO₂e/k$ or kgCO₂e/k€. Some figures are reported in Table 9.10. For example, a business air travel, whose cost is equal to $1 000, induces a scope 3 emissions of 1 970 kgCO₂e according to the scope 3 evaluator tool.

<table><caption>Table 9.10: Examples of monetary scope 3 emission factors</caption><thead><tr><th>Category</th><th>S3E</th><th>ADEME</th><th>Category</th><th>S3E</th><th>ADEME</th></tr></thead><tbody><tr><td>Agriculture</td><td>2 500</td><td>2 300</td><td>Air transport</td><td>1 970</td><td>1 190</td></tr><tr><td>Construction</td><td>810</td><td>360</td><td>Education</td><td>310</td><td>120</td></tr><tr><td>Financial intermediation</td><td>140</td><td>110</td><td>Health and Social Work</td><td>300</td><td>500</td></tr><tr><td>Hotels and restaurants</td><td>560</td><td>320</td><td>Rubber and plastics</td><td>1 270</td><td>800</td></tr><tr><td>Telecommunications</td><td>300</td><td>170</td><td>Textiles</td><td>1 100</td><td>600</td></tr></tbody></table>

Source: Scope 3 Evaluator (S3E), https://quantis-suite.com/Scope-3-Evaluator & ADEME, https://bilans-ges.ademe.fr.

Ducoulombier (2021) highlights the importance of scope 3 emissions, but also the lack of data robustness. Since the reporting of these indirect emissions remains voluntary, we observe heterogeneous data in the CDP database with scope 3 items that are partially or not calculated. In this context, most of ESG data providers estimate scope 3 upstream and downstream values using statistical model or environmentally-extended input-output (EEIO) framework<fnref n="22" />. This means that the reported scope 3 emissions are rarely used.

**Remark 101** In order to distinguish the different scopes, we use the following notations: $SC_1$ for scope 1 emissions, $SC_2$ for scope 2 emissions and $SC_3 = SC_3^{up} + SC_3^{down}$ for scope 3 emissions, where $SC_3^{up}$ and $SC_3^{down}$ refer to upstream and downstream scope 3 emissions. The cumulative emissions are then denoted by $SC_{1-2} = SC_1 + SC_2$, $SC_{1-3} = SC_1 + SC_2 + SC_3^{up}$ and $SC_{1-3} = SC_1 + SC_2 + SC_3$.

### 9.1.4 Carbon emissions of investment portfolios

There are two main methods for measuring the carbon footprint of an investment portfolio. The first method if the financed emissions approach. In this case, the investor calculates the carbon emissions that are financed across both equity and debt. Generally, we use EVIC to estimate the value of the enterprise. It is “the sum of the market capitalization of ordinary and preferred shares at fiscal year end and the book values of total debt and minorities interests” (TEG, 2019b). Let W be the wealth invested in the company, the financed emissions are equal to:

$$ \mathcal{CE}(W) = \frac{W}{\text{EVIC}} \cdot \mathcal{CE} $$

In the case of a portfolio ($W_1, \dots, W_n$) where $W_i$ is the wealth invested in company $i$, we have:

$$ \mathcal{CE}(W) = \sum_{i=1}^{n} \mathcal{CE}_i(W_i) = \sum_{i=1}^{n} \frac{W_i}{\text{EVIC}_i} \cdot \mathcal{CE}_i \quad (9.3) $$

<footnote n="21">²¹The tool is available at https://quantis-suite.com/Scope-3-Evaluator.</footnote>

<footnote n="22">²²This model is studied in Section 8.4 on page 807.</footnote>
where EVIC$_i$ and $\mathcal{CE}_i$ are the enterprise value and carbon emissions of company i. It follows that $\mathcal{CE}(W)$ is expressed in tCO$_2$e.

