<figure><img src="image_450.png" /><figcaption>Figure 9.13: 2019 carbon emissions per GICS sector in GtCO₂e (scope 1, 2 & 3)</figcaption></figure>

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

In Figure 9.12, we observe that some sectors are highly impacted by the upstream scope 3 emissions. For instance, the ratio $\frac{SC_3^{up}}{SC_{1-2}}$ is greater than 2.5 for Consumer Discretionary, Consumer Staples and Health Care, and is close to 2 for Information Technology. Among the strategic sectors, Energy and Industrials are the most penalized whereas the upstream scope 3 emissions of Utilities is relatively small compared to its scope 1 emissions.

While the impact of the upstream scope 3 is significant, the impact of the downstream scope 3 is huge as demonstrated in Figure 9.13. Four sectors have very large downstream carbon emissions: Consumer Discretionary, Energy, Industrials and Materials. While Utilities has the most important contribution in terms of scope 1 and 2 since it represents 34.4% of carbon emissions, its contribution to scope 3 is relatively modest and is equal to 4.8%. Including or not scope 3, in particular the downstream carbon emissions, changes the whole picture of the breakdown between the sectors. Figure 9.14 is a visualisation of the sector contribution by considering the addition of several scopes. At each step, the contribution of Materials and Utilities decreases whereas it increases for Consumer Discretionary, Energy, Industrials and Information Technology. Among the most significant sectors<fnref n="24" />, the behavior of Consumer Staples is singular since its contribution increases when adding scope 2 and upstream scope 3, but decreases when considering downstream scope 3.

**Remark 103** When considering scope 3 emissions, double counting is a real issue. According to Table 9.11, the total carbon emissions is 17.6 GtCO₂e for scope 1 + 2, and 81.6 GtCO₂e for scope 1 + 2 + 3, while we estimate that the world emits about 36 GtCO₂e per year.

<footnote n="24">²⁴They correspond to sectors that have a contribution greater than 2%.</footnote>
<figure><img src="image_451.png" /></figure>

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

In Figure 9.15, we draw the histogram of carbon emissions and indicate the 5% and 95% percentile values. We need to use a logarithmic scale, because the range is between some tonnes of CO₂e to several dozen tonnes of CO₂e. This graph shows that it is difficult to compute the carbon footprint of a portfolio based on carbon emissions, because this metric is not homogeneous to the company size. This is why the carbon intensity metric is preferred in financial markets.

## 9.1.6 Negative emissions, avoided emissions, and carbon offsetting

Negative emissions, also known as carbon dioxide removal or CDR, is the process of removing CO₂ from the atmosphere. There are two main categories of negative emissions:

1. Natural climate solutions

Examples include forest restoration and afforestation<fnref n="25" />, reducing soil disturbance<fnref n="26" />, etc.

<footnote n="25">²⁵Afforestation is the process of creating a new forest (planting trees in an area where there was no forest in the past), while reforestation is the process of planting trees in areas where there was forest before.</footnote>

<footnote n="26">²⁶This is the practice of minimizing disturbance to the soil surface and structure, such as using minimum tillage or planting certain crops that protect the soil.</footnote>
<figure><img src="image_452.png" /><figcaption>Figure 9.15: Histogram of 2019 carbon emissions (logarithmic scale, tCO₂e)</figcaption></figure>

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

## 2. Negative emission technologies

Examples are direct air capture with carbon storage<fnref n="27" /> (DACCS), bioenergy with carbon capture and storage<fnref n="28" /> (BECCS), enhanced weathering<fnref n="29" />, ocean fertilization<fnref n="30" />, etc.

Tanzer and Ramírez (2019) gives a more formal definition of negative emissions by considering four minimum criteria for determining whether a technology induces negative emissions:

"[...] (1) Physical greenhouse gases are removed from the atmosphere. (2) The removed gases are stored out of the atmosphere in a manner intended to be permanent. (3) Upstream and downstream greenhouse gas emissions associated with the removal and storage process, such as biomass origin, energy use, gas fate, and co-product fate, are comprehensively estimated and included in the emission balance. (4) The total quantity of atmospheric greenhouse gases removed and permanently stored is greater than the total quantity of greenhouse gases emitted to the atmosphere." (Tanzer and Ramírez, 2019, page 1216)

In a series of three review papers, Jan Minx and his co-authors provided a comprehensive overview of negative emissions (Minx et al., 2018; Fuss et al., 2018; Nemet et al., 2018). They emphasized

<footnote n="27">²⁷This technology uses special filters to capture CO₂ directly from the air, while the captured CO₂ is then stored underground or used in other applications.</footnote>

<footnote n="28">²⁸This process involves capturing and storing the CO₂ emissions from burning biomass, such as wood or grasses.</footnote>

<footnote n="29">²⁹This process involves the application of finely ground minerals, such as olivine or basalt, to land surfaces. When these minerals react with atmospheric CO₂, they form harmless minerals and carbonates, trapping the carbon in a stable mineral form. The goal is to accelerate the natural process of weathering.</footnote>

<footnote n="30">³⁰This technology involves adding nutrients to the ocean, which can stimulate the growth of phytoplankton in the ocean, which then absorbs CO₂ through photosynthesis.</footnote>
that the efficiency, capacity, and cost of different technologies vary widely. A typical example is direct air capture technology<fnref n="31" />. The rationale for the technology is presented in the book published by National Academies of Sciences, Engineering, and Medicine (2019). There are two general types of DAC processes: DAC with liquid solvents (L-DAC) and DAC with solid sorbents (S-DAC). In an L-DAC process, there are four stages: absorption, regeneration, purification and separation. Each phase involves a chemical reaction:

$$ 
\begin{align*}
2 \text{KOH} + \text{CO}_2 & \rightarrow \text{H}_2\text{O} + \text{K}_2\text{CO}_3 \\
\text{CaO} + \text{H}_2\text{O} & \rightarrow \text{Ca}(\text{OH})_2 \\
\text{K}_2\text{CO}_3 + \text{Ca}(\text{OH})_2 & \rightarrow 2 \text{KOH} + \text{CaCO}_3 \\
\text{CaCO}_3 & \rightarrow \text{CaO} + \text{CO}_2
\end{align*}
$$

The goal is to use the liquid solvent KOH to react with atmospheric carbon dioxide CO₂ to produce pure CO₂ and calcium oxide CaO. In an S-DAC process, solid materials or sorbents, such as porous polymers or metal-organic frameworks, are used to adsorb CO₂. The costs associated with DAC technology include the initial investment to build the DAC system (e.g., air contractor, causticizer, calciner, and slaker), the price of solvents and sorbents, the electricity needs to perform the chemical reactions, and the cost of storage. The current price of removing a tonne of CO₂ is around $1000, which is high compared to the price of carbon traded on CO₂ markets. Another factor in assessing the relevance of DAC technologies is the measurement of carbon efficiency, which depends on the amount and carbon intensity of electricity used to remove atmospheric CO₂. Today, the carbon efficiency of the best DAC plans is less than 70%. This is, of course, the current situation and many improvements are expected in the coming years. For instance, IEA (2022b) estimated that DAC costs could fall below $100/tCO₂ by 2030.

Box 9.4: An example of DAC companies: Climeworks

Climeworks (https://climeworks.com) is a Swiss company founded in 2009 as a spin-off from ETH Zurich. It specializes in DAC technology and has established itself as a pioneer in this field with two other companies: Carbon Engineering (Canada) and Global Thermostat (USA). In September 2021, Climeworks inaugurates the world's first large-scale direct air capture and storage plant “Orca” in Iceland, with a capacity to capture 4 000 tonnes of CO₂ per year. The storage of CO₂ is carried out by the company Carbfix, which injects it deep underground, where it mineralizes and turns into stone. In June 2022, Climeworks announces a second, newest and largest direct air capture and storage facility, “Mammoth”, also in Iceland. It will have a nominal CO₂ capture capacity of up to 36 000 tonnes per year when fully operational.

A related concept to negative emissions is avoided emissions, often incorrectly referred to as Scope 4 emissions. According to Russell (2023), “comparative impacts are estimated as the difference between the total, attributional, life-cycle GHG inventories of a company's product (the assessed product) and an alternative (or reference) product that provides an equivalent function”:

$$ \mathcal{AE} = \mathcal{CE} (\text{reference product}) - \mathcal{CE} (\text{assessed product}) $$

Avoided emissions can be positive ($\mathcal{AE} \ge 0$) or negative ($\mathcal{AE} < 0$). For example, an electric car emits CO₂, especially when we consider the life cycle of the batteries, but electric cars do not emit

<footnote n="31">³¹DAC and DACCS are two interchangeable terms because carbon storage is implicit in all carbon dioxide capture, use, and storage (CCUS) technologies.</footnote>
greenhouse gases from burning gasoline. In this example, the reference product is the gasoline-
powered car and the assessed product is the electric car, and we expect the avoided emissions to
be positive. However, there are two issues in calculating avoided emissions. First, which car should
we choose to represent the gasoline car or the reference product? Second, what is the use of the
electric car? In fact, the avoided emissions depend on many factors, such as the carbon intensity of
the electricity, recycling assumptions, etc.

In addition to negative emissions and avoided emissions, carbon offsetting includes a third concept: carbon credits. Carbon credits are transferable financial instruments that represent one tonne of carbon dioxide or another greenhouse gas. They are traded on carbon markets where companies, governments and individuals can buy and sell credits to meet their emission reduction targets. The price of carbon credits can vary depending on supply and demand, as well as the type of project and the region in which it is located. There are two main types of carbon credit systems:

* Cap-and-trade systems

These systems place a limit on the total amount of GHG emissions that can be released from a given region or industry. Companies are allocated a certain number of carbon credits (emission allowances) and can buy or sell credits to meet their emissions targets. These government-regulated schemes make up the compliance carbon market.

* Voluntary carbon markets

These markets are not regulated by the government, and companies can voluntarily buy carbon credits to offset their emissions. Voluntary carbon markets are often used to offset emissions from activities not covered by cap-and-trade systems. In this case, the avoided emissions from a carbon offset (e.g., through the use of negative emission technologies) must be counted on the balance sheet of the buyer, not the seller, who is the developer of the project.

We can now give a precise definition of carbon offsetting. Carbon offsetting is when a company offsets its own carbon emissions by providing emission reductions outside of its own operations. This means that the company purchases a verified carbon credit in a voluntary carbon market that funds a negative emission project. Carbon offsetting does not involve avoided emissions because they concerns the company's own operations and are associated with a change in the company's business strategy. Carbon offsets also do not include carbon credits purchased in a cap-and-trade system because these carbon credits do not necessarily result in negative emissions. Because carbon offsetting involves reducing a company's carbon footprint, it is commonly associated with the race to net-zero emissions. However, we need to make a clear distinction between the two concepts. It is now accepted that some activities will continue to emit GHGs in 2050 due to a lack of carbon-free alternatives, even in the most stringent net-zero scenario. In such situations, carbon offsets must be used primarily by companies exposed to these hard-to-abate sectors, such as cement or airlines. In Figure 9.16 we reproduce the taxonomy of carbon offsets proposed by Allen et al. (2020). Based on our definition, only categories IV and V fall under the strict definition of carbon offsetting.

Allen et al. (2020) proposed a framework for assessing the relevance of carbon offsets to ensure that they contribute to a net-zero economy. The Oxford principles for net-zero aligned carbon offsetting are:

1. Cut emissions, use high quality offsets, and regularly revise offsetting strategy as best practice evolves

Companies' first priority is to reduce their own emissions, not to purchase carbon offsets. If they do, they need to buy offsets that ensure environmental integrity, high standards and certification in line with accounting practices. The largest GHG offset programs are the Verified
<figure><img src="image_453.png" /><figcaption>Figure 9.16: Taxonomy of carbon offsets</figcaption></figure>

Source: Allen et al. (2020, Figure 1, page 7).
Carbon Standard (VCS) and the Gold Standard (GS). The status of projects that meet these
standards can be tracked through official registries whose goal is to certify the ownership of
each negative emissions project. Specifically, ownership is transferred to the buyer of the
carbon credits and then canceled when the credits are sold. The purpose of registries is to
ensure that negative emissions are not counted twice.

2. Shift to carbon removal offsetting

There is clearly an imbalance between the supply of certified negative emissions projects and the projects needed to achieve net zero in the long term. Creating demand for carbon removal offsets today will send the necessary market signal to increase supply. Figure 9.17 shows the size of the voluntary carbon market (VCM). It has been multiplied by two after 2019. According to Ecosystem Marketplace (2023b), the cumulative volume has reached 2.3 GtCO2e with a value of $10 billion. This implies an average price of $4.35 per tonne of CO2. From 2021, the average price is more likely to be between $7 and $8 per tonne of CO2. The market is largely dominated by renewable energy projects and forestry & land use. Since 2020, projects on household & community devices have also been promoted. Although developing, the voluntary carbon market remains relatively small and immature, with many intermediaries and few end users<fnref n="32" />. For example, the energy sector is the main buyer of voluntary carbon credits, accounting for more than 50% of the market. However, the market is expected to reach between $10 billion and $40 billion by 2030, up from a record $2.1 billion in 2021 (BCG, 2023a).

<figure><img src="image_454.png" /><figcaption>Figure 9.17: Voluntary carbon market size by volume of traded carbon credits</figcaption></figure>

Source: Ecosystem Marketplace (2023b, Figure 2, page 8).

<footnote n="32"><sup>32</sup>In fact, it is concentrated in a few companies. According to Ecosystem Marketplace (2023a), the top 10 buyers in 2021 were Delta Air Lines, TotalEnergies, Shell, Volkswagen, Takeda Pharmaceuticals, Comcast, Diamondback Energy, La Poste, Telstra and Eni.</footnote>
### 3. Shift to long-lived storage

The issue of CO₂ storage and sequestration is an important one. As noted by Allen et al. (2020), “short-lived storage involves methods that have a higher risk of being reversed over decades. Long-lived storage refers to methods of storing carbon that have a low risk of reversal over centuries to millennia, such as storing CO₂ in geological reservoirs or mineralising carbon into stable forms. Short-lived storage offsets help buy time to reduce emissions and invest in long-lived storage, but they are not a long-term solution for achieving balance between sinks and sources.” Measuring the efficiency of a technology is not straightforward and is highly dependent on the lifetime of the project (Terlouw et al., 2021) and the system boundary. Figure 9.18 shows an example taken from Tanzer and Ramírez (2019). Chiquier et al. (2022) proposed to evaluate the efficiency of carbon dioxide removal by considering the amount of CO₂ stored (or removed) and the amount of CO₂ leaked (or emitted) over the supply chain:

$$ \eta(t) = \frac{\text{CO}_2^{\text{stored}}(t) - \text{CO}_2^{\text{leaked}}(t)}{\text{CO}_2^{\text{stored}}(t)} $$

The metric $\eta(t)$ depends on the lifetime $t$ expressed in years. In general, it is a decreasing function of time $t$, which means that the efficiency is maximum at the beginning of the project. In the case of an afforestation/reforestation project implemented in 2020 in the UK, Chiquier et al. (2022) estimates $\eta(10) = 87.1\%$, $\eta(30) = 98.8\%$, $\eta(100) = 98.9\%$, and $\eta(1000) = 61.9\%$. Here, the CDR efficiency increases in the beginning because the forest establishment emits CO₂ and the trees are young. Then the trees grow and the efficiency is close to 100% between 30 and 100 years. In the long term, the efficiency decreases due to the risk of forest fires. A summary of key features for each CDR pathway is provided in Table 9.12.

<table><caption>Table 9.12: Summary of key features for each CDR pathway</caption><thead><tr><th>CDR</th><th>η (100)</th><th>η (1000)</th><th>Timing</th><th>Permanence</th></tr></thead><tbody><tr><td>Afforestation</td><td>63 to 99%</td><td>31 to 95%</td><td>Decades</td><td>Very low</td></tr><tr><td>Reforestation</td><td>63 to 99%</td><td>31 to 95%</td><td>Decades</td><td>Very low</td></tr><tr><td>BECCS</td><td>52 to 87%</td><td>78 to 87%</td><td>Immediate to decades</td><td>High/very high</td></tr><tr><td>Biochar</td><td>20 to 39%</td><td>-3 to 5%</td><td>Immediate</td><td>Low/very low</td></tr><tr><td>DACCS</td><td>-5 to 90%</td><td>-5 to 90%</td><td>Immediate</td><td>Very high</td></tr><tr><td>Enhanced weathering</td><td>17 to 92%</td><td>51 to 92%</td><td>Immediate to decades</td><td>High/very high</td></tr></tbody></table>

Source: Chiquier et al. (2022, Table 1, page 4400).

### 4. Support the development of net-zero aligned offsetting

The fourth principle is to promote carbon offsetting. To develop this market, companies can enter into long-term agreements, form sector-specific alliances, support the restoration and protection of natural and semi-natural ecosystems, and incorporate these principles into regulation.

Carbon accounting for negative emission technologies is still an open question (Brander et al., 2021; Kaplan et al., 2023). It is closely related to the issue of certification and credibility of CDR projects. Accounting for carbon offsets also challenges the economic incentives of these projects from the perspective of buyers of carbon credits. Microsoft's experience described in *Nature* is an interesting testimony from an end user and provides some insights to improve the ecosystem of negative emission technologies and carbon offsetting (Joppa et al., 2021).
<figure><img src="image_455.png" /><figcaption>Figure 9.18: Perceived CO₂ emissions of a simplified steel production system when viewed from different system boundaries</figcaption></figure>

The dashed line in each sub-figure represents the system boundaries used to estimate the total CO₂ emissions in the upper right corner of each figure. The system design and numbers used are greatly simplified for illustrative purposes. (a-c) show the gate-to-gate CO₂ emissions of a steel mill, considering only the CO₂ produced at the mill itself for normal production (a), with the use of carbon capture and storage (b), and with the use of bioenergy with carbon capture and storage (c). (d) extends the system boundaries to include photosynthetic uptake of the exact amount of CO₂ released by combustion, assuming the charcoal is carbon neutral. (e) is a simplified cradle-to-grave system that includes in its boundaries the CO₂ absorbed by the wood that is lost in the charcoal production process, the CO₂ emissions from biomass harvesting and transportation, the CO₂ emissions from charcoal production, and the CO₂ emissions from CO₂ storage. (f) is a variant where biomass production has significant indirect land use change emissions. (g) is a variant where geological storage of CO₂ results in the production and combustion of fossil fuels whose CO₂ emissions exceed the CO₂ stored.

Source: Tanzer and Ramírez (2019, Figure 2, page 1214).
## 9.2 Carbon intensity

While carbon emissions measure the carbon footprint in an absolute value, the carbon intensity is a relative metric of the carbon footprint. The underlying idea is to normalize the carbon emissions by a size or activity unit. For instance, we can measure the carbon footprint of countries by tCO₂e per capita, watching television by CO₂e emissions per viewer-hour, washing machines by kgCO₂e per wash, cars by kgCO₂e per kilometer driven, companies by ktCO₂e per $1 mn revenue, etc. We distinguish two types of carbon intensity: carbon intensities whose activity units are physical and carbon intensities whose activity units are monetary.

### 9.2.1 Physical intensity ratios

The product carbon footprint (PCF) measures the relative carbon emissions of a product throughout its life cycle. This approach, which is called life cycle assessment (LCA), distinguishes two methods:

* Cradle-to-gate refers to the carbon footprint of a product from the moment it is produced (including the extraction of raw materials) to the moment it enters the store;

* In contrast, cradle-to-grave covers the entire life cycle of a product, including the use-phase and recycling.

Below, we report some examples of product carbon footprint computed by ADEME.

<table><caption>Table 9.13: Examples of product carbon footprint (in kgCO₂e per unit)</caption><thead><tr><th>Product</th><th>Category</th><th>Cradle-to-gate</th><th>Cradle-to-grave</th></tr></thead><tbody><tr><td rowspan="2">Screen</td><td>21.5 inches</td><td>222</td><td>236</td></tr><tr><td>23.8 inches</td><td>248</td><td>265</td></tr><tr><td rowspan="3">Computer</td><td>Laptop</td><td>156</td><td>169</td></tr><tr><td>Desktop</td><td>169</td><td>189</td></tr><tr><td>High performance</td><td>295</td><td>394</td></tr><tr><td rowspan="2">Smartphone</td><td>Classical</td><td>16</td><td>16</td></tr><tr><td>5 inches</td><td>33</td><td>32</td></tr><tr><td rowspan="2">Oven</td><td>Built-in electric</td><td>187</td><td>319</td></tr><tr><td>Professional (combi steamer)</td><td>734</td><td>12 676</td></tr><tr><td rowspan="2">Washing machine</td><td>Capacity 5kg</td><td>248</td><td>468</td></tr><tr><td>Capacity 7kg</td><td>275</td><td>539</td></tr><tr><td rowspan="2">Shirt</td><td>Coton</td><td>10</td><td>13</td></tr><tr><td>Viscose</td><td>9</td><td>12</td></tr><tr><td rowspan="2">Balloon</td><td>Football</td><td>3.4</td><td>5.1</td></tr><tr><td>Basket-ball</td><td>3.6</td><td>5.9</td></tr></tbody></table>

Source: Lhotellier et al. (2018, Annex 4, pages 212-215).

The previous analysis can be extended to corporate carbon footprint (CCF). For instance, we can measure the CCF of a cement manufacturer by the amount of GHG emissions per tonne of cement. In the airline sector, the main traffic metric is the revenue passenger kilometers (RPK), which is calculated by multiplying the number of paying passengers by the distance traveled. Therefore, the CCF of airlines can be measured by the amount of GHG emissions per RPK (Table 9.14).
<table><caption>Table 9.14: Physical carbon intensity per production unit</caption><thead><tr><td>Sector</td><td>Unit</td><td>Description</td></tr></thead><tbody><tr><td>Transport sector (aviation)</td><td>CO<sub>2</sub>e/RPK</td><td>Revenue passenger kilometers</td></tr><tr><td>Transport sector (shipping)</td><td>CO<sub>2</sub>e/RTK</td><td>Revenue tonne kilometers</td></tr><tr><td>Industry (cement)</td><td>CO<sub>2</sub>e/t cement</td><td>Tonne of cement</td></tr><tr><td>Industry (steel)</td><td>CO<sub>2</sub>e/t steel</td><td>Tonne of steel</td></tr><tr><td>Electricity</td><td>CO<sub>2</sub>e/MWh</td><td>Megawatt hour</td></tr><tr><td>Buildings</td><td>CO<sub>2</sub>e/SQM</td><td>Square meter</td></tr></tbody></table>

### 9.2.2 Monetary intensity ratios

From a financial point of view, it does not make sense to compare and aggregate the carbon emissions of a large cap company with the carbon emissions of a small cap company. Carbon intensity is then a more relevant metric. ESG analysts can then compare companies that belong to the same activity sector by using physical intensity ratios. For example, they can compare all the cement manufacturers, because they can normalize the carbon emissions by the volume of cement production. In a similar way, they can compare all the airline companies, because they can normalize the carbon emissions by the RPK metric. Nevertheless, the physical intensity ratios are not relevant when we consider a portfolio that is invested in several sectors. How to compare a cement-based carbon intensity with a RPK-based carbon intensity? How to aggregate the two metrics? Until now, nobody has the answer.

Therefore, portfolio managers will use monetary intensity ratios, which are defined as:

$$CI = \frac{CE}{Y}$$

where CE is the company's carbon emissions and Y is a monetary variable measuring its activity. For instance, we can use revenues, sales, etc. to normalize carbon emissions:

* Revenue:

$$CI^{Revenue} = \frac{CE}{Revenue}$$

* Sales:

$$CI^{Sales} = \frac{CE}{Sales}$$

* Enterprise value including cash:

$$CI^{EVIC} = \frac{CE}{EVIC}$$

* Market value:

$$CI^{MV} = \frac{CE}{MV}$$

Even the previous carbon emission metrics based on EVIC and market value can be viewed as carbon intensity metrics.
If we consider the EVIC-based approach, the carbon intensity of the portfolio is given by:

$$
\begin{align*}
\mathcal{CI}^{\text{EVIC}}(w) &= \frac{\mathcal{CE}^{\text{EVIC}}(W)}{W} \\
&= \frac{1}{W} \sum_{i=1}^{n} \frac{W_i}{\text{EVIC}_i} \cdot \mathcal{CE}_i \\
&= \sum_{i=1}^{n} \frac{W_i}{W} \cdot \frac{\mathcal{CE}_i}{\text{EVIC}_i} \\
&= \sum_{i=1}^{n} w_i \cdot \mathcal{CI}_i^{\text{EVIC}}
\end{align*}
$$

where $w = (w_1, \dots, w_n)$ is the vector of portfolio weights. We notice that the carbon intensity satisfies the additivity property. In a similar way, we obtain:

$$
\mathcal{CI}^{\text{MV}}(w) = \sum_{i=1}^{n} w_i \cdot \mathcal{CI}_i^{\text{MV}}
$$

Let us now consider the revenue-based carbon intensity (also called the economic carbon intensity).
We denote by $Y_i$ the revenue of issuer $i$. The carbon intensity of the portfolio becomes:

$$
\mathcal{CI}^{\text{Revenue}}(w) = \frac{\mathcal{CE}(w)}{Y(w)}
$$

where $\mathcal{CE}(w)$ measures the carbon emissions of the portfolio:

$$
\mathcal{CE}(w) = \sum_{i=1}^{n} W_i \cdot \frac{\mathcal{CE}_i}{\text{MV}_i} = W \sum_{i=1}^{n} \frac{w_i}{\text{MV}_i} \cdot \mathcal{CE}_i
$$

and $Y(w)$ is the total revenue of the portfolio:

$$
Y(w) = \sum_{i=1}^{n} W_i \cdot \frac{Y_i}{\text{MV}_i} = W \sum_{i=1}^{n} \frac{w_i}{\text{MV}_i} \cdot Y_i
$$

We deduce that:

$$
\begin{align*}
\mathcal{CI}^{\text{Revenue}}(w) &= \frac{\sum_{i=1}^{n} \frac{w_i}{\text{MV}_i} \cdot \mathcal{CE}_i}{\sum_{i=1}^{n} \frac{w_i}{\text{MV}_i} \cdot Y_i} \\
&= \sum_{i=1}^{n} w_i \cdot \omega_i \cdot \mathcal{CI}_i^{\text{Revenue}}
\end{align*}
$$

where $\omega_i$ is the ratio between the revenue per market value of company $i$ and the weighted average
revenue per market value of the portfolio:

$$
\omega_i = \frac{\frac{Y_i}{\text{MV}_i}}{\sum_{k=1}^{n} w_k \cdot \frac{Y_k}{\text{MV}_k}}
$$
Except when all the companies have the same revenue per market value ratio, we deduce that the revenue-based carbon intensity does not satisfy the additivity property since we have $\mathcal{CI}^{\text{Revenue}}(w) \neq \sum_{i=1}^{n} w_i \cdot \mathcal{CI}_i^{\text{Revenue}}$. In order to avoid this problem, we generally use the weighted average carbon intensity (WACI) of the portfolio:

$$ \mathcal{CI}^{\text{Revenue}}(w) = \sum_{i=1}^{n} w_i \cdot \mathcal{CI}_i^{\text{Revenue}} \qquad (9.5) $$

This method is the standard approach in portfolio management.

