European Union

In December 2019, the European Commission announced the European Green Deal, a framework and roadmap to make Europe the first climate neutral continent. The European Green Deal includes a wide range of initiatives to tackle the challenges of climate change, halt environmental degradation, promote sustainable growth and ensure a just transition. The goal set out in the European Green Deal has been translated into law in the European Climate Law, which was formally adopted by the EU Council in June 2021. The objective of the European Climate Law is to set the long-term direction for achieving the 2050 climate neutrality target across all policy areas in a socially equitable and cost-effective manner, and includes the following components:

* Set a 2030 target to reduce net greenhouse gas emissions by at least 55% compared to 1990 levels;

* Establish a system to monitor progress and take further action if necessary;

* Establish the European Scientific Advisory Board on Climate Change to provide independent scientific advice;

* Define the EU-wide greenhouse gas emission reduction path for 2030–2050;

* Provide predictability for investors and other economic actors;

* Ensure that the transition to climate neutrality is irreversible;

* Commit to negative emissions after 2050.

In July 2021, the European Commission presented the Fit for 55 package, which aims to meet the first quantitative target of the European Climate Change Law, namely to reduce Europe's net greenhouse gas emissions by at least 55% by 2030 (Figure 8.78). The package consists of several legislative proposals. As of 1 January 2024, 12 proposals have been adopted<fnref n="85" />:

1. Reform of the emissions trading system (ETS, April 2023)

The EU emissions trading system tackles climate change by putting a price on carbon emissions. Companies operating in certain sectors must buy allowances each year to cover their greenhouse gas emissions. These allowances are limited in number and decrease each year, providing a financial incentive to reduce emissions. To prevent certain industries from being disadvantaged by carbon leakage (the shifting of emissions to unregulated areas), they receive free allowances to maintain their competitiveness. The EU ETS covers around 40% of total emissions produced in the EU and has significantly reduced emissions from covered sectors by 41% since its launch in 2005. It covers around 10,000 companies in sectors such as electricity and heat production, energy-intensive industries (including oil refineries, steel production, cement, glass and paper manufacturing) and commercial aviation for flights within the European Economic Area. The reform extends the scope of the EU ETS to maritime transport and increases the 2030 emission reduction target from 43% to 62% compared to the 2005 baseline. This implies a faster reduction in the emissions cap, with an annual reduction of 4.3% between 2024 and 2027 and 4.4% between 2028 and 2030, compared to the current rate of 2.2%. As a result, fewer allowances will be available on the market between 2024 and 2030. In addition, there are plans to gradually phase out free allowances for certain sectors, coinciding with the introduction of the carbon border adjustment mechanism (CBAM).

<footnote n="85">⁸⁵All the information below has been taken from the European Council website (Figure 8.79). By clicking on each icon, we can access an infographic explaining the content of a particular regulation. The text of each bullet point is a plain language version of each infographic.</footnote>
<figure><img src="image_360.png" /><figcaption>Figure 8.78: European Commission communication on the European Green Deal, 14 July 2021</figcaption></figure>

Source: European Commission (2021), https://twitter.com/EU_Commission/status/1415289818565816321.
2. New emissions trading system for building and road transport fuels (April 2023)

A new emissions trading system, known as ETS 2, has been set up to regulate CO₂ emissions from fuel combustion in buildings, road transport and other additional sectors (small industries). The main objectives are to stimulate investment in building renovation and to promote low-emission mobility. ETS 2 is due to start in 2027. Unlike the existing EU ETS, ETS 2 will use a cap-and-trade mechanism that focuses on upstream emissions, targeting fuel suppliers rather than end users such as households or vehicle owners. The emissions cap for ETS 2 aims to achieve a 42% reduction by 2030 compared to 2005 levels.

3. Social climate fund (April 2023)

It is estimated that more than 34 million people in the EU live in energy poverty. In addition, the implementation of the new emissions trading system for buildings, road transport and other sectors may indirectly affect certain new categories of individuals and businesses. The main objective of the social climate fund is to mitigate energy poverty and improve access to low-emission mobility and transport in the EU. Revenues from the sale of allowances under the ETS 2 are allocated to the social climate fund. Member States can use the fund to support vulnerable households, small businesses and transport users through various measures and investments aimed at improving energy efficiency in buildings, renovating buildings, decarbonizing heating and cooling systems, promoting low-emission mobility and transport, and providing temporary and limited direct income support.

4. Revision of the effort sharing regulation (ESR, March 2023)

The effort sharing regulation sets targets for each Member State to reduce greenhouse gas emissions from several economic sectors that currently account for about 60% of total EU emissions<fnref n="86" />: agriculture, buildings, road transport, small industry and waste. With the introduction of the Fit for 55 package, the buildings and road transport sectors will be covered by both the ESR and ETS 2. Under the updated regulations, the EU aims to achieve a 40% reduction in greenhouse gas emissions from the ESR sectors by 2030 compared to 2005 levels, an increase from the previous target of 29%. To achieve this, the revised effort sharing regulation sets new binding 2030 targets for each Member State and sets annual national emission limits leading progressively to these national 2030 targets<fnref n="87" />. It also introduces a number of flexibilities to help Member States meet their targets<fnref n="88" />.

5. Revision of the regulation on land use, forestry and agriculture (LULUCF, March 2023)

The LULUCF regulation, adopted in 2018, requires EU Member States to ensure that emissions from land use, land-use change, and forestry are offset by equivalent removals in the period 2021–2030 (the “no debit” rule). Activities covered by the rules include those related to land use and forestry, which can occur in areas such as grasslands, agricultural land, and forests. Carbon sequestration is the process by which forests and land absorb CO₂, with EU forests currently absorbing nearly 10% of the EU’s total annual greenhouse gas emissions. The new rules include an increased carbon sequestration target of 310 MtCO₂e by 2030, compared to the current target<fnref n="89" /> of 225 MtCO₂e. Binding targets for Member States are based on recent

<footnote n="86">⁸⁶The remaining 40% of GHG emissions are covered by the EU ETS.</footnote>

<footnote n="87">⁸⁷Table 8.29 shows the national reduction targets by 2030 under the ESR. It compares the original 2018 levels with the revised 2023 levels. For example, Austria's original target was to reduce GHG emissions by 36%. The revised target is now 48%.</footnote>

<footnote n="88">⁸⁸There are 3 flexibility rules: (1) if a country emits less than its annual limit, it can use part of the surplus in the following year; (2) if a country emits more than its annual allocation, it can borrow from the following year's allocation; (3) countries can buy and sell surplus allocations among themselves.</footnote>

<footnote n="89">⁸⁹In 2019, net carbon removals in the EU were already 195 MtCO₂e, with the following breakdown: -329.4 forest,</footnote>
<table><caption>Table 8.29: National reduction targets under the ESR (revised 2023 vs. original 2018)</caption><thead><tr><th>Country</th><th>2018</th><th>2023</th><th>Country</th><th>2018</th><th>2023</th><th>Country</th><th>2018</th><th>2023</th></tr></thead><tbody><tr><td>Austria</td><td>36.0%</td><td>48.0%</td><td>France</td><td>37.0%</td><td>47.5%</td><td>Malta</td><td>19.0%</td><td>19.0%</td></tr><tr><td>Belgium</td><td>35.0%</td><td>47.0%</td><td>Germany</td><td>38.0%</td><td>50.0%</td><td>Netherlands</td><td>36.0%</td><td>48.0%</td></tr><tr><td>Bulgaria</td><td>0.0%</td><td>10.0%</td><td>Greece</td><td>16.0%</td><td>22.7%</td><td>Poland</td><td>7.0%</td><td>17.7%</td></tr><tr><td>Croatia</td><td>7.0%</td><td>16.7%</td><td>Hungary</td><td>7.0%</td><td>17.7%</td><td>Portugal</td><td>17.0%</td><td>28.7%</td></tr><tr><td>Cyprus</td><td>24.0%</td><td>32.0%</td><td>Ireland</td><td>30.0%</td><td>42.0%</td><td>Romania</td><td>2.0%</td><td>12.7%</td></tr><tr><td>Czechia</td><td>14.0%</td><td>26.0%</td><td>Italy</td><td>33.0%</td><td>43.7%</td><td>Slovakia</td><td>12.0%</td><td>22.7%</td></tr><tr><td>Denmark</td><td>39.0%</td><td>50.0%</td><td>Latvia</td><td>6.0%</td><td>17.0%</td><td>Slovenia</td><td>15.0%</td><td>27.0%</td></tr><tr><td>Estonia</td><td>13.0%</td><td>24.0%</td><td>Lithuania</td><td>9.0%</td><td>21.0%</td><td>Spain</td><td>26.0%</td><td>37.7%</td></tr><tr><td>Finland</td><td>39.0%</td><td>50.0%</td><td>Luxembourg</td><td>40.0%</td><td>50.0%</td><td>Sweden</td><td>40.0%</td><td>50.0%</td></tr></tbody></table>

Source: European Commission (2023), https://climate.ec.europa.eu/eu-action/effort-sharing-member-states-emission-targets.

removal levels and the potential for further increases. The accounting rules are simplified and a two-phase approach is introduced, with phase 1 maintaining the current system until 2025 and phase 2 introducing a new EU-level target for net removals by 2030.

## 6. Revised regulation on CO₂ emissions from new cars and vans (March 2023)

Cars and vans are responsible for around 15% of the EU's total CO₂ emissions. To address this, the proposed regulation increases the CO₂ emission reduction targets for 2030 and sets a new 100% target for 2035. This means that all new cars and vans put on the EU market after 2035 would be zero-emission vehicles. The proposed agenda for CO₂ emission reductions for new cars and vans is as follows:

(a) 2021–2024: limit value of 95 gCO₂/km for cars and 147 gCO₂/km for vans;

(b) 2025–2029: 15% reduction for both cars and vans (resulting in 81 gCO₂/km for cars and 125 gCO₂/km for vans);

(c) 2030–2034: 55% reduction for cars and 50% for vans (resulting in 43 gCO₂/km for cars and 73.5 gCO₂/km for vans);

(d) 2035+: 100% reduction for both cars and vans (resulting in 0 gCO₂/km).

In addition to the transition to zero-emission vehicles and cleaner air, this regulation aims to strengthen the technological leadership and competitiveness of the EU automotive industry<fnref n="90" />

## 7. Carbon border adjustment mechanism (CBAM, April 2023)

The carbon border adjustment mechanism is a tool to address carbon leakage, where industries with high GHG emissions relocate production outside the EU to areas with less stringent climate policies. CBAM works by requiring EU importers to purchase CBAM allowances for products manufactured outside the EU to offset the price advantage resulting from the lack of ETS allowances. This mechanism aligns with the EU's goal of climate neutrality by working alongside the EU ETS to promote emission reductions in high-emitting industries inside and

<footnote>+41.1 cropland, +44.4 settlement, +13.1 grassland, +1.7 other land, +16.9 wood products, +16.9 wetlands and +0.4 other.</footnote>

<footnote n="90">⁹⁰The EU automotive industry accounts for more than 7% of the EU's GDP, provides 12.7 million jobs or 6.6% of all jobs in the EU, is the world's second largest producer of motor vehicles after China, and is the largest research and development sector in the EU.</footnote>
outside the EU. It also encourages other countries to adopt carbon pricing policies. Initially, the CBAM will cover sectors with high emissions and a high risk of carbon leakage, including iron and steel, cement, fertilizers, aluminum, electricity, hydrogen production, and certain products upstream and downstream of these sectors. Over time, the scope of the CBAM will be expanded to include additional sectors.

8. Renewable energy directive (RED III, October 2023)

In 2009, the first renewable energy directive (RED I) set a minimum target of 20% for the use of renewable energy by 2020. This target was exceeded, with nearly 22% of energy consumed in the EU coming from renewable sources in 2021. Building on this achievement, RED II was introduced in 2018, aiming for a 32% share of renewable energy by 2030. In October 2023, the European Council adopted the third iteration of the EU Renewable Energy Directive (RED III), setting a target of 42.5% renewable energy by 2030. This new target will nearly double the EU's share of renewable energy compared to 2020. In addition, RED III sets specific sectoral targets for 2030:

(a) Buildings: Target of 49% renewable energy;

(b) Industry: Increase the use of renewable energy by 1.6% per year;

(c) Hydrogen in industry: Target 42% from renewable fuels of non-biological origin (60% by 2035);

(d) Heating and cooling: Increase the use of renewable energy by 0.8% per year until 2026, followed by an annual increase of 1.1% until 2030;

(e) Transport: Member States have the option of either reducing the emission intensity of transport fuels by 14.5% or ensuring that the share of renewable energy in transport is at least 29%; In addition, the combination of advanced biofuels and renewable fuels of non-biological origin should reach 5.5%, with at least 1% coming from renewable fuels of non-biological origin (mainly hydrogen).

9. Energy efficiency directive (EED, July 2023)

The EU energy efficiency directive has led to significant energy savings of nearly one-third compared to 2007 consumption projections for 2030. The revised legislation makes it mandatory for the EU as a whole to reduce final energy consumption. Specifically, the new target requires a mandatory 11.7% reduction in EU final energy consumption by 2030 compared to 2020 projections. This builds on the previous achievement of an average reduction of 29% (compared to 2007 estimates for 2030). In particular, the previous targets aimed at a 32.5% reduction in both primary and final consumption. The revised strategy introduces two new targets: an indicative target of a 40.6% reduction in primary consumption and a mandatory target of a 38% reduction in final consumption. Member States will also have to step up their energy savings efforts between 2024 and 2030, with an average annual saving of 1.49% in final energy consumption, rising to 1.9% by 2030. Key sectors such as buildings, transport and industry will be targeted for energy savings. The public sector will contribute by reducing final consumption by 1.9% per year, excluding public transport and the armed forces, and by renovating buildings to improve energy performance at a rate of 3% per year.

10. Alternative fuels infrastructure regulation (AFIR, July 2023)

The aim of the AFIR is to ensure that there is sufficient infrastructure for cars, trucks, ships and planes to charge or refuel with alternative fuels such as hydrogen or liquefied methane. This infrastructure should be well distributed across the Union to alleviate concerns about
limited range. Although currently only about 5% of the total fleet, there are over 13.4 million alternative fuel cars and vans in the EU and it is estimated that this could increase tenfold by 2050. The expected changes will affect road transport, ports and airports. For road transport, charging stations will be installed every 60 kilometers on major roads for vehicles under 3.5 tonnes by 2025 and for heavier trucks by 2030. These stations will increase their capacity annually to accommodate the growing number of registered vehicles. For trucks over 3.5 tonnes, there will be two charging stations per secure parking area by 2027, increasing to four by 2030. Charging stations will also be installed in urban areas, with exceptions for low-traffic roads. Hydrogen refuelling stations will be installed every 200 kilometers on major roads by 2030, with at least one in every urban area. Each station will be able to provide 1 tonne of hydrogen per day at 700 bar pressure. Liquefied methane refuelling stations will be strategically located along major roads to facilitate the use of methane-powered vehicles across the EU. In ports, at least 90% of container and passenger ships in the busiest seaports will have access to shore-side electricity. Most inland waterway ports will have shore-side electricity by 2030. For airports, electricity will be available for all aircraft stands near terminals by 2025 and for all remote stands by 2030. Exceptions may be made for remote stands at airports with less than 10 000 flights per year.

11. ReFuelEU aviation regulation (October 2023)

The ReFuelEU aviation regulation is a new legislation aimed at reducing emissions from aviation, which currently accounts for 14.4% of EU transport emissions (2018 data). The regulation requires aviation fuel suppliers at EU airports to gradually increase the proportion of sustainable fuels they distribute, with a particular focus on synthetic fuels. The mandatory minimum share of sustainable aviation fuels will start at 2% in 2025, rising to 6% in 2030, 20% in 2035, 34% in 2040 and 42% in 2045. The ultimate goal is to reach 70% by 2050. It also requires EU airports to develop the necessary infrastructure for the delivery, storage and refuelling of sustainable aviation fuels. In addition, the regulation includes the implementation of an EU labeling system to inform consumers about the environmental performance of flights. This system is designed to empower consumers to make informed choices and encourage the uptake of greener flight options.

12. FuelEU maritime regulation (July 2023)

Like the ReFuelEU aviation regulation, the FuelEU maritime regulation is a new regulation for ships over 5 000 gross tonnes calling at European ports<fnref n="91" /> (with exceptions such as fishing vessels). The objective is to reduce the GHG intensity of the energy used on board by 2% in 2025, 6% in 2030, 14.5% in 2035, 31% in 2040 and 62% in 2045. The ultimate goal is to achieve 80% by 2050.

In addition to these 12 adopted proposals, three new proposals have already been provisionally agreed by the co-legislators:

* Energy performance of buildings directive (EPBD)

Buildings in the EU are significant energy consumers, accounting for 40% of final energy consumption and 36% of energy-related GHG emissions, with around 75% of existing buildings inefficient and in need of major renovation. New regulations require all new public buildings to be zero emission by 2028, followed by all new construction by 2030. Existing buildings will also see significant changes. Non-residential buildings will have to meet minimum energy

<footnote n="91">⁹¹Ships over 5 000 gross tonnes represent 55% of all ships and are responsible for 90% of CO₂ emissions from the maritime sector, which accounts for 13.5% of EU transport emissions (2018 data).</footnote>
standards set by Member States, with targets to improve the energy performance of 16% of
the worst performing buildings by 2030 and 26% by 2033. Similarly, residential buildings will
be required to achieve a 16% reduction in energy consumption by 2030 and a 20-22% reduction
by 2035, primarily through renovation of the least efficient buildings, with the ultimate goal
of achieving zero emissions from all buildings by 2050. Exceptions will be made for certain
building types, such as historic buildings and those used for religious activities. In addition, the
use of solar energy will be expanded, with requirements for solar panels in new construction and
incentives for retrofitting existing buildings. The EU also plans to provide financial incentives
and administrative support to facilitate these transformative efforts.

* EU methane regulation for the energy sector

Globally, 60% of methane emissions are caused by human activities, with wetlands, agriculture, energy and waste activities, in descending order of importance, being the main sources. The new EU regulations include several key measures: (1) implementing robust procedures for measuring and reporting methane emissions in the energy sector, including independent audits and the establishment of public inventories for inactive wells and coal mines; (2) requiring leak detection and repair to reduce emissions within the EU, as well as limiting methane releases at energy production facilities through measures such as venting and flaring, and requiring Member States to adopt mitigation plans; (3) addressing energy imports by requiring the tracking of emissions associated with imported energy.

* Updated EU rules to decarbonise gas markets and promote hydrogen

Renewable and low-carbon gases refer to gaseous fuels that have a reduced carbon footprint compared to fossil fuels. Renewable gases, derived from organic sources such as biogas and biomethane, or from non-biological renewable sources using electricity, such as renewable hydrogen and synthetic methane, offer environmentally sustainable alternatives. Low carbon gases, while not derived from renewable energy, emit at least 70% fewer greenhouse gas emissions over their life cycle than fossil natural gas. The proposed hydrogen and decarbonized gas market package aims to revise and introduce new regulations to reduce the carbon impact of the gas market. The aim is to phase out fossil natural gas and promote the introduction of renewable and low-carbon gases in the EU by 2030 and beyond. Key objectives of the new rules include: creating a hydrogen market with a target of 40 gigawatts of renewable hydrogen electrolyser capacity and 10 million tonnes of renewable hydrogen production by 2030; integrating renewable and low-carbon gases into the existing gas grid; ensuring consumer engagement and protection through simplified processes for switching energy suppliers; and increasing security of supply while limiting dependence on gas imports from Russia and Belarus through increased cooperation and restrictions.