A second method is to use the ownership approach (Le Guenedal and Roncalli, 2022). In this case, we break down the carbon emissions between the stockholders of the company. Equation (9.3) becomes:

$$ \mathcal{CE}(W) = \sum_{i=1}^{n} \frac{W_i}{MV_i} \cdot \mathcal{CE}_i = \sum_{i=1}^{n} \varpi_i \cdot \mathcal{CE}_i \quad (9.4) $$

where $MV_i$ is the market value of company i and $\varpi_i$ is the ownership ratio of the investor. Let $W = \sum_{i=1}^{n} W_i$ be the portfolio value. The portfolio weight of asset i is given by:

$$ w_i = \frac{W_i}{W} $$

We deduce that:

$$ \varpi_i = \frac{W_i}{MV_i} = \frac{w_i \cdot W}{MV_i} $$

and:

$$ \mathcal{CE}(W) = \sum_{i=1}^{n} \frac{w_i \cdot W}{MV_i} CE_i = W \left( \sum_{i=1}^{n} w_i \cdot \frac{\mathcal{CE}_i}{MV_i} \right) = W \left( \sum_{i=1}^{n} w_i \cdot \mathcal{CI}_i^{MV} \right) $$

where $\mathcal{CI}_i^{MV}$ is the market value-based carbon intensity of company i:

$$ \mathcal{CI}_i^{MV} = \frac{\mathcal{CE}_i}{MV_i} $$

Since $\mathcal{CE}(W)$ is a linear function of $W$, the carbon footprint of the portfolio is generally computed with $W = \$1 mn$ and is expressed in tCO$_2$e (per $ mn invested).

**Remark 102** The second approach is valid only for equity portfolios. To compute the market value (or the total market capitalization), we use the following approximation:

$$ \text{MV} = \frac{\text{MC}}{\mathcal{FP}} $$

where MC and $\mathcal{FP}$ are the free float market capitalisation and percentage of the company.

**Example 38** We consider a $100 mn investment portfolio with the following composition: $63.1 mn in company A, $16.9 mn in company B and $20.0 mn in company C. The data are the following:

<table><thead><tr><th rowspan="2">Issuer</th><th colspan="3">Market capitalization (in $ bn)</th><th rowspan="2">Debt<br>(in $ bn)</th><th rowspan="2">FP<br>(in %)</th><th rowspan="2">SC<sub>1-2</sub><br>(in ktCO<sub>2</sub>e)</th></tr><tr><th>31/12/2021</th><th>31/12/2022</th><th>31/01/2023</th></tr></thead><tbody><tr><td>A</td><td>12.886</td><td>10.356</td><td>10.625</td><td>1.112</td><td>99.8</td><td>756.144</td></tr><tr><td>B</td><td>7.005</td><td>6.735</td><td>6.823</td><td>0.000</td><td>39.3</td><td>23.112</td></tr><tr><td>C</td><td>3.271</td><td>3.287</td><td>3.474</td><td>0.458</td><td>96.7</td><td>454.460</td></tr></tbody></table>

As of 31 January 2023, the EVIC value for company A is equal to:

$$ \text{EVIC}_A = \frac{10\,356}{0.998} + 1\,112 = \$11\,489 \text{ mn} $$

We deduce that the financed emissions are equal to:

$$ \mathcal{CE}_A (\$63.1 mn) = \frac{63.1}{11\,489} \times 756.144 = 4.153 \text{ ktCO}_2\text{e} $$
If we assume that the investor has no bond in the portfolio, we can use the ownership approach:

$$ \bar{\omega}_A = \frac{63.1}{(10625/0.998)} = 59.2695 \text{ bps} $$

The carbon emissions of the investment in company A is then equal to:

$$ \mathcal{CE}_A (\$63.1 mn) = 59.2695 \times 10^{-4} \times 756.144 = 4.482 \text{ ktCO}_2\text{e} $$

Finally, we obtain the following results<fnref n="23" />:

<table><thead><tr><th></th><th>Financed emissions</th><th>Carbon emissions</th></tr></thead><tbody><tr><td>Company A</td><td>4.153</td><td>4.482</td></tr><tr><td>Company B</td><td>0.023</td><td>0.022</td></tr><tr><td>Company C</td><td>2.356</td><td>2.530</td></tr><tr><td>Portfolio</td><td>6.532</td><td>7.034</td></tr></tbody></table>