**Remark 104** Carbon intensity is additive when we consider a given issuer:

$$ \begin{aligned} \mathcal{CI}_i(\mathcal{SC}_{1-3}) &= \frac{\mathcal{CE}_i(\mathcal{SC}_1) + \mathcal{CE}_i(\mathcal{SC}_2) + \mathcal{CE}_i(\mathcal{SC}_3)}{Y_i} \\ &= \mathcal{CI}_i(\mathcal{SC}_1) + \mathcal{CI}_i(\mathcal{SC}_2) + \mathcal{CI}_i(\mathcal{SC}_3) \end{aligned} $$

**Example 39** We assume that $\mathcal{CE}_1 = 5 \times 10^6 \text{ CO}_2\text{e}$, $Y_1 = \$0.2 \times 10^6$, $MV_1 = \$10 \times 10^6$, $\mathcal{CE}_2 = 50 \times 10^6 \text{ CO}_2\text{e}$, $Y_2 = \$4 \times 10^6$ and $MV_2 = \$10 \times 10^6$. We invest $W = \$10 \text{ mn}$.

We deduce that:

$$ \mathcal{CI}_1 = \frac{5 \times 10^6}{0.2 \times 10^6} = 25.0 \text{ tCO}_2\text{e/}$ mn

and $\mathcal{CI}_2 = 12.5 \text{ tCO}_2\text{e/}$ mn. Since we have:

$$ \left\{ \begin{aligned} \mathcal{CE}(w) &= W \left( w_1 \frac{\mathcal{CE}_1}{MV_1} + w_2 \frac{\mathcal{CE}_2}{MV_2} \right) \\ Y(w) &= W \left( w_1 \frac{Y_1}{MV_1} + w_2 \frac{Y_2}{MV_2} \right) \\ \mathcal{CI}(w) &= w_1 \mathcal{CI}_1 + w_2 \mathcal{CI}_2 \end{aligned} \right. $$

We obtain the following results:

<table><thead><tr><th>w<sub>1</sub></th><th>w<sub>2</sub></th><th>CE(w)<br>(×10<sup>6</sup> CO<sub>2</sub>e)</th><th>Y(w)<br>(×$10<sup>6</sup>)</th><th>CE(w)<br>Y(w)</th><th>CI(w)</th></tr></thead><tbody><tr><td>0%</td><td>100%</td><td>50.00</td><td>4.00</td><td>12.50</td><td>12.50</td></tr><tr><td>10%</td><td>90%</td><td>45.50</td><td>3.62</td><td>12.57</td><td>13.75</td></tr><tr><td>20%</td><td>80%</td><td>41.00</td><td>3.24</td><td>12.65</td><td>15.00</td></tr><tr><td>30%</td><td>70%</td><td>36.50</td><td>2.86</td><td>12.76</td><td>16.25</td></tr><tr><td>50%</td><td>50%</td><td>27.50</td><td>2.10</td><td>13.10</td><td>18.75</td></tr><tr><td>70%</td><td>30%</td><td>18.50</td><td>1.34</td><td>13.81</td><td>21.25</td></tr><tr><td>80%</td><td>20%</td><td>14.00</td><td>0.96</td><td>14.58</td><td>22.50</td></tr><tr><td>90%</td><td>10%</td><td>9.50</td><td>0.58</td><td>16.38</td><td>23.75</td></tr><tr><td>100%</td><td>0%</td><td>5.00</td><td>0.20</td><td>25.00</td><td>25.00</td></tr></tbody></table>

We notice that the weighted average carbon intensity can be very different than the economic carbon intensity. Let us assume that we buy the two companies, implying that $W = \$20 \text{ mn}$, $w_1 = 50\%$ and $w_2 = 50\%$. In this case, we obtain $\mathcal{CE}(w) = 55 \times 10^6$ and $Y(w) = \$4 \times 10^6$. The economic carbon intensity is then equal to $55/4 = 13.10$ while the WACI is 18.75.

**Remark 105** For sovereign issuers, the economic carbon intensity is measured in mega-tonnes of CO$_2\text{e}$ per million dollars of GDP while the physical carbon intensity unit is tCO$_2\text{e}$ per capita.
9.2.3 Statistics

Some CCF values are provided in Table 9.15. These figures illustrate some issues in the computation of the carbon footprint at the issuer level. First, it is obvious that it is important to take into account scope 3 to have the real picture of the carbon footprint of an issuer. Indeed, we notice that some issuers have a low scope 1, because they have more or less outsourced the manufacturing of their products. Since a part of the production is located in upstream scope 3, we can not make a fair comparison between issuers if we only consider scope 1 and 2. We face a similar issue with the distribution of the products. The magnitude of some scope 3 carbon intensities raises also the question of their computation. Indeed, while scope 1 and 2 are mandatory to report, there is no obligation for a company to report its scope 3. Moreover, while there is one unique figure for scope 1 and 2 emissions in the CDP reporting files, scope 3 emissions are split into 15 categories, and it is extremely rare that a company reports all scope 3 categories. This explains that the frequency of estimated values is larger for scope 3.

<figure><img src="image_456.png" /><figcaption>Figure 9.19: Histogram of 2019 carbon intensities (logarithmic scale, tCO₂e/$ mn)</figcaption></figure>

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

In Figure 9.19, we show the distribution of carbon intensities. Since the range may be very large (from zero to several thousand), we use a logarithmic scale. Moreover, the dotted vertical lines indicate the 5th and 95th percentiles. We observe that the distribution support is very large for scope 1, 2 and 3 downstream. In this case, there are many extreme points with very low and very high carbon intensities. Therefore, we will see that it is relatively easy to reduce the carbon footprint of a portfolio. Now, if we focus on upstream scope 3, we obtain another story, because the range is not so large. Indeed, we do not have issuers with very low carbon intensity. Therefore, incorporating upstream scope 3 will change the nature of portfolio decarbonization, which will become more difficult.
<table><caption>Table 9.15: Examples of 2019 carbon emissions and intensities</caption><thead><tr><th rowspan="2">Company</th><th colspan="4">Carbon emissions (in tCO<sub>2</sub>e)</th><th rowspan="2">Revenue<br>(in $ mn)</th><th colspan="4">Intensity (in tCO<sub>2</sub>e/$ mn)</th></tr><tr><th>SC<sub>1</sub></th><th>SC<sub>2</sub></th><th>SC<sub>3</sub><sup>up</sup></th><th>SC<sub>3</sub><sup>down</sup></th><th>SC<sub>1</sub></th><th>SC<sub>2</sub></th><th>SC<sub>3</sub><sup>up</sup></th><th>SC<sub>3</sub><sup>down</sup></th></tr></thead><tbody><tr><td>Airbus</td><td>576 705</td><td>386 674</td><td>12 284 183</td><td>23 661 432</td><td>78 899</td><td>7.3</td><td>4.9</td><td>155.7</td><td>299.9</td></tr><tr><td>Allianz</td><td>46 745</td><td>224 315</td><td>3 449 234</td><td>3 904 000</td><td>135 279</td><td>0.3</td><td>1.7</td><td>25.5</td><td>28.9</td></tr><tr><td>Alphabet</td><td>111 283</td><td>5 118 152</td><td>7 142 566</td><td></td><td>161 857</td><td>0.7</td><td>31.6</td><td>44.1</td><td></td></tr><tr><td>Amazon</td><td>5 760 000</td><td>5 500 000</td><td>20 054 722</td><td>10 438 551</td><td>280 522</td><td>20.5</td><td>19.6</td><td>71.5</td><td>37.2</td></tr><tr><td>Apple</td><td>50 549</td><td>862 127</td><td>27 624 282</td><td>5 470 771</td><td>260 174</td><td>0.2</td><td>3.3</td><td>106.2</td><td>21.0</td></tr><tr><td>BNP Paribas</td><td>64 829</td><td>280 789</td><td>1 923 307</td><td>1 884</td><td>78 244</td><td>0.8</td><td>3.6</td><td>24.6</td><td>0.0</td></tr><tr><td>Boeing</td><td>611 001</td><td>871 000</td><td>9 878 431</td><td>22 959 719</td><td>76 559</td><td>8.0</td><td>11.4</td><td>129.0</td><td>299.9</td></tr><tr><td>BP</td><td>49 199 999</td><td>5 200 000</td><td>103 840 194</td><td>582 639 687</td><td>276 850</td><td>177.7</td><td>18.8</td><td>375.1</td><td>2 104.5</td></tr><tr><td>Caterpillar</td><td>905 000</td><td>926 000</td><td>15 197 607</td><td>401 993 744</td><td>53 800</td><td>16.8</td><td>17.2</td><td>282.5</td><td>7 472.0</td></tr><tr><td>Danone</td><td>722 122</td><td>944 877</td><td>28 969 780</td><td>4 464 773</td><td>28 308</td><td>25.5</td><td>33.4</td><td>1 023.4</td><td>157.7</td></tr><tr><td>Enel</td><td>69 981 891</td><td>5 365 386</td><td>8 726 973</td><td>53 774 821</td><td>86 610</td><td>808.0</td><td>61.9</td><td>100.8</td><td>620.9</td></tr><tr><td>Exxon</td><td>111 000 000</td><td>9 000 000</td><td>107 282 831</td><td>594 131 943</td><td>255 583</td><td>434.3</td><td>35.2</td><td>419.8</td><td>2 324.6</td></tr><tr><td>JPMorgan Chase</td><td>81 655</td><td>692 299</td><td>3 101 582</td><td>15 448 469</td><td>115 627</td><td>0.7</td><td>6.0</td><td>26.8</td><td>133.6</td></tr><tr><td>Juventus</td><td>6 665</td><td>15 739</td><td>35 842</td><td>77 114</td><td>709</td><td>9.4</td><td>22.2</td><td>50.6</td><td>108.8</td></tr><tr><td>LVMH</td><td>67 613</td><td>262 609</td><td>11 853 749</td><td>942 520</td><td>60 083</td><td>1.1</td><td>4.4</td><td>197.3</td><td>15.7</td></tr><tr><td>Microsoft</td><td>113 414</td><td>3 556 553</td><td>5 977 488</td><td>4 003 770</td><td>125 843</td><td>0.9</td><td>28.3</td><td>47.5</td><td>31.8</td></tr><tr><td>Nestle</td><td>3 291 303</td><td>3 206 495</td><td>61 262 078</td><td>33 900 606</td><td>93 153</td><td>35.3</td><td>34.4</td><td>657.6</td><td>363.9</td></tr><tr><td>Netflix</td><td>38 481</td><td>145 443</td><td>1 900 283</td><td>2 192 255</td><td>20 156</td><td>1.9</td><td>7.2</td><td>94.3</td><td>108.8</td></tr><tr><td>NVIDIA</td><td>2 767</td><td>65 048</td><td>2 756 353</td><td>1 184 981</td><td>11 716</td><td>0.2</td><td>5.6</td><td>235.3</td><td>101.1</td></tr><tr><td>PepsiCo</td><td>3 552 415</td><td>1 556 523</td><td>32 598 029</td><td>14 229 956</td><td>67 161</td><td>52.9</td><td>23.2</td><td>485.4</td><td>211.9</td></tr><tr><td>Pfizer</td><td>734 638</td><td>762 840</td><td>4 667 225</td><td>133 468</td><td>51 750</td><td>14.2</td><td>14.7</td><td>90.2</td><td>2.6</td></tr><tr><td>Roche</td><td>288 157</td><td>329 541</td><td>5 812 735</td><td>347 437</td><td>64 154</td><td>4.5</td><td>5.1</td><td>90.6</td><td>5.4</td></tr><tr><td>Samsung Electronics</td><td>5 067 000</td><td>10 998 000</td><td>33 554 245</td><td>60 978 947</td><td>197 733</td><td>25.6</td><td>55.6</td><td>169.7</td><td>308.4</td></tr><tr><td>TotalEnergies</td><td>40 909 135</td><td>3 596 127</td><td>49 817 293</td><td>456 993 576</td><td>200 316</td><td>204.2</td><td>18.0</td><td>248.7</td><td>2 280.0</td></tr><tr><td>Toyota</td><td>2 522 987</td><td>5 227 844</td><td>66 148 020</td><td>330 714 268</td><td>272 608</td><td>9.3</td><td>19.2</td><td>242.6</td><td>1 213.2</td></tr><tr><td>Volkswagen</td><td>4 494 066</td><td>5 973 894</td><td>65 335 372</td><td>354 913 446</td><td>282 817</td><td>15.9</td><td>21.1</td><td>231.0</td><td>1 254.9</td></tr><tr><td>Walmart</td><td>6 101 641</td><td>13 057 352</td><td>40 651 079</td><td>32 346 229</td><td>514 405</td><td>11.9</td><td>25.4</td><td>79.0</td><td>62.9</td></tr></tbody></table>

Source: Trucost (2022) & Barahhou et al. (2022).
<table><caption>Table 9.16: Carbon intensity in tCO₂e/$ mn per GICS sector and sector contribution in % (MSCI World, June 2022)</caption><thead><tr><th rowspan="2">Sector</th><th rowspan="2">b<sub>i</sub><br>(in %)</th><th colspan="4">Carbon intensity</th><th colspan="4">Risk contribution</th></tr><tr><th>SC<sub>1</sub></th><th>SC<sub>1-2</sub></th><th>SC<sub>1-3</sub><sup>up</sup></th><th>SC<sub>1-3</sub></th><th>SC<sub>1</sub></th><th>SC<sub>1-2</sub></th><th>SC<sub>1-3</sub><sup>up</sup></th><th>SC<sub>1-3</sub></th></tr></thead><tbody><tr><td>Communication Services</td><td>7.58</td><td>2</td><td>28</td><td>134</td><td>172</td><td>0.14</td><td>1.31</td><td>3.30</td><td>1.31</td></tr><tr><td>Consumer Discretionary</td><td>10.56</td><td>23</td><td>65</td><td>206</td><td>590</td><td>1.87</td><td>4.17</td><td>6.92</td><td>6.21</td></tr><tr><td>Consumer Staples</td><td>7.80</td><td>28</td><td>55</td><td>401</td><td>929</td><td>1.68</td><td>2.66</td><td>10.16</td><td>7.38</td></tr><tr><td>Energy</td><td>4.99</td><td>632</td><td>698</td><td>1 006</td><td>6 823</td><td>24.49</td><td>21.53</td><td>16.33</td><td>34.37</td></tr><tr><td>Financials</td><td>13.56</td><td>13</td><td>19</td><td>52</td><td>244</td><td>1.33</td><td>1.58</td><td>2.28</td><td>3.34</td></tr><tr><td>Health Care</td><td>14.15</td><td>10</td><td>22</td><td>120</td><td>146</td><td>1.12</td><td>1.92</td><td>5.54</td><td>2.12</td></tr><tr><td>Industrials</td><td>9.90</td><td>111</td><td>130</td><td>298</td><td>1 662</td><td>8.38</td><td>7.83</td><td>9.43</td><td>16.38</td></tr><tr><td>Information Technology</td><td>21.08</td><td>7</td><td>23</td><td>112</td><td>239</td><td>1.13</td><td>3.03</td><td>7.57</td><td>5.06</td></tr><tr><td>Materials</td><td>4.28</td><td>478</td><td>702</td><td>1 113</td><td>2 957</td><td>15.89</td><td>18.57</td><td>15.48</td><td>12.93</td></tr><tr><td>Real Estate</td><td>2.90</td><td>22</td><td>101</td><td>167</td><td>571</td><td>0.48</td><td>1.81</td><td>1.57</td><td>1.65</td></tr><tr><td>Utilities</td><td>3.21</td><td>1 744</td><td>1 794</td><td>2 053</td><td>2 840</td><td>43.47</td><td>35.59</td><td>21.41</td><td>9.24</td></tr><tr><td>MSCI World</td><td></td><td>130</td><td>163</td><td>310</td><td>992</td><td></td><td></td><td></td><td></td></tr><tr><td>MSCI World EW</td><td></td><td>168</td><td>211</td><td>391</td><td>1 155</td><td></td><td></td><td></td><td></td></tr></tbody></table>

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

Let $b = (b_1, \dots, b_n)$ be the weights of the assets that belong to a benchmark. Its weighted average carbon intensity is given by $CI(b) = \sum_{i=1}^n b_i \cdot CI_i$ where $CI_i$ is the carbon intensity of asset $i$. If we focus on the carbon intensity for a given sector, we use the following formula:

$$CI(Sector_j) = \frac{\sum_{i \in Sector_j} b_i \cdot CI_i}{\sum_{i \in Sector_j} b_i}$$

In Table 9.16, we report the carbon intensity of the MSCI World index and its sectors. We also compute the risk contribution of each sector as follows: $RC(Sector_j) = (\sum_{i \in Sector_j} b_i \cdot CI_i) / CI(b)$. We obtain 130 tCO₂e/$ mn for scope 1, 163 tCO₂e/$ mn if we include scope 2, 310 tCO₂e/$ mn if we add upstream scope 3, and finally 992 tCO₂e/$ mn if we consider the full scope 3. We notice a large cap bias because the MSCI World equally-weighted portfolio shows higher figures. We also observe a high discrepancy between sectors. Low-carbon sectors are Communication Services, Financials, Health Care and Information Technology, whereas high-carbon sectors are Energy, Materials and Utilities. In terms of risk contribution, Consumer Services represents 7.58% of the nominal allocation, but only 0.14% of the carbon allocation if we consider scope 1. If we focus on the first two scopes, Utilities is the main contributor, followed by Energy and Materials. By including upstream scope 3 emissions, the contribution of Consumer Staples becomes significant. We also notice that the Utilities contribution has strongly been reduced whereas the Industrials contribution increases when we consider the three scopes.

**Remark 106** The question of double-counting is less important when we consider carbon intensities, especially monetary measures. Indeed, the carbon intensity can be seen as a scoring system, and portfolio managers generally use carbon intensity in a relative way, and not in an absolute way. For instance, they do not target a given carbon intensity. Their goal is more reducing the carbon intensity relatively to a benchmark, without analyzing the absolute value of the benchmark itself. Moreover, the aggregation at the portfolio level is generally done thanks to the WACI measure, which indicates that the carbon intensity is more viewed as a score than a physical measure.
## 9.3 Dynamic risk measures

In this section, we present the basics for building dynamic carbon metrics that are very useful when defining net-zero investment portfolios and assessing the decarbonization policy of issuers. The main tools are the carbon budget, the carbon trend and the carbon target. By combining these tools, we will be able to present the PAC framework which is the cornerstone of implied temperature ratings (ITR). It measures the participation, the ambition and the credibility of a company to reduce its carbon emissions.

### 9.3.1 Carbon budget

#### Definition

The carbon budget defines the amount of GHG emissions that a country, a company or an organization produces over the time period $[t_0, t]$. From a mathematical point of view, it corresponds to the signed area of the region bounded by the function $\mathcal{C}\mathcal{E}(t)$:

$$ \mathcal{CB}(t_0, t) = \int_{t_0}^{t} \mathcal{CE}(s) \, ds $$

The carbon budget can be computed with other functions than the carbon emissions. For instance, if the reference level is equal to $\mathcal{CE}^*(t)$ at time $t$, we obtain:

$$ \mathcal{CB}^*(t_0, t) = \int_{t_0}^{t} \mathcal{CE}^*(s) \, ds $$

Therefore, we can easily compute the excess (or net) carbon budget since we have:

$$ \int_{t_0}^{t} (\mathcal{CE}(s) - \mathcal{CE}^*(s)) \, ds = \mathcal{CB}(t_0, t) - \mathcal{CB}^*(t_0, t) $$

If the reference level is constant — $\mathcal{CE}^*(t) = \mathcal{CE}^*$, the previous formula becomes:

$$ \int_{t_0}^{t} (\mathcal{CE}(s) - \mathcal{CE}^*) \, ds = \mathcal{CB}(t_0, t) - \mathcal{CE}^*(t - t_0) $$

**Example 40** In Table 9.17, we report the historical data of carbon emissions from 2010 to 2020. Moreover, the company has announced its carbon targets for the years until 2050.

<table><caption>Table 9.17: Carbon emissions in MtCO₂e</caption><thead><tr><th>t</th><th>2010</th><th>2011</th><th>2012</th><th>2013</th><th>2014</th><th>2015</th><th>2016</th><th>2017</th></tr></thead><tbody><tr><td>C<sub>E</sub>(t)</td><td>4.800</td><td>4.950</td><td>5.100</td><td>5.175</td><td>5.175</td><td>5.175</td><td>5.175</td><td>5.100</td></tr><tr><td>t</td><td>2018</td><td>2019</td><td>2020</td><td>2025*</td><td>2030*</td><td>2035*</td><td>2040*</td><td>2050*</td></tr><tr><td>C<sub>E</sub>(t)</td><td>5.025</td><td>4.950</td><td>4.875</td><td>4.200</td><td>3.300</td><td>1.500</td><td>0.750</td><td>0.150</td></tr></tbody></table>

The asterisk * indicates that the company has announced a carbon target for this year.

We consider the carbon pathway given in Example 40 and report different carbon budgets in Figure 9.20. The first panel (top/left) corresponds to the carbon budget that was spent by the company from 2010 to 2020. The second panel (top/right) is the targeted carbon budget that is estimated or planned by the company for the period between 2020 and 2035. The last two panels (bottom/left and bottom/right) considers a constant reference level, which may be for example the
average target of the industry. If we assume that the reference level $\mathcal{CE}^*$ is equal to 3 MtCO$_2$e (bottom/left panel), we notice that the net carbon budget is the difference between two areas. From January 2020 to October 2030, the carbon emissions are greater than $\mathcal{CE}^*$ and this period has a positive contribution to the carbon budget. On the contrary, the period from November 2030 to December 2035 has a negative contribution. On average, the net carbon budget is positive. In the case where the reference level $\mathcal{CE}^*$ is equal to 4 MtCO$_2$e (bottom/right panel), the excess carbon budget is negative.

<figure><img src="image_457.png" /><figcaption>Figure 9.20: Past, expected and net carbon budgets (Example 40)</figcaption></figure>

## Computation of the carbon budget (numerical solution)

We consider the equally-spaced partition $\{[t_0, t_0 + \Delta t], \dots, [t - \Delta t, t]\}$ of $[t_0, t]$. Let $m = \frac{t - t_0}{\Delta t}$ be the number of intervals. We set $\mathcal{CE}_k = \mathcal{CE}(t_0 + k\Delta t)$. The right Riemann approximation is:

$$ \mathcal{CB}(t_0, t) = \int_{t_0}^{t} \mathcal{CE}(s) \, ds \approx \sum_{k=1}^{m} \mathcal{CE}(t_0 + k\Delta t) \Delta t = \Delta t \sum_{k=1}^{m} \mathcal{CE}_k $$

If we use the left Riemann sum, we obtain:

$$ \mathcal{CB}(t_0, t) \approx \Delta t \sum_{k=0}^{m-1} \mathcal{CE}_k $$

Finally, the midpoint rule is given by:

$$ \mathcal{CB}(t_0, t) \approx \Delta t \sum_{k=1}^{m} \mathcal{CE} \left( t_0 + \frac{k}{2} \Delta t \right) $$
In the case of a yearly partition, the previous formulas are simplified since we have $\Delta t = 1$. For instance, the left Riemann sum becomes:

$$ \mathcal{CB}(t_0, t) = \sum_{k=0}^{m-1} \mathcal{CE}_k = \mathcal{CE}(t_0) + \dots + \mathcal{CE}(t-1) $$

If we consider Example 40, the carbon budget from 1st January 2010 to 1st January 2020 is equal to:

$$ \begin{aligned} \mathcal{CB}(2010, 2020) &= 4.8 + 4.95 + 5.1 + 5.175 + 5.175 + 5.175 + 5.175 + 5.1 + 5.025 + 4.95 \\ &= 50.625 \text{ MtCO}_2\text{e} \end{aligned} $$

**Remark 107** Instead of Riemann sums, we can use more sophisticated methods such as trapezoidal and Simpson's rules (Roncalli, 2020a, Section A.1.2.3, pages 1037-1041). We can also interpolate the carbon emissions with spline functions and then implement a Gaussian quadrature.