**Remark 84** One proposal did not materialise. The European Commission originally tabled a proposal to revise the Energy Taxation Directive (ETD) as part of the Fit for 55 package. The ETD entered into force in 2003 and sets the framework for the taxation of energy products used as electricity, motor and aviation fuels and most heating fuels. The reforms announced in July 2021 should include the introduction of tax rates, expressed in euros per gigajoule, based on the environmental impact of energy products rather than their volume, and a broadening of the tax base to include energy contents and processes that were previously excluded. To be adopted, the proposal requires unanimity in the European Council after consultation of the European Parliament and the European Economic and Social Committee. In November 2023, the European Council concluded that the positions of delegations remained divergent on several key issues and that the reform of the Energy Taxation Directive was abandoned.
<figure><img src="image_361.png" /><figcaption>Figure 8.79: What is included in the Fit for 55 package?</figcaption></figure>

Source: European Council (2024), https://www.consilium.europa.eu/en/policies/green-deal/fit-for-55-the-eu-plan-for-a-green-transition.

## France

EU regulations apply to each EU member state, but each EU member state also has its own climate-related regulations and policies. One of the problems is that it is sometimes difficult to distinguish between national regulations that simply mirror an EU regulation and national regulations that fall outside the scope of the EU regulation. This is because a European regulation transposed into national law often combines several pieces of European legislation. Here are some examples from France:

* Energy Transition Act of 2015 (Loi relative à la transition énergétique pour la croissance verte du 17 août 2015)

* Energy and Climate Act of 2019 (Loi relative à l'énergie et au climat du 8 novembre 2019)

* Mobility Orientation Act of 2019 (Loi d'orientation des mobilités du 24 décembre 2019)

* Anti-Waste Act for a Circular Economy of 2020 (Loi relative à la lutte contre le gaspillage et à l'économie circulaire du 10 février 2020)

* National low carbon strategy (Stratégie nationale bas-carbone du 21 avril 2020 or SNBC)

* Climate and Resilience Act of 2021 (Loi climat et résilience du 22 août 2021)

* Environnemental Regulation (Réglementation environnementale RE2020)

The Energy Transition Act provides France with specific targets and action plans to implement the Paris Climate Agreement. Initially introduced in 2015 as a unique French regulation, its significance
expanded with the European Union's adoption of the Paris Agreement. Consequently, the Act now aligns with the EU's objectives regarding the Paris Climate Agreement. The national low carbon strategy (SNBC) is the roadmap of France to reduce its GHG emissions. It details the targets and instruments for each of the sectors concerned: buildings, agriculture, transport, forest wood and soils, energy production and industry. Again, it can be seen as a mix of several regulations that make up the Fit for 55 package. The RE2020 environmental regulation will come into force on 1 January 2022. As it can be seen as a proper French law on climate change, it borrows many elements from EU legislation on buildings.

## Germany

Germany has enacted several important climate change-related laws and regulations to promote environmental sustainability and reduce greenhouse gas emissions. Here are some of the most important laws and regulations with their German equivalents<fnref n="92" />:

*   **Klimaschutzgesetz (climate change act)**

Introduced in 2019, this law establishes binding targets for reducing greenhouse gas emissions in Germany, with the aim of achieving climate neutrality by 2050. The legislation specifies annual CO₂ reduction targets and emission limits for all sectors, with ongoing monitoring to facilitate corrective action if necessary.

*   **Klimaschutzprogramm 2030 (climate action programme 2030)**

The Klimaschutzprogramm 2030, a key component of the Germany climate change act, focuses on achieving the 2030 emission reduction targets and serves as Germany's contribution to the Fit for 55 package. Some key measures include the implementation of a national emissions trading system for smaller emitters, financial incentives for energy-efficient building renovation, subsidies for the purchase of electric vehicles and the development of charging infrastructure, and a coal phase-out plan with a 2038 deadline for the end of coal-fired power generation.

*   **Erneuerbare-Energien-Gesetz (EEG or renewable energy sources act)**

The Erneuerbare-Energien-Gesetz promotes renewable electricity generation to reduce greenhouse gas emissions and achieve energy independence. Enacted in 2000 and regularly updated to reflect market changes and technological advances, the EEG prioritizes the integration of renewable energy sources such as wind, solar, biomass, hydropower, and geothermal into the national grid. Key elements include feed-in tariffs (which guarantee fixed prices for electricity generated from renewable sources), priority grid access, grid expansion and modernization, and financing mechanisms.

*   **Energieeinsparverordnung (EnEV or energy saving ordinance)**

The Energieeinsparverordnung sets minimum energy performance standards for both new and existing buildings to reduce their energy consumption and associated greenhouse gas emissions. Key aspects of the EnEV include energy performance standards for insulation, heating systems, ventilation, and the use of renewable energy sources. It also includes provisions for energy certification, energy audits and technical standards.

*   **Energiewirtschaftsgesetz (EnWG or energy industry act)**

The Energiewirtschaftsgesetz regulates the energy sector. Enacted in 1935 and extensively revised over the years, the EnWG governs various aspects of energy production, distribution and consumption with the goal of ensuring a reliable, affordable and sustainable energy supply.

<footnote n="92">⁹²Source: www.bundesregierung.de/breg-de/schwerpunkte/klimaschutz.</footnote>
## United Kingdom

The Climate Change Act 2008 is the cornerstone of the UK's strategy for tackling climate change. It mandates the reduction of carbon dioxide and other greenhouse gas emissions. The Act sets out a comprehensive framework for achieving these targets and underlines the UK's commitment to urgent international action on climate change. Key features of the Climate Change Act include setting a target to significantly reduce the UK's greenhouse gas emissions by 2050; establishing the Committee on Climate Change (CCC) to provide an evidence-based and independent assessment of emissions targets; a commitment to reduce greenhouse gas emissions to at least 100% below 1990 levels (net zero) by 2050, including emissions from the devolved administrations; the implementation of legally binding carbon budgets to limit greenhouse gas emissions over five-year periods, with advice from the CCC on appropriate levels; and the production of a UK climate change risk assessment (CCRA) every five years to assess current and future climate risks and opportunities, accompanied by a national adaptation programme (NAP) in response. In addition, the Climate Change Levy (CCL) is a tax on the use of energy in industry, commerce and the public sector. The CCL is paid either at the main rates for electricity, gas and solid fuels such as coal, lignite, coke and petroleum coke, or at the carbon price support (CPS) rates for gas, LPG, coal and other solid fossil fuels to encourage industry to use low carbon technology to generate electricity. The UK has its own emissions trading scheme and an equivalent European-style CBAM mechanism is planned to be implemented by 2027.

## Other countries

There are many climate change-related laws around the world. For example, the Climate Change Laws of the World database<fnref n="93" />, developed in collaboration with the Grantham Research Institute at LSE and the Sabin Center at Columbia Law School, contains more than 5,000 laws, policies, and UNFCCC submissions from every country. Below are some key regulations for some other countries:

* Australia: national greenhouse and energy reporting scheme (2007), emissions reduction fund (2014), safeguard mechanism (2015), renewable energy target scheme (2021), national climate resilience and adaptation strategy (2021–2025).

* Canada: Canadian environmental protection act (1999), pan-Canadian framework on clean growth and climate change (2016), greenhouse gas pollution pricing act (2018), Canadian net-zero emissions accountability act (2021), clean fuel standard (2022).

* China: renewable energy law (2005), energy conservation law (2007), carbon emissions trading management regulations (2021), 14th five-year plan for green development (2021–2025).

* India: national action plan on climate change (2008), energy conservation act (2001), renewable energy sources act (2015), national programme on climate change & human health (2019), green hydrogen/ammonia policy (2022).

* Japan: act on promotion of global warming countermeasures (1998), energy conservation act (1979), climate change adaptation act (2018), green growth strategy through achieving carbon neutrality in 2050 (2021), renewable energy act (2022).

<footnote n="93">⁹³The website is https://climate-laws.org.</footnote>
## 8.3 Integrated assessment models

Integrated assessment models (IAMs) can be defined as approaches that link main features of society and economy with the biosphere and atmosphere into a common modeling framework. They generally couple a macroeconomic model with a climate risk model in order to simulate the economic impacts of climate change. IAMs are used by policy markers to analyze the economic cost of climate change and the impact of climate action. According to Nordhaus (2017b), the most important applications are:

1. Making projections of economic variables (e.g., GDP) that take into account global warming;

2. Calculating the impacts of alternative assumptions;

3. Tracing the effects and estimating the costs and benefits of alternative climate policies;

4. Estimating the uncertainty of future economic pathways.

For instance, the US government uses three IAMs to estimate the social cost of greenhouse gases, that is the value of avoiding one tonne of GHG emissions (IWG, 2021). These models are Dynamic Integrated Climate and Economy (DICE), Climate Framework for Uncertainty, Negotiation, and Distribution (FUND) and Policy Analysis of the Greenhouse Gas Effect (PAGE). The number of IAMs has grown rapidly, especially these last years with the increasing awareness of scientists to take into account climate change into economic modeling. Already in the late 1990s, Kelly and Kolstad (1999) count 21 major integrated assessment models. Today, current IAMs may be very complex and integrate social dimensions (e.g., inequality, education, health, food security), industry dimensions (e.g., sectors, infrastructure, rare earth elements), biodiversity dimensions (e.g., species, ecosystem, food), etc. In this section, we first present the DICE model, which is certainly one of the simplest IAMs, but also the most famous. Even if DICE is a highly stylized reference point, it is an excellent educational tool for understanding the economics and physics of IAMs.

### 8.3.1 The DICE model

There are many versions of DICE. William Nordhaus began to develop DICE with a simple energy/climate model in the 1970s (Nordhaus, 1977). The current format of the model can already be found in Nordhaus (1992). Since this publication in Science, William Nordhaus has multiplied the research projects on DICE. Some of them were published in academic journals, others in many books<fnref n="94" />, but most of them were unpublished. In what follows, we use the presentation and the notations of Nordhaus and Sztorc (2013), which corresponds to the user's manual of the DICE 2013R software. We also extensively refer to the comprehensive survey of Le Guenedal (2019).

#### The 2013 model

DICE uses a standard neoclassical model of economic growth known as the Ramsey-Cass-Koopmans model. It is an extension of the Solow model when the saving rate is not constant but endogenous. The social planner maximizes then the welfare utility function and determines the optimal path of saving rates to increase global consumption in the future. Nordhaus introduces the impact of climate change as a negative externality that hurts the economy and the output. In order to mitigate the cost

<footnote n="94">⁹⁴The most famous are *Managing the Global Commons: The Economics of Climate Change* (1994), *A Question of Balance: Weighing the Options on Global Warming Policies* (2008), *The Climate Casino: Risk, Uncertainty, and Economics for a Warming World* (2013) and *The Spirit of Green* (2021).</footnote>
of climate physical risks, the social planner can increase climate investments via a control variable on climate transition risks. Therefore, the general equilibrium depends on two decision variables: the saving rate and the climate control variable.

**Production and consumption functions** Nordhaus and Sztorc (2013) assume that the gross production Y (t) is given by the Cobb-Douglas function:

$$Y(t) = A(t) K(t)^{\gamma} L(t)^{1-\gamma}$$

where A(t) is the total productivity factor (or technological progress), K(t) is the capital input and L(t) is the labor input. The parameter $\gamma \in ]0, 1[$ measures the elasticity of the capital factor:

$$\gamma = \frac{\partial \ln Y(t)}{\partial \ln K(t)} = \frac{\partial Y(t)}{\partial K(t)} \frac{K(t)}{Y(t)}$$

Traditional economic models do not make the distinction between the production Y (t) and the net output Q(t) because we have the identity $Y(t) = Q(t)$. Nevertheless, physical and transition climate risks generate losses:

$$Q(t) = \Omega_{\text{climate}}(t) Y(t) \le Y(t)$$

where $\Omega_{\text{climate}}(t) \in ]0, 1[$ is the loss percentage of the production. Q(t) is then the net output when taking into account negative externalities of climate change. The saving rate s(t) is assumed to be time-dependent and is a control variable. Using the classical identities $Q(t) = C(t) + I(t)$ and $I(t) = s(t)Q(t)$ where I(t) is the investment, the expression of the consumption C(t) is then equal to:

$$\begin{aligned} C(t) &= (1 - s(t)) Q(t) \\ &= (1 - s(t)) \Omega_{\text{climate}}(t) A(t) K(t)^{\gamma} L(t)^{1-\gamma} \end{aligned} \qquad (8.19)$$

In order to introduce the time dependence and complete the economic model, the authors assume that the dynamics of the state variables are:

$$\left\{ \begin{aligned} A(t) &= (1 + g_A(t)) A(t - 1) \\ K(t) &= (1 - \delta_K) K(t - 1) + I(t) \\ L(t) &= (1 + g_L(t)) L(t - 1) \end{aligned} \right. \qquad (8.20)$$

where $g_A(t)$ is the growth rate of the technological progress, $\delta_K$ is the depreciation rate of the capital stock and $g_L(t)$ is the growth rate of the labor factor. The two growth rates $g_A(t)$ and $g_L(t)$ decline over time, implying that:

$$g_A(t) = \frac{1}{1 + \delta_A} g_A(t - 1)$$

and:

$$g_L(t) = \frac{1}{1 + \delta_L} g_L(t - 1)$$

where $\delta_A \ge 0$ and $\delta_L \ge 0$.

**Example 28** The world population was equal to 7.725 billion in 2019 and 7.805 billion in 2020. At the beginning of the 1970s, we estimate that the annual growth rate was equal to 2.045%. According to the United Nations, the global population could surpass 10 billion by 2100.
In 2020, the annual growth rate was equal to:

$$g_L (2020) = \frac{L (2020)}{L (2019)} - 1 = \frac{7.805}{7.725} - 1 = 1.036\%$$

Since we have $g_L(t) = \left(\frac{1}{1 + \delta_L}\right)^{t-t_0} g_L(t_0)$, we deduce that:

$$\delta_L = \left( \frac{g_L(t_0)}{g_L(t)} \right)^{1/(t-t_0)} - 1$$

An estimate of $\delta_L$ is then:

$$\delta_L = \left( \frac{g_L (1970)}{g_L (2020)} \right)^{1/30} - 1 = 2.292\%$$

In Figure 8.80, we report the dynamics of the labor input $L(t)$ by assuming that $L(2020) = 7.805$, $g_L(2020) = 1.036\%$ and $\delta_L = 2.292\%$. We estimate that the global population could reach 11.4 billion by 2100. In order to measure the sensitivity to the model parameters, we have also estimated the world population when $\delta_L$ is equal to 1.50% and 3.25%. The range is 10.5 and 12.6 billion at the end of this century.

<figure><img src="image_362.png" /><figcaption>Figure 8.80: Evolution of the labor input $L(t)$</figcaption></figure>

In Figure 8.81, we compare the different probabilistic projections, which are computed by the Department of Economic and Social Affairs (DESA) of the United Nations<fnref n="95" />. We notice that the previous calibration is close to the median estimation. Of course, we can use these data in order to better calibrate the DICE model. For instance, if we have a time series of $g_L(t)$, we can estimate $\delta_L$

<footnote n="95">⁹⁵The data source are available at https://population.un.org/wpp.</footnote>
<figure><img src="image_363.png" /><figcaption>Figure 8.81: Projection of the world population</figcaption></figure>

Source: United Nations (2022), World Population Prospects, https://population.un.org/wpp.

<figure><img src="image_364.png" /><figcaption>Figure 8.82: Population growth rate</figcaption></figure>

Source: United Nations (2022), https://population.un.org/wpp & Author's calculations.
by considering the AR(1) process: $g_L(t) = \phi g_L(t-1) + \varepsilon(t)$. We have $\hat{\delta}_L = (1 - \hat{\phi}) / \hat{\phi}$. Another method consists in using the following relationship:

$$g_L(t) = \left( \frac{1}{1 + \delta_L} \right)^{t-t_0} g_L(t_0)$$

We deduce that $\ln g_L(t) = \ln g_L(t_0) - (t - t_0) \ln(1 + \delta_L)$. We can then estimate the parameter $\delta_L$ by considering the linear regression: $\ln g_L(t) = \beta_0 + \beta_1 (t - t_0) + \varepsilon(t)$. It follows that $\hat{\delta}_L = e^{-\hat{\beta}_1} - 1$. Using the historical population growth rates between 1970 and 2021 provided by the United Nations (Figure 8.82), we obtain $\hat{\phi} = 0.985$, $\hat{\beta}_0 = 0.724$ and $\hat{\beta}_1 = -0.014$. The estimated value of $\delta_L$ is then equal to 1.529% for the AR(1) process and 1.369% for the log-linear model. In Figure 8.82, we observe a high decline of the population growth rate in the last years, since the last value $g_L$ (2021) was equal to 0.82%.

<table><caption>Table 8.30: Average productivity growth rate (in %)</caption><thead><tr><th>Country</th><th>1960-1970</th><th>1970-1980</th><th>1980-1990</th><th>1990-2000</th><th>2000-2010</th><th>2010-2020</th></tr></thead><tbody><tr><td>AUS</td><td>1.02</td><td>0.07</td><td>-0.23</td><td>1.02</td><td>0.36</td><td>0.13</td></tr><tr><td>BRA</td><td>2.39</td><td>2.05</td><td>-1.04</td><td>-1.12</td><td>-0.17</td><td>-1.63</td></tr><tr><td>CAN</td><td>2.18</td><td>0.38</td><td>-0.25</td><td>0.21</td><td>-0.21</td><td>0.40</td></tr><tr><td>CHN</td><td>-0.03</td><td>-0.06</td><td>-0.04</td><td>-0.41</td><td>2.24</td><td>-0.35</td></tr><tr><td>FRA</td><td>3.59</td><td>1.63</td><td>1.12</td><td>0.61</td><td>-0.11</td><td>0.02</td></tr><tr><td>DEU</td><td>2.33</td><td>1.63</td><td>0.75</td><td>1.52</td><td>0.01</td><td>0.74</td></tr><tr><td>IND</td><td>2.37</td><td>-1.22</td><td>1.06</td><td>1.04</td><td>0.70</td><td>1.89</td></tr><tr><td>ITA</td><td>3.71</td><td>1.66</td><td>-0.19</td><td>-0.20</td><td>-1.32</td><td>-0.34</td></tr><tr><td>JPN</td><td>4.05</td><td>0.77</td><td>1.09</td><td>-0.22</td><td>-0.15</td><td>0.69</td></tr><tr><td>ZAF</td><td>2.37</td><td>0.30</td><td>-0.84</td><td>-1.11</td><td>0.50</td><td>-1.20</td></tr><tr><td>GBR</td><td>0.50</td><td>0.72</td><td>0.75</td><td>0.42</td><td>0.12</td><td>0.08</td></tr><tr><td>USA</td><td>1.00</td><td>0.42</td><td>0.46</td><td>0.73</td><td>0.65</td><td>0.56</td></tr></tbody></table>

Source: Penn World Table 10.01 (Feenstra et al., 2015) & Author's calculations.

Concerning the total factor productivity (TFP), also called the multifactor productivity (MFP), Nordhaus and Sztorc (2013) estimated that $g_A(2010) = 1.53\%$ and $\delta_A = 0.12\%$. If we would like to recalibrate these figures, there are currently four main databases that provide TFP/MFP statistics<fnref n="96" />: the OECD Productivity Statistics, the EUKLEMS-INTANProd database, the Conference Board Total Economy Database, and the Penn World Table (PWT), which is certainly the most known framework among economists (Feenstra et al., 2015). We use this last database and consider a sample of twelve countries. The TFP index is reported in Figure 8.83. We notice a high discrepancy between countries and also between periods (see Table 8.30). Based on these statistics, $g_A(2020)$ is closer to 0.75% if we include developing countries. If we assume that the TFP growth rate has been divided by a factor $d$ in $n$ years, we have $\delta_A = \sqrt[n]{d} - 1$. Some examples of the growth rate path $g_A(t)$ are given in Figure 8.84.