### 9.1.5 Statistics

In what follows, we use the analysis done by Barahhou et al. (2022). We consider the Trucost dataset of carbon emissions as of 01/06/2022 and analyze the distribution of carbon emissions in 2019 for around 15 000 companies. We prefer to use the year 2019 instead of the year 2020, because the covid-19 crisis had a significant impact on the carbon footprint. In Figure 9.11, we have reported the scope 1 and 2 carbon emissions per GICS sector. We notice that including scope 2 has a limited impact, except for some low-carbon sectors such as Consumer Services, Information Technology and Real Estate. In Table 9.11, we have calculated the breakdown of carbon emissions. Scope 1 and 2 emissions represent 17.6 GtCO₂e, and the most important sector contributors are Utilities (34.4%), Materials (31.4%), Energy (14.0%) and Industrials (10.0%). This means that these 4 strategic sectors explain about 90% of scope 1 and 2 carbon emissions.

<table><caption>Table 9.11: Breakdown (in %) of carbon emissions in 2019</caption><thead><tr><th>Sector</th><th>SC<sub>1</sub></th><th>SC<sub>2</sub></th><th>SC<sub>1-2</sub></th><th>SC<sub>3</sub><sup>up</sup></th><th>SC<sub>3</sub><sup>down</sup></th><th>SC<sub>3</sub></th><th>SC<sub>1-3</sub></th></tr></thead><tbody><tr><td>Communication Services</td><td>0.1</td><td>5.1</td><td>0.8</td><td>1.5</td><td>0.2</td><td>0.4</td><td>0.5</td></tr><tr><td>Consumer Discretionary</td><td>1.7</td><td>9.7</td><td>2.9</td><td>14.1</td><td>10.2</td><td>10.8</td><td>9.1</td></tr><tr><td>Consumer Staples</td><td>2.3</td><td>6.7</td><td>2.9</td><td>18.6</td><td>1.6</td><td>4.4</td><td>4.1</td></tr><tr><td>Energy</td><td>15.0</td><td>8.5</td><td>14.0</td><td>14.1</td><td>40.1</td><td>36.0</td><td>31.2</td></tr><tr><td>Financials</td><td>0.7</td><td>1.8</td><td>0.9</td><td>2.6</td><td>1.8</td><td>2.0</td><td>1.7</td></tr><tr><td>Health Care</td><td>0.3</td><td>1.7</td><td>0.5</td><td>2.6</td><td>0.2</td><td>0.6</td><td>0.6</td></tr><tr><td>Industrials</td><td>10.2</td><td>8.9</td><td>10.0</td><td>15.6</td><td>24.2</td><td>22.8</td><td>20.0</td></tr><tr><td>Information Technology</td><td>0.6</td><td>6.8</td><td>1.5</td><td>4.9</td><td>2.3</td><td>2.7</td><td>2.5</td></tr><tr><td>Materials</td><td>29.8</td><td>40.7</td><td>31.4</td><td>20.2</td><td>13.5</td><td>14.6</td><td>18.2</td></tr><tr><td>Real Estate</td><td>0.3</td><td>2.8</td><td>0.6</td><td>1.1</td><td>1.0</td><td>1.0</td><td>0.9</td></tr><tr><td>Utilities</td><td>39.0</td><td>7.3</td><td>34.4</td><td>4.7</td><td>4.8</td><td>4.8</td><td>11.2</td></tr><tr><td>Total (in GtCO<sub>2</sub>e)</td><td>15.1</td><td>2.6</td><td>17.6</td><td>10.3</td><td>53.7</td><td>64.0</td><td>81.6</td></tr></tbody></table>

Source: Trucost (2022) & Barahhou et al. (2022).

<footnote n="23">²³For the financed emissions, we use the data as of 31 December 2022 while the ownership ratio is based on the current data (as of 31 January 2023). In this example, the data as of 31 December 2021 are never used.</footnote>
<figure><img src="image_448.png" /><figcaption>Figure 9.11: 2019 carbon emissions per GICS sector in GtCO₂e (scope 1 & 2)</figcaption></figure>

Source: Trucost (2022) & Barahhou et al. (2022).

<figure><img src="image_449.png" /><figcaption>Figure 9.12: 2019 carbon emissions per GICS sector in GtCO₂e (scope 1, 2 & 3 upstream)</figcaption></figure>

Source: Trucost (2022) & Barahhou et al. (2022).