### Computation of the carbon budget (analytical solution)

**Constant reduction rate** If we use a constant linear reduction rate $\mathcal{R}(t_0, t) = \mathcal{R}(t - t_0)$, we obtain the following analytical expression:

$$ \mathcal{CB}(t_0, t) = \int_{t_0}^{t} (\mathcal{CE}(t_0) - \mathcal{R}(s - t_0)) \, ds = (t - t_0) \mathcal{CE}(t_0) - \frac{(t - t_0)^2}{2} \mathcal{R} \quad (9.6) $$

In the case of a constant compound reduction rate:

$$ \mathcal{CE}(t) = (1 - \mathcal{R})^{(t-t_0)} \mathcal{CE}(t_0) $$

we obtain:

$$ \begin{aligned} \mathcal{CB}(t_0, t) &= \mathcal{CE}(t_0) \int_{t_0}^{t} (1 - \mathcal{R})^{(s-t_0)} \, ds \\ &= \mathcal{CE}(t_0) \left[ \frac{(1 - \mathcal{R})^{(s-t_0)}}{\ln(1 - \mathcal{R})} \right]_{t_0}^{t} \\ &= \frac{(1 - \mathcal{R})^{(t-t_0)} - 1}{\ln(1 - \mathcal{R})} \mathcal{CE}(t_0) \end{aligned} \quad (9.7) $$

If we assume that $\mathcal{CE}(t) = e^{-\mathcal{R}(t-t_0)}\mathcal{CE}(t_0)$, we have:

$$ \mathcal{CB}(t_0, t) = \mathcal{CE}(t_0) \left[ - \frac{e^{-\mathcal{R}(t-t_0)}}{\mathcal{R}} \right]_{t_0}^{t} = \mathcal{CE}(t_0) \frac{(1 - e^{-\mathcal{R}(t-t_0)})}{\mathcal{R}} \quad (9.8) $$

**Remark 108** If the carbon emissions increase at a positive growth rate $g$, we set $\mathcal{R} = -g$.

According to IPCC (2018), the probability that the temperature $\mathcal{T}$ remains below 1.5°C by 2050 depends on a carbon budget. They estimated that the remaining carbon budget $\mathcal{CB}(2019, t)$ is 580 GtCO$_2$e for a 50% probability of limiting warming to 1.5°C, 420 GtCO$_2$e for a 66% probability and 300 GtCO$_2$e for a 83% probability. In Figure 9.21, we have computed $\mathcal{CB}(2019, t)$ by setting $\mathcal{CE}(2019) = 36 \text{ GtCO}_2\text{e}$ and assuming a constant compound reduction rate $\mathcal{R}$. If nothing is done, the probability to reach 1.5°C by 2035 is close to 50% since we obtain $\mathcal{CB}(2019, 2035) = 571.41 \text{ GtCO}_2\text{e}$. With a reduction rate of 7%, we have a probability of 65% to limit global warming to 1.5°C by 2050. This explains that many reduction targets are calibrated on this figure, for example the Paris aligned benchmarks (CTB and PAB).
<figure><img src="image_458.png" /><figcaption>Figure 9.21: Probability to reach 1.5°C</figcaption></figure>

**Linear function** If we assume that $\mathcal{CE}(t) = \beta_0 + \beta_1 t$, we deduce that:

$$ \begin{aligned} \mathcal{CB}(t_0, t) &= \int_{t_0}^{t} (\beta_0 + \beta_1 s) \, ds \\ &= \left[ \beta_0 s + \frac{1}{2} \beta_1 s^2 \right]_{t_0}^{t} \\ &= \beta_0 (t - t_0) + \frac{1}{2} \beta_1 (t^2 - t_0^2) \end{aligned} \qquad (9.9) $$

We can extend this formula to a piecewise linear function. We assume that $\mathcal{CE}(t)$ is known for $t \in \{t_0, t_1, \dots, t_m\}$ and $\mathcal{CE}(t)$ is linear between two consecutive dates:

$$ \mathcal{CE}(t) = \mathcal{CE}(t_{k-1}) + \frac{\mathcal{CE}(t_k) - \mathcal{CE}(t_{k-1})}{t_k - t_{k-1}} (t - t_{k-1}) \quad \text{if } t \in [t_{k-1}, t_k] $$

We notice that this equation can be written as:

$$ \mathcal{CE}(t) = \underbrace{\frac{t_k}{t_k - t_{k-1}} \mathcal{CE}(t_{k-1}) - \frac{t_{k-1}}{t_k - t_{k-1}} \mathcal{CE}(t_k)}_{\beta_{0,k}} + \underbrace{\frac{\mathcal{CE}(t_k) - \mathcal{CE}(t_{k-1})}{t_k - t_{k-1}} t}_{\beta_{1,k}} $$

We deduce that:

$$ \mathcal{CB}(t_0, t) = \sum_{k=1}^{k(t)} \int_{t_{k-1}}^{t_k} \mathcal{CE}(s) \, ds + \int_{t_{k(t)}}^{t} \mathcal{CE}(s) \, ds $$
where $k(t) = \{\max k : t_k \le t\}$. Using Equation (9.9), we conclude that<fnref n="33" />:

$$ \mathcal{CB}(t_0, t) = \sum_{k=1}^{k(t)} \beta_{0,k}^{*} (t_k - t_{k-1}) + \frac{1}{2} \sum_{k=1}^{k(t)} \beta_{1,k} (t_k^2 - t_{k-1}^2) + \beta_{0,k(t)+1} (t - t_{k(t)}) + \frac{1}{2} \beta_{1,k(t)+1} (t^2 - t_{k(t)}^2) \quad (9.10) $$

If we consider Example 40 and assume that the carbon emissions are linear between two consecutive years, the carbon budget from 1st January 2010 to 1st January 2020 is equal to 50.662 MtCO₂e, which is close to the value 50.625 MtCO₂e obtained previously. When we consider the carbon targets, the Riemann sums are not appropriate because the targets are measured every five or ten years. Since the objective is that the company reduces continuously its carbon emissions, it is better to compute the carbon budget by assuming a piecewise linear function. In our example, we obtain **CB** (2020, 2035) = 53.437 MtCO₂e with the following decomposition<fnref n="34" />: **CB** (2020, 2025) = 22.687 MtCO₂e (42.46%), **CB** (2025, 2030) = 18.750 MtCO₂e (35.09%) and **CB** (2030, 2035) = 12.000 MtCO₂e (22.46%). We also have **CB** (2020, 2050) = 63.562 MtCO₂e, implying that the first period 2020–2035 represents 84.07% of the carbon emissions.

<table><caption>Table 9.18: IEA NZE scenario (in GtCO₂e)</caption><thead><tr><th>Sector</th><th>2010</th><th>2011</th><th>2012</th><th>2013</th><th>2014</th><th>2015</th><th>2016</th><th>2017</th><th>2018</th><th>2019</th></tr></thead><tbody><tr><td>Electricity</td><td>12.4</td><td>13</td><td>13.3</td><td>13.5</td><td>13.6</td><td>13.3</td><td>13.3</td><td>13.5</td><td>14</td><td>13.8</td></tr><tr><td>Buildings</td><td>2.89</td><td>2.81</td><td>2.78</td><td>2.9</td><td>2.84</td><td>2.87</td><td>2.91</td><td>2.95</td><td>2.98</td><td>3.01</td></tr><tr><td>Transport</td><td>7.01</td><td>7.13</td><td>7.18</td><td>7.37</td><td>7.5</td><td>7.72</td><td>7.88</td><td>8.08</td><td>8.25</td><td>8.29</td></tr><tr><td>Industry</td><td>8.06</td><td>8.47</td><td>8.57</td><td>8.71</td><td>8.78</td><td>8.71</td><td>8.56</td><td>8.52</td><td>8.72</td><td>8.9</td></tr><tr><td>Other</td><td>1.87</td><td>1.89</td><td>1.91</td><td>1.96</td><td>1.87</td><td>1.89</td><td>1.89</td><td>1.92</td><td>1.92</td><td>1.91</td></tr><tr><td>Gross emissions</td><td>32.2</td><td>33.3</td><td>33.7</td><td>34.4</td><td>34.5</td><td>34.5</td><td>34.5</td><td>35</td><td>35.9</td><td>35.9</td></tr><tr><td>BECCS/DACCS</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Net emissions</td><td>32.2</td><td>33.3</td><td>33.7</td><td>34.4</td><td>34.5</td><td>34.5</td><td>34.5</td><td>35</td><td>35.9</td><td>35.9</td></tr></tbody></table>

<table><thead><tr><th>Sector</th><th>2020</th><th>2025</th><th>2030</th><th>2035</th><th>2040</th><th>2045</th><th>2050</th></tr></thead><tbody><tr><td>Electricity</td><td>13.5</td><td>10.8</td><td>5.82</td><td>2.12</td><td>-0.08</td><td>-0.31</td><td>-0.37</td></tr><tr><td>Buildings</td><td>2.86</td><td>2.43</td><td>1.81</td><td>1.21</td><td>0.69</td><td>0.32</td><td>0.12</td></tr><tr><td>Transport</td><td>7.15</td><td>7.23</td><td>5.72</td><td>4.11</td><td>2.69</td><td>1.5</td><td>0.69</td></tr><tr><td>Industry</td><td>8.48</td><td>8.14</td><td>6.89</td><td>5.25</td><td>3.48</td><td>1.8</td><td>0.52</td></tr><tr><td>Other</td><td>1.91</td><td>1.66</td><td>0.91</td><td>0.09</td><td>-0.46</td><td>-0.82</td><td>-0.96</td></tr><tr><td>Gross emissions</td><td>33.9</td><td>30.3</td><td>21.5</td><td>13.7</td><td>7.77</td><td>4.3</td><td>1.94</td></tr><tr><td>BECCS/DACCS</td><td>0</td><td>-0.06</td><td>-0.32</td><td>-0.96</td><td>-1.46</td><td>-1.8</td><td>-1.94</td></tr><tr><td>Net emissions</td><td>33.9</td><td>30.2</td><td>21.1</td><td>12.8</td><td>6.32</td><td>2.5</td><td>0.00</td></tr></tbody></table>

Source: IEA (2021, Figure 2.3, page 55).

$$ \mathcal{CB}(t_0, t) = \sum_{k=1}^{k(t)} (\mathcal{CE}(t_{k-1}) t_k - \mathcal{CE}(t_k) t_{k-1}) + \frac{1}{2} \sum_{k=1}^{k(t)} (\mathcal{CE}(t_k) - \mathcal{CE}(t_{k-1})) (t_k + t_{k-1}) $$

$$ \mathcal{CB}(t_0, t_2) = \int_{t_0}^{t_2} \mathcal{CE}(s) \, ds = \int_{t_0}^{t_1} \mathcal{CE}(s) \, ds + \int_{t_1}^{t_2} \mathcal{CE}(s) \, ds = \mathcal{CB}(t_0, t_1) + \mathcal{CB}(t_1, t_2) $$

<footnote n="33">³³When $t$ belongs to the set $\{t_0, t_1, \dots, t_m\}$, we can simplify this expression as follows:</footnote>

<footnote n="34">³⁴We use the Chasles property of the Riemann integral:</footnote>
**Example 41** We consider the net-zero emissions (NZE) scenario provided by the International Energy Agency (IEA, 2021). We remind that it is a normative scenario that shows a pathway for the global energy sector to achieve net-zero CO₂e emissions by 2050. For each important sector, IEA gives the past trajectory of carbon emissions and the decarbonization pathway that could be achievable (Table 9.18). This net-zero scenario has been calibrated with a carbon budget of approximately 500 GtCO₂e.

<figure><img src="image_459.png" /><figcaption>Figure 9.22: CO₂ emissions by sector in the IEA NZE scenario (in GtCO₂e)</figcaption></figure>

Source: IEA (2021) & Author's calculations.

In Figure 9.22, we show the decarbonization pathway of each sector and the global economy. We notice the importance of the technologies bioenergy with carbon capture and storage (BECCS) and direct air carbon capture with carbon storage (DACCS). Indeed, they will help to compensate the remaining 2 GtCO₂e of carbon emissions. In Table 9.19, we have computed the carbon budget CB (2019, t) for each sector and the global system. We notice that the sectors Electricity, Industry and Transport represent about 85% of the global budget.

<table><caption>Table 9.19: Carbon budget in the IEA NZE scenario (in GtCO₂e)</caption><thead><tr><th>t</th><th>Electricity</th><th>Buildings</th><th>Transport</th><th>Industry</th><th>Other</th><th>Gross emissions</th></tr></thead><tbody><tr><td>2025</td><td>74.4</td><td>50.2</td><td>43.7</td><td>16.2</td><td>10.8</td><td>195.4</td></tr><tr><td>2030</td><td>115.9</td><td>87.8</td><td>76.0</td><td>26.8</td><td>17.3</td><td>324.9</td></tr><tr><td>2040</td><td>140.9</td><td>140.0</td><td>117.6</td><td>39.1</td><td>18.8</td><td>466.6</td></tr><tr><td>2045</td><td>139.9</td><td>153.2</td><td>128.1</td><td>41.6</td><td>15.6</td><td>496.8</td></tr><tr><td>2050</td><td>138.2</td><td>159.0</td><td>133.6</td><td>42.7</td><td>11.2</td><td>512.4</td></tr></tbody></table>

Source: IEA (2021) & Author's calculations.
### 9.3.2 Carbon trend

#### Linear trend model

Le Guenedal et al. (2022) defined the carbon trend by considering the linear trend model:

$$ \mathcal{CE}(t) = \beta_0 + \beta_1 t + u(t) \qquad (9.11) $$

where $u(t) \sim \mathcal{N}(0, \sigma_u^2)$. We estimate the parameters $\beta_0$ and $\beta_1$ with the least squares method and a sample of observations. Therefore, the projected carbon trajectory is given by:

$$ \mathcal{CE}^{\text{Trend}}(t) = \hat{\mathcal{CE}}(t) = \hat{\beta}_0 + \hat{\beta}_1 t \qquad (9.12) $$

Let {$t_{\text{First}}, t_{\text{First}} + 1, \dots, t_{\text{Last}}$} be the set of observation dates. We interpret $\mathcal{CE}^{\text{Trend}}(t)$ as follows:

* If $t < t_{\text{First}}$, $\mathcal{CE}^{\text{Trend}}(t)$ is the back-calculated value of carbon emissions before the first observation date (retropolation);

* If $t_{\text{First}} \le t \le t_{\text{Last}}$, $\mathcal{CE}^{\text{Trend}}(t)$ is the predicted value of carbon emissions and can be compared with the observed value of carbon emissions (out-of-sample estimation);

* If $t > t_{\text{Last}}$, $\mathcal{CE}^{\text{Trend}}(t)$ is the forecast value of carbon emissions and could be compared with the future value of carbon emissions when this later will be available (out-of-sample estimation);

This model is very simple and the underlying idea is to extrapolate the past trajectory. Nevertheless, we can derive several metrics that are useful to compare the existing track record of the issuer with its willingness to really reduce its carbon emissions.

Equation (9.12) is not easy to interpret, because the intercept $\hat{\beta}_0$ corresponds to the estimated value $\hat{\mathcal{CE}}(0)$ at time $t = 0$. Then, it is convenient to use another base year $t_0$, implying that Equation (9.11) becomes:

$$ \mathcal{CE}(t) = \beta'_{0} + \beta'_{1}(t - t_{0}) + u(t) \qquad (9.13) $$

In this case, the carbon trend is given by:

$$ \mathcal{CE}^{\text{Trend}}(t) = \hat{\beta}'_{0} + \hat{\beta}'_{1}(t - t_{0}) \qquad (9.14) $$

We can show that the two models (9.12) and (9.14) are equivalent and give the same value $\hat{\mathcal{CE}}(t)$. Indeed, we have the following relationships:

$$ \left\{ \begin{array}{l} \beta'_{0} = \beta_{0} + \beta_{1}t_{0} \\ \beta'_{1} = \beta_{1} \end{array} \right. $$

The new parameterization does not change the slope of the trend, but only the constant $\hat{\beta}'_0$ which is now equal to $\hat{\mathcal{CE}}(t_0)$.

**Remark 109** The previous approach can be extended to the carbon intensity measure $\mathcal{CI}(t)$.

**Example 42** In Table 9.20, we report the evolution of scope 1 + 2 carbon emissions for company A.
<table><caption>Table 9.20: Carbon emissions in MtCO₂e (company A)</caption><thead><tr><th>Year</th><th>2007</th><th>2008</th><th>2009</th><th>2010</th><th>2011</th><th>2012</th><th>2013</th></tr></thead><tbody><tr><th><i>CE</i> (t)</th><td>57.8</td><td>58.4</td><td>57.9</td><td>55.1</td><td>51.6</td><td>48.3</td><td>47.1</td></tr><tr><th>Year</th><th>2014</th><th>2015</th><th>2016</th><th>2017</th><th>2018</th><th>2019</th><th>2020</th></tr><tr><th><i>CE</i> (t)</th><td>46.1</td><td>44.4</td><td>42.7</td><td>41.4</td><td>40.2</td><td>41.9</td><td>45.0</td></tr></tbody></table>

Using the carbon emissions given in Table 9.20, we obtain the following estimates<fnref n="35" />: $\hat{\beta}_0 = 2.970.43$ and $\hat{\beta}_1 = -1.4512$. If we consider the regression model (9.13), the results become $\hat{\beta}'_0 = 57.85$ and $\hat{\beta}'_1 = -1.4512$ if the base year $t_0$ is set to 2007, and $\hat{\beta}'_0 = 38.99$ and $\hat{\beta}'_1 = -1.4512$ if the base year $t_0$ is set to 2020. We verify that all the figures are coherent:

$$ \begin{aligned} \mathcal{CE}^{\text{Trend}}(t) &= 38.99 - 1.4512 \times (t - 2020) \\ &= 2970.43 - 1.4512 \times t \end{aligned} $$

We notice that the trend model is more intuitive if we use the base year $t_0 = 2020$. The estimated carbon emissions is equal to 38.99 MtCO₂e in 2020 and we observe a reduction of 1.4512 MtCO₂e every year. For instance, the forecast value for the year 2025 is:

$$ \mathcal{CE}^{\text{Trend}}(2025) = 38.99 - 1.4512 \times 5 = 31.73 \text{ MtCO}_2\text{e} $$

We have reported the in-sample estimated values and out-of-sample forecast values in Figure 9.23. When $t$ is set to the year 2020, we observe that there is a gap between the observed value $\mathcal{CE}(t)$ and the estimated value $\hat{\mathcal{CE}}(t)$ because $\mathcal{CE}(2020) = 45.0 \gg \hat{\mathcal{CE}}(2020) = 38.99$. We deduce that the current carbon emissions are greater than the figure given by the trend, meaning that the company has made less effort in recent years compared to the past history. We can then rescale the trend model by imposing that the last value $\mathcal{CE}(t_{\text{Last}})$ is equal to the estimated value $\hat{\mathcal{CE}}(t_{\text{Last}})$. We deduce that:

$$ \hat{\beta}'_0 + \hat{\beta}'_1 (t_{\text{Last}} - t_0) = \mathcal{CE}(t_{\text{Last}}) \Leftrightarrow \hat{\beta}'_0 = \mathcal{CE}(t_{\text{Last}}) - \hat{\beta}'_1 (t_{\text{Last}} - t_0) $$

If $t_{\text{Last}} = t_0$, we obtain $\hat{\beta}'_0 = \mathcal{CE}(t_{\text{Last}})$. In our example, the rescaled model has the following expression:

$$ \mathcal{CE}^{\text{Trend}}(t) = 45 - 1.4512 \times (t - 2020) $$

In Figure 9.23, we verify that the rescaled model has the same slope as previously, but it is now coherent with the last observation of carbon emissions. In the sequel, we will always consider rescaled trend models, because they are more relevant.

## Log-linear trend model

Instead of a linear model, we can use a log-linear trend model:

$$ \ln \mathcal{CE}(t) = \gamma_0 + \gamma_1 (t - t_0) + v(t) \quad (9.15) $$

where $v(t) \sim N(0, \sigma_v^2)$. Again, we estimate the parameters $\gamma_0$ and $\gamma_1$ by ordinary least squares. Let $Y(t) = \ln \mathcal{CE}(t)$ be the logarithmic transform of the carbon emissions. We have:

$$ \hat{Y}(t) = \hat{\gamma}_0 + \hat{\gamma}_1 (t - t_0) $$

<footnote n="35">³⁵ $\hat{\sigma}_u$ is equal to 2.5844.</footnote>
<figure><img src="image_460.png" /><figcaption>Figure 9.23: Linear carbon trend (Example 42)</figcaption></figure>

and:

$$ 
\begin{aligned}
\hat{\mathcal{C}\mathcal{E}}(t) &= \exp(\hat{Y}(t)) \\
&= \exp(\hat{\gamma}_0 + \hat{\gamma}_1(t - t_0)) \\
&= \hat{\mathcal{C}\mathcal{E}}(t_0) \exp(\hat{\gamma}_1(t - t_0))
\end{aligned}
$$

where $\hat{\mathcal{C}\mathcal{E}}(t_0) = \exp(\hat{\gamma}_0)$. The estimator (9.15) does not take into account the variance bias of log-linear models. Indeed, the correct value of the mathematical expectation is equal to$^{36}$:

$$ 
\begin{aligned}
\mathbb{E}[\mathcal{C}\mathcal{E}(t)] &= \mathbb{E}[e^{Y(t)}] \\
&= \mathbb{E}[\mathcal{LN}(\gamma_0 + \gamma_1(t - t_0), \sigma_v^2)] \\
&= \exp\left(\gamma_0 + \gamma_1(t - t_0) + \frac{1}{2}\sigma_v^2\right)
\end{aligned}
$$

Therefore, we obtain:

$$ 
\begin{aligned}
\hat{\mathcal{C}\mathcal{E}}(t) &= \exp\left(\hat{\gamma}_0 + \hat{\gamma}_1(t - t_0) + \frac{1}{2}\hat{\sigma}_v^2\right) \\
&= \hat{\mathcal{C}\mathcal{E}}(t_0) \exp(\hat{\gamma}_1(t - t_0))
\end{aligned}
$$

where $\hat{\mathcal{C}\mathcal{E}}(t_0) = \exp(\hat{\gamma}_0 + \frac{1}{2}\hat{\sigma}_v^2)$. Again, we can rescale the trend model such that $\hat{\mathcal{C}\mathcal{E}}(t_{\mathcal{L}ast}) = \mathcal{C}\mathcal{E}(t_{\mathcal{L}ast})$. It follows that:

$$ \hat{\gamma}_0 + \frac{1}{2}\hat{\sigma}_v^2 = \ln \mathcal{C}\mathcal{E}(t_{\mathcal{L}ast}) - \hat{\gamma}_1(t_{\mathcal{L}ast} - t_0) $$

<footnote>$^{36}$We remind that $\mathbb{E}[X] = \exp(\mu + \frac{1}{2}\sigma^2)$ if $X \sim \mathcal{LN}(\mu, \sigma^2)$ (see Section A.2.1 on page 1153).</footnote>
If the base year $t_0$ is equal to the last year $t_{Last}$, the forecast value is equal to:

$$ \mathcal{C}\mathcal{E}^{Trend}(t) = \mathcal{C}\mathcal{E}(t_0) \exp(\hat{\gamma}_1(t - t_0)) \qquad (9.16) $$

**Remark 110** While the slope of the trend is measured in CO₂e in the linear trend model, it is measured in % in the log-linear model. In fact, we estimate an absolute trend in the former model and a relative trend in the later model. From Equation (9.15), we have:

$$ \frac{\partial \mathcal{C}\mathcal{E}(t)}{\partial t} = \frac{\exp(\gamma_0 + \gamma_1(t - t_0) + v(t))}{\partial t} = \gamma_1 \mathcal{C}\mathcal{E}(t) $$

We verify that the slope $\gamma_1$ is the relative variation of carbon emissions:

$$ \frac{\partial \mathcal{C}\mathcal{E}(t)}{\partial t} = \frac{\partial \ln \mathcal{C}\mathcal{E}(t)}{\partial t} = \gamma_1 $$

Using Example 42 and the 2020 base year, we obtain the following results: $\hat{\gamma}_0 = 3.6800$, $\hat{\gamma}_1 = -2.95\%$ and $\hat{\sigma}_v = 0.0520$. It follows that $\hat{\mathcal{C}}\mathcal{E}(2020)$ takes the value 39.65 MtCO₂e without the correction of the variance bias and 39.70 MtCO₂e with the correction of the variance bias. Using the parameterization (9.16), we compare the estimated log-linear trend with the estimated linear trend in Figure 9.24. We notice that the log-linear trend is convex and the future reduction rate of carbon emissions are less important than those obtained with the linear model.