For the investment $I(t)$, the capital stock $K(t)$ and the gross output $Y(t)$, we can use the previous databases such as the Penn World Table, or the investment and capital stock dataset (ICSD)

<footnote n="96">⁹⁶We can download them at www.oecd.org/sdd/productivity-stats, https://euklems-intanprod-llee.luiss.it, www.conference-board.org/data/economydatabase/total-economy-database-productivity and www.rug.nl/ggdc/productivity/pwt.</footnote>
<figure><img src="image_365.png" /><figcaption>Figure 8.83: Total factor productivity index (base 100 = 1960)</figcaption></figure>

Source: Penn World Table 10.01 (Feenstra et al., 2015) & Author's calculations.

<figure><img src="image_366.png" /><figcaption>Figure 8.84: Dynamics of the TFP growth rate</figcaption></figure>
of the International Monetary Fund<fnref n="97" />. We have reported the evolution of the variables<fnref n="98" /> in Figure 8.85. In 2019, we obtain $I$ (2019) = $30.625$ tn, $K$ (2019) = $318.773$ tn and $Y$ (2019) = $124.418$ tn. We have also calculated the annual depreciation rate<fnref n="99" />:

$$ \delta_K(t) = \frac{K(t-1) - K(t) + I(t)}{K(t-1)} $$

and we obtain $\delta_K$ (2019) = 6.25%. We have now all the elements to calibrate the initial value of $A(t)$. For that, we assume that $\gamma$ takes the typical value of 0.30 and we obtain:

$$ A(2019) = \frac{Y(t)}{K(t)^{\gamma} L(t)^{1-\gamma}} = \frac{124.418}{318.773^{0.30} \times 7.725^{0.70}} = 5.276 $$

We can now solve the macroeconomic model in the absence of climate change effects. In this case, $\Omega_{\text{climate}}(t) = 1$ and $Q(t) = Y(t)$. By assuming that the saving rate $s(t)$ is constant<fnref n="100" />, we obtain the simulations<fnref n="101" /> in Figure 8.86. We recall that the variables $C(t)$, $I(t)$, $K(t)$ and $Y(t)$ are expressed in trillions of constant 2017 international dollars. If $s(t) = 25\%$, the GDP in 2100 would be equal to $507.47$ tn. It would be reduced to $408.65$ tn if $s(t) = 15\%$.

**Cost function of climate change** We have seen that the net output is reduced because of climate change. The survival function is given by:

$$ \Omega_{\text{climate}}(t) = \Omega_D(t) \Omega_{\Lambda}(t) = \frac{1}{1+D(t)} (1-\Lambda(t)) $$

where $D(t) \ge 0$ is the climate damage function (physical risk) and $\Lambda(t) \ge 0$ is the mitigation or abatement cost (transition risk). The costs $D(t)$ result from natural disasters and climatic events, such as wildfires, storms, and floods, whereas the costs $\Lambda(t)$ are incurred from reducing GHG emissions and result from policy for financing the transition to a low-carbon economy. Nordhaus and Sztorc (2013) assume that $D(t)$ depends on the atmospheric temperature $T_{\text{AT}}(t)$:

$$ D(t) = \psi_1 T_{\text{AT}}(t) + \psi_2 T_{\text{AT}}(t)^2 $$

where $\psi_1 \ge 0$ and $\psi_2 \ge 0$ are two parameters. $T_{\text{AT}}(t)$ measures the global mean surface temperature and corresponds to the temperature increase in °C from 1900. Therefore, $\mathcal{L}_D(t) = 1 - \Omega_D(t) = 1 - (1+D(t))^{-1}$ represents the fraction of net output that is lost because of the global warming. For the abatement cost function, the authors consider that it depends on the control variable $\mu(t)$:

$$ \Lambda(t) = \theta_1(t) \mu(t)^{\theta_2} $$

where $\theta_1(t) \ge 0$ and $\theta_2 \ge 0$ are two parameters. For the emission-control rate, we have $\mu(t) \in [0, 1]$. Finally, we deduce that the global impact of climate change is equal to:

$$ \Omega_{\text{climate}}(t) = \frac{1 - \theta_1(t) \mu(t)^{\theta_2}}{1 + \psi_1 T_{\text{AT}}(t) + \psi_2 T_{\text{AT}}(t)^2} \quad (8.21) $$

<footnote n="97">⁹⁷The database is available at https://data.imf.org.</footnote>

<footnote n="98">⁹⁸$I(t)$ is the sum of the general government investment (variable igov\_rppp), the private investment (variable ipriv\_rppp) and public-private partnership investment (variable ippp\_rppp); $K(t)$ is the sum of general government capital stock (variable kgov\_rppp), private capital stock (variable kpriv\_rppp) and public-private partnership capital stock (variable kppp\_rppp); $Y(t)$ corresponds to the gross domestic product (variable gdp\_rppp). All the variables are expressed in trillions of constant 2017 international dollars.</footnote>

<footnote n="99">⁹⁹Since the trajectory of $\delta_K(t)$ is erratic, we have preferred to report the 3-year moving average of $\delta_K(t)$.</footnote>

<footnote n="100">¹⁰⁰We recall that $s(t)$ is a control variable.</footnote>

<footnote n="101">¹⁰¹We use the following initial values: $L(2019) = 7.725$ bn, $g_L = 1\%$, $\delta_L = 1.5\%$, $I(2019) = 30.625$ tn, $K(2019) = 318.773$ tn, $\delta_K = 6.25\%$, $A(2019) = 5.276$, $g_A(2019) = 0.75\%$, $\delta_A = 0.12\%$, $\gamma = 0.30$ and $Y(2019) = 124.418$ tn.</footnote>
<figure><img src="image_367.png" /><figcaption>Figure 8.85: Historical estimates of I (t), K (t), Y (t) and δ<sub>K</sub> (t)</figcaption></figure>

Source: IMF Investment and Capital Stock Dataset (2021) & Author's calculations.

<figure><img src="image_368.png" /><figcaption>Figure 8.86: Simulation of the DICE macroeconomic module</figcaption></figure>
We notice that $\Omega_{\text{climate}}(t)$ depends on two variables: the atmospheric temperature $\mathcal{T}_{\text{AT}}(t)$ and the control variable $\mu(t)$. In the DICE model, $\mathcal{T}_{\text{AT}}(t)$ is a state variable while $\mu(t)$ is a decision variable. $\Omega_{\text{climate}}(t)$ is a then a decreasing function of $\mathcal{T}_{\text{AT}}(t)$ and $\mu(t)$ and we have the following properties:

$$ 
\left\{ 
\begin{array}{l}
\Omega_{\text{climate}}(t) = 1 \Leftrightarrow \mathcal{T}_{\text{AT}}(t) = 0 \wedge \mu(t) = 0 \\
\lim_{\mathcal{T}_{\text{AT}}(t) \to \infty} \Omega_{\text{climate}}(t) = 0 \\
\lim_{\mu(t) \to 1} \Omega_{\text{climate}}(t) = \frac{1 - \theta_1(t)}{1 + \psi_1 \mathcal{T}_{\text{AT}}(t) + \psi_2 \mathcal{T}_{\text{AT}}(t)^2}
\end{array}
\right.
$$

Nordhaus and Sztorc (2013) used $\psi_1 = 0$ and $\psi_2 = 2.67 \times 10^{-3}$ for the specification of $D(t)$. In Figure 8.87, we have represented the corresponding loss function $\mathcal{L}_D(\mathcal{T}_{\text{AT}}) := \mathcal{L}_D(t) = 1 - \Omega_D(t)$ as a function of the temperature increase $\mathcal{T}_{\text{AT}}$. For instance, we obtain $\mathcal{L}_D(1^{\circ}\text{C}) = 0.27\%$, $\mathcal{L}_D(2^{\circ}\text{C}) = 1.06\%$, $\mathcal{L}_D(5^{\circ}\text{C}) = 6.26\%$ and $\mathcal{L}_D(10^{\circ}\text{C}) = 21.07\%$ (dashed green line). We may think that these figures are underestimated in 2023. Already, fifteen years ago, the magnitude of these losses has created a scientific debate among economists. For instance, Hanemann (2008) found a higher loss function if the temperature reaches 2.5°C. Indeed, he estimated an annual loss of $113 bn in the US versus $28 bn for Nordhaus and Boyer (2000) (base year = 1990). Therefore, we can show that the parameter $\psi_2$ must take the value $1.135 \times 10^{-3}$ to obtain the ratio 113/28:

$$ D(t) = 11.35 \times 10^{-3} \times \mathcal{T}_{\text{AT}}(t)^2 $$

In a series of research papers (Weitzman, 2009, 2010, 2012), Martin Weitzman investigated several functional forms of the climate damage function. Using a stochastic modeling of the temperature $\mathcal{T}_{\text{AT}}$, Weitzman (2012) estimated the following climate damage function:

$$ D(t) = \left( \frac{\mathcal{T}_{\text{AT}}(t)}{20.46} \right)^2 + \left( \frac{\mathcal{T}_{\text{AT}}(t)}{6.081} \right)^{6.754} $$

The first term corresponds to the Nordhaus specification<fnref n="102" />, whereas the second term corresponds to “a tipping point where the damages function changes dramatically around the iconic global warming level of 6°C” (Weitzman, 2012, page 235). If we fit the Hanemann damage function using the Weitzman function, we obtain an alternative specification with higher damage for low temperature increase:

$$ D(t) = 1.35 \times 10^{-2} \times \mathcal{T}_{\text{AT}}(t)^2 + 6.0287 \times 10^{-7} \times \mathcal{T}_{\text{AT}}(t)^{7.5} $$

While the previous damage functions are based on the functional form $D(t) = \sum_{k=1}^{m} \psi_k \mathcal{T}_{\text{AT}}(t)^{\alpha_k}$, Weitzman (2009) suggested the exponential quadratic damage function<fnref n="103" />: $\Omega_D(t) = \exp(-\beta \mathcal{T}_{\text{AT}}(t)^2)$. Pindyck (2012) proposed then the following parameterization:

$$ \Omega_D(t) = e^{-\beta' \mathcal{T}_{\text{AT}}(t)^2 G} $$

where $G \sim \mathbf{F}_G$ is a random variable that follows a displaced gamma distribution<fnref n="104" /> $\mathcal{DG}(\alpha, \beta, \theta)$ where $\alpha = 4.5$, $\beta = 4.69 \times 10^{-5}$ and $\theta = -7.46 \times 10^{-5}$. Following Daniel et al. (2016), we set $\beta' = 13.97$ and $G = \mathbf{F}_G^{-1}(99.99\%)$. We have reported all these alternative loss functions<fnref n="105" /> in Figure 8.87. We notice a high level of disagreement between these different research works (Howard and Sterner, 2017; Tol, 2022).

<footnote n="102">¹⁰²We have $1/20.46^2 \approx 2.4 \times 10^{-3}$, which was the value used by Nordhaus (2008).</footnote>

<footnote n="103">¹⁰³See Pindyck (2013, page 867) and Wagner and Weitzman (2015).</footnote>

<footnote n="104">¹⁰⁴If $X \sim \mathcal{DG}(\alpha, \beta, \theta)$, then $X - \theta \sim \mathcal{G}(\alpha, \beta)$.</footnote>

<footnote n="105">¹⁰⁵We also include the piecewise function $D(t) = 0.011\mathcal{T}_{\text{AT}}(t) + 1\{\mathcal{T}_{\text{AT}}(t) > 3^{\circ}\text{C}\} \cdot (0.036 - 0.011)(\mathcal{T}_{\text{AT}}(t) - 3^{\circ}\text{C})$, which has been estimated by Newbold and Marten (2014).</footnote>
<figure><img src="image_369.png" /><figcaption>Figure 8.87: Loss function due to climate damage costs</figcaption></figure>

For the abatement cost function, Nordhaus (2018b) assumed that $\theta_1(t) = 0.0741 \times 0.98^{t-t_0}$ where $t_0$ is the base year of the abatement technology and $\theta_2 = 2.6$. The explanation is the following:

> "The interpretation here is that at zero emissions for the first period ($t = t_0$), abatement is 7.41% of output. That percentage declines at 2% per year. The abatement cost function is highly convex, reflecting the sharp diminishing returns to reducing emissions. [...] The model assumes the existence of a backstop technology, which is a technology that produces energy services with zero GHG emissions ($\mu = 1$). The backstop price in 2020 is $550 per ton of CO2e, and the backstop cost declines at 0.5% per year. Additionally, it is assumed that there are no negative emissions technologies initially, but that negative emissions are available after 2150." (Nordhaus, 2018b, page 357).

Therefore, we can also define the cost of mitigation efforts as the following alternative form (Kellett et al., 2019):

$$ \theta_1(t) \propto p_b(t_0) (1 - \delta_b)^{t-t_0} \sigma(t) $$

where $p_b(2020) = \$550/\text{tCO}_2\text{e}$, $\delta_b = 0.5\%$ and $\sigma(t)$ is the carbon intensity of economic activity, which is equal to the ratio of anthropogenic GHG emissions to economic output $Y(t)$. In Figure 8.88, we show the abatement cost function $\Lambda(t)$ for different values of the control rate $\mu(t)$, which is the second control variable with the saving rate $s(t)$.

**Remark 85** There is a large literature on abatement costs, especially about the concept of marginal abatement cost (MAC). Since this concept is at the core of the transition risk (Gillingham and Stock, 2018) and is highly related to the carbon tax (Daniel et al., 2019), it will be developed in Section 10.2.3 on page 1003.
<figure><img src="image_370.png" /><figcaption>Figure 8.88: Abatement cost function</figcaption></figure>

**Temperature modeling** We now turn to the physics of the DICE model. As noticed by Nordhaus (2017b), IAMs are “highly simplified representations of complex economic and geophysical realities”, and can not compete with general circulation models<fnref n="106" /> (GCM). Nevertheless, these models are based on basic physics of temperature forecasts, produce plausible temperature trajectories and “do a reasonable job capturing uncertainty about the equilibrium climate sensitivity” (Calel and Stainforth, 2017a, page 1202).

The GHG emissions $\mathcal{CE}(t)$ depends on the production $Y(t)$ and the land-use emissions $\mathcal{CE}_{\text{Land}}(t)$:

$$ \begin{aligned} \mathcal{CE}(t) &= \mathcal{CE}_{\text{Industry}}(t) + \mathcal{CE}_{\text{Land}}(t) \\ &= (1 - \mu(t))\sigma(t)Y(t) + \mathcal{CE}_{\text{Land}}(t) \end{aligned} \qquad (8.22) $$

where $\sigma(t)$ is the impact of the production on GHG emissions. $\mathcal{CE}_{\text{Industry}}(t)$ corresponds to the anthropogenic emissions due to industrial activities. We have $\mathcal{CE}_{\text{Land}}(t) = (1 - \delta_{\text{Land}}) \mathcal{CE}_{\text{Land}}(t - 1)$. The control variable $\mu(t)$ measures the impact of climate change mitigation policies:

$$ \underbrace{\mathcal{CE}_{\text{Land}}(t)}_{\mu(t)=1} \le \mathcal{CE}(t) \le \underbrace{\sigma(t)Y(t) + \mathcal{CE}_{\text{Land}}(t)}_{\mu(t)=0} $$

$\mu(t) = 1$ corresponds to the case where mitigation policies have eliminated the effects of climate change, while $\mu(t) = 0$ indicates that no specific policy has been put in place. Since $\mu(t)$ is an endogenous variable, it must be viewed as an effort rate that the economy must bear to limit global

<footnote n="106">¹⁰⁶They simulate general circulation of planetary atmosphere and oceans, using the Navier-Stokes equations on a rotating sphere and thermodynamic terms for energy sources (radiation, latent heat). They are used for weather forecasting, understanding the climate, and forecasting climate change.</footnote>
warming. Concerning the ratio $\sigma(t)$, it measures the relationship between the carbon emissions due to industrial activities and the gross output in the absence of mitigation policies ($\mu(t) = 0$):

$$ \sigma(t) = \frac{\mathcal{CE}(t) - \mathcal{CE}_{\text{Land}}(t)}{Y(t)} = \frac{\mathcal{CE}_{\text{Industry}}(t)}{Y(t)} $$

Therefore, it can be interpreted as an emission factor or the anthropogenic carbon intensity of the economy. This parameter is integrated into the model as follows:

$$ \sigma(t) = (1 + g_{\sigma}(t))\sigma(t - 1) \qquad (8.23) $$

where:

$$ g_{\sigma}(t) = \frac{1}{1 + \delta_{\sigma}} g_{\sigma}(t - 1) $$

and $\delta_{\sigma}$ is the decline rate of the GHG emissions growth.

<figure><img src="image_371.png" /><figcaption>Figure 8.89: Physical carbon pump</figcaption></figure>

Source: ocean-climate.org.

In order to relate the carbon emissions $\mathcal{CE}(t)$ and the atmospheric temperature $T_{\text{AT}}(t)$, the DICE model uses a reduced form of a general circulation model describing the evolution of GHG concentrations in three carbon-sink reservoirs: the atmosphere AT, the upper ocean UP and the deep (or lower) ocean LO. As we will see on page 1126, the ocean hold fifty times more CO$_2$ than the atmosphere. Indeed, a fraction of carbon from the atmosphere is absorbed by the upper ocean and transported by ocean circulation patterns. It is then stored in the deep ocean or re-radiated
from the upper ocean to the atmosphere influencing regional temperatures. The sequestration of carbon in the deep ocean depends on the temperature of cold waters, implying that a small part of the carbon stored in the deep ocean may return to the upper ocean. This physical carbon pump is illustrated in Figure 8.89. Taking into account this global heat transfer scheme, Nordhaus and Sztorc (2013) assume that the dynamics of carbon concentrations<fnref n="107" /> are:

$$ egin{cases} 
CC_{AT}(t) = \phi_{1,1}CC_{AT}(t-1) + \phi_{1,2}CC_{UP}(t-1) + \phi_{1}C\mathcal{E}(t) \\
CC_{UP}(t) = \phi_{2,1}CC_{AT}(t-1) + \phi_{2,2}CC_{UP}(t-1) + \phi_{2,3}CC_{LO}(t-1) \\
CC_{LO}(t) = \phi_{3,2}CC_{UP}(t-1) + \phi_{3,3}CC_{LO}(t-1) 
\end{cases} $$

where $\phi_{i,j}$ represents the flow parameters between carbon-sink reservoirs<fnref n="108" />, and $\phi_1$ is the mass percentage of carbon in CO₂. Let $CC = (CC_{AT}, CC_{UP}, CC_{LO})$ be the vector of the three-reservoir layers. The dynamics of $CC$ is then a vector autoregressive process:

$$ CC(t) = \Phi_{CC}CC(t-1) + B_{CC}C\mathcal{E}(t) \quad (8.24) $$

where $B_{CC} = (\phi_1, 0, 0)$ and:

$$ \Phi_{CC} = \begin{pmatrix} \phi_{1,1} & \phi_{1,2} & 0 \\ \phi_{2,1} & \phi_{2,2} & \phi_{3,2} \\ 0 & \phi_{3,2} & \phi_{3,3} \end{pmatrix} $$

The next step consists in linking accumulated carbon emissions in the atmosphere and global warming at the Earth's surface through increases in radiative forcing:

$$ \mathcal{F}_{\text{RAD}}(t) = \frac{\eta}{\ln 2} \ln \left( \frac{CC_{\text{AT}}(t)}{CC_{\text{AT}}(1750)} \right) + \mathcal{F}_{\text{EX}}(t) \quad (8.25) $$

where $\mathcal{F}_{\text{RAD}}(t)$ is the change in total radiative forcing of GHG emissions since 1750 (expressed in W/m²), $\eta$ is the temperature forcing parameter and $\mathcal{F}_{\text{EX}}(t)$ is the exogenous forcing. Finally, the climate system for temperatures is characterized by a two-layer system:

$$ egin{cases} 
\mathcal{T}_{\text{AT}}(t) = \mathcal{T}_{\text{AT}}(t-1) + \xi_1 (\mathcal{F}_{\text{RAD}}(t) - \xi_2 \mathcal{T}_{\text{AT}}(t-1) - \xi_3 (\mathcal{T}_{\text{AT}}(t-1) - \mathcal{T}_{\text{LO}}(t-1))) \\ 
\mathcal{T}_{\text{LO}}(t) = \mathcal{T}_{\text{LO}}(t-1) + \xi_4 (\mathcal{T}_{\text{AT}}(t-1) - \mathcal{T}_{\text{LO}}(t-1)) 
\end{cases} $$

where $\mathcal{T}_{\text{AT}}(t)$ and $\mathcal{T}_{\text{LO}}(t)$ are respectively the mean surface temperature and the temperature of the deep ocean. This system of equations means that “higher radiative forcing warms the atmospheric layer, which then warms the upper ocean, gradually warming the deep ocean” (Nordhaus and Sztorc, 2013, page 17). The authors simplify the relationships by only considering a two-box model: the atmosphere and the upper ocean. According to Calel and Stainforth (2017a), $\xi_1$ measures the speed of adjustment parameter for atmospheric temperature, $\xi_2$ is the ratio of increased forcing from CO₂ doubling to the climate sensitivity, $\xi_3$ is the heat loss coefficient from atmosphere to oceans and $\xi_4$ is the heat gain coefficient by deep oceans. Let $\mathcal{T} = (\mathcal{T}_{\text{AT}}, \mathcal{T}_{\text{LO}})$ be the temperature vector. We have:

$$ \mathcal{T}(t) = \Xi_{\mathcal{T}}\mathcal{T}(t-1) + B_{\mathcal{T}}\mathcal{F}_{\text{RAD}}(t) \quad (8.26) $$

where $B_{\mathcal{T}} = (\xi_1, 0)$ and:

$$ \Xi_{\mathcal{T}} = \begin{pmatrix} 1 - \xi_1(\xi_2 + \xi_3) & \xi_1\xi_3 \\ \xi_4 & 1 - \xi_4 \end{pmatrix} = \begin{pmatrix} \xi'_{1,1} & \xi'_{1,2} \\ \xi'_{2,1} & \xi'_{2,2} \end{pmatrix} $$

<footnote n="107">¹⁰⁷The carbon concentration is expressed in grams of carbon, and not in gCO₂e. The conversion is 12/44 or 0.2727 gC/g CO₂.</footnote>

<footnote n="108">¹⁰⁸By construction, we have $\sum_{i=1}^3 \phi_{i,j} = 1$ for $j = 1, 2, 3$. $\phi_{i,i}$ is then the probability that a molecule remains in its carbon sink, while $\phi_{i,j}$ is the probability that a molecule stored in the carbon sink $j$ goes to the carbon sink $i$.</footnote>
Box 8.17: Correspondence between DICE and the two-box model

We have previously seen that the two-box model of climate change is based on the heat diffusion between the atmosphere and the deep ocean:

$$ 
\left\{ 
\begin{aligned}
c_{\text{AT}} \frac{d\mathcal{T}_{\text{AT}}(t)}{dt} &= \mathcal{F}_{\text{RAD}}(t) - \lambda \mathcal{T}_{\text{AT}}(t) - \beta (\mathcal{T}_{\text{AT}}(t) - \mathcal{T}_{\text{LO}}(t)) \\
c_{\text{LO}} \frac{\mathcal{T}_{\text{LO}}(t)}{dt} &= \beta (\mathcal{T}_{\text{AT}}(t) - \mathcal{T}_{\text{LO}}(t))
\end{aligned}
\right. 
$$

where $c_{\text{AT}}$ is the effective heat capacity of the atmosphere (including the upper ocean and the land surface), $c_{\text{LO}}$ is the effective heat capacity of the deep ocean, $\lambda$ is the feedback parameter and $\beta$ is the heat transfer coefficient between the upper and lower oceans. Using a forward Euler discretization, Calel and Stainforth (2017a) found that:

$$ 
\left\{ 
\begin{aligned}
\mathcal{T}_{\text{AT}}(t) &= \mathcal{T}_{\text{AT}}(t-1) + \frac{1}{c_{\text{AT}}} (\mathcal{F}_{\text{RAD}}(t-1) - \lambda \mathcal{T}_{\text{AT}}(t-1) - \\
&\qquad \beta (\mathcal{T}_{\text{AT}}(t-1) - \mathcal{T}_{\text{LO}}(t-1))) \Delta t \\
\mathcal{T}_{\text{LO}}(t) &= \mathcal{T}_{\text{LO}}(t-1) + \frac{\beta}{c_{\text{LO}}} (\mathcal{T}_{\text{AT}}(t-1) - \mathcal{T}_{\text{LO}}(t-1)) \Delta t
\end{aligned}
\right. 
$$

where $\Delta t$ is the length of time step (expressed in seconds). Therefore, we have the following relationshipsª:

$$ 
\left\{ 
\begin{aligned}
\xi_1 &= \frac{1}{c_{\text{AT}}} \Delta t \\
\xi_2 &= \lambda \\
\xi_3 &= \beta \\
\xi_4 &= \frac{\beta}{c_{\text{LO}}} \Delta t
\end{aligned}
\right. 
\quad \text{and} 
\quad 
\left\{ 
\begin{aligned}
\xi'_{1,1} &= 1 - \frac{\lambda + \beta}{c_{\text{AT}}} \Delta t \\
\xi'_{1,2} &= \frac{\beta}{c_{\text{AT}}} \Delta t \\
\xi'_{2,1} &= \frac{\beta}{c_{\text{LO}}} \Delta t \\
\xi'_{2,2} &= 1 - \frac{\beta}{c_{\text{LO}}} \Delta t
\end{aligned}
\right. 
$$

Below, we report the parameter and initial values used by the 2013 version<fnref n="109" />:

* $t_0 = 2010$, $\Delta t = 5$ years;

* Equation (8.22): $\mathcal{CE}(t_0) = 36.91 \text{ GtCO}_2\text{e}$, $\mu(t_0) = 3.9\%$, $\sigma(t_0) = 0.5491 \text{ GtCO}_2\text{e/}$ $\text{tn}$, $Y(t_0) = \$63.69 \text{ tn}$, $\mathcal{CE}_{\text{Industry}}(t_0) = 33.61 \text{ GtCO}_2\text{e}$, $\delta_{\text{Land}} = 0.2$, and $\mathcal{CE}_{\text{land}}(t_0) = 3.3 \text{ GtCO}_2\text{e}$;

* Equation (8.23): $\sigma(t_0) = 0.5491 \text{ GtCO}_2\text{e/}$ $\text{tn}$, $g_\sigma(t_0) = 1\%$ and $\delta_\sigma = 0.1\%$;

* Equation (8.24): $\mathcal{CC}_{\text{AT}}(t_0) = 830.4$, $\mathcal{CC}_{\text{UP}}(t_0) = 1527$, $\mathcal{CC}_{\text{LO}}(t_0) = 10010$, $\phi_1 = 0.2727 \text{ gC/gCO}_2$, $\mathcal{CE}(t_0) = 36.91 \text{ GtCO}_2\text{e}$, and the carbon cycle diffusion matrix is equal to:

$$ \Phi_{\mathcal{CC}} = \begin{pmatrix} 91.20\% & 3.83\% & 0 \\ 8.80\% & 95.92\% & 0.03\% \\ 0 & 0.25\% & 99.97\% \end{pmatrix} $$

<footnote>ªCalel and Stainforth (2017a) noticed that the only difference is that the DICE model uses $\mathcal{F}_{\text{RAD}}(t)$ instead of $\mathcal{F}_{\text{RAD}}(t-1)$, which is not a problem if the time step is small.</footnote>

<footnote n="109">¹⁰⁹They can be found in Kellett et al. (2019, Table 2 & Appendices A and B).</footnote>
* Equation (8.25): $\mathcal{F}_{\text{RAD}}(t_0) = 2.14 \text{ W/m}^2$, $\eta = 3.8 \text{ W/m}^2$, $\mathcal{CC}_{\text{AT}}(t_0) = 830.4$, $\mathcal{CC}_{\text{AT}}(1750) = 588 \text{ GtC}$, $\mathcal{F}_{\text{EX}}(t)$ is the linear interpolation function between $\mathcal{F}_{\text{EX}}(t_0) = 0.25 \text{ W/m}^2$ and $\mathcal{F}_{\text{EX}}(2100) = 0.70 \text{ W/m}^2$ for $t \in [t_0, 2100]$, and is equal to $\mathcal{F}_{\text{EX}}(2100)$ for $t \ge 2100$;

* Equation (8.26): $\mathcal{T}_{\text{AT}}(t_0) = 0.8^{\circ}\text{C}$, $\mathcal{T}_{\text{LO}}(t_0) = 0.0068^{\circ}\text{C}$, $\xi_1 = 0.098$, $\mathcal{F}_{\text{RAD}}(t_0) = 2.14 \text{ W/m}^2$, and the temperature diffusion matrix is given by$^{110}$:

$$ \Xi_{\tau} = \begin{pmatrix} 86.30 & 0.8624 \\ 2.5 & 97.5 \end{pmatrix} \times 10^{-2} $$

From Equation (8.26), we can compute the steady-state temperatures:

$$ \begin{aligned} \mathcal{T}(\infty) &= (I_2 - \Xi_{\tau})^{-1} B_{\tau} \mathcal{F}_{\text{RAD}}(\infty) \\ &= \frac{\mathcal{F}_{\text{RAD}}(\infty)}{\xi_2} \mathbf{1}_2 \\ &= \begin{pmatrix} 0.76316 \\ 0.76316 \end{pmatrix} \mathcal{F}_{\text{RAD}}(\infty) \end{aligned} $$

We deduce that:

$$ \Delta \mathcal{T}_{\text{AT}}(\infty) = \frac{1}{\xi_2} \Delta \mathcal{F}_{\text{RAD}}(\infty) = 0.76316 \times \Delta \mathcal{F}_{\text{RAD}}(\infty) $$

A variation of $\pm 1 \text{ W/m}^2$ implies a variation of $\pm 1^{\circ}\text{C}$ of the atmospheric temperature. Therefore, in order to limit global warming, we need to reduce radiative forcing and the carbon concentration $\mathcal{CC}_{\text{AT}}(t)$ in the atmosphere (Equation 8.25). This is done by emitting lower carbon emissions (Equation 8.24). In Figure 8.90, we show the impulse response analysis$^{111}$ by considering a reduction of one GtCO$_2$e. We notice that the carbon concentration in the atmosphere is reduced because the carbon is stored in the upper ocean and then in the deep ocean. To achieve carbon emissions reduction, we have three choices (Equation 8.22):

1. We can reduce the production $Y(t)$;

2. We can reduce the carbon intensity $\sigma(t)$ of industrial activities;

3. We can increase the mitigation effort $\mu(t)$ and accelerate the transition to a low-carbon economy;

In order to illustrate the climate module, we run Equations (8.22–8.26) with the parameter and initial values used by the 2013 version. For that, we need to define $Y(t)$ and $\mu(t)$, which are the two external variables of the temperature modeling system. By assuming that $Y(t) = Y(t_0)$ and $\mu(t) = \mu(t_0)$, we obtain the results in Table 8.31. If we consider the following functions $Y(t) = (1 + g_Y)^{t-t_0} Y(t_0)$ and $\mu(t) = \min(e^{g_\mu(t-t_0)}\mu(t_0), 1)$, the trajectories of $Y(t)$, $\mu(t)$, $\mathcal{T}_{\text{AT}}(t)$ and $\mathcal{T}_{\text{LO}}(t)$ those given in Figure 8.91. We observe that we can limit global warming if we limit the economic growth $g_Y$ or we dramatically increase the mitigation effort $\mu(t)$. The nightmare climate-economic scenario is obtained when the economic growth is high and there is no mitigation effort (Figure 8.92).

$$ \Xi_{\tau} = \begin{pmatrix} 97.2592 & 0.1725 \\ 0.5 & 99.5 \end{pmatrix} \times 10^{-2} $$

and $\xi_1 = 0.0196$ because $c_{\text{AT}} = 51.0204$, $c_{\text{LO}} = 17.6$, $\lambda = 1.3103$ and $\beta = 0.088$.

<footnote>$^{110}$We have $\xi_1 = 0.098$, $\xi_2 = 3.8/2.9$, $\xi_3 = 0.088$, and $\xi_4 = 0.025$ (Calel and Stainforth, 2017a, page 1203). The DICE model uses a 5-year time step. If we prefer to use $\Delta t = 1$ year, we have:</footnote>

<footnote>$^{111}$Following Roncalli (2020a, page 646), the matrix of impulse responses after $t$ years is $\Phi_{\text{CC}}^t$, while the matrix of cumulated responses is $I_3 + \sum_{k=1}^t \Phi_{\text{CC}}^k$.</footnote>
<figure><img src="image_372.png" /><figcaption>Table 8.31: Output of the DICE climate module ($Y(t) = Y(t_0)$, $\mu(t) = \mu(t_0)$)</figcaption></figure>

<table><thead><tr><th>t</th><th>Cε (t)</th><th>σ (t)</th><th>CC<sub>AT</sub> (t)</th><th>F<sub>RAD</sub> (t)</th><th>T<sub>AT</sub> (t)</th><th>T<sub>LO</sub> (t)</th></tr></thead><tbody><tr><td>2010</td><td>36.91</td><td>0.55</td><td>830.4</td><td>2.14</td><td>0.800</td><td>0.007</td></tr><tr><td>2015</td><td>36.25</td><td>0.55</td><td>825.7</td><td>2.14</td><td>0.900</td><td>0.027</td></tr><tr><td>2020</td><td>36.06</td><td>0.56</td><td>821.9</td><td>2.14</td><td>0.986</td><td>0.048</td></tr><tr><td>2025</td><td>35.97</td><td>0.57</td><td>818.9</td><td>2.14</td><td>1.061</td><td>0.072</td></tr><tr><td>2030</td><td>35.98</td><td>0.57</td><td>816.6</td><td>2.15</td><td>1.127</td><td>0.097</td></tr><tr><td>2035</td><td>36.05</td><td>0.58</td><td>814.9</td><td>2.16</td><td>1.186</td><td>0.122</td></tr><tr><td>2040</td><td>36.18</td><td>0.58</td><td>813.9</td><td>2.18</td><td>1.238</td><td>0.149</td></tr><tr><td>2045</td><td>36.36</td><td>0.59</td><td>813.3</td><td>2.20</td><td>1.286</td><td>0.176</td></tr><tr><td>2050</td><td>36.58</td><td>0.59</td><td>813.3</td><td>2.23</td><td>1.329</td><td>0.204</td></tr><tr><td>2055</td><td>36.82</td><td>0.60</td><td>813.6</td><td>2.26</td><td>1.370</td><td>0.232</td></tr><tr><td>2060</td><td>37.09</td><td>0.61</td><td>814.4</td><td>2.29</td><td>1.408</td><td>0.261</td></tr><tr><td>2065</td><td>37.39</td><td>0.61</td><td>815.4</td><td>2.32</td><td>1.445</td><td>0.289</td></tr><tr><td>2070</td><td>37.70</td><td>0.62</td><td>816.8</td><td>2.35</td><td>1.480</td><td>0.318</td></tr><tr><td>2075</td><td>38.02</td><td>0.62</td><td>818.4</td><td>2.39</td><td>1.514</td><td>0.347</td></tr><tr><td>2080</td><td>38.36</td><td>0.63</td><td>820.3</td><td>2.43</td><td>1.547</td><td>0.376</td></tr><tr><td>2085</td><td>38.71</td><td>0.64</td><td>822.4</td><td>2.46</td><td>1.580</td><td>0.406</td></tr><tr><td>2090</td><td>39.06</td><td>0.64</td><td>824.7</td><td>2.50</td><td>1.612</td><td>0.435</td></tr><tr><td>2095</td><td>39.43</td><td>0.65</td><td>827.1</td><td>2.55</td><td>1.645</td><td>0.464</td></tr><tr><td>2100</td><td>39.80</td><td>0.66</td><td>829.7</td><td>2.59</td><td>1.677</td><td>0.494</td></tr></tbody></table>
<figure><img src="image_373.png" /><figcaption>Figure 8.92: The nightmare climate-economic scenario ($g_Y = 0\%$, $\mu(t) = 0$)</figcaption></figure>

<figure><img src="image_374.png" /></figure>
**Social welfare maximization** The last step of the DICE model is to define the social preferences and the objective function to optimize. The model assumes that the climate policy $\mu(t)$ is chosen to maximize the social welfare function $W$, which is the discounted sum of the generational utilities:

$$W(s(t), \mu(t)) = \sum_{t=t_0+1}^{T} \frac{L(t) \mathcal{U}(c(t))}{(1+\rho)^{t-t_0}}$$

where $\rho$ is the (generational) discount rate and $c(t) = C(t)/L(t)$ is the consumption per capita. $\mathcal{U}(c)$ is the CRRA utility function:

$$\mathcal{U}(c) = \frac{c^{1-\alpha} - 1}{1 - \alpha}$$

where $\alpha \ge 0$ is the constant elasticity of marginal utility of consumption, which can be interpreted as the generational inequality aversion<fnref n="112" /> in the context of the Ramsey-Cass-Koopmans model. When $\alpha$ is close to zero, there is low risk aversion to intergenerational inequality. When $\alpha$ is equal to one, the social welfare becomes egalitarian between generations. When $\alpha \to \infty$, we obtain an infinite total utility.

The optimal control problem is then given by:

$$(s^*(t), \mu^*(t)) = \arg\max_{\mu(t) \in [0, 1]} W(s(t), \mu(t)) \quad (8.27)$$

$$\text{s.t.} \quad \begin{cases} \text{Equations (8.19-8.26)} \\ \mu(t) \in [0, 1] \\ s(t) \in [0, 1] \end{cases}$$

We reiterate that the optimization problem has two control variables: the saving rate $s(t)$ and the climate mitigation policy $\mu(t)$. The first control variable $s(t)$ is common to many economic models of optimal growth. It manages the substitution between the present consumption and the present investment, which determines the future consumption. The second control variable $\mu(t)$ is a new variable that manages the substitution between the present climate damages and the future climate damages. The two variables do not have the same status, because $s(t)$ is generally chosen by households, which is not the case of $\mu(t)$.

**Remark 86** The 2013 version of the DICE model assumes that $\alpha = 1.45$ and $\rho = 1.5\%$.

## Social cost of carbon

Above, we have presented the mathematical objective function of the DICE model. Nevertheless, the final objective of the DICE model is not to estimate the optimal pathway of $\mu(t)$, which is a conceptual measure of the mitigation effort. The main goal is to compute the social cost of carbon, which is the central pillar in the cost-benefit analysis of climate change policies:

> "The most important single economic concept in the economics of climate change is the social cost of carbon (SCC). This term designates the economic cost caused by an additional tonne of carbon dioxide emissions or its equivalent. In a more precise definition, it is the change in the discounted value of economic welfare from an additional unit of COtwo-equivalent emissions. The SCC has become a central tool used in climate change policy, particularly in the determination of regulatory policies that involve greenhouse gas emissions." (Nordhaus, 2017a, page 1518).

<footnote n="112">¹¹²When $\alpha \to 1$, $\mathcal{U}(c) = \ln c$.</footnote>
From a mathematical viewpoint, the social cost of carbon is then defined as:

$$ \text{SCC}(t) = \frac{\frac{\partial W(t)}{\partial \mathbf{CE}(t)}}{\frac{\partial W(t)}{\partial C(t)}} = \frac{\partial C(t)}{\partial \mathbf{CE}(t)} $$

It is expressed in $/tCO₂. This measure can be extended to other greenhouse gases. For instance, IWG (2021) also estimated this figure for the Methane and the Nitrous Oxide. They found a magnitude order of $200 for CO₂, $6000 for CH₄, and $70000 for N₂O.