<figure><img src="image_461.png" /><figcaption>Figure 9.24: Log-linear carbon trend (Example 42)</figcaption></figure>
<table><caption>**Example 43** We consider several historical trajectories of scope 1 carbon emissions:</caption><thead><tr><th>#</th><th>2010</th><th>2011</th><th>2012</th><th>2013</th><th>2014</th><th>2015</th><th>2016</th><th>2017</th><th>2018</th><th>2019</th><th>2020</th></tr></thead><tbody><tr><td>1</td><td>10.0</td><td>11.1</td><td>10.5</td><td>12.5</td><td>13.0</td><td>14.8</td><td>16.0</td><td>16.5</td><td>17.0</td><td>17.5</td><td>19.8</td></tr><tr><td>2</td><td>10.0</td><td>10.2</td><td>10.5</td><td>11.0</td><td>10.8</td><td>10.8</td><td>13.0</td><td>12.5</td><td>13.5</td><td>13.6</td><td>13.6</td></tr><tr><td>3</td><td>10.0</td><td>9.9</td><td>9.5</td><td>9.0</td><td>9.3</td><td>8.8</td><td>8.7</td><td>8.5</td><td>9.0</td><td>8.0</td><td>8.2</td></tr><tr><td>4</td><td>10.0</td><td>9.5</td><td>9.0</td><td>9.0</td><td>8.3</td><td>8.1</td><td>7.7</td><td>6.5</td><td>7.0</td><td>6.1</td><td>6.2</td></tr></tbody></table>

For each historical trajectory, we estimate the two models and report the estimated trends in Figure 9.25. When the slope is positive, the log-linear trend is systematically above the linear trend in the long run ($t \to \infty$). When the slope is negative, we observe the opposite phenomenon. In this last case, the linear trend $\hat{\mathcal{C}\mathcal{E}}(\infty)$ tends to $-\infty$ while the log-linear trend $\hat{\mathcal{C}\mathcal{E}}(\infty)$ tends to 0. The fact that carbon emissions are negative may be disturbing. From a theoretical viewpoint, it is not impossible because of the impact of negative emissions (due to negative emissions, carbon credits or carbon removal methods for instance). Nevertheless, it is extremely rare especially if we take into account scope 3 emissions. From a modeling viewpoint, it is then better to impose that carbon emissions are positive.

<figure><img src="image_462.png" /><figcaption>Figure 9.25: Log-linear vs. linear carbon trend (Example 43)</figcaption></figure>

It seems then that the log-linear model is more relevant when the trend is negative. If we consider companies with a positive trend, the log-linear model may produce exploding carbon emissions. Let us compute the ratio of expected growth rates between $t_0$ and $t$. For the linear model, the expected growth rate is:

$$ \frac{\hat{\mathcal{C}\mathcal{E}}(t) - \mathcal{C\mathcal{E}}(t_0)}{\mathcal{C\mathcal{E}}(t_0)} = \frac{\mathcal{C\mathcal{E}}(t_0) + \hat{\beta}_1(t - t_0) - \mathcal{C\mathcal{E}}(t_0)}{\mathcal{C\mathcal{E}}(t_0)} = \frac{\hat{\beta}_1(t - t_0)}{\mathcal{C\mathcal{E}}(t_0)} $$
while the log-linear model gives:

$$ \frac{\widehat{\mathcal{CE}}(t) - \mathcal{CE}(t_0)}{\mathcal{CE}(t_0)} = \frac{\mathcal{CE}(t_0) \exp(\hat{\gamma}_1(t - t_0)) - \mathcal{CE}(t_0)}{\mathcal{CE}(t_0)} = \exp(\hat{\gamma}_1(t - t_0)) - 1 $$

Therefore, the ratio of expected growth rates is proportional to:

$$ \begin{aligned} \frac{\exp(\hat{\gamma}_1(t - t_0)) - 1}{\hat{\beta}_1(t - t_0)} &\approx \frac{1}{\hat{\beta}_1(t - t_0)} \sum_{n=1}^{\infty} \frac{\hat{\gamma}_1^n (t - t_0)^n}{n!} \\ &= \frac{\hat{\gamma}_1}{\hat{\beta}_1} \sum_{n=0}^{\infty} \frac{\hat{\gamma}_1^n (t - t_0)^n}{(n + 1)!} \\ &\approx \frac{\hat{\gamma}_1}{\hat{\beta}_1} \left( 1 + \frac{1}{2}\hat{\gamma}_1(t - t_0) + \frac{1}{6}\hat{\gamma}_1^2(t - t_0)^2 + \frac{1}{24}\hat{\gamma}_1^3(t - t_0)^3 + \dots \right) \end{aligned} $$

We deduce that the exploding effect cannot be avoided. This is why we must be very careful when we consider the log-linear model. One possible solution is to use the linear model when $\hat{\gamma}_1$ is greater than a threshold and the log-linear model otherwise. In Table 9.21, we compute the multiplication factor $M(\gamma_1, t) = e^{\gamma_1 t}$ for different values of the growth rate $\gamma_1$ and different time horizon. For instance, if we target the year 2050, we have $t \approx 30$ years, we notice that carbon emissions are multiplied by a factor greater than 10 if the growth rate is 8%. Since there are many uncertainties about data collection and data computation, an historical growth rate of 8% may be explained by several factors:

* The company has really increased its carbon emissions by 8%;

* The company has underestimated its carbon emissions in the past and is more conservative today;

* The company has changed the reporting perimeter;

* Etc.

Applying a factor of 10 with a 30-year time horizon may then be not realistic. This is why we must be careful when estimating the carbon trend and analyze the outlier companies.

<table><caption>Table 9.21: Multiplication factor $M(\gamma_1, t)$</caption><thead><tr><th rowspan="2">t<br>(in years)</th><th colspan="10">γ<sub>1</sub></th></tr><tr><th>1%</th><th>2%</th><th>3%</th><th>4%</th><th>5%</th><th>6%</th><th>7%</th><th>8%</th><th>9%</th><th>10%</th></tr></thead><tbody><tr><td>1</td><td>1.01</td><td>1.02</td><td>1.03</td><td>1.04</td><td>1.05</td><td>1.06</td><td>1.07</td><td>1.08</td><td>1.09</td><td>1.11</td></tr><tr><td>5</td><td>1.05</td><td>1.11</td><td>1.16</td><td>1.22</td><td>1.28</td><td>1.35</td><td>1.42</td><td>1.49</td><td>1.57</td><td>1.65</td></tr><tr><td>10</td><td>1.11</td><td>1.22</td><td>1.35</td><td>1.49</td><td>1.65</td><td>1.82</td><td>2.01</td><td>2.23</td><td>2.46</td><td>2.72</td></tr><tr><td>15</td><td>1.16</td><td>1.35</td><td>1.57</td><td>1.82</td><td>2.12</td><td>2.46</td><td>2.86</td><td>3.32</td><td>3.86</td><td>4.48</td></tr><tr><td>20</td><td>1.22</td><td>1.49</td><td>1.82</td><td>2.23</td><td>2.72</td><td>3.32</td><td>4.06</td><td>4.95</td><td>6.05</td><td>7.39</td></tr><tr><td>25</td><td>1.28</td><td>1.65</td><td>2.12</td><td>2.72</td><td>3.49</td><td>4.48</td><td>5.75</td><td>7.39</td><td>9.49</td><td>12.18</td></tr><tr><td>30</td><td>1.35</td><td>1.82</td><td>2.46</td><td>3.32</td><td>4.48</td><td>6.05</td><td>8.17</td><td>11.02</td><td>14.88</td><td>20.09</td></tr></tbody></table>

**Remark 111** The expected growth rate is related to the concept of carbon momentum that will be presented later on page 927.
Stochastic trend model

Following Roncalli (2020a), the linear trend model can be written as:

$$
\begin{cases}
y(t) = \mu(t) + u(t) \\
\mu(t) = \mu(t-1) + \beta_1
\end{cases}
$$

where $u(t) \sim \mathcal{N}(0, \sigma_u^2)$. In this case, we have $y(t) = \beta_0 + \beta_1 t + u(t)$ where $\beta_0 = \mu(t_0) - \beta_1 t_0$. A way to introduce a stochastic trend is to add a noise $\eta(t)$ in the trend equation: $\mu(t) = \mu(t-1)+\beta_1+\eta(t)$ where $\eta(t) \sim \mathcal{N}(0, \sigma_\eta^2)$. Let us now assume that the slope of the trend is also stochastic:

$$
\left\{
\begin{aligned}
y(t) &= \mu(t) + u(t) \\
\mu(t) &= \mu(t-1) + \beta_1(t-1) + \eta(t) \\
\beta_1(t) &= \beta_1(t-1) + \zeta(t)
\end{aligned}
\right.
$$

where $\zeta(t) \sim \mathcal{N}(0, \sigma_{\zeta}^2)$. This model is called the local linear trend (LLT) model (Roncalli, 2020a, page 653). Using the Kalman filter (KF), we can estimate both the stochastic trend $\mu(t)$ and the stochastic slope $\beta_1(t)$.

Let us come back to Example 42. We estimate the parameters $(\sigma_u, \sigma_\eta, \sigma_\zeta)$ by maximizing the Whittle log-likelihood function (Roncalli, 2020a, pages 686-687). We obtain $\hat{\sigma}_u = 0.7022$, $\hat{\sigma}_\eta = 0.7019$ and $\hat{\sigma}_\zeta = 0.8350$. We deduce that the standard deviation of the stochastic slope variation $\beta_1(t) - \beta_1(t-1)$ is equal to 0.8350 MtCO$_2$e. This indicates that there is a high uncertainty in the trend computation. Then, we run the Kalman filter to estimate $\hat{\mu}(t)$ and $\hat{\beta}_1(t)$. In Table 9.22, we report these values and compare them with the estimates using the rolling least squares (RLS). We notice that the time-varying slope produced by the Kalman filter may be very different from the one produced by the method of least squares$^{37}$. In particular, the magnitude of the variability is not the same. This is normal since the rolling least squares estimate a global slope from the beginning of the sample to time $t$ whereas the Kalman filter estimates a local slope for the period $[t-1, t]$. Therefore, $\hat{\beta}_i(t)$ is an estimator of the average slope in the case of the linear trend model and an estimator of the marginal slope in the case of the local linear trend model.

**Remark 112** With the stochastic trend model, Le Guenedal et al. (2022) introduce the concept of carbon velocity, which measures the normalized slope change between $t-h$ and $t$:

$$
\mathbf{v}^{(h)}(t) = \frac{\hat{\beta}_1(t) - \hat{\beta}_1(t-h)}{h}
$$

The rationale for this measure is the following. A commitment to reduce carbon emissions implies a negative trend: $\hat{\beta}_1(t) < 0$. Nevertheless, it can take many years for a company to change the sign of the trend slope if it has a bad track record. Therefore, we can use the velocity to verify that the company is making significant efforts in the recent period. In this case, we must have $\mathbf{v}^{(h)}(t) < 0$ for low values$^{38}$ of $h$. In the case of the local linear trend model, we notice that the one-step velocity is equal to the innovation of the slope:

$$
\mathbf{v}^{(1)}(t) = \hat{\beta}_1(t) - \hat{\beta}_1(t-1) = \hat{\zeta}(t)
$$

<footnote>$^{37}$For instance, we have $\hat{\beta}_1(2020) = -1.4512$ MtCO$_2$e with the least squares method and $\hat{\beta}_1(2020) = +1.7701$ MtCO$_2$e with the Kalman filter.</footnote>

<footnote>$^{38}$Generally, $h$ is equal to 1, 2 or 3 years.</footnote>
<table><caption>Table 9.22: Kalman filter estimation of the stochastic trend (Example 42)</caption><thead><tr><th>t</th><th>Cε (t)</th><th>β̂<sub>1</sub> (t)<br>(RLS)</th><th>β<sub>1</sub> (t)<br>(KF)</th><th>μ (t)<br>(KF)</th></tr></thead><tbody><tr><td>2007</td><td>57.80</td><td></td><td>0.0000</td><td>57.80</td></tr><tr><td>2008</td><td>58.40</td><td></td><td>0.2168</td><td>58.25</td></tr><tr><td>2009</td><td>57.90</td><td>0.0500</td><td>-0.0441</td><td>58.00</td></tr><tr><td>2010</td><td>55.10</td><td>-0.8600</td><td>-1.3941</td><td>55.56</td></tr><tr><td>2011</td><td>51.60</td><td>-1.5700</td><td>-2.6080</td><td>52.01</td></tr><tr><td>2012</td><td>48.30</td><td>-2.0200</td><td>-3.1288</td><td>48.47</td></tr><tr><td>2013</td><td>47.10</td><td>-2.0929</td><td>-2.2977</td><td>46.82</td></tr><tr><td>2014</td><td>46.10</td><td>-2.0321</td><td>-1.5508</td><td>45.85</td></tr><tr><td>2015</td><td>44.40</td><td>-1.9817</td><td>-1.5029</td><td>44.38</td></tr><tr><td>2016</td><td>42.70</td><td>-1.9406</td><td>-1.5887</td><td>42.73</td></tr><tr><td>2017</td><td>41.40</td><td>-1.8891</td><td>-1.4655</td><td>41.36</td></tr><tr><td>2018</td><td>40.20</td><td>-1.8329</td><td>-1.3202</td><td>40.15</td></tr><tr><td>2019</td><td>41.90</td><td>-1.6824</td><td>0.1339</td><td>41.41</td></tr><tr><td>2020</td><td>45.00</td><td>-1.4512</td><td>1.7701</td><td>44.45</td></tr></tbody></table>

Carbon momentum

Le Guenedal et al. (2022) define the long-term carbon momentum as the growth rate of carbon emissions. In the case of the linear trend model, we have:

$$CM^{Long}(t) = \frac{\hat{\beta}_1(t)}{CE(t)}$$

while it is directly equal to $\hat{\gamma}_1(t)$ in the case of the log-linear trend model:

$$CM^{Long}(t) = \hat{\gamma}_1(t)$$

Le Guenedal et al. (2022) also define the short-term carbon momentum as the one-year carbon velocity:

$$CM^{Short}(t) = \frac{v^{(1)}(t)}{CE(t)}$$

If we apply this concept to the log-linear model, we obtain $CM^{Short}(t) = v^{(1)}(t)$ where:

$$v^{(h)}(t) = \frac{\hat{\gamma}_1(t) - \hat{\gamma}_1(t-h)}{h}$$

In the case of the stochastic trend model, we have $CM^{Short}(t) = \hat{\zeta}(t)$.

**Remark 113** Carbon momentum plays a key role when we will define net-zero investment portfolios, because it is highly related to the concept of self-decarbonization<fnref n="39" />.

<footnote n="39">³⁹See Section 11.3 on page 1083.</footnote>
Application

Table 9.23 and 9.24 give some statistics about carbon momentum. It reproduces the results obtained by Barahhou et al. (2022) by considering the issuers of the MSCI World index. Since it is difficult to obtain at least 5-year historical data, we focus on the scopes $SC_1$, $SC_{1-2}$ and $SC_{1-3}^{up}$, and we do not consider the scope $SC_{1-3}$. If we use the linear trend model, the median value of $CM^{Long}(t)$ is equal to 0% for scope 1, 1.6% when we include the scope 2, and 2.3% when we add the upstream scope 3. The carbon momentum is negative for only 29.4% of issuers when we consider $SC_{1-3}^{up}$. This means that a majority of issuers have a positive carbon trend. For instance, about 10% of issuers have a carbon momentum greater than 10%! If we consider carbon intensity instead of carbon emission, we obtain another story. Indeed, issuers with a negative trend dominate issuers with a positive trend. Therefore, it is easier to build a self-decarbonized portfolio when we consider the carbon intensity measure. If we estimate the carbon momentum with the log-linear trend model, results are slightly different. For instance, 19.2% of issuers have a carbon momentum $SC_{1-3}^{up}$ greater than 10% versus 8.0% with the linear trend model.

<table><caption>Table 9.23: Statistics (in %) of carbon momentum $CM^{Long}(t)$ (MSCI World index, 1995 – 2021, linear trend)</caption><thead><tr><th rowspan="2">Statistics</th><th colspan="3">Carbon emissions</th><th colspan="3">Carbon intensity</th></tr><tr><th>$SC_1$</th><th>$SC_{1-2}$</th><th>$SC_{1-3}^{up}$</th><th>$SC_1$</th><th>$SC_{1-2}$</th><th>$SC_{1-3}^{up}$</th></tr></thead><tbody><tr><td>Median</td><td>0.0</td><td>1.6</td><td>2.3</td><td>-4.8</td><td>-2.4</td><td>-1.3</td></tr><tr><td>Negative</td><td>49.9</td><td>41.1</td><td>29.4</td><td>76.0</td><td>69.6</td><td>75.6</td></tr><tr><td>Positive</td><td>50.1</td><td>58.9</td><td>70.6</td><td>24.0</td><td>30.4</td><td>24.4</td></tr><tr><td>&lt; -10%</td><td>23.4</td><td>15.8</td><td>5.8</td><td>36.0</td><td>25.0</td><td>5.7</td></tr><tr><td>&lt; -5%</td><td>32.1</td><td>22.2</td><td>10.6</td><td>48.6</td><td>36.7</td><td>13.4</td></tr><tr><td>&gt; +5%</td><td>22.9</td><td>27.5</td><td>23.6</td><td>6.2</td><td>7.3</td><td>2.7</td></tr><tr><td>&gt; +10%</td><td>9.2</td><td>9.5</td><td>8.0</td><td>2.3</td><td>2.6</td><td>1.0</td></tr></tbody></table>

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

<table><caption>Table 9.24: Statistics (in %) of carbon momentum $CM^{Long}(t)$ (MSCI World index, 1995 – 2021, log-linear trend)</caption><thead><tr><th rowspan="2">Statistics</th><th colspan="3">Carbon emissions</th><th colspan="3">Carbon intensity</th></tr><tr><th>$SC_1$</th><th>$SC_{1-2}$</th><th>$SC_{1-3}^{up}$</th><th>$SC_1$</th><th>$SC_{1-2}$</th><th>$SC_{1-3}^{up}$</th></tr></thead><tbody><tr><td>Median</td><td>-0.1</td><td>1.7</td><td>2.8</td><td>-3.6</td><td>-1.9</td><td>-1.2</td></tr><tr><td>Negative</td><td>50.6</td><td>40.3</td><td>29.0</td><td>76.3</td><td>69.0</td><td>75.8</td></tr><tr><td>Positive</td><td>49.4</td><td>59.7</td><td>71.0</td><td>23.7</td><td>31.0</td><td>24.2</td></tr><tr><td>&lt; -10%</td><td>13.6</td><td>8.0</td><td>2.8</td><td>20.8</td><td>12.3</td><td>2.1</td></tr><tr><td>&lt; -5%</td><td>26.6</td><td>16.9</td><td>7.5</td><td>42.3</td><td>29.0</td><td>8.4</td></tr><tr><td>&gt; +5%</td><td>29.8</td><td>35.9</td><td>37.1</td><td>9.0</td><td>10.1</td><td>4.0</td></tr><tr><td>&gt; +10%</td><td>16.9</td><td>19.4</td><td>19.2</td><td>4.0</td><td>4.1</td><td>1.6</td></tr></tbody></table>

Source: Trucost database (2022) & Barahhou et al. (2022).
### 9.3.3 Participation, ambition and credibility for an alignment strategy

In this section, we define the three pillars that help to evaluate a company’s alignment strategy with respect to a given climate scenario, e.g., the net-zero emissions scenario. These three pillars are participation, ambition and credibility. They form the PAC framework, and they can be quantified using the tool of carbon budget.

#### Carbon target and decarbonization scenario

In addition to the historical pathway of carbon emissions, the PAC framework requires two other time series:

* The reduction targets announced by the company;

* The market-based sector scenario associated to the company that defines the decarbonization pathway.

Carbon reduction targets are defined by companies at a scope emissions level with different time horizons$^{40}$. For instance, the issuer can commit to reduce its scope 1 emissions by 50% over a period of 20 years and its scope 3 emissions by 30% over a period of 10 years. Even if the time frame of carbon reduction targets goes to 60 years, most of reduction targets concern the next twenty years. In the CDP database, we observe that most targets are underway or new, and a large proportion of companies set targets to reduce emissions by less than 50% from their base year. We also notice that some targets are reported over multiple scopes$^{41}$ and we can have multiple release dates. Therefore, it is important to transform these heterogeneous figures into a unique reduction pathway with one base year $t_0$:

$$ \mathbb{CT} = \{ \mathbf{R}^{\mathit{Target}}(t_0, t_k), k = 1, \dots, n_T \} $$

where $n_T$ is the number of targets and $\mathbf{R}^{\mathit{Target}}(t_0, t_k)$ is the reduction rate between $t_0$ and $t_k$ for the $k^{th}$ target.

Concerning the market-based scenario, we generally use sector scenarios provided by IPCC, IEA or IIASA. In some circumstances, we can take global scenarios, but only when we do not have the choice because there is no appropriate scenario for the sector. Again, the decarbonization scenario is defined as a set of reduction rates:

$$ \mathbb{CS} = \{ \mathbf{R}^{\mathit{Scenario}}(t_0, t_k), k = 1, \dots, n_S \} $$

where $n_S$ is the number of scenario data points. The reduction rate is calculated as follows:

$$ \mathbf{R}^{\mathit{Scenario}}(t_0, t_k) = 1 - \frac{\mathcal{C}\mathcal{E}^{\mathit{Scenario}}(t_k)}{\mathcal{C}\mathcal{E}^{\mathit{Scenario}}(t_0)} $$

where $t_0$ is the base year and $\mathcal{C}\mathcal{E}^S(t_k)$ is the value of carbon emissions at time $t_k$ in the market-based scenario. For instance, if we consider the IEA NZE scenario (see Table 9.18 on page 918), we obtain the results given in Table 9.25. Carbon emissions have been floored at zero in order to verify that the reduction rate is always less than or equal to 100%. We notice that the Electricity sector must decarbonize very quickly: -20% in 2025, -57% in 2030 and -84% in 2035. The carbon emissions reduction of the Industry and Transport sectors is delayed and really begins after 2025. If we consider the global scenario, the reduction rate is set to 10% in 2025 and increases by 5% every year until 2035.

<footnote>$^{40}$Carbon reduction targets can be found in the CDP database.</footnote>

<footnote>$^{41}$For instance, the target can concern only one scope, scope 1 + 2 or all scopes.</footnote>
<table><caption>Table 9.25: Reduction rates of the IEA NZE scenario (base year = 2020)</caption><thead><tr><th>Year</th><th>Electricity</th><th>Industry</th><th>Transport</th><th>Buildings</th><th>Other</th><th>Global</th></tr></thead><tbody><tr><td>2025</td><td>20.0</td><td>4.0</td><td>-1.1</td><td>15.0</td><td>13.1</td><td>10.6</td></tr><tr><td>2030</td><td>56.9</td><td>18.8</td><td>20.0</td><td>36.7</td><td>52.4</td><td>36.6</td></tr><tr><td>2035</td><td>84.3</td><td>38.1</td><td>42.5</td><td>57.7</td><td>95.3</td><td>59.6</td></tr><tr><td>2040</td><td>100.0</td><td>59.0</td><td>62.4</td><td>75.9</td><td>100.0</td><td>77.1</td></tr><tr><td>2045</td><td>100.0</td><td>78.8</td><td>79.0</td><td>88.8</td><td>100.0</td><td>87.3</td></tr><tr><td>2050</td><td>100.0</td><td>93.9</td><td>90.3</td><td>95.8</td><td>100.0</td><td>94.3</td></tr></tbody></table>

Source: IEA (2021) & Author's calculations.

## Definition of the PAC framework

The PAC framework has been introduced by Le Guenedal et al. (2022) and is based on the relative positioning of three carbon trajectories: (1) the historical trajectory and its trend, (2) the carbon targets and (3) the decarbonization scenario. It helps to answer several operational questions.

1. First, is the trend of the issuer in line with the scenario? In this case, we would like to know if the company has already reduced its carbon emissions. While the reduction targets correspond to future intentions, the carbon trend measures the past efforts of the company.

2. Is the commitment of the issuer to fight climate change ambitious? In particular, we would like to know if the target trajectory is above, below or in line with the market-based scenario, which is appropriate for the sector of the issuer. This is an important topic, because achieving the net-zero emissions scenario can only be possible if there are no free riders.

3. Finally, a third question is critical and certainly the most important issue. Is the target setting of the company relevant and robust? Indeed, we may wonder if the target trajectory is a too ambitious promise and a form of greenwashing or, on the contrary, a plausible decarbonization pathway.

Therefore, the assessment of the company's targets has three dimensions or pillars: (historical) participation, ambition and credibility. They form the PAC framework.

**Example 44** We consider again Example 42. Company A has announced the following targets: $R^{\text{Target}}(2020, 2025) = 40\%$, $R^{\text{Target}}(2020, 2030) = 50\%$, $R^{\text{Target}}(2020, 2035) = 75\%$, $R^{\text{Target}}(2020, 2040) = 80\%$ and $R^{\text{Target}}(2020, 2050) = 90\%$. Since company A is an utility corporation, we propose to use the IEA NZE scenario for the sector Electricity.