The DICE model can be applied to different scenarios. For instance, we can consider the baseline scenario ($\mu(t) = \mu(t_0)$), the optimal scenario (social welfare maximization) or the 2°C scenario ($T_{AT}(t) \le 2°C$). In Figures 8.93–8.96, we report the results found by Le Guenedal (2019) when we use the 2013 and 2016 versions of the model<fnref n="113" />. The difference between the two versions is mainly explained by the update of the parameter values (Kellett et al., 2019, Table 2). With the 2013 version, the control variable $\mu(t)$ remains below one when considering the optimal scenario, the temperature $T_{AT}(t)$ crosses 2°C around 2050 and the social cost of carbon is less than $150 (Figure 8.93). With the 2016 version, the optimal scenario is a little bit different since we need to make more efforts in terms of mitigation and the SCC in greater than $150 after 2050 (Figure 8.95). The 2°C scenario implies higher values of $\mu(t)$ and SCC(t) for the 2013 version (Figure 8.94). Unfortunately, we observe that this scenario is no longer feasible with the 2016 version<fnref n="114" /> (Figure 8.96).

<table><caption>Table 8.32: Global SCC under different scenario assumptions (in $/tCO₂)</caption><thead><tr><th>Scenario</th><th>2015</th><th>2020</th><th>2025</th><th>2030</th><th>2050</th><th>CAGR</th></tr></thead><tbody><tr><td>Baseline</td><td>31.2</td><td>37.3</td><td>44.0</td><td>51.6</td><td>102.5</td><td>3.46%</td></tr><tr><td>Optimal</td><td>30.7</td><td>36.7</td><td>43.5</td><td>51.2</td><td>103.6</td><td>3.54%</td></tr><tr><td>2.5°C-max</td><td>184.4</td><td>229.1</td><td>284.1</td><td>351.0</td><td>1006.2</td><td>4.97%</td></tr><tr><td>2.5°C-mean</td><td>106.7</td><td>133.1</td><td>165.1</td><td>203.7</td><td>543.3</td><td>4.76%</td></tr></tbody></table>

Source: Nordhaus (2017a, Table 1, page 1520).

In Table 8.32, we report the SCC estimated for standard DICE models computed by Nordhaus (2017a). The baseline scenario corresponds to the current policy, while the optimal scenario is the given by the optimized control path (Equation 8.27). Under the baseline scenario assumption, the SCC value is $31.2/tCO₂ in 2015 and reaches $102.5/tCO₂ in 2050, implying a compound annual growth rate of 3.46%. The optimal scenario gives SCC figures that are very close to the baseline scenario. Nordhaus (2017a) also evaluated two alternative scenarios: the 2.5°C-max scenario constraints the temperature to be below 2.5°C, whereas the 2.5°C-mean imposes an average temperature of 2.5°C for the next 100 years. The impact of these two alternative scenarios is significant. In this case, the social cost of carbon can reach the value $1000/tCO₂ in 2050.

**Remark 87** The social cost of carbon is also known as the carbon tax, the carbon price or the shadow price of carbon emissions. Since it is one of the main policy tools to manage the transition risk, it will be extensively studied in Chapter 10.

<footnote n="113">¹¹³These simulations are computed with the Matlab implementation of MPC-DICE, which replicates the functionality of the DICE-2013R/2016R model (Faulwasser et al., 2018). The source code is available at https://github.com/cmkellett/MPC-DICE. These results can also be generated using the two other Maltab implementations provided by Kellett et al. (2016) and Lemoine (2020).</footnote>

<footnote n="114">¹¹⁴See Nordhaus (2017b, page 17).</footnote>
<figure><img src="image_375.png" /><figcaption>Figure 8.93: Optimal welfare scenario (DICE 2013R)</figcaption></figure>

Source: Le Guenedal (2019).

<figure><img src="image_376.png" /><figcaption>Figure 8.94: 2°C scenario (DICE 2013R)</figcaption></figure>

Source: Le Guenedal (2019).
<figure><img src="image_377.png" /><figcaption>Figure 8.95: Optimal welfare scenario (DICE 2016R)</figcaption></figure>

Source: Le Guenedal (2019).

<figure><img src="image_378.png" /><figcaption>Figure 8.96: 2°C scenario (DICE 2016R)</figcaption></figure>

Source: Le Guenedal (2019).
## The Stern-Nordhaus controversy

In 2007, Nicholas Stern published a report called *The Economics of Climate Change: The Stern Review*, that was commissioned by the British government (Stern, 2007). The Stern Review called for sharp and immediate action to stabilize greenhouse gases because “*the benefits of strong, early action on climate change outweighs the costs.*” According to Weitzman (2007), the analysis supporting this conclusion is based on two main arguments. The first one concerns the high value of the discount rate, which is used by IAMs and the DICE model. The second argument is more ethical and related to the large uncertainty about the future of the world. Therefore, the Stern Review proposes to use $\rho = 0.10\%$. The publication of the book has created turmoil in the scientific community, and many economists have participated to the debate. For instance, we can find dozens of reviews and critiques of the Stern Review that have been published in academic journals (*Journal of Economic Literature*, *Science*, *Energy Policy*, *Climate Change*, etc.).

<figure><img src="image_379.png" /><figcaption>Figure 8.97: Discounted value of $10</figcaption></figure>

In order to illustrate the impact of $\rho$, we report the discounted value of $100 in Figure 8.97. Since the time horizon of the economic analysis is relatively long (greater or equal to 100 years), we obtain very different discounted values. For instance, if $\rho$ is equal to 2% (or 0.1%), the well-being of someone 100yr from now would be valued 86% (or 10%) less than the well-being of someone living today (Hänsel et al., 2020). In this context, it is obvious that $\rho$ is certainly the most important parameter when computing the social cost of carbon.

The time (or generational) discount rate $\rho$ used in the DICE model is also called the pure rate of time preference and is related to the Ramsey rule<fnref n="115" />:

$$r = \rho + \alpha g_c$$

<footnote n="115">¹¹⁵See Box 8.18.</footnote>
Box 8.18: The Ramsey rule

We consider the optimization problem$^{a}$:

$$
\begin{align*}
v(k(t)) &= \max_{c(t)} \int_{0}^{\infty} e^{-\rho t} \mathcal{U}(c(t)) \, dt \\
\text{s.t.} \quad &dk(t) = rk(t) \, dt - c(t) \, dt
\end{align*}
$$

where $c(t) = C(t)/L(t)$ and $k(t) = K(t)/L(t)$ are the per capita consumption and capital, and $r$ is the real interest rate. The Bellman equation is:

$$
\max_{c(t)} \left\{ \mathcal{U}(c(t)) - \rho v(k(t)) + \frac{\partial v(k(t))}{\partial k} (rk(t) - c(t)) \right\} = 0
$$

or:

$$
\rho v(k(t)) = \max_{c(t)} \{ \mathcal{U}(c(t)) - v'(k(t)) c(t) \} + rk(t) v'(k(t))
$$

If we assume that the utility function is CRRA — $\mathcal{U}(c) = c^{1-\alpha}/(1-\alpha)$, we obtain:

$$
\max_{c} \{ \mathcal{U}(c) - v'(k) c \} \Leftrightarrow v'(k) = \mathcal{U}'(c) = c^{-\alpha}
$$

implying that $c = v'(k)^{-1/\alpha}$ and:

$$
\max_c \{ \mathcal{U}(c) - v'(k) c \} = \frac{c^{1-\alpha}}{1-\alpha} - c^{1-\alpha} = \frac{\alpha}{1-\alpha} c^{1-\alpha} = \frac{\alpha}{1-\alpha} v'(k)^{-(1-\alpha)/\alpha}
$$

The Bellman equation becomes then:

$$
\rho v(k) = \frac{\alpha}{1-\alpha} v'(k)^{-(1-\alpha)/\alpha} + rkv'(k)
$$

We can verify that the optimal solution of the previous equation has the following expres-
sion:

$$
v^*(k(t)) = \left( \frac{\rho - r(1-\alpha)}{\alpha} \right)^{-\alpha} \frac{k(t)^{1-\alpha}}{1-\alpha}
$$

The optimal consumption is then equal to:

$$
c^*(t) = \left( \frac{\partial v^*(k(t))}{\partial k} \right)^{-1/\alpha} = \frac{\varrho - r(1-\alpha)}{\alpha} k(t)
$$

It follows that:

$$
dk^*(t) = rk^*(t) \, dt - c^*(t) \, dt = \left( r - \frac{\varrho - r(1-\alpha)}{\alpha} \right) k^*(t) \, dt = \underbrace{\frac{r-\varrho}{\alpha} k^*(t)}_{g_k} \, dt
$$

In the Ramsey model, the optimal growth of capital is also equal to the optimal growth of consumption ($g_k = g_c$). Finally, we obtain the Ramsey rule:

$$
g_c = \frac{\partial_t c(t)}{c(t)} = \frac{r - \varrho}{\alpha}
$$

<footnote><sup>a</sup>This derivation of the Ramsey rule has been suggested by Peter Tankov.</footnote>
where *r* is the real interest rate, *g*<sub>c</sub> = ∂c(t)/c(t) is the growth rate of per capita consumption, and *α* is the consumption elasticity of the utility function (or the elasticity of substitution)<fnref n="116" />. The Ramsey rule can be interpreted as a condition for the optimality of intertemporal consumption choice. Therefore, *r* can be interpreted as the consumption discount rate, the social discount rate or the discount rate on goods (Nordhaus, 2019). There are then two ways to use the Ramsey rule. From the previous relationship, we can deduce the optimal value of *ρ* from the historical estimates of *r* and *g*<sub>c</sub>: ρ = *r* − *αg*<sub>c</sub>. For example, if the real interest rate is equal to 4%, *g*<sub>c</sub> = 2% and *α* = 1, we obtain ρ = 2%. Nordhaus (2019) justified the historical (or descriptive) approach as follows:

> "This approach assumes that investments to slow climate change must compete with investments in other areas. The benchmark should therefore reflect the opportunity cost of investment. The descriptive approach yields a market rate of return in the neighborhood of 5% per year when risks are appropriately included."

According to Nordhaus (2007), the assumption $\rho = 0.10\%$ is not consistent with historical obser-
vations, because this implies a low goods discount rate, around 1% per year, while the descriptive
approach yields a market rate of return in the neighborhood of 5% per year. The first approach
contrasts with the normative (or prescriptive) approach, which consists in fixing directly $\rho$ for ethical
reasons. This is the position of Stern and Taylor (2007), who claims that there are “many reasons for
thinking that market rates and other approaches that illustrate observable market behavior cannot
be seen as reflections of an ethical response to the issues at hand. There is no real economic market
that reveals our ethical decisions on how we should act together on environmental issues in the
very long term. Most long-term capital markets are very thin and imperfect.” In fact, as shown by
Gollier (2010, 2013), the Ramsey rule is not always valid when the uncertainty on economic growth
is high. In particular, there is no reason to use the same rate to discount environmental impacts
and monetary benefits.

<table><caption>Table 8.33: Global SCC under different discount rate assumptions</caption>
  <thead>
    <tr>
      <th>Discount rate</th>
      <th>2015</th>
      <th>2020</th>
      <th>2025</th>
      <th>2030</th>
      <th>2050</th>
      <th>CAGR</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Stern</td>
      <td>197.4</td>
      <td>266.5</td>
      <td>324.6</td>
      <td>376.2</td>
      <td>629.2</td>
      <td>3.37%</td>
    </tr>
    <tr>
      <td>Nordhaus</td>
      <td>30.7</td>
      <td>36.7</td>
      <td>43.5</td>
      <td>51.2</td>
      <td>103.6</td>
      <td>3.54%</td>
    </tr>
    <tr>
      <td>2.5%</td>
      <td>128.5</td>
      <td>140.0</td>
      <td>152.0</td>
      <td>164.6</td>
      <td>235.7</td>
      <td>1.75%</td>
    </tr>
    <tr>
      <td>3%</td>
      <td>79.1</td>
      <td>87.3</td>
      <td>95.9</td>
      <td>104.9</td>
      <td>156.6</td>
      <td>1.97%</td>
    </tr>
    <tr>
      <td>4%</td>
      <td>36.3</td>
      <td>40.9</td>
      <td>45.8</td>
      <td>51.1</td>
      <td>81.7</td>
      <td>2.34%</td>
    </tr>
    <tr>
      <td>5%</td>
      <td>19.7</td>
      <td>22.6</td>
      <td>25.7</td>
      <td>29.1</td>
      <td>49.2</td>
      <td>2.65%</td>
    </tr>
  </tbody>
</table>

Source: Nordhaus (2017a, Table 1, page 1520).

In Table 8.33, we report the estimated values of SCC, which was obtained by Nordhaus (2017a) under different discount rate assumptions. We better understand the call for sharp and immediate action by Nicholas Stern, since the social cost of carbon was equal to $197.4/tCO₂ in 2015 when the discount rate is set to 0.1%!

<footnote n="116">¹¹⁶Hänsel et al. (2020, page 783) interpret the parameter α as measuring intertemporal inequality aversion:</footnote>

<footnote>> "Due to diminishing marginal utility, the idea is that an additional $1 is worth more to a poor person than to a rich one. In a growing economy, citizens in the future will be richer and their lower marginal utility motivates discounting. Suppose the economy grows at 2%. People living in 100yr will be seven times richer. If inequality aversion is the only reason for discounting, if α = 1 (or 1.45), which corresponds to the values of the median expert (Nordhaus), the value of $1 in 100yr is only 14 (or 6) cents."</footnote>
### 8.3.2 Other models

In Table 8.34, we report a short list of IAMs. For instance, Grubb et al. (2021) counts 28 major climate economic models<fnref n="117" />. They make the distinction between stylized simple IAMs and complex IAMs. While the purpose of the first category is focused on the optimal path of the economy based on policy optimization, the second category generally aims to evaluate the impact of a climate scenario or a given policy on the economy.

<table><caption>Table 8.34: Main integrated assessment models</caption><thead><tr><th>Model</th><th>Reference</th><th>Name</th></tr></thead><tbody><tr><td colspan="3">Stylized simple models</td></tr><tr><td>DICE</td><td>Nordhaus and Sztorc (2013)</td><td>Dynamic Integrated Climate-Economy</td></tr><tr><td>FUND</td><td>Anthoff and Tol (2014)</td><td>Climate Framework for Uncertainty, Negotiation and Distribution</td></tr><tr><td>PAGE</td><td>Hope (2011)</td><td>Policy Analysis of the Greenhouse Effect</td></tr><tr><td colspan="3">Complex models</td></tr><tr><td>AIM/CGE</td><td>Fujimori et al. (2017)</td><td>Asia-Pacific Integrated Model/Computable General Equilibrium</td></tr><tr><td>GCAM</td><td>Calvin et al. (2019)</td><td>Global Change Assessment Model</td></tr><tr><td>GLOBIOM</td><td>Havlik et al. (2018)</td><td>Global Biosphere Management Model</td></tr><tr><td>IMACLIM-R</td><td>Sassi et al. (2010)</td><td>Integrated Model to Assess Climate Change</td></tr><tr><td>IMAGE</td><td>Stehfest et al. (2014)</td><td>Integrated Model to Assess the Greenhouse Effect</td></tr><tr><td>MAGICC</td><td>Meinshausen et al. (2011)</td><td>Model for the Assessment of Greenhouse Gas Induced Climate Change</td></tr><tr><td>MAgPIE</td><td>Dietrich et al. (2019)</td><td>Model of Agricultural Production and its Impact on the Environment</td></tr><tr><td>MESSAGEix</td><td>Huppmann et al. (2019)</td><td>Model for Energy Supply Strategy Alternatives and their General Environmental Impact</td></tr><tr><td>REMIND</td><td>Aboumahboub et al. (2020)</td><td>Regional Model of INvestments and Development</td></tr><tr><td>WITCH</td><td>Bosetti et al. (2006)</td><td>World Induced Technical Change Hybrid</td></tr></tbody></table>

Source: Grubb et al. (2021, Table A1, pages 24-26) & Author's research.

The FUND model is available at www.fund-model.org. A Julia implementation can be found at https://github.com/fund-model/MimiFUND.jl. The PAGE model requires Microsoft Excel and the @RISK Excel plug-in, but Moore et al. (2018) propose a Julia open-source implementation at https://github.com/anthofflab/MimiPAGE2009.jl and http://anthofflab.berkeley.edu/MimiPAGE2009.jl/stable. The AIM/CGE is hosted at www-iam.nies.go.jp/aim/index.html. The GCAM model is available at http://jgcri.github.io/gcam-doc/index.html (documentation and open-source code). GLOBIOM has been developed using GAMS<fnref n="118" />. The source code is not available, but a graphical user interface and a R interface is provided at https://iiasa.

<footnote n="117">¹¹⁷ A more exhaustive list can be found in www.iamcdocumentation.eu, where a reference card is provided for each model.</footnote>

<footnote n="118">¹¹⁸ GAMS (General Algebraic Modeling System) is a high level modeling system for mathematical programming and optimization. It consists of a language compiler and a range of optimizer solvers, and requires a user license to be used (www.gams.com).</footnote>
github.io/GLOBIOM. IMACLIM-R is not public. This is also the case of IMAGE, but we can
download many data items, such as SSP scenario data, spatial data global land-use maps, sce-
nario results and renewable energy data (www.pbl.nl/en/image/about-image). A compiled ver-
sion of MAGGIC can be download at https://magicc.org. An online version is also available at
https://live.magicc.org. The documentation of MAgPIE can be found at www.pik-potsdam.de/
en/institute/departments/activities/land-use-modelling/magpie, while the link https://
github.com/magpiemodel/magpie contains the model code and the ⚙️ implementation. For MES-
SAGEix, the documentation is available at https://docs.messageix.org/en/stable, while the
python programs can be found at https://github.com/iiasa/message_ix. The documentation
of REMIND can be found at www.pik-potsdam.de/en/institute/departments/transformation-
pathways/models/remind, while the link https://github.com/remindmodel/remind contains the
implementation. Finally, the website www.witchmodel.org is dedicated to the WITCH model.
The code is open-source (https://github.com/witch-team/witchmodel), but requires a GAMS li-
cense.

Stylized IAMs

Stylized integrated assessment models use the same structure as DICE (Figure 8.98). The FUND model has five main components: (1) population, production and income, (2) emissions, abatements and costs, (3) atmosphere and climate, (4) impacts (agriculture, forestry, water resources, energy consumption, sea level rise, ecosystems, human health, extreme weather, mortality), (5) optimization of the utility function, where the social welfare is a mixture of per capita income, damage of climate change and air pollution, and emission reduction costs Tol (1997). It also considers 16 regions. For its part, PAGE has two main modules (Hope, 2011; Moore et al., 2018). Like the DICE model, the climate sensitivity is derived from a stylized GCM, but it explicitly introduces a sea level rise component and also considers CH₄ and N₂O forcing cycles, and not only the CO₂ cycle<fnref n="119" />. The economic module has four components: (1) population, production and income, (2) damage costs, (3) climate policy action and (4) consumption utility maximization. It is also possible to perform a regional analysis<fnref n="120" />.

<figure><img src="image_380.png" /><figcaption>Figure 8.98: Stylized integrated assessment models</figcaption></figure>

A comprehensive comparison of DICE, FUND and PAGE can be found in Diaz and Moore (2017a,b), and Rose et al. (2017a,b). It appears that parameter values and functional forms are not always the same (Diaz and Moore (2017a, Table 1, page 776), Rose et al. (2017b, Table S2, pages 2-5)), and damage estimates projected by the three models are dramatically different (Diaz

<footnote n="119"><sup>119</sup>We also find these additions in the FUND model</footnote>

<footnote n="120">¹²⁰The default regions are Africa, China, Europe, Latin America, other OECD countries, Russia, South-East Asia and the United States.</footnote>
<figure><img src="image_381.png" /><figcaption>Figure 8.99: Histogram of the 150 000 US Government SCC estimates for 2020 with a 3% discount rate</figcaption></figure>

The figure combines the 50 000 2020 3% discount rate estimates from each of the three US Government models to illustrate their influence on the aggregate histogram that determines the official USG SCC for 2020 at 3%, which is equal to $42 (average) and $123 (95th percentile).