We have reported the different pathways of company A in Figure 9.26. We notice that the announced targets are below the carbon trend except in 2050. A comparison between the targets and the global scenario indicates that company A is more ambitious than the average firm. Nevertheless, the comparison is less favorable when we consider the decarbonization scenario of the corresponding sector. In order to quantify the relative the relative positioning of these trajectories, we compute the carbon budgets with the different pathways. If we consider the time horizon 2035, the carbon budget of the targets is slightly lower than the carbon budget of the decarbonization scenario (388 vs. 407 MtCO₂e). This indicates a true ambition to reduce its carbon emissions in line with what the market expects. Nevertheless, we observe a high carbon budget based on the trend model (512 MtCO₂e). This questions the credibility of the targets, even if the company has done some efforts in the past.
<table><caption>Table 9.26: Comparison of carbon budgets (base year = 2020, Example 44)</caption><thead><tr><th>Year</th><th>Trend (linear)</th><th>Trend (log-linear)</th><th>Target</th><th>Scenario (global)</th><th>Scenario (electricity)</th></tr></thead><tbody><tr><td>2025</td><td>207</td><td>209</td><td>180</td><td>213</td><td>203</td></tr><tr><td>2030</td><td>377</td><td>390</td><td>304</td><td>385</td><td>341</td></tr><tr><td>2035</td><td>512</td><td>546</td><td>388</td><td>502</td><td>407</td></tr><tr><td>2040</td><td>610</td><td>680</td><td>439</td><td>573</td><td>425</td></tr><tr><td>2045</td><td>671</td><td>796</td><td>478</td><td>613</td><td>425</td></tr><tr><td>2050</td><td>697</td><td>896</td><td>506</td><td>634</td><td>425</td></tr></tbody></table>

## Assessment of the PAC pillars

These three pillars depend on the carbon trajectories $CE(t)$, $CE^{Trend}(t)$, $CE^{Target}(t)$ and $CE^{Scenario}(t)$, where $CE(t)$ is the time series of historical carbon emissions, $CE^{Trend}(t)$ and $CE^{Target}(t)$ are the estimated carbon emissions deduced from the trend model and the targets, and $CE^{Scenario}(t)$ is the market-based decarbonization scenario. Generally, the participation only depends on the past observations and corresponds to the track record analysis of historical carbon emissions. The ambition compares the target trajectory on one side and the scenario or the trend on the other side. Indeed, we measure to what extent companies are willing to reverse their current carbon emissions and have objectives that match the scenario. Finally, we can measure the credibility of the targets by comparing the current trend of carbon emissions and the reduction targets or by analyzing the recent dynamics of the track record.

We note $t_{First}$ as the first date, $t_{Last}$ as the last reporting date and $t_{Scenario}$ as the target date of the decarbonization scenario. In Figure 9.27, we illustrate the underlying ideas of the PAC pillars. Let us consider the first three panels. They show the historical carbon emissions of different companies. It is obvious that the company in the top-left panel has a positive participation to slow global warming, whereas the participation of the company in the top-center panel is negative. In the top-right panel, we give three examples that are mixed. In this case, we do not observe a clear pattern: downward or upward trend of carbon emissions. Therefore, the company's participation can be measured by the metrics that are related to the carbon trend. The next three panels in Figure 9.27 illustrate the ambition pillar. In this case, we directly compare the carbon targets of the company and the market-based risk scenario. The companies belong to the same sector, implying that the decarbonization scenario is the same for the middle-left, middle-center and middle-right panels. The middle-left panel shows an ambitious company since its carbon targets are lower than the market-based scenario. In other words, the company has announced that it will make a greater effort than is expected by the market. On the contrary, the company in the middle-center panel is less ambitious, because it plans to reduce its carbon emissions at a slower pace. Finally, the middle-right panel presents two mixed situations. The first one concerns a company that has high ambitions at the beginning of the period [$t_{Last}, t_{Scenario}$] but it has not disclosed its ambitions for the end of the period. The company's ambition in the short term is then counterweighted by the absence of ambition in the long run. The second example is about a company that concentrates its ambition in the long run. These two examples question the true willingness of these companies to substantially reduce their carbon emissions. Finally, the credibility pillar is illustrated in the last three panels in Figure 9.27. In this case, we compare the carbon emissions trend and the targets communicated by the company. The bottom-left panel corresponds to a credible company, since it has announced more or less a reduction trajectory that is in line with what it has done in the
<figure><img src="image_463.png" /><figcaption>Figure 9.26: Carbon trend, targets and NZE scenario of company A</figcaption></figure>

Source: IEA (2021) & Author's calculations.

past. This is not the case of the company in the bottom-center panel. Clearly, it has announced a reduction of its carbon emissions, but it has continuously increased them in the past. Again, the bottom-right panel presents a mixed situation. The company has announced a reduction trajectory that is not very far from the past trend, but there are two issues. The first one is that it has increased its carbon emissions in the short term, implying that we can have some doubts about the downward trend. The second issue is that it accelerates its objective of carbon emissions reduction at the end of the period $[t_{Last}, t_{Scenario}]$ in order to meet the requirements of the market-based scenario, but its efforts are not very substantial in the short term.

Temperature scoring system

Le Guenedal et al. (2022) derive many metrics to measure the three dimensions. They can be classified into four main families. The gap metrics measure the differences between two trajectories or carbon budgets$^{42}$. The duration metrics calculate the time which is necessary to achieve a given objective$^{43}$. The velocity metrics assess the short-term dynamics of the carbon emissions. Finally,

$$ \text{Gap}(t, t^{\text{Scenario}}) = \mathbf{CB}^{\text{Trend}}(t, t^{\text{Scenario}}) - \mathbf{CB}^{\text{Scenario}}(t, t^{\text{Scenario}}) $$

$$ \mathcal{D}^{\text{Trend}} = \inf \{ t : \mathcal{CE}^{\text{Trend}}(t) \le 0 \} $$

<footnote>$^{42}$An example is the difference between the trend budget and the scenario budget:</footnote>

<footnote>$^{43}$For instance, the trend duration is the time horizon for achieving zero carbon emissions:</footnote>
<figure><img src="image_464.png" /><figcaption>Figure 9.27: Illustration of the participation, ambition and credibility pillars</figcaption></figure>

Source: Le Guenedal et al. (2022).

the last category computes the short-term reduction that the company must implement to satisfy the objective imposed by the market-based scenario<fnref n="44" />. Each pillar can then be studied using different metrics. Like the ESG risk, we can use a scoring system in order to analyze the *PAC* pillars and build three scores: the participation score *PS*ᵢ, the ambition score *AS*ᵢ and the credibility score *CS*ᵢ. In Figure 9.28, we have represented several configurations of the *PAC* scoring system. If the three scores *PS*ᵢ, *AS*ᵢ and *CS*ᵢ are high and greater than 0.5 (which is the median value of a *q*-score), the company is both ambitious and credible and has already made some efforts to reduce its carbon emissions (Panel (a)). On the contrary, in Panel (b), we have a company, whose three scores are below the median. These two extreme cases are very frequent. Nevertheless, we can also obtain a more balanced scoring. For instance, Panel (c) corresponds to a company that has substantially reduced its past emissions but has announced weak reduction targets. Therefore, its ambition score is low, but its credibility score is high. It may be a company that does not talk a lot about its climate change policy, but its track record has demonstrated that it is committed. Finally, Panel (d) represents the scoring of a company with very high ambition, but it has continuously increased its carbon emissions in the past. Therefore, we can suspect a type of greenwashing. These examples

<footnote n="44">⁴⁴The burn-out scenario refers to a sudden and violent reduction of carbon emissions such that the gap is equal to zero.</footnote>
show that the three dimensions are correlated. For instance, we can assume a positive correlation
between participation and credibility, and a negative correlation between ambition and credibility.
Indeed, high credibility can only be obtained if participation is high or ambition is weak. Similarly,
low credibility can be associated with excessively high ambition or weak participation, implying that
the correlation between participation and credibility is unclear.

<figure><img src="image_465.png" /><figcaption>Figure 9.28: The PAC scoring system</figcaption></figure>

Source: Le Guenedal et al. (2022).

**Remark 114** While the PAC framework is the backbone to analyze the decarbonization commitment of the company’s climate strategy (like the ESG pillars for analyzing the extra-financial risks of the company), the PAC scoring system is similar to the ESG scoring system. Most of implied temperature ratings are based on the participation, ambition and credibility pillars.

9.3.4 Illustration

We consider the analysis done by Le Guenedal et al. (2022). The base year is 2013 and they rebase the trajectories by the carbon emissions *CE* (2013). The market-based scenario is the IEA *NZE* scenario. Company *B* is a US based multinational technology conglomerate. Carbon emissions and targets are reported in Figure 9.29. Company *B* is a particularly relevant example of the expectations from investors in the *NZE* context. Indeed, participation switched favorably after 2018 with a significant reduction in the scope 3 emissions from the use of sold products. The credibility is confirmed with the 2020 data point as the duration *D*^Trend^ drops under the duration of the *NZE* scenario. Figure 9.30 illustrates Company *C* which is a major US airline. Its participation switched favorably in 2020. The credibility has not switched even with the significant drop in 2020, since the duration *D*^Trend^ remains larger than the *NZE* time horizon. In fact, the reduction of *CE* (2020) sourced from both scope 1 and scope 3 emissions is related to the drop in activity due to the Covid-19 crisis. Company
D is a European multinational company which supplies industrial resources and services to various industries (Figure 9.31). The company has a clear ambition and has embraced the NZE context. However, the metrics indicate that in terms of participation, the trend has not been negative and has deteriorated in previous years. We stress here that although Company D pays attention to its carbon intensity policy, it has not been active on the absolute carbon emissions level.

<figure><img src="image_466.png" /><figcaption>Figure 9.29: Carbon emissions, trend, targets and NZE scenario (Company B)</figcaption></figure>

Source: CDP database (2021), IEA (2021) & Le Guenedal et al. (2022).

On a global basis, we observe an increase of carbon emissions between 2013 and 2019, and a plateau in 2020, which is probably due to emissions reduction related to the Covid-19 crisis (Figure 9.32). Carbon targets are in line with the NZE scenario until 2025. After this date, we clearly see that the targeted reduction rates are lower than the NZE required reduction rates. We notice that the reduction targets are more or less in line with the NZE scenario. However, and more strikingly, there is a huge gap between the upward trend between 2013 and 2020 and what has been announced by the companies. Figure 9.32 perfectly illustrates the interest of the *PAC* framework, since we observe inconsistencies between the ambition of these issuers on one side, and their participation and credibility on another side. The sector analysis is interesting, because we observe some large differences between the sectors in Figure 9.33. First, the trajectory of carbon emissions is highly dependent on the sector. Electricity is the sector that has been making the greatest effort whereas we observe a large increase in carbon emissions for the Industry sector. The impact of the Covid-19 pandemic on the transport sector is particularly striking. Second, there are small differences between the reduction targets, except for the Transport sector that is slightly less ambitious.
<figure><img src="image_467.png" /><figcaption>Figure 9.30: Carbon emissions, trend, targets and NZE scenario (Company C)</figcaption></figure>

Source: CDP database (2021), IEA (2021) & Le Guenedal et al. (2022).

<figure><img src="image_468.png" /><figcaption>Figure 9.31: Carbon emissions, trend, targets and NZE scenario (Company D)</figcaption></figure>

Source: CDP database (2021), IEA (2021) & Le Guenedal et al. (2022).
<figure><img src="image_469.png" /><figcaption>Figure 9.32: Carbon emissions, trend, targets and NZE scenario (median analysis, global universe)</figcaption></figure>

Source: CDP database (2021), IEA (2021) & Le Guenedal et al. (2022).

<figure><img src="image_470.png" /><figcaption>Figure 9.33: Carbon emissions, trend, targets and NZE scenario (median analysis, sector universe)</figcaption></figure>

Source: CDP database (2021), IEA (2021) & Le Guenedal et al. (2022).
## 9.4 Greenness measures

Until now, we have focused on carbon metrics to measure the “brownness” of organizations or products. We now consider “greenness” measures, whose objective is to assess the positive contribution to limit global warming. In some sense, brownness and greenness measures are related. Nevertheless, we cannot deduce one measure from another one. Let us define the brown and green intensities of a company as the proportion of brown and green activities. We note them **BI** and **GI**. By construction, we have **BI** ∈ [0, 1], **GI** ∈ [0, 1] and 0 ≤ **BI** + **GI** ≤ 1. Most of the time, we have **BI** + **GI** ≠ 1, meaning that we cannot deduce the green intensity from the brown intensity. While the carbon footprint is a well-defined concept, greenness is then more difficult to assess. In fact, it is a multi-faceted concept. For instance, if one company changes its business model so that its new products are carbon efficient, we can measure the company’s greenness based on the avoided emissions generated by the change of the business model. For other companies, the greenness can be evaluated by estimating the R&D amount dedicated to green projects. Therefore, we observe a big difference between carbon and greenness metrics. Indeed, while it makes sense to compute the carbon footprint of all companies, the greenness may be indefinite for some companies, because they have no vocation to participate in the transition to a low-carbon economy. Therefore, Figure 9.34 illustrates that we cannot classify all activities into these two categories, since there are many activities that are neither brown nor green. Some companies are then neutral and are not exposed to the green business. These remarks argue in favor of considering simple and homogeneous measures of greenness.

<figure><img src="image_471.png" /><figcaption>Figure 9.34: Brown and green activities at the company level</figcaption></figure>

### 9.4.1 Green taxonomy

The purpose of a green taxonomy is to define what is green, and its objective is to inform investors about the greenness of their investments. Therefore, they can evaluate whether these levels satisfy or not their expectations. A green taxonomy is all the more important as we observe a strong development of green sentiment among investors (Brière and Ramelli, 2021). In this context, the investor may want to assess the proportion of his portfolio that is invested in environmentally sustainable assets. Therefore, a green taxonomy is necessary for both asset owners and managers.

The most famous example is the European green taxonomy, which has been already presented on page 33. We remind that the EU taxonomy for sustainable activities is “a classification system, establishing a list of environmentally sustainable economic activities.” These economic activities must have a substantive contribution to at least one of the following six environmental objectives: (1) climate change mitigation; (2) climate change adaptation; (3) sustainable use and protection of water and marine resources; (4) transition to a circular economy; (5) pollution prevention and control; (6) protection and restoration of biodiversity and ecosystems. Moreover, a business activity must also meet two other criteria to qualify as sustainable. First, the activity must do no significant
harm to the other environmental objectives (DNSH constraint) and second, it must comply with minimum social safeguards (MS constraint). Figure 9.35 summarizes the different steps.

<figure><img src="image_472.png" /><figcaption>Figure 9.35: EU taxonomy for sustainable activities</figcaption></figure>

**Remark 115** The EU taxonomy is not finalized and only concerns the first two objectives as of today (January 2023).

## 9.4.2 Green revenue share

### Relationship between the green intensity and the green revenue share

There are several ways to compute the green intensity. This is why we observe some significant differences between data providers. One method is to translate the 3-step approach of the EU taxonomy into the following equation:

$$ \mathcal{GI} = \frac{GR}{TR} \cdot (1 - P) \cdot 1\{\mathcal{S} \geq \mathcal{S}^*\} $$

where $GR$ is the green revenue deduced from the six environmentally sustainable objectives, $TR$ is the total revenue, $P$ is the penalty coefficient reflecting the DNSH constraint, $S$ is the minimum safeguard score and $S^*$ is the threshold. The first term is a proxy of the turnover KPI and corresponds to the green revenue share:

$$ \mathcal{GRS} = \frac{GR}{TR} $$

By construction, we have $0 \leq \mathcal{GRS} \leq 1$. This measure is then impacted by the DNSH coefficient. If the penalty coefficient is equal to zero, the green activities of the issuer do not significantly harm the other objectives and we have $\mathcal{GI} = \mathcal{GRS}$. Otherwise, the green intensity satisfies $0 \leq \mathcal{GI} = \mathcal{GRS} \cdot (1 - P) \leq \mathcal{GRS}$. Finally, the indicator function $1\{\mathcal{S} \geq \mathcal{S}^*\}$ is a binary all-or-nothing variable. It is equal to one if the firm complies with minimum social safeguards. Otherwise, the green intensity is equal to zero if the firm doesn't pass this materiality test. It follows that an upper bound of the green intensity is the green revenue share since we have $\mathcal{GI} \leq \mathcal{GRS}$.
Box 9.5: EU green taxonomy

The EU taxonomy is described in the Delegated Act on the climate objectives of 4 June 2021. For each activity, three items are provided: the description of the activity, the technical screening criteria, and the DNSH compliance. Let us consider Transport of CO₂ (page 100 of the Delegated Act). We have the following information:

1. Description of the activity

This concerns the transport of captured CO₂ via all modes; the construction and operation of CO₂ pipelines and retrofit of gas networks where the main purpose is the integration of captured CO₂. The economic activities in this category could be associated with several NACE codes, in particular F42.21 and H49.50. An economic activity in this category is an enabling activity.

2. Technical screening criteria

This activity has a substantial contribution to climate change mitigation:

(a) The CO₂ transported from the installation where it is captured to the injection point does not lead to CO₂ leakages above 0.5% of the mass of CO₂ transported.

(b) The CO₂ is delivered to a permanent CO₂ storage site that meets the criteria for underground geological storage of CO₂ set out in Section 5.12 of this Annex; or to other transport modalities, which lead to permanent CO₂ storage site that meet those criteria.

(c) Appropriate leak detection systems are applied and a monitoring plan is in place, with the report verified by an independent third party.

(d) The activity may include the installation of assets that increase the flexibility and improve the management of an existing network.

3. Do no significant harm

Three out of five categories are concerned: (2) climate change adaptation: the activity complies with the criteria set out in Appendix A to this Annex; (3) sustainable use and protection of water and marine resources: The activity complies with the criteria set out in Appendix B to this Annex. (4) transition to a circular economy: N/A; (5) pollution prevention and control: N/A; (6) protection and restoration of biodiversity and ecosystems: the activity complies with the criteria set out in Appendix D to this Annex.

This example shows that the revenues generated by the transport of CO₂ are not necessarily green, because the technical screening criteria imply that CO₂ leakages must be below 0.5% of the mass of CO₂ transported. We also observe that some criteria are generic while others are specific to an activity. For instance, the life cycle GHG emissions from the generation of electricity (whatever the electricity source) must be lower than 100 gCO₂e per kWh. Some criteria also concern the activity efficiency. For example, the power density of the electricity generation facility must be above 5 Watt per m² for the hydropower sector. Concerning the DNSH compliance criteria, they may for instance imply that the activity does use persistent organic pollutants, ether, mercury, substances that deplete the ozone layer, certain hazardous substances, etc. In a similar way, an environmental impact assessment (EIA) must be conducted for sites located near biodiversity-sensitive areas.
**Example 45** We consider a company in the hydropower sector which has five production sites. Below, we indicate the power density efficiency, the GHG emissions, the DNSH compliance with respect to the biodiversity and the corresponding revenue:

<table><thead><tr><th>Site</th><th>#1</th><th>#2</th><th>#3</th><th>#4</th><th>#5</th></tr></thead><tbody><tr><td>Efficiency (in Watt per m<sup>2</sup>)</td><td>3.2</td><td>3.5</td><td>3.3</td><td>5.6</td><td>4.2</td></tr><tr><td>GHG emissions (in gCO<sub>2</sub>e per kWh)</td><td>35</td><td>103</td><td>45</td><td>12</td><td>36</td></tr><tr><td>Biodiversity DNSH compliance</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td></td></tr><tr><td>Revenue (in $ mn)</td><td>103</td><td>256</td><td>89</td><td>174</td><td>218</td></tr></tbody></table>

The total revenue is equal to:

$$ \mathcal{TR} = 103 + 256 + 89 + 174 + 218 = \$840 \text{ mn} $$

We notice that the fourth site does not pass the technical screening, because the power density is above 5 Watt per m². The second site does not also comply because it has a GHG emissions greater than 100 gCO₂e per kWh. We deduce that the green revenue is equal to:

$$ \mathcal{GR} = 103 + 89 + 218 = \$410 \text{ mn} $$

We conclude that the green revenue share is equal to 48.8%. According to the EU green taxonomy, the green intensity is lower because the last site is close to a biodiversity area and has a negative impact. Therefore, we have:

$$ \mathcal{GI} = \frac{103 + 89}{840} = 22.9\% $$

## Statistics

In Table 9.27, we report the descriptive statistics of green revenue share calculated by Barahhou et al. (2022) with the MSCI database. For each category<fnref n="45" />, they have computed:

* The frequency $F(x) = \text{Pr}\{\mathcal{GRS} > x\}$;

* The statistical quantile $Q(\alpha) = \inf\{x: \text{Pr}\{\mathcal{GRS} \le x\} \ge \alpha\}$;

* The arithmetic average $n^{-1} \sum_{i=1}^{n} \mathcal{GRS}_i$ and the weighted mean $\mathcal{GRS}(b) = \sum_{i=1}^{n} b_i \mathcal{GRS}_i$ where $b_i$ is the weight of issuer i in the MSCI ACWI IMI benchmark.

For instance, 9.82% of issuers have a green revenue share that concerns alternative energy. This figure becomes less than 1% if we consider a green revenue share greater than 50%. The average value is equal to 1.36% whereas the weighted value is equal to 0.77%. This indicates a small cap bias. For energy efficiency, the average is lower than the weighted mean, implying a bias towards big companies. If we consider the total green revenue share, 27.85% have a positive figure and only 3.17% have a figure greater than 50%. The 90% quintile is equal to 11.82%. Therefore, we notice a high positive skewness for the distribution. The green revenue share is then located in a small number of companies.

**Remark 116** Barahhou et al. (2022) estimated that the green revenue share of the MSCI World index and the Bloomberg Global Investment Grade Corporate Bond index are respectively equal to 5.24% and 3.49% in June 2022. This is not a high figure, because the economy is today far to be green. These results are confirmed by Alessi and Battiston (2022), who estimated “a greenness of about 2.8% for EU financial markets” according to the existing EU green taxonomy<fnref n="46" />.

<footnote n="45">⁴⁵The MSCI taxonomy uses 6 categories: (1) alternative energy, (2) energy efficiency, (3) green building, (4) pollution prevention and control, (5) sustainable agriculture and (6) sustainable water.</footnote>

<footnote n="46">⁴⁶This concerns the first two categories, which are the most important.</footnote>
<table><caption>Table 9.27: Statistics in % of green revenue share (MSCI ACWI IMI, June 2022)</caption><thead><tr><th rowspan="2">Category</th><th colspan="4">Frequency F (x)</th><th colspan="4">Quantile Q (α)</th><th colspan="2">Mean</th></tr><tr><th>0</th><th>25%</th><th>50%</th><th>75%</th><th>75%</th><th>90%</th><th>95%</th><th>Max</th><th>Avg</th><th>Wgt</th></tr></thead><tbody><tr><td>(1)</td><td>9.82</td><td>1.47</td><td>0.96</td><td>0.75</td><td>0.00</td><td>0.00</td><td>2.85</td><td>100.00</td><td>1.36</td><td>0.77</td></tr><tr><td>(2)</td><td>14.10</td><td>1.45</td><td>0.65</td><td>0.31</td><td>0.00</td><td>1.25</td><td>6.12</td><td>100.00</td><td>1.39</td><td>3.50</td></tr><tr><td>(3)</td><td>4.84</td><td>1.68</td><td>1.02</td><td>0.31</td><td>0.00</td><td>0.00</td><td>0.00</td><td>100.00</td><td>1.16</td><td>0.51</td></tr><tr><td>(4)</td><td>4.79</td><td>0.30</td><td>0.10</td><td>0.06</td><td>0.00</td><td>0.00</td><td>0.00</td><td>99.69</td><td>0.32</td><td>0.22</td></tr><tr><td>(5)</td><td>1.00</td><td>0.39</td><td>0.20</td><td>0.09</td><td>0.00</td><td>0.00</td><td>0.00</td><td>98.47</td><td>0.26</td><td>0.10</td></tr><tr><td>(6)</td><td>4.75</td><td>0.28</td><td>0.11</td><td>0.05</td><td>0.00</td><td>0.00</td><td>0.00</td><td>99.98</td><td>0.29</td><td>0.14</td></tr></tbody><tfoot><tr><td>Total</td><td>27.85</td><td>5.82</td><td>3.17</td><td>1.68</td><td>0.42</td><td>11.82</td><td>30.36</td><td>100.00</td><td>4.78</td><td>5.24</td></tr></tfoot></table>

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

### 9.4.3 Green capex

Green capEx refers to investments in physical assets that contribute to environmental sustainability or climate change mitigation. These long-term investments in physical assets help reduce a company's environmental footprint and typically include renewable energy infrastructure, energy-efficient buildings, sustainable manufacturing equipment, heat recovery systems, green manufacturing infrastructure, clean transportation, and water recycling and treatment systems. While green revenue shares reflect a company's current environmental footprint, green capex is generally considered an indicator of a company's future environmental footprint, making it a forward-looking measure of the company's greenness.