Source: (Rose et al., 2017a, page 3).

and Moore, 2017a, Figure 2, page 777). In this context, the social cost of carbon reaches different levels depending on the model (Figure 8.99). In particular, Rose et al. (2017a) confirmed the ordering/raking PAGE ≻ DICE ≻ FUND that most academic studies have found when computing the social cost of carbon:

> “We find significant variation in component-level behavior between models driven by model-specific structural and implementation elements, some resulting in artificial differences in results. These elements combine to produce model-specific tendencies in climate and damage responses that contribute to differences observed in SCC outcomes — producing PAGE SCC distributions with longer and fatter right tails and higher averages, followed by DICE with more compact distributions and lower averages, and FUND with distributions that include net benefits and the lowest averages.” (Rose et al., 2017a, page 1).

The previous stylized models have inspired many research projects and IAMs. This is especially true for DICE. For instance, the RESPONSE model of Pottier et al. (2015) and the ENTICE model of Popp (2004) are two variants of DICE. We can also mention the RICE model developed by Nordhaus and Yang (1996), which is a regional extension of the DICE framework with 12 regions. The RICE model introduces an important feature, which is the use of Negishi welfare weights with multiple countries. Those weights are given by the relationship $ω_k(t) ∝ 1/μ'(c_k(t))$ where k is the
country index. In the case of the CRRA utility function $\mathcal{U}(c) = c^{1-\alpha}/(1-\alpha)$, we obtain:

$$ \omega_k(t) = \frac{c_k(t)^\alpha}{\sum_{k'=1}^{n_C} c_{k'}(t)^\alpha} $$

where $n_C$ is the number of countries. Then, the social welfare function becomes:

$$ W(s_1(t), \dots, s_{n_C}(t), \mu_1(t), \dots, \mu_{n_C}(t), \dots) = \sum_{k=1}^{n_C} \sum_{t=t_0+1}^{T} \omega_k(t) \frac{L_k(t) \mathcal{U}(c_k(t))}{(1 + \varrho_k)^{t-t_0}} $$

Negishi welfare weights are generally used to find a market equilibrium by optimising the weighted objective function (or Negishi social welfare function). However, these weights have been criticized to be inequitable, since the current wealth distribution would change. For instance, Stanton (2011) pointed out that the choice of weights are related to the regional inequality issue:

> "In a global climate policy debate fraught with differing understandings of right and wrong, the importance of making transparent the ethical assumptions used in climate-economics models cannot be overestimated. [...] Negishi weights freeze the current distribution of income between world regions; without this constraint, IAMs that maximize global welfare would recommend an equalization of income across regions as part of their policy advice. With Negishi weights in place, these models instead recommend a course of action that would be optimal only in a world in which global income redistribution cannot and will not take place."

This question of regional inequality is central when considering climate change and the transition risk. For instance, Dennig et al. (2015) choose $\omega_k(t) = 1$ in their Nested Inequalities Climate-Economy (NICE) model, which is a direct extension of RICE, because they restrict consumption redistribution between regions.

## Complex IAMs

As their name suggest, complex IAMs are complex. While stylized simple IAMs are generally developed by one or two academics, complex IAMs are built by an institution with a big research team and are continuously improved and updated$^{121}$. For that reason, it seems a pointless exercise to present one of them in details. For instance, GCAM has been developed at Pacific Northwest National Laboratory for over 30 years. The source code is today about 120 Mo, there are 910 CSV data files for a total of 230 Mo. Therefore, it is better to understand how they work and how they can be used. In Figure 8.100, we have reported the global structure of GCAM. In this type of complex models, the economic component is a small part of the structure. Furthermore, these big complex models are generally split into elementary complex models. In Figure 8.101, we have reported the structure of GLOBIOM. This model is specialized in agriculture, bioenergy and forestry. This model can communicate with MESSAGEix and other models as shown in Figure 8.102. With GCAM and MESSAGE, REMIND is another key complex IAM (Figure 8.103). Again, we notice that REMIND can be connected with MAgPIE, which is specialized in the land use. We also remark that it can take

<footnote>$^{121}$AIM/CGE, IMAGE, GCAM, IMACLIM-R, MESSAGE-GLOBIOM, REMIND-MAgPIE, WITCH are respectively hosted by the National Institute for Environmental Studies (NIES, Japan), the PBL Environmental Assessment Agency (Netherlands), the Pacific Northwest National Laboratory (PNNL, US), the Centre international de recherche sur l'environnement et le développement (CIRED, France), the International Institute for Applied Systems Analysis (IIASA, Austria), the Potsdam Institute for Climate Impact Research (PIK, Germany) and the European Institute on Economics and the Environment (RFF-CMCC-EIEE, Italy).</footnote>
<figure><img src="image_382.png" /><figcaption>Figure 8.100: Linkages between the major systems in GCAM</figcaption></figure>

Source: Calvin et al. (2019).

<figure><img src="image_383.png" /><figcaption>Figure 8.101: The main land-use sectors of GLOBIOM</figcaption></figure>

Source: https://iiasa.github.io/GLOBIOM.
<figure><img src="image_384.png" /><figcaption>Figure 8.102: Overview of the IIASA IAM framework</figcaption></figure>

Source: https://docs.messageix.org/projects/global/en/latest/overview/index.html.
<figure><img src="image_385.png" /><figcaption>Figure 8.103: The Remind-MAgPIE framework</figcaption></figure>

Source: www.pik-potsdam.de/en/institute/departments/transformation-pathways/models/remind
input data from MAGICC, exactly like MESSAGEix, and it can also feed MAGICC with output data.

Even if these models can be used in an optimization mode, whose objective is to compute optimal paths and perform a cost-benefit analysis, they are mainly exploited in order to evaluate a climate policy or to simulate climate/economic trajectories with respect to a given scenario. Like any models, even if they are extensively used by policy markers and the scientific community, they have often been criticised.

## Criticisms of integrated assessment models

According to Pindyck (2017), “IAM-based analyses of climate policy create a perception of knowledge and precision that is illusory and can fool policymakers into thinking that the forecasts the models generate have some kind of scientific legitimacy.” In particular, Robert Pindyck considers that the most important flaws are the following:

* Certain inputs, such as the discount rate, are arbitrary, but they can have a big impact on the model outputs, such as the social cost of carbon;

* There is a lot of uncertainty about climate sensitivity and the temperature trajectory;

* Modeling damage functions is arbitrary, because we have little data to estimate the relationship between an increase in temperature and gross domestic product;

* IAMs are unable to consider tail risk, i.e., a tipping point or the likelihood/possible impact of a catastrophic climate outcome.

In Table 8.35, we report another list of criticisms found by Gambhir et al. (2019). We can multiply this type of criticism, and it is relatively essay to do it. We know that all models have flaws and are a simplification of reality. Nevertheless, to repeat the words of George box, “*all models are wrong, but some are useful.*” For instance, we notice that Pindyck (2017) only referenced DICE, FUND and PAGE, while Gambhir et al. (2019) focused on complex IAMs. In both cases, criticisms are general and dot not concern a specific model. In this context, it is not always obvious to form an opinion on a model versus another one. For complex IAMs, it is almost impossible. As well, we notice that most comparative studies concern the stylized simple models.

### 8.3.3 Scenarios

As we have already explained, there are two ways to use IAMs: policy optimization and policy/scenario evaluation/simulation. In this section, we focus on the second approach, and more precisely on the scenario simulation. We distinguish two types of simulation outputs: shared socioeconomic pathways (SSPs) and macroeconomic pathways. Before presenting them, we briefly present climate scenarios, which are the input of the evaluation process.

<figure><img src="image_386.png" /><figcaption>Figure 8.104: Scenario evaluation</figcaption></figure>
<table><caption>Table 8.35: Main integrated assessment models</caption><thead><tr><td>Criticism category</td><td>Example of specific criticism</td></tr></thead><tbody><tr><td>Lack of transparency</td><td>Lack of documentation</td></tr><tr><td>Inappropriate input assumptions</td><td>Low share of solar PV in mitigation modelling exercises</td></tr><tr><td>Lack of clarity on model inputs versus outputs</td><td>Energy demand reduction as a result of model choice or modeler input?</td></tr><tr><td>Reliance of mitigation costs on baseline assumptions</td><td>Significant differences in costs of achieving mitigation</td></tr><tr><td>Inadequate representation of innovation processes</td><td>Spillover between low-carbon technologies</td></tr><tr><td>Lack of representation of behavioural and economic systems</td><td>Customer behavior changes</td></tr><tr><td>Lack of assessment of real-world feasibility</td><td>Feasibility of pathways given full consideration of social, political and technical barriers</td></tr><tr><td>Lack of interaction with other policy goals</td><td>Lack of consideration of mitigation pathways in light of other policy goals such as energy security</td></tr><tr><td>Lack of representation of fine temporal and geographical scale</td><td>Geographical dispersion of electricity systems</td></tr><tr><td>Over-reliance on negative emissions technologies</td><td>Unrealistically high levels of negative emissions technologies</td></tr></tbody></table>

Source: Adapted from Gambhir et al. (2019, Table 1, page 10).

## Climate scenarios

It is important to understand the different climate scenarios that are used to simulate economic scenarios. In what follows, we focus on four main scenario databases:

* The representative concentration pathways (RCPs) that can be found in the IPCC AR5 (IPCC, 2013);

* The decarbonization pathway scenarios developed by the IEA (IEA, 2017);

* The 1.5°C scenarios proposed by the IPCC SR15 (IPCC, 2018) and the IEA (IEA, 2021);

* The new scenarios for the future published in the IPCC AR6 (IPCC, 2022).

**The RCP scenarios** A representative concentration pathway (RCP) is a greenhouse gas concentration trajectory adopted by IPCC (2013). The AR5 scenario database comprises 31 models and 1184 scenarios. Among these different scenarios, four pathways have been selected to represent four reference scenarios:

1. RCP 2.6: GHG emissions start declining by 2020 and go to zero by 2100 (IMAGE);

2. RCP 4.5: GHG emissions peak around 2040, and then decline (MiniCAM);

3. RCP 6.0: GHG emissions peak around 2080, and then decline (AIM);

4. RCP 8.5: GHG emissions continue to rise throughout the 21st century (MESSAGE).
<figure><img src="image_387.png" /><figcaption>Figure 8.105: Total radiative forcing (in W/m²)</figcaption></figure>

Source: https://tntcat.iiasa.ac.at/RcpDb.

<figure><img src="image_388.png" /><figcaption>Figure 8.106: Greenhouse gas concentration trajectory</figcaption></figure>

Source: https://tntcat.iiasa.ac.at/RcpDb.
<figure><img src="image_389.png" /><figcaption>Figure 8.107: Greenhouse gas emissions trajectory</figcaption></figure>

Source: https://tntcat.iiasa.ac.at/RcpDb.

<figure><img src="image_390.png" /><figcaption>Figure 8.108: Total GHG emissions trajectory (in GtCO₂e)</figcaption></figure>

Source: https://tntcat.iiasa.ac.at/RcpDb.
The name of each scenario refers to the value of the radiative forcing by 2050 (Figure 8.105). Figure 8.106 shows the GHG concentration trajectory for CO₂, CH₄ and N₂O gases. We also report the CO₂e concentration, which is the aggregation of the several GHG concentrations. In Figure 8.107, we show the emissions trajectories for the three main greenhouse gases. For CO₂, we distinguish between two categories: industry and land-use change. Finally, we give the total GHG emissions<fnref n="122" /> trajectories in Figure 8.107. We notice that the RCP scenario database do not provide a temperature scenario, because the relationship between GHG emissions and temperature depends on several factors. This is why we can only provide a temperature range for each RCP scenario (Figure 8.108).

**The IEA scenarios** As energy represents the largest portion of GHG emissions, the most common way to reduce global emissions is to impose a global shift in the energy supply or demand. To encourage this shifting, the international energy agency has developed energy pathways. These scenarios are not a forecast of the future but explore the different possibilities across the energy system. For instance, IEA (2017) presented three pathways for energy sector development to 2060. The reference technology scenario (RTS) was a baseline scenario that took into account existing climate-related commitments by countries, while the 2°C scenario (2DS) corresponded to a rapid decarbonisation pathway in line with international policy goals. The beyond 2°C scenario (B2DS) was a variant of the 2DS, where the energy sector reached carbon neutrality by 2060 to limit future temperature increases to 1.75°C by 2100. The previous three scenarios were replaced by two new scenarios in 2020. The sustainable development scenario (SDS) “sets out the major changes that would be required to reach the key energy-related goals of the UN sustainable development agenda”, including a universal access to modern energy by 2030. This scenario is consistent with reaching net zero emissions by around 2070. The previous RTS scenario becomes the stated policies scenario (STEPS). Finally, the last edition of the IEA’s Energy Technology Perspectives (ETP) only focuses on the NZE<fnref n="123" /> and the announced pledges scenario (APS), which assumes that governments will meet all the climate-related commitments they have announced.

The international energy agency evaluates these different scenarios using its Global Energy and Climate (GEC) model<fnref n="124" />. This model covers 26 regions individually with dedicated bottom-up modelling for final energy demand, energy transformation and energy supply. Using assumptions on population, GDP, fossil fuel prices and resources (crude oil, natural gas and steam coal), CO₂ prices, electricity and technology costs, the GEC model predicts primary energy demand and supply by sources (oil, coal, natural gas, nuclear, biomass and waste, hydro, geothermal, wind, solar, hydrogen), electricity generation and capacity, CO₂ and CH₄ emissions, investment needs and costs, and materials and critical minerals demand. For instance, we report the CO₂ emissions estimated in ETP 2017 in Figure 8.109. We notice that we can not achieve the degC2 scenario when considering the country commitments (RTS).

$$ \begin{aligned} \text{Total GHG emissions in GtCO}_2\text{e} = & \text{CO}_2 \text{ emissions in PgC} \times \frac{44}{12} + \\ & \text{CH}_4 \text{ emissions in Tg} \times \frac{28}{1000} + \\ & \text{N}_2\text{O emissions in Tg} \times \frac{265}{1000} \end{aligned} $$

This formula uses the AR5 GWP values (see Section 9.1.1 on page 870), the mass percentage of carbon in carbon dioxide, which is equal to 12/44 = 27.27% (because carbon has an atomic mass of 12 and oxygen has an atomic mass of 16, implying that CO₂ has an atomic mass of 44), and the petagram/teragram conversion factor (1 Pg = 1000 Tg).

<footnote n="122">¹²²They have been calculated as follows:</footnote>

<footnote n="123">¹²³See page 919.</footnote>

<footnote n="124">¹²⁴Its description is available at www.iea.org/reports/global-energy-and-climate-model.</footnote>
<figure><img src="image_391.png" /><figcaption>Figure 8.109: Direct CO₂ emissions (in Gt)</figcaption></figure>

Source: IEA (2017, Figures 1.6 and 1.9).

**The 1.5°C scenarios** In 2018, IPCC issued a special report on the impacts of global warming of 1.5°C above pre-industrial levels and related global greenhouse gas emission pathways. According to the report, “limiting global warming to 1.5°C, compared with 2°C, could reduce the number of people both exposed to climate-related risks and susceptible to poverty by up to several hundred million by 2050.” While the goal of the climate Paris Agreement was to limit global temperature increase to well below 2°C, the IPCC reaffirms then that states must pursue efforts to limit the temperature increase to 1.5°C, because the cost-benefit analysis of the 1.5°C scenario versus 2°C scenario shows that we can reach some tipping points in the second scenario that will generate too much costs for the society, governments and people.

The IPCC SR15 database contains 177 scenarios computed with 24 models<fnref n="125" />. We filter the database in order to keep climate scenarios, whose explicit objective is 1.5°C or well below 2°C. We obtain 114 scenarios, whose CO2 emissions trajectory is reported in Figure 8.110. For each date, we estimate the mean and the standard deviation and compute the 50% and 95% confidence interval in Figure 8.111. We notice that the emissions are close to zero by 2050 on average. This is why the 11.5°C scenario is generally associated to the net zero emissions by 2050 (NZE) scenario. We must be careful because these figures correspond to expected values, meaning that there is a significant probability that the temperature increase can be greater than 1.5°C even if the scenario occurs. For instance, we have reported the global mean temperature of the 114 scenarios, which has been computed with the MAGICC 6 model (Figure 8.112). We observe that the temperature can be

<footnote n="125">¹²⁵AIM/CGE (2.0 & 2.1), C-ROADS-5.005, GCAM 4.2, GENeSYS-MOD 1.0, IEA 2017 models (ETP & WEM), IMAGE (3.0.1 & 3.0.2), MERGE-ETL 6.0, MESSAGE (V.3, GLOBIOM & ix-GLOBIOM), POLES (ADVANCE, CD-LINKS & EMF33), REMIND (1.5 & 1.7), REMIND-MAgPIE (1.5 & 1.7-3.0), Shell World Energy Model 2018, and WITCH-GLOBIOM (3.1, 4.2 & 4.4).</footnote>
<figure><img src="image_392.png" /><figcaption>Figure 8.110: IPCC 1.5°C scenarios of CO₂ emissions</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/iamc-1.5c-explorer.

<figure><img src="image_393.png" /><figcaption>Figure 8.111: Confidence interval of the average IPCC 1.5°C scenario</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/iamc-1.5c-explorer.
<figure><img src="image_394.png" /><figcaption>Figure 8.112: IPCC 1.5°C scenarios of the global mean temperature</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/iamc-1.5c-explorer.

greater than 1.5°C by 2100, since only 56% of the 114 scenarios produce an expected temperature below this threshold. In Figures 8.113 and 8.114, we also give the evolution of the exceedance probabilities. We notice that the probability to observe a temperature greater than 1.5°C by 2050 is greater than 60%, while it is equal to only 15% if the threshold is 2°C.

**The AR6 scenarios** IPCC (2022) has updated the RCP scenarios. The new dataset<fnref n="126" /> contains 188 models, 1 389 scenarios, 244 countries and regions, and 1 791 variables, which can be split into six main categories:

* Agriculture: agricultural demand, crop, food, livestock, production, etc.

* Capital cost: coal, electricity, gas, hydro, hydrogen, nuclear, etc.

* Energy: capacity, efficiency, final energy, lifetime, OM cost, primary/secondary energy, etc.

* GHG impact: carbon sequestration, concentration, emissions, forcing, temperature, etc.

* Natural resources: biodiversity, land cover, water consumption, etc.

* Socio-economic variables: capital formation, capital stock, consumption, discount rate, employment, expenditure, export, food demand, GDP, Gini coefficient, import, inequality, interest rate, investment, labour supply, policy cost, population, prices, production, public debt, government revenue, taxes, trade, unemployment, value added, welfare, etc.

For example, we have reported the distribution of some output variables on page 790.

<footnote n="126">¹²⁶The amount of data is huge (3 Go).</footnote>
<figure><img src="image_395.png" /><figcaption>Figure 8.113: Confidence interval of the exceedance probability $Pr\{T > 1.5°C\}$ (MAGICC 6)</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/iamc-1.5c-explorer.

<figure><img src="image_396.png" /><figcaption>Figure 8.114: Confidence interval of the exceedance probability $Pr\{T > 2°C\}$ (MAGICC 6)</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/iamc-1.5c-explorer.
<figure><img src="image_397.png" /><figcaption>Figure 8.115: Histogram of some AR6 output variables by 2100</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/ar6.

<figure><img src="image_398.png" /><figcaption>Figure 8.116: Histogram of some AR6 output variables by 2100</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/ar6.
## Shared socioeconomic pathways

Shared socioeconomic pathways are a set of five narrative scenarios proposed by O’Neill et al. (2014). These narratives describe alternative economic developments that are related to challenges for climate change adaptation and mitigation (Figure 8.117). According to (O’Neill et al., 2014, page 389), “the SSPs describe plausible alternative trends in the evolution of society and natural systems over the 21st century at the level of the world and large world regions; They consist of two elements: a narrative storyline and a set of quantified measures of development. SSPs are reference pathways in that they assume no climate change or climate impacts, and no new climate policies.”