<table><caption>Table 9.28: Carbon intensity and green intensity by sector (MSCI EMU, June 2024)</caption><thead><tr><th rowspan="2">Sector</th><th colspan="4">Carbon intensity</th><th colspan="3">Green intensity</th></tr><tr><th>SC<sub>1</sub></th><th>SC<sub>1-2</sub></th><th>SC<sub>1-3</sub><sup>up</sup></th><th>SC<sub>1-3</sub></th><th>Turnover</th><th>Capex</th><th>Opex</th></tr><tr><td></td><td colspan="4">(in tCO<sub>2</sub>e/$ mn)</td><td colspan="3">(in %)</td></tr></thead><tbody><tr><td>Communication Services</td><td>2.6</td><td>32.2</td><td>73.6</td><td>133.4</td><td>1.1</td><td>0.5</td><td>0.2</td></tr><tr><td>Consumer Discretionary</td><td>6.7</td><td>19.5</td><td>170.9</td><td>617.7</td><td>2.0</td><td>7.9</td><td>7.4</td></tr><tr><td>Consumer Staples</td><td>19.4</td><td>34.3</td><td>296.0</td><td>471.2</td><td>0.0</td><td>7.9</td><td>4.2</td></tr><tr><td>Energy</td><td>220.7</td><td>237.8</td><td>462.5</td><td>2120.9</td><td>3.9</td><td>19.1</td><td>8.7</td></tr><tr><td>Financials</td><td>1.2</td><td>4.7</td><td>25.8</td><td>684.5</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Health Care</td><td>11.9</td><td>28.1</td><td>106.5</td><td>128.9</td><td>0.0</td><td>1.1</td><td>0.0</td></tr><tr><td>Industrials</td><td>20.7</td><td>30.7</td><td>147.3</td><td>3383.6</td><td>9.1</td><td>10.7</td><td>12.1</td></tr><tr><td>Information Technology</td><td>3.6</td><td>14.8</td><td>106.5</td><td>653.2</td><td>0.3</td><td>0.4</td><td>0.3</td></tr><tr><td>Materials</td><td>477.4</td><td>780.9</td><td>1026.4</td><td>1418.3</td><td>1.7</td><td>5.5</td><td>3.1</td></tr><tr><td>Real Estate</td><td>48.1</td><td>90.7</td><td>119.3</td><td>515.0</td><td>17.5</td><td>35.0</td><td>17.2</td></tr><tr><td>Utilities</td><td>433.3</td><td>497.7</td><td>601.0</td><td>911.8</td><td>31.6</td><td>75.4</td><td>55.7</td></tr></tbody><tfoot><tr><td>MSCI EMU</td><td>65.6</td><td>95.5</td><td>209.3</td><td>1162.5</td><td>4.1</td><td>9.2</td><td>7.2</td></tr></tfoot></table>

Source: MSCI, Trucost & Author's calculations.

### 9.4.4 Green intensity versus carbon intensity

Green intensity does not convey the same information as carbon intensity. For example, one might think that companies with a high green intensity have a low carbon intensity. This is not currently the case. Table 9.28 shows the carbon and green intensities of sectors in the MSCI EMU Index at the end of June 2024. We find that high carbon sectors also have high green intensities. On average,
utilities stocks in the MSCI EMU stocks have a Scope 1 of 433 tonnes of CO2 per million dollars of revenue, while the green turnover, capex and opex<sup>47</sup> are 31.6%, 75.4% and 55.7%, respectively. These results show that some carbon-intensive sectors also have high green footprints. This is really one of the big issues in implementing a net zero investment policy, because reducing the carbon footprint of a portfolio is not the same as increasing the green footprint of the portfolio or improving the greenness of the economy.

**Remark 117** *Other metrics used to assess a company's green footprint include green R&D investment and the green-to-brown ratio, which compares environmentally beneficial activities to environmentally harmful activities.*

## 9.5 Exercises

<footnote n="47"><sup>47</sup>We use the three measures of green intensity defined in the EU taxonomy: green turnover, green capex (capital expenditure) and green opex (operating expenditure). The data correspond to the figures reported by the companies.</footnote>
<figure><img src="image_473.png" /></figure>
# Chapter 10

## Transition Risk Modeling

The transition to a low-carbon economy is one of the most significant economic shifts of the twenty-first century. It is changing the way societies produce, consume, and value goods and services. Unlike physical climate risks, which stem from the direct impacts of climate change, such as extreme weather events and rising sea levels, transition risks stem from decarbonizing the economy and achieving climate stabilization goals. These risks affect virtually every sector, including energy production, heavy industry, transportation, and real estate. They have significant implications for investors, firms, and policymakers.

Transition risk operates through several interconnected channels. First, regulatory interventions such as carbon pricing, emissions standards, and mandates to phase out fossil fuels alter the relative costs of production and consumption. These changes reduce the economic viability of carbon-intensive activities. Second, technological disruption accelerates this shift as clean energy alternatives — from renewable electricity to electric vehicles — become cost-competitive with, or cheaper than, traditional carbon-based technologies. This reshapes competitive dynamics across industries. Third, evolving consumer preferences and stakeholder expectations increasingly favor sustainable products and penalize firms perceived as climate laggards. These preferences and expectations can generate market and reputational pressures for corporate transformation. Together, these forces reshape competitive landscapes and asset valuations throughout the global economy.

This chapter provides a comprehensive analysis of transition risk across its major dimensions. Section One establishes the economic foundation by examining how externalities create market failures and exploring the implications for welfare of correcting them with different policy instruments. It also discusses shadow pricing methodologies for valuing non-market goods and the broader application of cost-benefit analysis in assessing climate policies. Section Two analyzes carbon pricing mechanisms by comparing the theoretical properties and practical performance of carbon taxes, emissions trading systems, and carbon credit programs across jurisdictions. Additionally, it explores how firms implement internal carbon pricing. Section Three examines the concept of stranded assets, distinguishing between stranded resources, stranded capital, and stranded financial assets across sectors. The analysis reveals which industries are most exposed and how these risks could spread through financial systems. Finally, Section Four outlines decarbonization pathways, detailing the necessary transformations, materials, and financing required to achieve net-zero emissions by 2050¹.

<footnote n="1">¹One important dimension of transition risk — shifts in consumer preferences and demand patterns — receives less systematic treatment in this chapter. Although these shifts are recognized as significant drivers, modeling them is methodologically challenging and has not been widely explored in academic literature, despite being a key focus in regulatory and technological channels.</footnote>
# 10.1 Economic analysis of negative externalities

From an economic perspective, climate change is fundamentally linked to the concept of negative externalities. This is because the market price of goods and services that produce greenhouse gases reflects only their private production costs and not their broader social costs, leading to a market failure. Specifically, individuals and firms have insufficient economic incentive to reduce their emissions because they do not fully internalize the costs of the damage they cause to the climate. The transition from a carbon-intensive to a low-carbon economy requires a shift toward mechanisms that internalize these externalities, such as carbon pricing, regulation, and investment in clean technologies, thereby aligning private costs with social costs and promoting more sustainable economic behavior.

## 10.1.1 Welfare analysis of negative externalities

### Definition

The concept of externalities was first introduced by Alfred Marshall, who recognized that individual economic activities could have unintended side effects on others not directly involved in the transaction. However, it was Arthur Pigou who systematically developed the modern framework for analyzing externalities. In his seminal work (Pigou, 1920), he formalized the distinction between private and social costs and laid the foundation for what is now known as the Pigouvian tax. Here is the definition we will use: An externality is a cost or benefit resulting from the economic activities of an individual or organization that affects a third party who has neither consented to nor been compensated for these effects and that occurs outside normal market pricing mechanisms. This definition highlights why externalities represent a market failure and captures the economic significance of externalities. They occur when private costs/benefits differ from social costs/benefits because certain impacts are not reflected in market transactions.

### Walrasian equilibrium

The profit of a producer is defined as the difference between his revenue and his costs:

$$ \pi = \mathcal{R}(q) - \mathcal{C}(q) $$

where $\mathcal{R}(q) = pq$ is the producer's revenue, $\mathcal{C}(q)$ is the producer's cost, $p$ is the price of the good, and $q$ is the quantity of the good produced. Maximizing the profit is equivalent to finding the optimal quantity $q^*$ such that the marginal cost equals the price:

$$ \frac{\partial \mathcal{C}(q)}{\partial q} = \frac{\partial \mathcal{R}(q)}{\partial q} = p $$

This condition characterizes the producer's supply decision in perfectly competitive markets, expressed as the supply function $q = S(p)$. The market supply curve for the good is obtained by horizontally aggregating the individual supply functions of all producers. The producer surplus is the difference between the amount a producer receives for a good (the price) and the minimum amount they would be willing to accept to produce it (the marginal cost):

$$ W_{PS}(q^*) - W_{PS}(0) = \int_{0}^{q^*} (p - \partial_q \mathcal{C}(q)) dq = \int_{0}^{q^*} (p - S^{-1}(q)) dq $$

where $S^{-1}(q)$ is the inverse supply function. We use the notation $W_{PS}$ because the producer surplus can be interpreted as a measure of producer welfare. In fact, it represents the economic benefit to producers of participating in the market.
Consumer theory is fundamentally based on preferences, which are typically represented by a utility function. In this framework, the consumer's objective is to maximize utility subject to a budget constraint:

$$ 
\begin{array}{l}
q^* = \arg\max \mathcal{U}(q) \\
\text{s.t.} \quad pq \le b
\end{array}
$$

where $b$ is the consumer's budget. The associated Lagrange function is $\mathcal{L}(q; \lambda) = \mathcal{U}(q) + \lambda(pq - b)$, while the Kuhn-Tucker condition is $\min(\lambda, b - pq) = 0$. The first-order condition is:

$$ \frac{\partial \mathcal{U}(q)}{\partial q} + \lambda p = 0 $$

Maximizing the utility function is equivalent to finding the optimal quantity $q^*$ such that marginal utility is proportional to price. Similar to production theory, this condition characterizes the consumer's demand decision in perfectly competitive markets, expressed as the demand function $q = D(p)$. The market demand curve for the good is obtained by horizontally aggregating the individual demand functions of all consumers. Consumer surplus is the difference between the total amount consumers are willing to pay for a good and the amount they actually pay:

$$ W_{CS}(q^*) - W_{CS}(0) = \int_{0}^{q^*} (D^{-1}(q) - p) \, dq $$

where $D^{-1}(q)$ is the inverse demand function that gives the price at which quantity $q$ is demanded.

These results can be easily generalized when there are $n$ goods<fnref n="2" />. In the previous analysis, both the producer and the consumer are price takers, i.e., they consider the price to be exogenous and do not have sufficient market power to influence it. In this case, the price is given by the market equilibrium between supply and demand. This competitive equilibrium, denoted by the vector $(q^*, p^*)$, is also called a Walrasian equilibrium because it satisfies the following conditions:

* each producer maximizes his profits;

* each consumer maximizes his preferences or utility function under a budget constraint;

* supply equals demand.

The Walrasian equilibrium is illustrated in Figure 10.1. This concept represents the outcome of a perfectly competitive market, where prices adjust to equate supply and demand across all markets simultaneously. The theory of Walrasian price formation assumes the absence of any distortions and relies on the ideal conditions of perfect competition — many buyers and sellers, full information, and no transaction costs. Under these assumptions, markets reach an efficient allocation of resources through decentralized decision-making. However, this framework breaks down in the presence of market failures, such as monopoly or oligopoly. In such cases, economic agents (firms or consumers) possess market power, allowing them to influence prices. Instead, alternative equilibrium concepts are used to analyze such situations. The main ones include Cournot equilibrium (firms compete by choosing quantities), Bertrand equilibrium (firms compete by setting prices), monopoly equilibrium, etc.

<footnote n="2">²In this case, the cost function and the utility function become $\mathcal{C}(q_1, \dots, q_n)$ and $\mathcal{U}(q_1, \dots, q_n)$. From the producer's optimization problem, we obtain $\partial_{q_i} \mathcal{C}(q_1, \dots, q_n) = p_i$. We also have $\partial_{q_i} \mathcal{U}(q_1, \dots, q_n) = \lambda p_i$. We can then derive the supply and demand functions: $q_i = S_i(p_1, \dots, p_n)$ and $q_i = D_i(p_1, \dots, p_n)$. They generally satisfies the following properties: $\partial_{p_i} S_i(p_1, \dots, p_n) \ge 0$ (supply curves slope upward) and $\partial_{p_i} D_i(p_1, \dots, p_n) \le 0$ (demand curves slope downward).</footnote>
<figure><img src="image_474.png" /><figcaption>Figure 10.1: Walrasian equilibrium</figcaption></figure>

Social efficiency refers to an economic state in which resources are allocated in such a way as to maximize the total welfare of society — this is known as Pareto optimality. Total welfare, or total social surplus, is the combined value of consumer surplus and producer surplus:

$$ 
\begin{align*}
W(q^*) - W(0) &= W_{PS}(q^*) - W_{PS}(0) + W_{CS}(q^*) - W_{CS}(0) \\
&= \int_{0}^{q^*} (p^* - S^{-1}(q)) dq + \int_{0}^{q^*} (D^{-1}(q) - p^*) dq \\
&= \int_{0}^{q^*} (D^{-1}(q) - S^{-1}(q)) dq
\end{align*}
$$

On the graph, this surplus is represented by the area between the demand curve and the supply curve (Figure 10.1). According to the first fundamental theorem of welfare economics<fnref n="3" />, a competitive equilibrium in which supply equals demand leads to an outcome that is socially efficient. Consider the solution $(q', p')$ such that $q' < q^*$. Using Figure 10.1, it is easy to see that we can increase the total surplus by the area corresponding to the teal triangle. This means that the vector $(q', p')$ does not maximize the total welfare of society. The teal triangle is called deadweight loss and represents the lost economic surplus due to market distortions that prevent the economy from achieving its optimal allocation of resources.

## Market equilibrium with negative production externalities

To illustrate how the previous economic framework is affected by externalities, we follow Gruber (2005) and introduce the following notations:

<footnote n="3">³When markets are perfectly competitive, information is complete, all economic agents behave rationally, and there are no externalities, the resulting private market equilibrium is Pareto efficient.</footnote>
* PC (q) is the private direct cost, representing the total cost to firms of producing q units;

* EC (q) is the external indirect cost imposed on society when firms produce q units;

* SC (q) := PC (q) + EC (q) is the social cost of producing q units;

* PB (q) is the private direct benefit to consumers from buying q units;

* EB (q) is the external indirect benefit to society when consumers buy q units;

* SB (q) := PB (q) + EB (q) is the social benefit of buying q units.

Using these definitions, we can derive the expression for the social marginal cost:

$$SMC(q) = \frac{\partial SC(q)}{\partial q} = PMC(q) + EMC(q)$$

where PMC is the private marginal cost and EMC is the external marginal cost. Similarly, the social marginal benefit is the sum of the private marginal benefit and the external marginal benefit:

$$SMB(q) = \frac{\partial SB(q)}{\partial q} = PMB(q) + EMB(q)$$

Assuming that firms do not consider their externalities, competitive equilibrium is reached when private marginal cost (supply) equals private marginal benefit (demand). This implies that the firm will produce a quantity $q^*$ that satisfies the equation $PMC(q) = PMB(q)$. The socially optimal equilibrium occurs when social marginal cost equals social marginal benefit, resulting in a quantity $q^*$ that satisfies the equation $SMC(q) = SMB(q)$. If we assume negative production externalities, then $EMC(q) \ge 0$ and $EMB(q) = 0$. It follows that:

$$\begin{cases} q^* > q^* \\ p^* < p^* \end{cases}$$

We conclude that negative production externalities lead to overproduction of $q^* - q^*$, because competitive equilibrium price is underestimated by $p^* - p^*$.

Total welfare is equal to the total private surplus plus external benefits minus external costs, or the difference between social benefits and social costs:

$$W(q) = SB(q) - SC(q)$$

Let us consider the Walrasian equilibrium $q^*$. Since we have $SMB(q^*) = PMC(q^*)$, we obtain:

$$\begin{aligned} dW(q^*) &= SMB(q^*) dq - PMC(q^*) dq - EMC(q^*) dq \\ &= -EMC(q^*) dq \end{aligned}$$

Since $EMC(q^*) > 0$, it follows that the total welfare increases when production is reduced ($dq < 0$). We conclude that the first welfare theorem fails in the presence of positive production externalities.

**Example 46** We assume that the total cost function of producing cars is $C(q) = 1+5q+\frac{1}{4}q^2$, while the utility function of buying cars is $U(q) = 5+20q-\frac{1}{2}q^2$. Cars produce pollution, which imposes an external cost on society $\mathcal{E}(q) = 3.6q+0.2q^2$.
<figure><img src="image_475.png" /><figcaption>Figure 10.2: Market equilibrium in the presence of negative production externalities</figcaption></figure>

Source: Gruber (2005, Figure 5.2, page 124) & Author's calculations.

Since we have $C'(q) = 5 + \frac{1}{2}q$, we deduce that the supply function is:

$$p = \text{PMC}(q) = 5 + \frac{1}{2}q$$

Similarly, we have $U'(q) = 20 - q$ and the demand function is:

$$p = \text{PMB}(q) = 20 - q$$

The Walrasian equilibrium is obtained by solving the equation $\text{PMC}(q) = \text{PMB}(q)$. We check that $q^* = 10$ and $p^* = \text{PMC}(q^*) = \text{PMB}(q^*) = 10$. The social welfare is<fnref n="4" />:

$$ 
\begin{aligned}
W(q^*) &= \text{PB}(q^*) - \text{PC}(q^*) - \text{EC}(q^*) \\
&= \mathcal{U}(q^*) - \mathcal{C}(q^*) - \mathcal{E}(q^*) \\
&= \left(5 + 20q^* - \frac{1}{2}(q^*)^2\right) - \left(1 + 5q^* + \frac{1}{4}(q^*)^2\right) - \left(3.6q^* + 0.2(q^*)^2\right) \\
&= 155 - 76 - 56 \\
&= 23
\end{aligned}
$$

$$W(q^*) = W(0) + \int_{0}^{q^*} (\text{PMB}(q) - \text{PMC}(q) - \text{EMC}(q)) \, dq = 4 + \left[ 11.4q - 0.95q^2 \right]_{0}^{10} = 4 + 19 = 23$$

because $\text{PMB}(q) - \text{PMC}(q) - \text{EMC}(q) = 20 - q - 5 - \frac{1}{2}q - 3.6q - 0.4q = 11.4 - 1.9q$.

<footnote n="4">⁴We verify that:</footnote>
If we reduce production by one unit, we get $q = 9$. We deduce that $W(9) = 144.5 - 66.25 - 48.6 = 29.65$ and $W(9) > W(10)$. By reducing production, we then improve social welfare. The social optimum is reached when PMB($q$) = SMC($q$). We obtain $q^* = 6$ and $p^* = 14$. The social welfare is:

$$W(q^*) = PB(q^*) - PC(q^*) - EC(q^*) = 107 - 40 - 28.8 = 38.2$$

The Walrasian optimum ($q^*, p^*$) and the social optimum ($q^*, p^*$) are shown in Figure 10.2. The different components of the total surplus are given below:

<table><thead><tr><th>Component</th><th>Surface</th><th>Calculus</th><th>Value</th></tr></thead><tbody><tr><td>$W(0)$</td><td></td><td></td><td>4</td></tr><tr><td>Consumer surplus</td><td>blue</td><td>0.5 × 10 × (20 − 10)</td><td>50</td></tr><tr><td>Producer surplus</td><td>green</td><td>0.5 × 10 × (10 − 5)</td><td>25</td></tr><tr><td>External cost</td><td>violet</td><td>10 × 3.6 + 0.5 × 10 × (7.6 − 3.6)</td><td>56</td></tr><tr><td>Deadweight loss</td><td>orange</td><td>0.5 × (10 − 6) × (17.6 − 10)</td><td>15.2</td></tr></tbody></table>

We verify that:

$$W(q^*) = 4 + 50 + 25 - 56 = 23$$

and:

$$W(q^*) = W(q^*) + \text{Deadweight loss} = 23 + 15.2 = 38.2$$

We understand why the deadweight loss is the loss of economic efficiency when the social optimum is not achieved. For instance, in the example above, the deadweight loss amounts to 15.2, indicating that society could gain an additional 15.2 in welfare if negative externalities were properly accounted for in the economic calculus.

## 10.1.2 Solutions to negative externalities

Gruber (2005, Chapter 5) distinguishes two types of solutions to negative externalities. The first type corresponds to private sector solutions and is mainly related to the work of Ronald Coase, whose seminal theorem suggests that, under specific conditions, economic agents can negotiate and reach efficient outcomes without government intervention. The second type corresponds to public sector solutions and consists of three approaches: a corrective or Pigouvian tax, which imposes a charge on activities that generate negative externalities in order to align private costs with social costs; subsidies, which provide incentives to reduce harmful activities; and government regulation (quotas) which establishes binding legal limits on activities that produce negative externalities.

### Private-sector solutions

Private sector solutions aim to address negative externalities without government intervention. Examples include reputational and social sanctions, activism, self-regulation and private bargaining. For example, firms may voluntarily reduce harmful behaviour due to social pressure or reputational risk, and industries may set standards and rules to avoid government regulation. However, the most influential private sector solution in economic theory is the bargaining approach formulated in Coase (1960):

> "In microeconomics, the market is innocent until proven guilty (and, similarly, the government is often guilty until proven innocent!). An excellent application of this principle can be found in a classic work by Ronald Coase, a professor at the Law School at the University of Chicago, who asked in 1960: Why won't the market simply compensate the affected parties for externalities?" (Gruber, 2005, page 130).
Box 10.1: Negative vs. positive externalities

Negative externalities occur when the production or consumption of goods and services imposes costs on third parties that are not reflected in market prices. This typically leads to overconsumption or overproduction because the true costs to society are underestimated. In economic terms, this happens when:

* The social marginal cost exceeds the private marginal cost
  For example, companies that emit air pollution during the production of goods do not pay directly for the environmental damage or health impacts caused by their emissions. These costs are borne by society in the form of respiratory diseases, environmental damage and climate change.

* The private marginal benefit exceeds the social marginal benefit
  This is often the case with demerit goods such as alcohol and cigarettes. While individuals may derive personal satisfaction from their consumption, society bears additional costs in terms of public health expenditure, lost productivity and social disorder (e.g. drink-driving, addiction-related crime).

Other examples of negative externalities include carbon emissions, noise pollution from construction that affects the mental health of nearby residents, and the overuse of antibiotics that creates resistant bacteria. In these cases, the market fails to allocate resources efficiently, resulting in a loss of welfare for society.

<table><thead><tr><th>Externality</th><th>Production</th><th>Consumption</th></tr></thead><tbody><tr><td>Negative</td><td>SMC > PMC</td><td>SMB < PMB</td></tr><tr><td>None</td><td>SMC = PMC</td><td>SMB = PMB</td></tr><tr><td>Positive</td><td>SMC < PMC</td><td>SMB > PMB</td></tr></tbody></table>

Positive externalities, on the other hand, occur when the production or consumption of a good or service provides benefits to third parties that are not reflected in the market price. Such goods then tend to be underproduced or underconsumed. This happens when:

* The social marginal cost is lower than the private marginal cost
  An example is medical research, which is costly for firms to undertake but generates widespread public benefits through improved health outcomes and reduced future health care costs.

* The social marginal benefit exceeds the private marginal benefit
  An example is vaccination. While individuals are protected from disease, society benefits even more from the herd immunity that helps prevent outbreaks. Similarly, honey production not only yields honey, but also improves pollination, which increases agricultural yields and biodiversity.

Other examples of positive externalities include education (individuals gain knowledge, but society benefits from a more productive workforce), public transport (reduces congestion for all road users), scientific research (generates knowledge that can have many applications), and more generally innovations. These examples illustrate how positive externalities can lead to underinvestment in socially beneficial activities, just as negative externalities can lead to overinvestment in socially costly activities.
Coase (1960) suggests that externalities are not outside market mechanisms and can be internalized through private negotiations. The private bargaining approach is commonly formulated as the Coase theorem:

**Coase Theorem — Part 1 (Efficiency Property)**

If property rights are clearly defined and transaction costs are zero, private bargaining between the party creating the externality and the party affected by the externality will lead to an efficient allocation of resources.

**Coase Theorem — Part 2 (Invariance Property)**

The final allocation of resources will be the same regardless of the initial assignment of property rights, as long as bargaining is costless and someone is assigned those rights.

The Coase theorem is often illustrated by well-known examples that highlight conflicts over externalities, such as the cattle rancher and the crop farmer, the fisherman and the steel mill, or the laundry affected by a polluting factory. These examples help to illustrate how, under certain conditions<fnref n="5" />, private bargaining can lead to efficient outcomes without government intervention, and the solution is Pareto optimal.

Consider the case of a firm that pollutes a river that is otherwise used by local residents for recreation or domestic purposes. If the individuals have property rights in the river, they can demand compensation from the firm for the right to pollute. In this case, the firm bears the cost of reducing the pollution, either by paying the individuals directly or by investing in cleaner technology to avoid paying. Conversely, if the firm owns the right to pollute the river, individuals can choose to pay the firm to reduce pollution. In this case, individuals bear the cost of reducing pollution. In both scenarios, pollution is reduced and resources are allocated more efficiently through voluntary negotiation — this illustrates part 1 of the Coase theorem. However, while the outcome in terms of efficiency is the same, the distribution of costs and benefits differs depending on how property rights are initially allocated — this illustrates part 2 of the Coase theorem. This reflects a crucial distinction between efficiency and equity. The Coase theorem tells us that the outcome will be efficient, but not necessarily fair, because that depends on who has the property rights.

The previous example satisfies the Coase theorem because the river is owned by either the polluting firm or the local residents. However, if the river is owned by a third party, the Coase theorem may not apply. This highlights that the Coase theorem is only valid under specific conditions — typically involving small and localized externalities that can be internalized through market bargaining. This is not the case with air pollution, for example, because there are no clearly defined property rights over the atmosphere. In fact, Gruber (2005) identifies four major limitations to the Coase solution: (1) the assignment problem, (2) the holdout problem, (3) the free rider problem, and (4) transaction costs and negotiation difficulties. The assignment problem arises when externalities affect a large number of agents, making it difficult to assign property rights clearly and to determine the appropriate compensation for damages. For example, pollution of a river that affects an entire region would require coordination among numerous individuals, each with different levels of exposure and damage. Furthermore, the pollution may come from multiple sources, complicating the attribution of responsibility. The valuation of damages from such externalities is often contentious

<footnote n="5">⁵According to Deryugina et al. (2021), the Coase theorem implies five key assumptions: no wealth effect (quasilinearity in the numeraire makes the externality independent of budgets and side-payment), perfect information (agents know each other’s indirect utility functions), rationality (agents maximize utility), no endowment effect (utility functions are smooth in the status quo, and economic agents behave the same whether or not they have the right to be free of externalities), and zero transaction costs (the bargain can be made without incurring costs).</footnote>
and imprecise, as affected parties may have different estimates based on personal, economic, or political interests. This lack of consensus is an obstacle to effective negotiation. The holdout problem arises when one or more parties strategically delay or obstruct a negotiated agreement in an effort to secure a greater share of the benefits or extract greater compensation. This behavior can undermine efficient outcomes even when a mutually beneficial solution is within reach. For example, if a factory requires the consent of all downstream residents to maintain a limited level of pollution, a single individual may refuse to agree unless offered disproportionately high compensation. Such strategic holdouts become more likely as the number of stakeholders increases, greatly complicating collective bargaining efforts. The free rider problem arises when individuals who benefit from pollution abatement have little incentive to contribute to its cost<fnref n="6" />. In situations where multiple affected residents must collectively compensate a factory to reduce its emissions, each person may reason that his own contribution is negligible and choose to withhold payment, expecting others to cover the cost. This self-interested refusal to cooperate may lead to an undersupply of pollution abatement and prevent the achievement of the socially optimal outcome predicted by the Coase theorem. Finally, transaction costs, such as legal fees, information costs, or logistical barriers, may be prohibitively high, especially in cases involving many stakeholders or complex valuation. Taken together, these four factors limit the real-world applicability of the Coase theorem, especially when dealing with large-scale environmental externalities.

**Remark 118** From a historical perspective, the private-sector solution proposed by Coase (1960) came after the public-sector approach developed by Pigou (1920). In many ways, the Coase theorem can be interpreted as a response by the Chicago School of economic thought (which includes influential economists such as Milton Friedman, George Stigler, and Gary Becker) to the Pigouvian tax advocated by the Cambridge tradition (represented by Marshall, Pigou, and Keynes). While the Cambridge tradition emphasizes market imperfections and calls for government intervention to correct externalities, the Chicago School is generally more skeptical of government involvement and emphasizes the potential for market-based solutions.

## Public-sector remedies

Given the large number of people affected, the significant transaction costs involved, and the complexity of valuation, a third party, typically the government, may be necessary to achieve the social optimum. This is particularly true for global warming and climate change, as these global externalities require cooperation at the international level, involving cooperation among countries, industries, consumers and producers, high-income and low-income households, and so on.

**Corrective taxation** A corrective (or Pigouvian) tax is a tax levied on activities that generate negative externalities. Consider Example 46 on page 949. We have seen that the competitive equilibrium $(q^*, p^*)$ is (10, 10), while the social optimum $(q^*, p^*)$ is (6, 14). Recall that the external cost function imposed on society is given by $\mathcal{E}(q) = 3.6q + 0.2q^2$. To reach the social optimum, the government can impose a tax on each unit produced equal to the external marginal cost evaluated at the social optimum:

$$ \tau = \text{EMC}(q^*) = \frac{\partial \mathcal{E}(q^*)}{\partial q} $$

In our example, we obtain $\tau = 3.6 + 0.4 \times 6 = \$6$. After imposing this tax, the supply function becomes $p = \text{PMC}^*(q) = \text{PMC}(q) + \tau = 11 + \frac{1}{2}q$. It follows that the new equilibrium $(q^*, p^*)$ is

<footnote n="6">⁶In other words, the free rider problem arises when individuals benefit from a negotiated solution without contributing to its costs.</footnote>
reached when $PMC^*(q) = PMB(q)$. The solution is $(q^*, p^*) = (6, 14)$, which corresponds to the social optimum$^7$. Figure 10.3 illustrates the implementation of the corrective tax, which shifts the supply function upward. The equilibrium implied by the corrective tax is the Coase solution, but in this case the externalities are paid by the producer, regardless of property rights.

<figure><img src="image_476.png" /><figcaption>Figure 10.3: Taxation as a solution to negative production externalities</figcaption></figure>

Source: Gruber (2005, Figure 5.6, page 135) & Author's calculations.

**Remark 119** To ensure that a Pigouvian tax is Pareto optimal, the external marginal cost (or the marginal damage function) must be known in order to set the appropriate tax level. For instance, if $\tau < EMC(q^*)$ or $\tau > EMC(q^*)$, the social equilibrium is not reached. Moreover, the equilibrium $(q^*, p^*)$ represents a short-term equilibrium, since taxes can influence production decisions in the long run.

**Subsidies** While the objective of taxes is to reduce production and consumption by increasing prices, the objective of subsidies is to encourage production and consumption by lowering prices. In both cases, taxes and subsidies aim to change the price system in order to influence supply and demand. In the first case, the tax-adjusted supply is given by:

$$p = PMC^*(q) = PMC(q) + EMC(q^*) \qquad (10.1)$$

$$11 + \frac{1}{2}q^* = 20 - q^* \Leftrightarrow \frac{3}{2}q^* = 9 \Leftrightarrow q^* = 6$$

and:

$$p^* = 11 + \frac{1}{2}q^* = 11 + \frac{1}{2} \times 6 = 14$$

<footnote>$^7$We have:</footnote>
where $q^*$ is the social optimum to achieve. In the second case, the subsidies-adjusted supply is given by:

$$p = \text{PMC}^*(q) = \text{PMC}(q) - \text{EMB}(q^*) \quad (10.2)$$

Subsidies are often used as positive incentives to encourage environmentally friendly behavior and the adoption of clean technologies in the fight against climate change. A common example is renewable energy subsidies, such as direct grants or tax incentives for installing solar panels.

**Quota** If the government knows the socially optimal level of a good that generates externalities — such as pollution — it can intervene directly by setting strict limits on production. This method, known as quantity regulation, involves ordering producers to produce exactly the optimal amount — no more, no less. In this way, the government ensures that the negative externality is limited to a level consistent with social welfare. Unlike taxes or subsidies, this approach does not rely on price signals to influence behavior. Instead, it imposes specific rules or caps, often backed by legal enforcement. Examples include caps on factory emissions.

**Command-and-control** Command-and-control regulation differs fundamentally from market-based instruments, such as taxes and tradable quotas. It relies on direct government intervention through specific rules, standards, and mandates to control behavior. Examples include emission standards (e.g., limits on vehicle emissions), bans on certain substances or activities, and prescriptive regulations that dictate the technologies or practices firms must adopt.

## Prices vs. quantities

Under standard economic conditions and in the absence of uncertainty, price-based instruments (such as taxes) are typically equivalent to quantity-based instruments (such as quotas). For example, the social optimum ($q^*, p^*$) can be achieved from the Walrasian equilibrium ($q^*, p^*$) by imposing a tax $\tau = p^* - p^*$ or by setting a quota $q \le q^*$. From a policy perspective, the choice between taxes and quotas is neutral in this deterministic framework. However, this equivalence was fundamentally challenged by the publication of the famous article “Prices vs. Quantities” (Weitzman, 1974). In this seminal contribution to environmental economics, Weitzman explores the relative merits of price-versus quantity-based regulatory instruments in the presence of uncertainty, particularly uncertainty about the costs and benefits of controlling externalities. He shows that under such uncertainty, the two instruments are no longer equivalent, and the choice between them becomes crucial. The key insight of the paper is that the relative slopes of the marginal cost and marginal benefit curves determine which policy instrument is more efficient:

* If the marginal benefit curve is relatively flat compared to the marginal cost curve, then a price instrument is preferred;

* If the marginal benefit curve is steeper than the marginal cost curve, then a quantity instrument is more efficient.

The research of Martin Weitzman has led to numerous extensions<fnref n="8" /> and academic/public debates. The comprehensive reviews by Hepburn (2006), Goulder and Parry (2008), and Stavins (2022) provide valuable insights into the challenges of selecting appropriate policy instruments such as taxes or quotas. These reviews also highlight how new instruments and hybrid solutions can potentially outperform these two traditional approaches, at least from a theoretical perspective, as demonstrated in the seminal works of Roberts and Spence (1976) and Pizer (2002).

<footnote n="8">⁸See, for example Adar and Griffin (1976), Stavins (1996), and Hoel and Karp (2002), to name a few.</footnote>
**The model** Weitzman (1974) assumes that a good is produced at cost $C(q)$ and yields the benefits (or utility) $B(q)$. The optimal production is given by the following optimization problem:

$$q^* = \arg \max \mathcal{B}(q) - \mathcal{C}(q)$$

The solution satisfies the first-order condition: $\partial_q \mathcal{B}(q^*) = \partial_q \mathcal{C}(q^*) = p^*$. If we assume that the social planner defines the optimal price $p^*$, the producer will maximize profits $p^*q - C(q)$ and the solution satisfies $p^* = \partial_q \mathcal{C}(q^*)$. If we assume that the social planner defines the optimal quantity $q^*$ to produce, the consumer will maximize benefits $\mathcal{B}(q) - pq$ and the solution satisfies $p^* = \partial_q \mathcal{B}(q^*)$. In this case, using price instruments is equivalent to using quantity instruments.

Weitzman then introduces uncertainty into the cost and benefit functions, which become $C(q; \varepsilon_C)$ and $\mathcal{B}(q; \varepsilon_B)$, where $\varepsilon_C$ and $\varepsilon_B$ are two random variables. In the case of the quantity instrument, the optimization problem with uncertainty becomes:

$$q_1^* = \arg \max \mathcal{E} [\mathcal{B}(q; \varepsilon_B) - \mathcal{C}(q; \varepsilon_C)]$$

The first-order condition is $\mathcal{E}[\partial_q \mathcal{B}(q_1^*; \varepsilon_B)] = \mathcal{E}[\partial_q \mathcal{C}(q_1^*; \varepsilon_C)]$. In the case of the price instrument, the producer's optimization problem is:

$$q^* = \arg \max p q - \mathcal{C}(q; \varepsilon_C)$$

The first-order condition is $\partial_q \mathcal{C}(q^*; \varepsilon_C) = p$, which implies the producer reaction function: $q^* = q(p, \varepsilon_C)$. The social planner will then choose the price $p_2^*$ that maximizes expected welfare:

$$p_2^* = \arg \max \mathcal{E} [\mathcal{B}(q(p, \varepsilon_C); \varepsilon_B) - \mathcal{C}(q(p, \varepsilon_C); \varepsilon_C)]$$

The first-order condition is:

$$\mathcal{E}[\partial_q \mathcal{B}(q(p_2^*, \varepsilon_C); \varepsilon_B) \partial_p q(p_2^*, \varepsilon_C)] = \mathcal{E}[\partial_q \mathcal{C}(q(p_2^*, \varepsilon_C); \varepsilon_C) \partial_p q(p_2^*, \varepsilon_C)] \\ = p_2^* \mathcal{E}[\partial_p q(p_2^*, \varepsilon_C)]$$

It follows that the ex-post output equilibrium $q_2^*$ depends on the ex-ante price $p_2^*$:

$$q_2^* := q_2^*(\varepsilon_C) = q(p_2^*, \varepsilon_C)$$

where:

$$p_2^* = \frac{\mathcal{E}[\partial_q \mathcal{B}(q(p_2^*, \varepsilon_C); \varepsilon_B) \partial_p q(p_2^*, \varepsilon_C)]}{\mathcal{E}[\partial_p q(p_2^*, \varepsilon_C)]}$$

Since $\varepsilon_C$ and $\varepsilon_B$ are unknown, we deduce that $p_2^* \neq \mathcal{E}[\partial_q \mathcal{B}(q_1^*; \varepsilon_B)]$ and $(q_1^*, p_1^*) \neq (q_2^*, p_2^*)$. Weitzman concludes:

> "In the presence of uncertainty, price and quantity instruments transmit central control in quite different ways. It is important to note that by choosing a specific mode for implementing an intended policy, the planners are at least temporarily locking themselves into certain consequences. The values of $\varepsilon_B$ and $\varepsilon_C$ are at first unknown and only gradually, if at all, become recognized through their effects. After the quantity $q_1^*$ is prescribed, producers will continue to generate that assigned level of output for some time even though in all likelihood $\partial_q \mathcal{B}(q_1^*; \varepsilon_B) \neq \partial_q \mathcal{C}(q_1^*; \varepsilon_C)$. In the price mode on the other hand, $q_2^*(\varepsilon_C)$ will be produced where except with negligible probability $\partial_q \mathcal{B}(q_2^*(\varepsilon_C); \varepsilon_B) \neq \partial_q \mathcal{C}(q_2^*(\varepsilon_C); \varepsilon_C)$. Thus neither instrument yields an optimum ex post." (Weitzman, 1974, page 482).
Weitzman proposes to assess the comparative advantage of the price instrument over the quantity instrument by measuring the difference in surplus between the two situations:

$$ 
\begin{aligned}
\Delta &= W(q_2^*) - W(q_1^*) \\
&= \mathbb{E}[(\mathbf{B}(q_2^*; \varepsilon_B) - \mathbf{C}(q_2^*; \varepsilon_C))] - \mathbb{E}[(\mathbf{B}(q_1^*; \varepsilon_B) - \mathbf{C}(q_1^*; \varepsilon_C))]
\end{aligned}
$$

Weitzman shows that the net welfare difference between using a price instrument and a quantity instrument is equal to<fnref n="9" />:

$$ \Delta = \frac{\sigma_C^2 \partial_q^2 \mathbf{B}(q)}{2 (\partial_q^2 \mathbf{C}(q))^2} + \frac{\sigma_C^2 - 2\sigma_{B,C}^2}{2\partial_q^2 \mathbf{C}(q)} \quad (10.3) $$

where:

* $\sigma_C^2 = \mathbb{E}[(\partial_q \mathbf{C}(q; \varepsilon_C) - \mathbb{E}[\partial_q \mathbf{C}(q; \varepsilon_C)])^2]$ is the variance of marginal costs;

* $\sigma_{B,C}^2 = \mathbb{E}[(\partial_q \mathbf{C}(q; \varepsilon_C) - \mathbb{E}[\partial_q \mathbf{C}(q; \varepsilon_C)])(\partial_q \mathbf{B}(q; \varepsilon_B) - \mathbb{E}[\partial_q \mathbf{B}(q; \varepsilon_B)])]$ is the covariance of marginal benefits and costs;

* $\partial_q^2 \mathbf{C}(q) > 0$ is the slope of the marginal cost function;

* $\partial_q^2 \mathbf{B}(q) < 0$ is the slope of the marginal benefit function.

Assuming that uncertainties on costs and benefits are independent — $\mathbb{E}[\varepsilon_B \varepsilon_C] = 0$ — Weitzman (1974) derives the following key result:

$$ \Delta = \frac{\sigma_C^2}{2\partial_q^2 \mathbf{C}(q)} \left( \frac{\partial_q^2 \mathbf{B}(q)}{\partial_q^2 \mathbf{C}(q)} + 1 \right) \quad (10.4) $$

Since $\partial_q^2 \mathbf{C}(q) > 0$ and $\partial_q^2 \mathbf{B}(q) < 0$, it follows that:

$$ \Delta \ge 0 \Leftrightarrow \partial_q^2 \mathbf{B}(q) \ge -\partial_q^2 \mathbf{C}(q) \Leftrightarrow |\partial_q^2 \mathbf{B}(q)| \le \partial_q^2 \mathbf{C}(q) $$

This means that if the marginal benefit curve is flatter than the marginal cost curve, then price is the preferred instrument. Otherwise, quantity is the better instrument<fnref n="10" />.

**Remark 120** In the producer/consumer market equilibrium framework, $\partial_q^2 \mathbf{C}(q)$ is the first derivative of the social marginal cost (SMC), while $\partial_q^2 \mathbf{B}(q)$ is the first derivative of the social marginal benefit (SMB) or the demand function. The following expression holds:

$$ \Delta = \frac{\sigma_C^2}{2\partial_q \text{SMC}(q)} \left( \frac{\partial_q \text{SMB}(q)}{\partial_q \text{SMC}(q)} + 1 \right) \quad (10.5) $$

If the functions are linear — that is, $\text{SMB}(q) = \alpha_B - \beta_B q$ and $\text{SMC}(q) = \alpha_C + \beta_C q$ — then we obtain:

$$ \Delta = \frac{\sigma_C^2}{2\beta_C} \left( 1 - \frac{\beta_B}{\beta_C} \right) \quad (10.6) $$

We check that $\Delta \ge 0 \Leftrightarrow \beta_B \le \beta_C$, i.e., when the slope of the SMB curve is less than the slope of the SMC curve.

**Remark 121** Stavins (1996, pages 223-224) analyzes the impact of the covariance term $\sigma_{B,C}^2$ in Equation (10.3) and shows that “a positive correlation between benefit and cost uncertainties tends to favor the quantity instrument (permits)’, while “a negative correlation always tends to favor the price instrument (taxes), ceteris paribus.”

<footnote n="9">⁹Here $q$ is equal to $q_1^*$ because Weitzman uses a Taylor expansion of the cost and benefit functions around $q_1^*$.</footnote>

<footnote n="10">¹⁰We have $\Delta \le 0 \Leftrightarrow \partial_q^2 \mathbf{C}(q) \le -\partial_q^2 \mathbf{B}(q) \Leftrightarrow \partial_q^2 \mathbf{C}(q) \le |\partial_q^2 \mathbf{B}(q)|$.</footnote>
**Deadweight loss analysis** We follow the analysis of Adar and Griffin (1976), who assume that the social marginal benefit and social marginal cost functions are linear, given by SMB (q) = αB - βBq and SMC (q) = αC + βCq. The social optimal equilibrium satisfies SMB (q) = SMC (q), which leads to the following solutions:

$$ 
\left\{ 
\begin{aligned}
q^* &= \frac{\alpha_B - \alpha_C}{\beta_B + \beta_C} \\
p^* &= \frac{\alpha_C \beta_B + \alpha_B \beta_C}{\beta_B + \beta_C}
\end{aligned}
\right. 
$$

If the social planner adopts the quantity instrument, the optimal quota is equal to:

$$ q_1^* = \frac{\alpha_B - \alpha_C}{\beta_B + \beta_C} $$

If instead the social planner uses the price instrument, the optimal price (including the corrective tax) is equal to:

$$ p_2^* = \frac{\alpha_C \beta_B + \alpha_B \beta_C}{\beta_B + \beta_C} $$

Now, suppose there is uncertainty about the marginal cost curve, that is, the social planner believes the cost curve is SMC (q) = αC + βCq, while the realized cost curve is SMC' (q) = α'C + β'Cq. Under this uncertainty, if the quantity instrument is chosen, the resulting price is:

$$ p'(q_1^*) = \text{SMC}'(q_1^*) = \alpha'_C + \beta'_C \left( \frac{\alpha_B - \alpha_C}{\beta_B + \beta_C} \right) \neq p^* = \frac{\alpha_C \beta_B + \alpha_B \beta_C}{\beta_B + \beta_C} $$

Similarly, if the price instrument is chosen, the resulting quantity is:

$$ q'(p_2^*) = \frac{\frac{\alpha_C \beta_B + \alpha_B \beta_C}{\beta_C + \beta_B} - \alpha'_C}{\beta'_C} \neq q^* = \frac{\alpha_B - \alpha_C}{\beta_B + \beta_C} $$

The two equilibria (q*_1, p'_1) and (q'_2, p*_2) are different. In fact, neither is optimal, since the optimal solution is:

$$ 
\left\{ 
\begin{aligned}
q' &= \frac{\alpha_B - \alpha'_C}{\beta_B + \beta'_C} \\
p' &= \frac{\alpha'_C \beta_B + \alpha_B \beta'_C}{\beta_B + \beta'_C}
\end{aligned}
\right. 
$$

But one equilibrium dominates the other, depending on the values of the slopes βB and βC.

Figure 10.4 illustrates the different equilibria<fnref n="11" /> in the case where βB > βC. Point A corresponds to the equilibrium (q*, p*), while point B corresponds to the equilibrium (q', p'). If the social planner adopts the quantity instrument, the resulting equilibrium is at point C = (q*_1, p'_1)). The associated deadweight loss is given by the area of the triangle ABC. If instead the social planner adopts the price instrument, the resulting equilibrium is at point E = (q' (p*_2), p*_2). The associated deadweight loss is given by the area of the triangle BDE. We observe that:

$$ \beta_B > \beta_C \Rightarrow \text{area}(ABC) < \text{area}(BDE) \Rightarrow q^* > p^* $$

because p'(q*_1) is closer to p* than q' (p*_2) is to q*. Since the deadweight loss associated with the price instrument exceeds the deadweight loss associated with the quantity instrument, the quota

<footnote n="11">¹¹We use the following numerical values: αB = 20, βB = 1, αC = 8, βC = 0.25, α'C = 10, and β'C = 0.25. The equilibria are A (9.6, 10.4), B (8, 12), C (9.6, 12.4), D (1.6, 18.4), and E (1.6, 10.4).</footnote>
<figure><img src="image_477.png" /><figcaption>Figure 10.5: Case (b): The marginal cost curve is steeper and there is uncertainty about the marginal cost</figcaption></figure>

<figure><img src="image_478.png" /></figure>
is preferable to the tax, and the quantity instrument is more efficient than the price instrument. Figure 10.5 illustrates the different equilibria<fnref n="12" /> in the case where $\beta_B < \beta_C$. We observe that:

$$ \beta_B < \beta_C \Rightarrow \text{area}(ABC) > \text{area}(BDE) \Rightarrow p^* > q^* $$

Therefore, the tax is better than the quota. We retrieve the key findings of Weitzman.

**An analytical example** We assume that the social marginal benefit and social marginal cost functions are defined as follows:

$$ \begin{cases} \text{SMB}(q) = \alpha_B - \beta_B q \\ \text{SMC}(q) = \alpha_C + \beta_C q + \varepsilon_C \end{cases} $$

where $\varepsilon_C$ is a discrete random variable with $p_i = \text{Pr}\{\varepsilon_C = \varepsilon_i\}$ and $i = 1, \dots, n$. We denote by $\mu_\varepsilon = \sum_{i=1}^n p_i \varepsilon_i$ and $\sigma_\varepsilon^2 = \sum_{i=1}^n p_i (\varepsilon_i - \mu_\varepsilon)^2$ the mean and variance of the uncertainty cost. The SMB curve is deterministic, while the SMC curve is stochastic due to the presence of $\varepsilon_C$. In most cases, we assume that there is information asymmetry between firms and the social planner. Firms have detailed knowledge of their production processes and internal costs, which allows them to understand their abatement costs more precisely than any external observer. Under perfect information, the market equilibrium $q_i^*$ for state $i$ can be derived from the condition:

$$ \begin{aligned} \text{SMB}(q) = \text{SMC}(q) &\Leftrightarrow \alpha_B - \beta_B q = \alpha_C + \beta_C q + \varepsilon_i \\ &\Leftrightarrow q_i^* = \frac{\alpha_B - \alpha_C - \varepsilon_i}{\beta_B + \beta_C} \end{aligned} $$

However, the social planner does not observe the abatement cost shock in advance. Therefore, when designing regulatory instruments, the social planner must rely on the expected values of the SMB and SMC curves to determine the optimal quantity or price:

$$ \begin{aligned} \mathbb{E}[\text{SMB}(q)] = \mathbb{E}[\text{SMC}(q)] &\Leftrightarrow \alpha_B - \beta_B q = \alpha_C + \beta_C q + \sum_{i=1}^n p_i \varepsilon_i \\ &\Leftrightarrow \left\{ \begin{aligned} q^* &= \frac{\alpha_B - \alpha_C - \sum_{i=1}^n p_i \varepsilon_i}{\beta_B + \beta_C} \\ p^* &= \alpha_C + \beta_C q^* + \sum_{i=1}^n p_i \varepsilon_i \end{aligned} \right. \end{aligned} $$

We consider the two policy scenarios:

* In the case of the quantity instrument, the social planner sets the fixed quantity $q^*$. For each state $i$, we have:

$$ q_{1,i}^* = q^* = \frac{\alpha_B - \alpha_C - \mu_\varepsilon}{\beta_B + \beta_C} $$

* In the case of the price instrument, the social planner sets the fixed price $p^*$. For each state $i$, we have $\text{SMC}(q) = \alpha_C + \beta_C q + \varepsilon_i = p^*$. We deduce that:

$$ \begin{aligned} q_{2,i}^* &= \frac{p^* - \alpha_C - \varepsilon_i}{\beta_C} \\ &= \frac{(\alpha_B - \alpha_C) \beta_C + \mu_\varepsilon \beta_B - \varepsilon_i (\beta_B + \beta_C)}{(\beta_B + \beta_C) \beta_C} \end{aligned} $$

<footnote n="12">¹²We use the following numerical values: $\alpha_B = 18$, $\beta_B = 0.5$, $\alpha_C = 1$, $\beta_C = 1$, $\alpha'_C = 7$, and $\beta'_C = 1$. The equilibria are A (11.33, 12.33), B (7.33, 14.33), C (11.33, 18.33), D (5.33, 15.33), and E (5.33, 12.33).</footnote>
We now evaluate the efficiency of each policy instrument by computing social welfare for each state. Social welfare is defined as the area between the marginal benefit and marginal cost curves:

$$
\begin{align*}
W_i(q) &= \int_0^q (\text{SMB}(x) - \text{SMC}(x)) \, dx \\
&= \int_0^q (\alpha_B - \beta_B x - \alpha_C - \beta_C x - \varepsilon_i) \, dx \\
&= \int_0^q (\alpha_B - \alpha_C - \varepsilon_i) \, dx - (\beta_B + \beta_C) \int_0^q x \, dx \\
&= (\alpha_B - \alpha_C - \varepsilon_i) q - \frac{1}{2} (\beta_B + \beta_C) q^2
\end{align*}
$$