<figure><img src="image_399.png" /><figcaption>Figure 8.117: The shared socioeconomic pathways</figcaption></figure>

Source: O’Neill et al. (2017).

The concept of SSP is clarified in O’Neill et al. (2017):

“The SSP narratives [are] a set of five qualitative descriptions of future changes in demographics, human development, economy and lifestyle, policies and institutions, technology, and environment and natural resources. [...] Development of the narratives drew on expert opinion to (1) identify key determinants of the challenges [to mitigation and adaptation] that were essential to incorporate in the narratives and (2) combine these elements in the narratives in a manner consistent with scholarship on their inter-relationships. The narratives are intended as a description of plausible future conditions at the level of large world regions that can serve as a basis for integrated scenarios of emissions and land use, as well as climate impact, adaptation and vulnerability analyses.”

In particular, they described the five pathways and developed the narrative storyline for each of them (Figure 8.118). SSP1 and SSP2 corresponds to smooth scenarios, which are characterized by
<figure><img src="image_400.png" /><figcaption>Figure 8.118: The shared socioeconomic pathways</figcaption></figure>

Source: O'Neill et al. (2017).

a relative easy transition toward a sustainable world. SSP1 implies a shift from historical patterns, less resource-intensive lifestyles, an improvement of natural resource efficiency and a global green growth strategy. SSP2 may be interpreted as a continuation of the economic development that was observed in the twentieth century. Emerging economies become rapidly developed and middle-income economies, global population growth is under control and moderate, but inequalities remains. SSP3 describes an increasingly individualist world with more nationalism and regional conflicts. In this situation, countries mainly focus on regional issues and less on international cooperation. This fragmented world implies high inequalities within and between countries. Inequality and poverty is also the central theme of SSP4, which corresponds to a multi-level society with a lot of issues (social cohesion, access to water, sanitation and health ware, education, etc). SSP5 corresponds to a high resilience of the economy, and consequently nothing is done to reduce inequalities and negative externalities. We observe that the SSPs can be related to ESG dimensions:

E The mitigation/adaptation trade-off is obviously an environmental issue, but the SSPs encompass other environmental narratives, e.g., land use, energy efficiency and green economy;

S The social dimension is the central theme of SSPs, and concerns demography, wealth, inequality & poverty, health, education, employment, and more generally the evolution of society. This explains that SSPs and SDGs are highly interconnected;

G Finally, the governance dimension is present though two major themes: international fragmentation or cooperation, and the political/economic system, including corruption, stability, rule of law, etc.
We notice that all these SSP themes can be found in a sovereign ESG framework<fnref n="127" />.

While O’Neill et al. (2014) have introduced the concept of SSP and given a very broad definition, the analysis proposed by O’Neill et al. (2017) is purely qualitative and descriptive. In fact, the first broad quantification of SSPs is done by Riahi et al. (2017), who have generated the five narrative scenarios with the following IAMs:

* SSP1: IMAGE (PBL)

* SSP2: MESSAGE-GLOBIOM (IIASA)

* SSP3: AIM/CGE (NIES)

* SSP4: GCAM (PNNL)

* SSP5: REMIND-MAGPIE (PIK) and WITCH-GLOBIOM (FEEM)

Results obtained by these authors are summarized in Figures 8.119–8.122. We only focus on baseline scenarios and we group the indicators into four main categories. In Figure 8.119, the demography projection shows three main pathways: population decline (SSP1, SSP5), population stabilization (SSP2, SSP4) and population growth (SSP5). If we consider the economic growth projections, we obtain another kind of breakdown (Figure 8.120). SSP5 corresponds to the highest GDP projections. It is following by SSP1 and SSP2, while SSP4 and SSP3 are the two laggard scenarios. In the last panel, we have reported the GDP per primary energy ratio. It shows that SSP1 has the highest efficiency. The case of SSP3 is interesting since the improvement of the energy efficiency follows the economic growth beyond 2040.

Figures 8.121 and 8.122 give an overview of the environmental narratives. As expected, SSP1 and SSP5 are the two extreme scenarios if we focus on climate change (temperature and CO₂ emissions). If we consider the land use, the opposite scenario become SSP1 and SSP3. In SSP1, the forest increases while cropland and pasture decrease or are stable. In SSP3, the forest dramatically decreases and there is 50% more cropland by 2100. Another interesting SSP indicator is the urban share (first panel in Figure 8.122). The urbanization of the world continues, but at different paces. By the end of century, the projected rate of urbanization reaches 60% for SSP3, 80% for SSP3 and more than 90% for the other SSPs. O’Neill et al. (2014) explained that “urbanization is constrained by slow economic growth and limited mobility across regions in SSP3, while urbanization is assumed to be rapid in both SSP1 and SSP5, which are associated with high income growth.”

To illustrate inequality issues, we can conduct the previous at a regional/country level. For example, Figure 8.123 shows demography and GDP/Capita growth projections in Africa and Asia. It is obvious that these two regions will not have the same economic development even if they follow the same shared socioeconomic pathway. This is particular true for the SSP4 scenario, for which we observe a population growth in Africa and a population decline in Asia. The impact of SSP4 on the GDP growth is also large in Africa with respect to the SSP2 scenario. The relationship between income inequalities and SSPs has been extensively studied by Rao et al. (2019). Using their dataset, we report in Figure 8.123 the boxplots of the Gini coefficient by 2100 for 184 countries, and compare these values with the 2020 estimates. We notice that income inequality within a country decreases for SSP1, SSP2 and SSP5 scenarios<fnref n="128" />, while it increases for SSP3 and SSP4. These results illustrate that inequalities across/within countries are important to understand the social cohesion dimension.

<footnote n="127">¹²⁷See Section 2.1.1 on page 54.</footnote>

<footnote n="128">¹²⁸Nevertheless, we also observe that there are more outlier countries for these scenarios.</footnote>
<figure><img src="image_401.png" /><figcaption>Figure 8.120: SSP economic projections</figcaption></figure>

<figure><img src="image_402.png" /></figure>
<figure><img src="image_403.png" /><figcaption>Figure 8.121: SSP environmental projections</figcaption></figure>

Source: https://tntcat.iiasa.ac.at/SspDb.

<figure><img src="image_404.png" /><figcaption>Figure 8.122: SSP land-use projections</figcaption></figure>

Source: https://tntcat.iiasa.ac.at/SspDb.
<figure><img src="image_405.png" /><figcaption>Figure 8.123: Example of SSP regional differences</figcaption></figure>

Source: https://tntcat.iiasa.ac.at/SspDb.

<figure><img src="image_406.png" /><figcaption>Figure 8.124: Gini coefficient projections by 2100</figcaption></figure>

Source: https://tntcat.iiasa.ac.at/SspDb.
Macroeconomic scenarios

The main objective of IAMs is to produce macroeconomic scenarios. For instance, the stylized simple models generate population, capital stock and GDP pathways, even though they are generally used only to compute the social cost of carbon. Indeed, these models are too simple to describe realistic economic scenarios. This is why these latter are simulated using complex IAMs. In the previous section dedicated to SSPs, we have already seen some output variables that are generated. In what follows, we focus on the NGFS scenarios, which have recently become a common standard. Today, the NGFS framework is an official tool that is used and recognized by central banks and supervisors, and also a reference point for financial institutions (investors, asset managers, and banks).

**NGFS scenarios framework** In 2020, the Network for Greening the Financial System (NGFS) has developed a set of scenarios for climate financial risk assessment<fnref n="129" />. The initial goal was to help central banks and supervisors exploring the possible impacts on the economy and the financial system. The NGFS scenarios framework consists in six scenarios located in the NGFS climate risk matrix given in Figure 8.125.

<figure><img src="image_407.png" /><figcaption>Figure 8.125: NGFS scenarios framework</figcaption></figure>

Source: Richters et al. (2022) & www.ngfs.net.

Below, we report the description of the six scenarios, which is done by NGFS (2022).

* Orderly scenarios assume climate policies are introduced early and become gradually more stringent. Both physical and transition risks are relatively subdued.

<footnote n="129">¹²⁹The third version of these scenarios was released in September 2022.</footnote>
#1 Net-zero 2050 limits global warming to 1.5°C through stringent climate policies and innovation, reaching global net zero CO2 emissions around 2050. Some jurisdictions such as the US, EU, UK, Canada, Australia and Japan reach net zero for all GHGs.

#2 Below 2°C gradually increases the stringency of climate policies, giving a 67% chance of limiting global warming to below 2°C.

* Disorderly scenarios explore higher transition risk due to policies being delayed or divergent across countries and sectors. For example, carbon prices are typically higher for a given temperature outcome.

#3 Divergent net zero reaches net zero around 2050 but with higher costs due to divergent policies introduced across sectors leading to a quicker phase out of oil use.

#4 Delayed transition assumes annual emissions do not decrease until 2030. Strong policies are needed to limit warming to below 2°C. Negative emissions are limited.

* Hot house world scenarios assume that some climate policies are implemented in some jurisdictions, but globally efforts are insufficient to halt significant global warming. The scenarios result in severe physical risk including irreversible impacts like sea-level rise.

#5 Nationally determined contributions (NDCs) includes all pledged targets even if not yet backed up by implemented effective policies.

#6 Current policies assumes that only currently implemented policies are preserved, leading to high physical risks.

In what follows, we use the following acronyms NZ, B2D, DNZ, D2D, NDC, CP when we refer to these 6 scenarios. Table 8.36 shows the impact of climate risk on each scenario. For the physical risk, NGFS uses both chronic and acute impacts. Chronic risks<fnref n="130" /> affect agriculture, labor productivity and natural capital. Acute risks or extreme weather events<fnref n="131" /> implies destruction, economic losses, lower insurance cover and business disruption. Transition risk takes several forms: policy risk, technology risk and coordination risk. The climate transition pathways are generated by three IAMs: GCAM, MESSAGEix-GLOBIOM and REMIND-MAgPIE, while the economic variables are produced by the NiGEM macroeconomic model<fnref n="132" /> in order to obtain harmonized metrics between scenarios and climate models. Moreover, a seventh scenario is produced by NiGEM without the IAM inputs and corresponds to the baseline (BSL) scenario, which does not take into account climate risk.

**Description of the database** The NGFS database<fnref n="133" /> is entirely described in Richters et al. (2022). It contains three EXCEL files. IAM_data contains the data generated by the three IAM models, Downscaled_data are the energy data produced by the previous models at the country level, while NiGEM_data corresponds to the data generated by the NiGEM model. In what follows, we only consider this last file. In order to select data, we first must specify the model and the scenario. The baseline scenario is obtained with the NiGEM model. For the six climate scenarios, we must choose one integrated assessment model (GCAM, MESSAGEix-GLOBIOM index[general]MESSAGE modelindex[general]GLOBIOM model or REMID-MAgPIE) and associate

<footnote n="130">¹³⁰Such as increased temperatures, rising sea levels, ocean acidification and seasonal precipitation.</footnote>

<footnote n="131">¹³¹Such as such as tropical cyclones, droughts, floods and wildfires.</footnote>

<footnote n="132">¹³²It is developed by the National Institute for Economic and Social Research (NIESR). A description can be found at www.niesr.ac.uk/nigem-macroeconomic-model.</footnote>

<footnote n="133">¹³³The economic scenarios can be downloaded at https://data.ene.iiasa.ac.at/ngfs.</footnote>
<table><caption>Table 8.36: Physical and transition risk level of NGFS scenarios</caption><thead><tr><th rowspan="2">Category</th><th rowspan="2">Scenario</th><th colspan="3">Physical risk</th><th colspan="3">Transition risk</th></tr><tr><th>Policy ambition</th><th>Policy reaction</th><th>Technology change</th><th>Carbon dioxide removal -</th><th>Regional policy variation+</th></tr></thead><tbody><tr><td rowspan="2">Orderly</td><td>Net Zero 2050</td><td>1.4°C</td><td>Immediate and smooth</td><td>Fast change</td><td>Medium-high use</td><td>Medium variation</td></tr><tr><td>Below 2°C</td><td>1.6°C</td><td>Immediate and smooth</td><td>Moderate change</td><td>Medium-high use</td><td>Low variation</td></tr><tr><td rowspan="2">Disorderly</td><td>Divergent Net Zero</td><td>1.4°C</td><td>Immediate but divergent across sectors</td><td>Fast change</td><td>Low-medium use</td><td>Medium variation</td></tr><tr><td>Delayed Transition</td><td>1.6°C</td><td>Delayed</td><td>Slow / Fast change</td><td>Low-medium use</td><td>High variation</td></tr><tr><td rowspan="2">Hot house world</td><td>Nationally Determined Contributions (NDCs)</td><td>2.6°C</td><td>NDCs</td><td>Slow change</td><td>Low-medium use</td><td>Medium variation</td></tr><tr><td>Current Policies</td><td>3°C +</td><td>Non-current policies</td><td>Slow change</td><td>Low use</td><td>Low variation</td></tr></tbody></table>

Lower risk, moderate risk and higher risk

Color coding indicates whether the characteristic makes the scenario more or less severe from a macro-financial risk perspective.

Source: NGFS (2022) & www.ngfs.net.
<table><caption>Table 8.37: NGFS database — Economic block</caption><thead><tr><th colspan="3">Scenarios</th></tr></thead><tbody><tr><td>1. Net Zero 2050 (NZ)</td><td>5. Notionally Determined Contribution (NDC)</td></tr><tr><td>2. Below 2°C (B2D)</td><td>6. Current Policies (CP)</td></tr><tr><td>3. Divergent Net Zero (DNZ)</td><td>7. Baseline (BSL)</td></tr><tr><td>4. Delayed Transition (DT)</td><td></td></tr></tbody></table>

<table><thead><tr><th colspan="2">Models</th></tr></thead><tbody><tr><td>GCAM 5.3+</td><td>NGFS</td></tr><tr><td>MESSAGEix-GLOBIOM 1.1-M-R12</td><td></td></tr><tr><td>NiGEM NGFS v1.22</td><td></td></tr><tr><td>REMIND-MAgPIE 3.0-4.4</td><td></td></tr><tr><td>NiGEM NGFS v1.22 / Downscaling</td><td></td></tr><tr><td>GCAM 5.3+</td><td>NGFS</td></tr><tr><td>MESSAGEix-GLOBIOM 1.1-M-R12</td><td></td></tr><tr><td>REMIND-MAgPIE 3.0-4.4</td><td></td></tr><tr><td>REMIND-MAgPIE 3.0-4.4</td><td></td></tr><tr><td>IntegratedPhysicalDamages (95th-high)</td><td></td></tr><tr><td>IntegratedPhysicalDamages (median)</td><td></td></tr></tbody></table>

<table><thead><tr><th colspan="2">Energy variables</th></tr></thead><tbody><tr><td>Coal price</td><td></td></tr><tr><td>Gas price</td><td></td></tr><tr><td>Oil price</td><td></td></tr><tr><td>Quarterly consumption of coal</td><td></td></tr><tr><td>Quarterly consumption of gas</td><td></td></tr><tr><td>Quarterly consumption of non-carbon</td><td></td></tr><tr><td>Quarterly consumption of oil</td><td></td></tr><tr><td>Total energy consumption</td><td></td></tr></tbody></table>

<table><thead><tr><th colspan="2">Economic variables</th></tr></thead><tbody><tr><td>Central bank intervention rate</td><td></td></tr><tr><td>Consumption (private sector)</td><td></td></tr><tr><td>Domestic demand</td><td></td></tr><tr><td>Effective exchange rate</td><td></td></tr><tr><td>Equity prices</td><td></td></tr><tr><td>Exchange rate</td><td></td></tr><tr><td>Exports (goods and services excluding MTIC)</td><td></td></tr><tr><td>Exports (goods and services)</td><td></td></tr><tr><td>Government consumption</td><td></td></tr><tr><td>Gross Domestic Product (GDP)</td><td></td></tr><tr><td>Gross domestic income</td><td></td></tr><tr><td>Gross operating surplus</td><td></td></tr><tr><td>House prices (residential)</td><td></td></tr><tr><td>Imports (goods and services excluding MTIC)</td><td></td></tr><tr><td>Imports (goods and services)</td><td></td></tr><tr><td>Inflation rate</td><td></td></tr><tr><td>Investment (government)</td><td></td></tr><tr><td>Investment (private sector)</td><td></td></tr><tr><td>Long term interest rate</td><td></td></tr><tr><td>Productivity (output per hour worked)</td><td></td></tr><tr><td>Real personal disposable income</td><td></td></tr><tr><td>Trend output for capacity utilisation</td><td></td></tr><tr><td>Unemployment rate</td><td></td></tr></tbody></table>
it with the NiGEM model. Once the model and the scenario are chosen, we can filter the data by region and/or variable. We have reported the name of the variables in Table 8.37. They are of two types. Energy variables concern price and consumption of coal, gas, oil and non-carbon (or renewables/green energy). Economic variables cover production, consumption, income, investment, interest rates, inflation and unemployment.

**Empirical results** We report the impact of NGFS scenarios on GDP in Tables 8.38, 8.39 and 8.40 when the time horizon is set to 2050 and the region corresponds to the world. The results depend on the selected model. For instance, if we consider the B2D scenario and the chronic physical risk, the GDP impact is equal to -3.09% with GCAM, -2.05% with MESSAGEix and -2.24% with REMIND. For each table, the impact is related to the selected scenario and the selected risk. We notice that the largest loss is obtained with the combined risk (physical and transition), but there is no individual risk that dominates the other. For example, the chronic physical risk has more impact for the B2D, DT, NDC and NZ scenarios while the transition risk dominates for the DNZ scenario. We also observe that the impacts of physical and transition risks are almost additive when they are combined. Another interesting remark concerns the magnitude of the GDP loss. If we consider GCAM and REMIND, there is no obvious ranking in terms of scenario severity. This is not the case of the MESSAGEix model, which assumes that DNZ and DT are the two most severe scenarios, followed by four other scenarios.

<table><caption>Table 8.38: Impact of climate change on the GDP loss by 2050 (GCAM)</caption><thead><tr><th>Risk</th><th>B2D</th><th>CP</th><th>DNZ</th><th>DT</th><th>NDC</th><th>NZ</th></tr></thead><tbody><tr><td>Chronic physical risk</td><td>-3.09</td><td>-5.64</td><td>-2.35</td><td>-3.28</td><td>-5.15</td><td>-2.56</td></tr><tr><td>Transition risk</td><td>-0.75</td><td></td><td>-3.66</td><td>-1.78</td><td>-0.89</td><td>-0.88</td></tr><tr><td>Combined risk</td><td>-3.84</td><td>-5.64</td><td>-6.00</td><td>-5.05</td><td>-6.03</td><td>-3.44</td></tr><tr><td>Combined + business confidence</td><td></td><td></td><td>-6.03</td><td>-5.09</td><td></td><td></td></tr></tbody></table>

Source: https://data.ene.iiasa.ac.at/ngfs.

<table><caption>Table 8.39: Impact of climate change on the GDP loss by 2050 (MESSAGEix-GLOBIOM)</caption><thead><tr><th>Risk</th><th>B2D</th><th>CP</th><th>DNZ</th><th>DT</th><th>NDC</th><th>NZ</th></tr></thead><tbody><tr><td>Chronic physical risk</td><td>-2.05</td><td>-5.26</td><td>-1.55</td><td>-2.64</td><td>-4.78</td><td>-1.59</td></tr><tr><td>Transition risk</td><td>-1.46</td><td></td><td>-10.00</td><td>-10.77</td><td>-1.39</td><td>-3.26</td></tr><tr><td>Combined risk</td><td>-3.51</td><td>-5.26</td><td>-11.53</td><td>-13.37</td><td>-6.16</td><td>-4.84</td></tr><tr><td>Combined + business confidence</td><td></td><td></td><td>-11.57</td><td>-13.40</td><td></td><td></td></tr></tbody></table>

Source: https://data.ene.iiasa.ac.at/ngfs.