If $q$ does not depend on $\varepsilon_i$, then the expected social welfare is equal to:

$$
\begin{align*}
\mathbb{E}[W(q)] &= \sum_{i=1}^{n} p_i W_i(q) \\
&= (\alpha_B - \alpha_C - \mu_\varepsilon) q - \frac{1}{2} (\beta_B + \beta_C) q^2
\end{align*}
$$

We deduce the following welfare functions:

• The welfare with the quantity instrument is obtained by replacing $q$ by $q_{1,i}^*$. We have:

$$
W_i(q_{1,i}^*) = \left( \frac{1}{2}(\alpha_B - \alpha_C) + \frac{1}{2}\mu_\epsilon - \epsilon_i \right) \left( \frac{\alpha_B - \alpha_C - \mu_\epsilon}{\beta_B + \beta_C} \right)
$$

and:

$$
\mathbb{E}[W(q_1^*)] = \frac{1}{2} \frac{(\alpha_B - \alpha_C - \mu_\epsilon)^2}{\beta_B + \beta_C}
$$

• The welfare with the price instrument is obtained by replacing $q$ by $q_{2,i}^*$. We have:

$$
W_i(q_{2,i}^*) = \frac{(\alpha_B - \alpha_C - \varepsilon_i)^2 \beta_C^2 - (\varepsilon_i - \mu_\varepsilon)^2 \beta_B^2}{2(\beta_B + \beta_C) \beta_C^2}
$$

and:

$$
\mathbb{E}[W(q_2^*)] = \frac{(\alpha_B - \alpha_C - \mu_\epsilon)^2 \beta_C^2 - \sigma_\epsilon^2 (\beta_B^2 - \beta_C^2)}{2(\beta_B + \beta_C) \beta_C^2}
$$

Finally, we deduce that:

$$
\begin{align*}
\Delta &= \mathbb{E}[W(q_2^*)] - \mathbb{E}[W(q_1^*)] \\
&= \frac{\sigma_{\varepsilon}^2 (\beta_C^2 - \beta_B^2)}{2 (\beta_B + \beta_C) \beta_C^2} \\
&= \frac{\sigma_{\varepsilon}^2 (\beta_C - \beta_B)}{2 \beta_C^2} \\
&= \frac{\sigma_{\varepsilon}^2}{2 \beta_C} \left(1 - \frac{\beta_B}{\beta_C}\right)
\end{align*}
$$

This matches the expression previously derived in Equation (10.6).
**Example 47** We assume the following parameters: $\alpha_B = 100$, $\beta_B = 4$, $\alpha_C = 20$ and $\beta_C = 5$. The probability distribution of the costs is:

<table><thead><tr><th><i>i</i></th><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><th><i>ε</i><sub><i>i</i></sub></th><td>10</td><td>25</td><td>60</td></tr><tr><th><i>p</i><sub><i>i</i></sub></th><td>40%</td><td>50%</td><td>10%</td></tr></tbody></table>

We have:

$$ \begin{aligned} \mu_{\varepsilon} &= 0.4 \times 10 + 0.5 \times 25 + 0.10 \times 60 \\ &= 21.50 \end{aligned} $$

and:

$$ \begin{aligned} \sigma_{\varepsilon}^{2} &= 0.4 \times 10^{2} + 0.5 \times 25^{2} + 0.10 \times 60^{2} - 21.50^{2} \\ &= 140.25 \end{aligned} $$

We calculate the market equilibrium and get:

$$ \begin{cases} q^* = \frac{100 - 20 - 21.5}{4 + 5} = 6.5 \\ p^* = 20 + 5 \times 6.5 + 21.5 = 74 \end{cases} $$

The social welfare is very different depending on the stochastic component of the cost. For example, in the case of the quota policy, the social welfare is 264.875, 167.375, and 4.875, respectively, when $\varepsilon_C$ takes the value $\varepsilon_1$, $\varepsilon_2$ and $\varepsilon_3$. As expected, social welfare is a decreasing function of $\varepsilon_i$. Since we have $\mathbb{E}[W(q_2^*)] = 192.930$ and $\mathbb{E}[W(q_1^*)] = 190.125$, we conclude that $\Delta$ is positive:

$$ \Delta = 192.930 - 190.125 = \frac{140.25}{2 \times 5} \times \left(1 - \frac{4}{5}\right) = 2.805 $$

Figure 10.6 shows a comparative analysis of quantity and price instruments with respect to the slope parameter $\beta_C$. There is an asymmetry around the line $\beta_C = \beta_B$. The area between the two expected social welfare values, $\mathbb{E}[W(q_1^*)]$ and $\mathbb{E}[W(q_2^*)]$, is substantial when $\beta_C < \beta_B$ and relatively small when $\beta_C > \beta_B$. This asymmetry arises because the parameters $\beta_B$ and $\beta_C$ affect the term $\Delta$ differently. Specifically, the numerator of $\Delta$ is a linear function of both $\beta_B$ and $\beta_C$, while the denominator is a quadratic function of only $\beta_C$. As a result, $\Delta$ decreases as $\beta_C$ increases, which explains the observed variation in expected welfare in Figure 10.6.

<table><caption>Table 10.1: Social welfare of quantity and price instruments (Example 47)</caption><thead><tr><th><i>i</i></th><th><i>q</i><sup>*</sup><sub><i>i</i></sub></th><th><i>q</i><sup>*</sup><sub>1,i</sub></th><th><i>W</i><sub><i>i</i></sub>(<i>q</i><sup>*</sup><sub>1,i</sub>)</th><th><i>q</i><sup>*</sup><sub>2,i</sub></th><th><i>W</i><sub><i>i</i></sub>(<i>q</i><sup>*</sup><sub>2,i</sub>)</th></tr></thead><tbody><tr><td>1</td><td>7.778</td><td>6.500</td><td>264.875</td><td>8.800</td><td>267.520</td></tr><tr><td>2</td><td>6.111</td><td>6.500</td><td>167.375</td><td>5.800</td><td>167.620</td></tr><tr><td>3</td><td>3.333</td><td>6.500</td><td>4.875</td><td>0.800</td><td>21.120</td></tr><tr><td colspan="2">E[W(<i>q</i><sup>*</sup>)]</td><td colspan="2">190.125</td><td colspan="2">192.930</td></tr></tbody></table>

**Remark 122** The analysis of Weitzman (1974) focuses on the cost uncertainty and not really on the benefit uncertainty. One way to show this is to consider the previous model with SMB($q$) = $\alpha_B - \beta_B q + \varepsilon_B$. In this case we get a complex formula of $\Delta$ that depends on both $\sigma_B^2$ and $\sigma_C^2$.
<figure><img src="image_479.png" /><figcaption>Figure 10.6: Comparative analysis between quantity and price instruments with respect to the slope parameter $\beta_C$ (Example 47)</figcaption></figure>

Double dividend

The double dividend hypothesis is a fundamental concept in environmental economics. It suggests using taxation to address negative externalities instead of quota-based policies. The double dividend refers to the idea that environmental taxes can yield two types of benefits or dividends:

1. The first dividend is an environmental benefit. By taxing pollution, the policy internalizes the external cost, thereby reducing harmful activities and improving environmental quality.

2. The second dividend is an economic efficiency gain. Revenue from environmental taxes can finance positive externalities, such as education, or reduce other distortionary taxes, such as income or payroll taxes. This can improve the overall efficiency of the tax system, leading to economic benefits beyond just the environmental improvement.

The idea of the double dividend hypothesis was first advanced by Tullock (1967). Until the 1970s, the two branches of tax theory — one tax intended to correct negative externalities (the Pigouvian tax) and the other intended to raise revenue by minimizing tax distortions (the Ramsey tax) — were considered separately (Jaeger, 2012). However, Sandmo (1975) studied the combination of these two tax problems. He demonstrated that in a second-best world where the government requires revenue and externalities exist, the optimal tax structure differs from those of pure Pigouvian and pure Ramsey taxes. The presence of other distortionary taxes affects the optimal environmental tax rate. About fifteen years after Sandmo's seminal contribution, and after a growing body of literature on the subject, Pearce (1991) coined the term double dividend:

> "While most taxes distort incentives, an environmental tax corrects a distortion, namely the externalities arising from the excessive use of environmental services. A carbon tax
would be set on the basis of the carbon content of fossil fuels. Given the widespread
use of these fuels, any tax would inevitably be revenue raising, even though the tax
works best if it is avoided through the introduction of low or zero carbon technologies.
Governments may then adopt a fiscally neutral stance on the carbon tax, using revenues
to finance reductions in incentive — distorting taxes such as income tax, or corporation
tax. This ‘double dividend’ feature of a pollution tax is of critical importance in the
political debate about the means of securing a ‘carbon convention’. Industry will resist
any new tax. Politicians are understandably nervous about introducing such taxes. But
the corporate and public acceptability of such a tax is greatly enhanced if the tax is
introduced as part of a ‘package’ of fiscally neutral measures.” (Pearce, 1991, page 940).

Goulder (1995) introduced two forms of the double dividend hypothesis:

*   **Weak form**
    When revenues from environmental taxes are used to reduce existing distortionary taxes, the economy can achieve cost savings compared to simply returning the revenue through lump-sum transfers<fnref n="13" />.

*   **Strong form**
    Revenue-neutral swaps replace distortionary taxes with environmental taxes, which can result in zero or even negative gross costs. This means there is no economic cost and a potential net gain before considering environmental benefits.

Goulder believed the weak double dividend hypothesis was valid but was skeptical of the strong double dividend hypothesis. He argued that environmental taxes could reduce certain distortions, but they could also create others. Since then, numerous economists have examined the double dividend hypothesis, but the evidence has not provided a definitive confirmation or refutation. While the existence of an environmental dividend is generally accepted, the existence and magnitude of the economic dividend remain contested, with empirical studies producing mixed and sometimes contradictory results (Freire-González, 2018).

**Remark 123** In practice, choosing between taxes and quotas is not mutually exclusive. For example, under conditions of significant uncertainty, a hybrid approach combining different instruments may be more effective (McKibbin and Wilcoxen, 2002; Hepburn, 2006).

10.1.3 Cost-benefit analysis

A cost-benefit analysis (CBA) is a systematic framework for evaluating the economic pros and cons of different choices. CBA is commonly used in areas such as public policy, infrastructure, health, and environmental decision-making. The central objective of CBA is to quantify and compare the total expected benefits and costs of a project or policy in monetary terms. By identifying, valuing, and aggregating all relevant impacts, CBA can determine whether the benefits of a proposed action outweigh its costs. If they do, the project is considered economically justified. Governments often use CBA to compare multiple policy options and select the one that will deliver the greatest net benefit to society. However, CBA is also widely used by private firms to guide investment decisions. Importantly, CBA is not a single model or rigid method but rather a set of tools and principles.

<footnote n="13">¹³In the context of environmental tax revenue recycling, a lump-sum transfer is the redistribution of revenue to individuals in a fixed and non-distortionary manner. This means that everyone receives the same amount, regardless of their income, behavior, or economic activity.</footnote>
<figure><img src="image_480.png" /><figcaption>Figure 10.7: Examples of CBA guidelines</figcaption></figure>

As a result, its application can vary significantly from one country to another, between different organizations, and across sectors. For instance, a CBA for environmental policies differs substantially from one for health interventions. This flexibility has led some to describe CBA as a ‘fourre-tout’ — a catch-all bag — rather than a strictly defined methodology. To ensure consistency and rigor, most CBAs are guided by formalized frameworks or official guidelines, which guarantee that costs and benefits are measured and weighed in a comparable and transparent manner. Figure 10.7 shows four examples of CBA official guidelines published by governments and international bodies<fnref n="14" />.

According to Mishan and Quah (2020), CBA has its roots to the mid-19th century. In 1844, Jules Dupuit was the first to link the value of public projects to users' willingness to pay, marking the beginning of CBA. Although Alfred Marshall formalized key economic concepts, such as utility and consumer surplus, in his *Principles of Economics* published in 1890, he did not explicitly refer to CBA. A crucial theoretical breakthrough occurred in the late 1930s when John Hicks and Nicholas Kaldor developed the compensation principle. This principle provides the foundation of welfare economics, which was lacking in early CBA applications. The Kaldor-Hicks efficiency criterion establishes that a policy change is economically efficient if the beneficiaries could theoretically compensate the losers and still be better off. This criterion allows economists to evaluate efficiency without requiring actual compensation or interpersonal utility comparisons. The practical application of CBA began with the U.S. Flood Control Act of 1936, which required that the benefits of flood control projects exceed their costs. Despite this formal adoption, early uses of CBA lacked a clear theoretical foundation in welfare economics and often failed to define costs and benefits rigor-

<footnote n="14">¹⁴The following detailed list of CBA guidelines illustrates how CBAs are formalized in developed countries: (1) Commonwealth of Australia (2006). *Handbook of Cost Benefit Analysis*. Financial Management Reference Material No. 6, January, 180 pages. (2) Treasury of New Zealand (2015). *Guide to Social Cost Benefit Analysis*. July, 78 pages. (3) European Commission (2015). *Guide to Cost-Benefit Analysis of Investment Projects — Economic Appraisal Tool for Cohesion Policy 2014–2020*. 364 pages. (4) Romijn, G. and Renes, G. (2015). *General Guidance for Cost-Benefit Analysis*. CPB Netherlands Bureau of Economic Policy Analysis and PBL Netherlands Environmental Assessment Agency, June, 184 pages. (5) New South Wales (NSW) Treasury (2017). NSW Government Guide to Cost-Benefit Analysis. TPP 17-03, March, 79 pages. (6) Atkinson, G., Braathen, N. A., Groom, B., and Mourato, S. (2018). *Cost-Benefit Analysis and the Environment: Further Developments and Policy Use*. OECD Publishing, June, 458 pages. (7) Treasury of New Zealand (2020). Guide to Cabinet's Impact Analysis Requirements. June, 46 pages. (8) European Commission (2022). *Economic Appraisal Vademecum 2021–2027 — General Principles and Sector Applications*. 98 pages. (9) HM Treasury (2022). The Green Book — Central Government Guidance on Appraisal and Evaluation. Government of the United Kingdom, 148 pages. (10) New South Wales (NSW) Treasury (2023). *Cost-Benefit Analysis Guidelines*. TPG23-08, February, 116 pages. (11) Office of Impact Analysis (2023). *Cost Benefit Analysis*. Australian Government, July, 19 pages. (12) US EPA (2024). *Guidelines for Preparing Economic Analyses*. Third edition, EPA-240-R-24-001, December, 411 pages.</footnote>
ously. Significant theoretical advances were made in the 1950s and 1960s, particularly through the work of economists such as Otto Eckstein, John Krutilla, and Roland McKean. These economists established a stronger foundation for CBA in welfare economics. During this time, the United States began applying CBA to new areas such as environmental regulation and education, while the United Kingdom used it to evaluate large-scale transportation projects such as motorways and underground railway lines. Despite its growing use in policy and project evaluation, CBA remained marginal in academic economics. By the mid-1970s, CBA had become a widely adopted tool in infrastructure and policy analysis, though methodological issues persisted.

## Welfare analysis

Welfare analysis is fundamental to cost-benefit analysis because it provides the theoretical and conceptual framework for evaluating the impact of policies, projects, or decisions on a society's overall well-being. It addresses issues such as utility aggregation, measuring social value, handling distributional effects, and establishing decision rules.

**Public goods, externalities and market failures** Public goods have two defining characteristics that distinguish them from private goods:

*   **Non-rivalrous**

One person's consumption of a public good does not reduce the amount available to others. For example, watching a public fireworks display does not diminish anyone else's enjoyment.

*   **Non-excludable**

It is difficult or impossible to prevent someone from consuming the good, even if they don't pay for it. For example, once national defense is provided, it protects all citizens, regardless of whether they paid their taxes.

Because public goods cannot effectively exclude users and one person's use does not diminish another's, these goods tend to be underprovided in free markets. Private firms have no incentive to produce public goods because they cannot easily charge users. This creates the classic free rider problem. Examples of public goods include bridges, clean air, infrastructure, national defense, roads, street lighting, and warning systems for natural disasters<fnref n="15" />.

In perfectly competitive markets, resources are efficiently allocated based on supply and demand. However, public goods violate key assumptions of competitive markets. Due to non-excludability, there is no clear pricing mechanism, and the marginal cost of providing an additional unit is often zero due to non-rivalry. Consequently, private firms cannot generate sufficient revenue to justify production, suggesting that profit maximization cannot be a goal or incentive. These factors result in market failures, which occur when the private market fails to produce the socially optimal quantity of a good or service. Furthermore, the provision of public goods often generates positive externalities, providing benefits that extend beyond the immediate users. For example, a well-educated population or widespread public health initiatives can improve the overall well-being of society, not just the individuals who receive education or medical care directly. Conversely, negative externalities can create the need for public goods. Pollution, for example, creates a demand for environmental regulations and public investment in cleanup systems. These dynamics highlight the relationship between public goods, externalities, and market failures.

<footnote n="15">¹⁵It's important to note that goods such as bridges and roads can have characteristics of both public and private goods because they can become both rivalrous, due to congestion, and excludable through tolls.</footnote>
A stable climate is considered a global public good. It satisfies the criteria of being both non-rivalrous<fnref n="16" /> and non-excludable<fnref n="17" />. However, maintaining a stable climate as a public good is challenging because of its global nature. This means that solutions require international cooperation and agreement. Greenhouse gas emissions are then a global negative externality. The case of biodiversity is more complex. For example, ecosystem services, such as pollination, water filtration, and carbon sequestration, often function as public goods. However, direct biodiversity benefits, such as pharmaceutical discoveries from plants, can be partially excludable through intellectual property. Additionally, some biodiversity benefits are local, such as watershed protection, while others are global, such as genetic diversity for crop resilience. This explains why some aspects of biodiversity may be club or private goods. For example, a safari park can exclude non-payers, and a fishery is both rivalrous and excludable with property rights.

**Willingness to pay and willingness to accept** Cost-benefit analysis involves measuring the costs and benefits — or more broadly, the gains and losses — associated with a policy, project, or regulation. To enable meaningful comparisons, whether between costs and benefits or across alternative projects, these gains and losses must be expressed in a common unit or ‘numéraire’. In CBA, this numéraire is usually money, which implies that preferences and utility must be translated into monetary terms. When private or market goods are involved, this translation is relatively straightforward because market prices can be used to evaluate costs and benefits. However, valuing public goods is challenging because they lack market prices and are not typically traded in conventional markets. In such cases, economists use the concepts of willingness to pay (WTP) and willingness to accept (WTA) to assign monetary values to changes in public goods.

According to Pearce et al. (2006), willingness to pay represents the maximum amount an individual is willing to pay for an improvement in environmental conditions from a baseline state $S_0$ to an improved state $S_1$, provided that the individual’s utility remains unchanged. In this framework, $S_0$ and $S_1$ represent the current and improved states of environmental quality, respectively. More generally, $S_0$ and $S_1$ can represent any initial and preferred states where $S_1 > S_0$. Let $\mathcal{U}(t_0, Y_0, S_0)$ represent the individual’s initial utility, well-being, or welfare given an income $Y_0$ and environmental state $S_0$. With the proposed improvement to $S_1$, the individual’s utility becomes $\mathcal{U}(t_1, Y_0, S_1)$. We assume the following preference relation:

$$S_1 > S_0 \Leftrightarrow \mathcal{U}(t_0, Y_0, S_1) \geq \mathcal{U}(t_0, Y_0, S_0)$$

However, as noted by Pearce et al. (2006), this change in utility cannot be directly observed or measured. Instead, an indirect monetary measure must be found that quantifies the improvement in terms of how much an individual would be willing to pay to achieve it. This monetary value is defined by the following equation:

$$\mathcal{U}(t_0, Y_0 - \text{WTP}, S_1) = \mathcal{U}(t_0, Y_0, S_0)$$

In this formulation, the individual adjusts the amount of money he is willing to pay until the utility in the improved state $S_1$, after deducting the willingness to pay from income, equals the original utility in state $S_0$. This measure of WTP is referred to as the individual’s compensating variation (CV). It reflects the monetary equivalent of the perceived benefit measured relative to the initial level of well-being at time $t_0$ (Pearce et al., 2006, page 45).

<footnote n="16">¹⁶The benefits of a stable climate, such as moderate temperatures, predictable weather patterns, and an absence of extreme events, are enjoyed by everyone simultaneously. One person’s enjoyment of a stable climate does not reduce anyone else’s ability to enjoy it.</footnote>

<footnote n="17">¹⁷It is practically impossible to exclude anyone on Earth from the benefits of a stable global climate. Efforts to mitigate climate change, such as reducing greenhouse gas emissions, benefit everyone, regardless of their contribution.</footnote>
Conversely, willingness to accept represents the minimum compensation required by an individual to tolerate environmental degradation. The individual is compensated until his utility remains unchanged, despite the decline in quality from the baseline state $S_0$ to the lower-quality state $S_1$:

$$ \mathcal{U}(t_1, Y_0 + \text{WTA}, S_0) = \mathcal{U}(t_1, Y_0, S_1) $$

where $S_1 \prec S_0$. This measure of WTA is referred to as the individual's equivalent variation (EV). It reflects the monetary equivalent of the perceived degradation measured relative to the final level of well-being at time $t_1$. Alternative definitions of WTP and WTA are also possible<fnref n="18" /> (see Table 10.2).

<table><caption>Table 10.2: Compensating and equivalent variation measures</caption><thead><tr><th>Reference time</th><th>t<sub>0</sub> (before change)</th><th>t<sub>1</sub> (after change)</th></tr><tr><th>Human welfare</th><th>Compensating variation</th><th>Equivalent variation</th></tr></thead><tbody><tr><td>Increase</td><td><b>U</b>(t<sub>0</sub>, Y<sub>0</sub> - WTP, S<sub>1</sub>) = <b>U</b>(t<sub>0</sub>, Y<sub>0</sub>, S<sub>0</sub>)</td><td><b>U</b>(t<sub>1</sub>, Y<sub>0</sub> + WTA, S<sub>0</sub>) = <b>U</b>(t<sub>1</sub>, Y<sub>0</sub>, S<sub>1</sub>)</td></tr><tr><td>Decrease</td><td><b>U</b>(t<sub>0</sub>, Y<sub>0</sub> + WTA, S<sub>1</sub>) = <b>U</b>(t<sub>0</sub>, Y<sub>0</sub>, S<sub>0</sub>)</td><td><b>U</b>(t<sub>1</sub>, Y<sub>0</sub> - WTP, S<sub>0</sub>) = <b>U</b>(t<sub>1</sub>, Y<sub>0</sub>, S<sub>1</sub>)</td></tr></tbody></table>

Source: Pearce et al. (2006, Table 1.1, page 46).

In general, willingness to pay and willingness to accept differ, except in certain special cases. One such case arises under quasi-linear utility, which takes the following form: $\mathcal{U}(Y, S) = V(S) + Y$, where $Y$ is the income and $S$ is the public good. The key property of quasi-linear utility functions is that the marginal utility of income is constant: $\partial_Y \mathcal{U}(Y, S) = 1$. This property eliminates the income effects associated with changes in the public good. As a result, under quasi-linear preferences, WTP equals WTA because the marginal rate of substitution between the public good and income remains constant, regardless of the individual's wealth. In this case, WTP and WTA both measure consumer surplus (Box 10.2). In the more general case, however, WTA is typically greater than WTP due to income effects. When individuals pay for a good, they become poorer, which may reduce the value they assign to it. Conversely, when they are compensated for a loss, they become richer, which may increase the value they associate with the good. However, it is generally accepted that the following approximation holds under certain conditions:

$$ \text{WTA} \approx W_{CS} \approx \text{WTP} $$

where $W_{CS}$ denotes consumer surplus. This approximation has been supported both empirically and theoretically. For instance, Willig (1976) demonstrates that:

$$ \frac{1}{2}\varepsilon - \frac{W_{CS}}{Y_0} < \left| \frac{\{\text{WTP, WTA}\} - W_{CS}}{W_{CS}} \right| < \frac{1}{2}\varepsilon + \frac{W_{CS}}{Y_0} $$

where $\varepsilon^-$ and $\varepsilon^+$ represent the minimum and maximum values of the income elasticity of demand, respectively, and $Y_0$ is initial income. Willig argues that the ratio $W_{CS}/Y_0$ is typically very small, which supports the validity of the approximation. Since the 1990s, however, this approximation has been increasingly challenged. Empirical and theoretical studies — including those by Hanemann (1991), Shogren et al. (1994), and Chapman et al. (2017) — have demonstrated that significant differences between WTP and WTA can arise, particularly in the presence of income effects, substitution asymmetries, or non-market goods.

<footnote n="18">¹⁸More generally, Pearce et al. (2006, page 46) define compensating variation as “the amount of Y that can be taken from an individual after a change such that he/she is as well off as they were before the change” while equivalent variation refers to “the amount of Y that would have to be given to the individual, if a change does not occur, to make him/her as well off as if the change did take place”. These different measures of welfare gains and losses are given in Table 10.2. By definition, CV and EV are Hicksian utility measures which are often used interchangeably with WTP and WTA (Exercise 10.5.1 on page 1038). However, they can differ in practice.</footnote>