<table><caption>Table 8.40: Impact of climate change on the GDP loss by 2050 (REMIND-MAgPIE)</caption><thead><tr><th>Risk</th><th>B2D</th><th>CP</th><th>DNZ</th><th>DT</th><th>NDC</th><th>NZ</th></tr></thead><tbody><tr><td>Chronic physical risk</td><td>-2.24</td><td>-6.05</td><td>-1.67</td><td>-2.65</td><td>-5.41</td><td>-1.76</td></tr><tr><td>Transition risk</td><td>-0.78</td><td></td><td>-3.01</td><td>-1.95</td><td>-0.33</td><td>-1.46</td></tr><tr><td>Combined risk</td><td>-3.02</td><td>-6.05</td><td>-4.68</td><td>-4.59</td><td>-5.73</td><td>-3.21</td></tr><tr><td>Combined + business confidence</td><td></td><td></td><td>-4.70</td><td>-4.63</td><td></td><td></td></tr></tbody></table>

Source: https://data.ene.iiasa.ac.at/ngfs.
By using the MESSAGEix-GLOBIOM model and selecting the combined risk, we obtain the results per country and region in Table 8.41. On average, the GDP loss by 2050 is between -3.51% (current policies) and -13.37% (delayed transition). This large difference is also observed when we compare countries and region. For example, if we focus on the NZ scenario, the impact is negative and equal to -17.11% for Russia while it is positive for developing Europe, South Korea and Switzerland. Therefore, we conclude that there is a high heterogeneity between countries (see also Figures 8.126 and 8.127).

<table><caption>Table 8.41: Impact of climate change on the GDP loss by 2050 (MESSAGEix-GLOBIOM)</caption><thead><tr><th>Region</th><th>B2D</th><th>CP</th><th>DNZ</th><th>DT</th><th>NDC</th><th>NZ</th></tr></thead><tbody><tr><td>Africa</td><td>-13.58</td><td>-7.50</td><td>-27.35</td><td>-29.37</td><td>-11.78</td><td>-18.36</td></tr><tr><td>Asia</td><td>-1.50</td><td>-7.29</td><td>-5.44</td><td>-8.76</td><td>-6.78</td><td>-1.38</td></tr><tr><td>Australia</td><td>-4.11</td><td>-3.90</td><td>-11.03</td><td>-11.74</td><td>-5.77</td><td>-5.19</td></tr><tr><td>Brazil</td><td>-4.43</td><td>-5.92</td><td>-13.15</td><td>-15.90</td><td>-6.67</td><td>-6.65</td></tr><tr><td>Canada</td><td>-1.02</td><td>-2.37</td><td>-15.07</td><td>-18.12</td><td>-4.33</td><td>-4.87</td></tr><tr><td>China</td><td>-2.33</td><td>-4.97</td><td>-5.13</td><td>-6.73</td><td>-4.67</td><td>-2.76</td></tr><tr><td>Developing Europe</td><td>-0.28</td><td>-3.11</td><td>-0.56</td><td>-7.38</td><td>-2.73</td><td>0.39</td></tr><tr><td>Europe</td><td>-1.02</td><td>-2.84</td><td>-9.64</td><td>-11.02</td><td>-4.01</td><td>-1.62</td></tr><tr><td>France</td><td>-1.15</td><td>-2.80</td><td>-8.35</td><td>-9.48</td><td>-3.68</td><td>-1.56</td></tr><tr><td>Germany</td><td>-0.77</td><td>-2.38</td><td>-8.58</td><td>-9.38</td><td>-3.63</td><td>-1.21</td></tr><tr><td>India</td><td>-3.45</td><td>-8.61</td><td>-16.43</td><td>-17.74</td><td>-8.71</td><td>-3.86</td></tr><tr><td>Italy</td><td>-0.15</td><td>-3.69</td><td>-9.23</td><td>-12.88</td><td>-4.85</td><td>-0.89</td></tr><tr><td>Japan</td><td>-1.26</td><td>-4.14</td><td>-7.16</td><td>-10.05</td><td>-4.61</td><td>-1.40</td></tr><tr><td>Latam</td><td>-4.35</td><td>-6.10</td><td>-12.70</td><td>-14.58</td><td>-6.97</td><td>-5.74</td></tr><tr><td>Middle East</td><td>-9.97</td><td>-7.98</td><td>-22.03</td><td>-21.96</td><td>-10.28</td><td>-15.24</td></tr><tr><td>Russia</td><td>-12.18</td><td>-2.26</td><td>-23.46</td><td>-23.80</td><td>-7.54</td><td>-17.11</td></tr><tr><td>South Africa</td><td>-2.02</td><td>-5.06</td><td>-7.24</td><td>-9.16</td><td>-5.38</td><td>-3.04</td></tr><tr><td>South Korea</td><td>0.11</td><td>-3.49</td><td>-3.23</td><td>-7.57</td><td>-3.33</td><td>0.12</td></tr><tr><td>Spain</td><td>-2.41</td><td>-3.81</td><td>-12.49</td><td>-12.89</td><td>-5.41</td><td>-3.30</td></tr><tr><td>Switzerland</td><td>2.32</td><td>-2.25</td><td>-9.47</td><td>-10.35</td><td>-2.18</td><td>2.30</td></tr><tr><td>United Kingdom</td><td>-0.86</td><td>-1.90</td><td>-6.50</td><td>-8.05</td><td>-2.56</td><td>-1.33</td></tr><tr><td>United States</td><td>-2.67</td><td>-4.38</td><td>-15.37</td><td>-17.66</td><td>-6.31</td><td>-4.36</td></tr><tr><td>World</td><td>-3.51</td><td>-5.26</td><td>-11.53</td><td>-13.37</td><td>-6.16</td><td>-4.84</td></tr></tbody></table>

Source: https://data.ene.iiasa.ac.at/ngfs.

The previous analysis can be extended to other economic variables. In Figure 8.128, we show the impact on inflation, unemployment, private investment, productivity, equity prices and public investment in the case of China. With respect to the baseline scenario, the delayed transition may induce more inflation (up to 6%) and unemployment (up to 0.8%) between 2030 and 2040, and less public and private investments. It may also strongly impact the equity market and the financial performance of stocks. Finally, the economy may be less productive. We have reported a similar analysis in the case of United States, France and United Kingdom in Figures 8.129, 8.130 and 8.131. Again, we notice that the impact on the economy depends on the country or region. Even if there is no winner for some scenarios, some countries will lose less than others. Climate change will then create new inequalities between regions. In particular, two regions seems to be especially vulnerable: Africa and the Middle East.
<figure><img src="image_408.png" /><figcaption>Figure 8.126: GDP impact by 2050 (% change from baseline) — Delayed transition scenario</figcaption></figure>
<figure><img src="image_409.png" /></figure>
<figure><img src="image_410.png" /><figcaption>Figure 8.128: Impact of climate scenarios on economics (% change from baseline) — China</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/ngfs & Author's calculations.

<figure><img src="image_411.png" /><figcaption>Figure 8.129: Impact of climate scenarios on economics (% change from baseline) — United States</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/ngfs & Author's calculations.
<figure><img src="image_412.png" /><figcaption>Figure 8.130: Impact of climate scenarios on economics (% change from baseline) — France</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/ngfs & Author's calculations.

<figure><img src="image_413.png" /><figcaption>Figure 8.131: Impact of climate scenarios on economics (% change from baseline) — United Kingdom</figcaption></figure>

Source: https://data.ene.iiasa.ac.at/ngfs & Author's calculations.
## 8.4 Environmentally-extended input-output model

An input-output model is a mathematical tool that represents the macroeconomic relationships between different units or industries. It can be used to model the supply chain of a product, the sectoral structure of an economy, the production network of a country or the foreign exchange between regions. Most of these models are monetary in nature. When applied to economic transactions between different sectors, they can be used to calculate the contribution or value added of each sector to the final output of an economy. Environmentally-extended input-output (EEIO) analysis is an extension of the input-output framework to include environmental externalities such as pollution and greenhouse gas emissions. In particular, we can use EEIO models to estimate upstream Scope 3 emissions, define a stress testing framework or calculate the value-at-risk of a portfolio (Desnos et al., 2023).

### 8.4.1 Input-output analysis

The input-output model was first introduced by Leontief (1936, 1941). It quantifies the interdependencies between different sectors in a single or multi-regional economy, based on the product flows between sectors (Miller and Blair, 2009). The underlying idea is to model the linkages between sectors and to describe the relationships from each of the producer/seller sectors to each of the purchaser/buyer sectors.

**The demand-pull quantity model**

Following Miller and Blair (2009), we consider $n$ different sectors and we note $Z_{i,j}$ the value of transactions from sector $i$ to sector $j$. We can interpret $Z_{i,j}$ in different ways:

1. It is the output that sector $i$ sells to sector $j$;

2. It is the input of sector $i$ required by sector $j$ for its production (or output).

Let $y_i$ be the final demand for products sold by sector $i$. This final demand is made up of the external sales to households, government purchases, and demand resulting from investment capacity and foreign trade. Then, the total production $x_i$ of sector $i$ is equal to:

$$ \underbrace{x_i}_{\text{Supply}} = \underbrace{\sum_{j=1}^{n} Z_{i,j} + y_i}_{\text{Demand}} \qquad (8.28) $$

In this equation, $x_i$ and $\sum_{j=1}^{n} Z_{i,j} + y_i$ are the supply and demand related to products of sector $i$, and $z_i = \sum_{j=1}^{n} Z_{i,j}$ represents intermediate demand. The interdependence relation between sectors is usually expressed as a ratio between $Z_{i,j}$ and $x_j$:

$$ A_{i,j} = \frac{Z_{i,j}}{x_j} $$

Let $A = (A_{i,j}) = Z \text{diag}(x)^{-1}$ be the input-output matrix of the technical coefficients $A_{i,j}$. In a matrix form, we have $x = Z\mathbf{1}_n + y$ and $Z \equiv A \text{diag}(x) = A \odot x^\top$, and we deduce that:

$$ x = Ax + y $$

where $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$. Assuming that final demand is exogenous, technical coefficients are fixed and output is endogenous, we obtain:

$$ x = (I_n - A)^{-1} y \qquad (8.29) $$
$\mathcal{L} = (I_n - A)^{-1}$ is known as the Leontief inverse (or multiplier) matrix and represents the amount of total output from sector $i$ that is required by sector $j$ to satisfy its final demand. Equation (8.29) describes a *demand-pull quantity* model.

**Example 29** We consider the basic economy given below:

<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">To</th><th rowspan="2">Final Demand y</th><th rowspan="2">Total Output x</th></tr><tr><th>Energy</th><th>Materials</th><th>Industrials</th><th>Services</th></tr></thead><tbody><tr><td rowspan="4">From</td><td>Energy</td><td>500</td><td>800</td><td>1 600</td><td>1 250</td><td>850</td><td>5 000</td></tr><tr><td>Materials</td><td>500</td><td>400</td><td>1 600</td><td>625</td><td>875</td><td>4 000</td></tr><tr><td>Industrials</td><td>250</td><td>800</td><td>2 400</td><td>1 250</td><td>3 300</td><td>8 000</td></tr><tr><td>Services</td><td>100</td><td>200</td><td>800</td><td>4 375</td><td>7 025</td><td>12 500</td></tr></tbody></table>

This basic economy has four sectors: energy, materials, industrials and services. In this economy, businesses in the energy sector buy $500 of goods and services from other businesses in the energy sector, $500 of goods and services from the materials sector, $250 of goods and services from the industrials sector, and $100 of goods and services from the services sector. The final demand for goods and services produced in the energy sector is equal to $850, while the total output of this sector is equal to $5 000.

We deduce that the matrix of technical coefficients is equal to:

$$A = Z \text{diag}(x)^{-1} = \begin{pmatrix} 10\% & 20\% & 20\% & 10\% \\ 10\% & 10\% & 20\% & 5\% \\ 5\% & 20\% & 30\% & 10\% \\ 2\% & 5\% & 10\% & 35\% \end{pmatrix}$$

It follows that the multiplier matrix is equal to:

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

We verify that:

$$x = \mathcal{L}y = \begin{pmatrix} 1.1881 & 0.3894 & 0.4919 & 0.2884 \\ 0.1678 & 1.2552 & 0.4336 & 0.1891 \\ 0.1430 & 0.4110 & 1.6303 & 0.3044 \\ 0.0715 & 0.1718 & 0.2993 & 1.6087 \end{pmatrix} \begin{pmatrix} 850 \\ 875 \\ 3 300 \\ 7 025 \end{pmatrix} = \begin{pmatrix} 5 000 \\ 4 000 \\ 8 000 \\ 12 500 \end{pmatrix}$$

Suppose we have a variation in final demand. From Equation (8.29), we obtain $\Delta x = \mathcal{L} \Delta y$. For instance, an increase of $10 in the final demand for services implies:

$$\Delta x = \mathcal{L} \Delta y = \begin{pmatrix} 1.1881 & 0.3894 & 0.4919 & 0.2884 \\ 0.1678 & 1.2552 & 0.4336 & 0.1891 \\ 0.1430 & 0.4110 & 1.6303 & 0.3044 \\ 0.0715 & 0.1718 & 0.2993 & 1.6087 \end{pmatrix} \begin{pmatrix} 0 \\ 0 \\ 0 \\ 10 \end{pmatrix} = \begin{pmatrix} 2.8842 \\ 1.8907 \\ 3.0444 \\ 16.0872 \end{pmatrix}$$

This means that energy production increases by $2.88, materials production increases by $1.89, and so on.
The cost-push price model

Let $m$ be number of primary inputs (e.g., labor, capital, etc.). Let $V = (V_{k,j})$ be the value added matrix where $V_{k,j}$ represents the amount of primary input $k$ required to produce the output of sector $j$. Since the total input of each sector is equal to its total output, we have $x_j = \sum_{i=1}^{n} Z_{i,j} + \sum_{k=1}^{m} V_{k,j}$. Therefore, $v_j = \sum_{k=1}^{m} V_{k,j} = x_j - \sum_{i=1}^{n} Z_{i,j}$ represents the other expenditure of sector $j$ or the total primary inputs used in sector $j$. We have $v = (v_1, \dots, v_n) = V^\top \mathbf{1}_m$. Let $p = (p_1, \dots, p_n)$ and $\psi = (\psi_1, \dots, \psi_m)$ be the vector of sector prices and primary inputs. $p_j$ and $\psi_k$ are then the prices per unit of sector $j$ and primary input $k$. As in the quantity model, the interdependence relationship between primary inputs and sectors is expressed as the ratio between $V_{k,j}$ and $x_j$:

$$B_{k,j} = \frac{V_{k,j}}{x_j}$$

We denote the input-output matrix of the technical coefficients by $B = (B_{k,j}) \equiv V \text{diag}(x)^{-1}$. Following Gutierrez (2008), the value of the output must be equal to the value of its inputs:

$$\underbrace{p_j x_j}_{\text{Value of the output}} = \underbrace{\sum_{i=1}^{n} Z_{i,j} p_i + \sum_{k=1}^{m} V_{k,j} \psi_k}_{\text{Value of the inputs}}$$

We deduce that:

$$\begin{aligned} p_j &= \sum_{i=1}^{n} \frac{Z_{i,j}}{x_j} p_i + \sum_{k=1}^{m} \frac{V_{k,j}}{x_j} \psi_k \\ &= \sum_{i=1}^{n} A_{i,j} p_i + \sum_{k=1}^{m} B_{k,j} \psi_k \end{aligned}$$

In a matrix form, we get $p = A^\top p + B^\top \psi$. $v = B^\top \psi$ is the vector of value added ratios. Finally, the output prices are equal to:

$$p = (I_n - A^\top)^{-1} v \qquad (8.30)$$

$\tilde{\mathcal{L}} = (I_n - A^\top)^{-1}$ is known as the dual inverse matrix and represents the amount of costs from sector $j$ that are passed on to sector $i$. Equation (8.30) describes a cost-push price model. By adding the income identity<fnref n="134" />, Gutierrez (2008) proposed the following complete version of the full basic input-output model:

$$\left\{ \begin{array}{l} x = (I_n - A)^{-1} y \\ v = V^\top \mathbf{1}_m \\ v = B^\top \psi \\ p = (I_n - A^\top)^{-1} v \\ x^\top v = y^\top p \end{array} \right. \qquad (8.31)$$

It mixes both the quantity and price models. In this system, $A$, $B$ and $V$ are the model parameters, $\psi$, $v$ and $y$ are the exogenous variables, and $x$ and $p$ are the endogenous variables. By changing the model parameters or the exogenous variables, we can measure the impacts $\Delta y$ and $\Delta v$ on the quantities and prices in the economy.

**Remark 88** The previous analysis was derived for physical input-output tables, where flows are expressed in product units. However, the analysis remains valid when monetary input-output tables

<footnote n="134">¹³⁴Since the input-output analysis assumes an equilibrium model, the total value of the revenues $y^\top p$ is equal to the total value of costs $x^\top v$.</footnote>
are considered. The only difference is the calculation of the primary cost vector. In a monetary input-output analysis, $\psi$ is set by construction to $\mathbf{1}_m$, which means that $v = B^\top \mathbf{1}_m = (v_1/x_1, \dots, v_n/x_n)$ and $p = \mathbf{1}_n$ (Miller and Blair, 2009, Section 2.6.3, pages 43-44).

**Example 30** We consider the previous basic economy. We assume that the value added is made up of two items: labour and capital. We have:

<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">To</th><th rowspan="2">Final Demand y</th><th rowspan="2">Total Output x</th></tr><tr><th>Energy</th><th>Materials</th><th>Industrials</th><th>Services</th></tr></thead><tbody><tr><td rowspan="4">From</td><td>Energy</td><td>500</td><td>800</td><td>1 600</td><td>1 250</td><td>850</td><td>5 000</td></tr><tr><td>Materials</td><td>500</td><td>400</td><td>1 600</td><td>625</td><td>875</td><td>4 000</td></tr><tr><td>Industrials</td><td>250</td><td>800</td><td>2 400</td><td>1 250</td><td>3 300</td><td>8 000</td></tr><tr><td>Services</td><td>100</td><td>200</td><td>800</td><td>4 375</td><td>7 025</td><td>12 500</td></tr><tr><td rowspan="2">Value added</td><td>Labour</td><td>3 000</td><td>800</td><td>1 000</td><td>3 000</td><td></td><td></td></tr><tr><td>Capital</td><td>650</td><td>1 000</td><td>600</td><td>2 000</td><td></td><td></td></tr><tr><td colspan="2">Income</td><td>5 000</td><td>4 000</td><td>8 000</td><td>12 500</td><td></td><td></td></tr></tbody></table>

The energy sector has a labour consumption of $3 000 and a total output of $5 000. By construction, the income of the sector is equal to the output of the sector. We deduce that the capital item (capital interest and net profit) is equal to $650.

We have:

$$V = \begin{pmatrix} 3\,000 & 800 & 1\,000 & 3\,000 \\ 650 & 1\,000 & 600 & 2\,000 \end{pmatrix}$$

and:

$$v = V^\top \mathbf{1}_2 = x - Z^\top \mathbf{1}_4 = \begin{pmatrix} 3\,650 \\ 1\,800 \\ 1\,600 \\ 5\,000 \end{pmatrix}$$

We deduce that:

$$B = \begin{pmatrix} 0.60 & 0.20 & 0.125 & 0.24 \\ 0.13 & 0.25 & 0.075 & 0.16 \end{pmatrix}$$

Since we have a monetary input-output table, the labour and capital costs are equal to the monetary unit, i.e., one dollar ($\psi_1 = \psi_2 = 1$). It follows that:

$$v = B^\top \mathbf{1}_2 = \begin{pmatrix} 0.73 \\ 0.45 \\ 0.20 \\ 0.40 \end{pmatrix}$$

The interpretation is as follows. For the energy sector, intermediate consumption is 27% and value added is 73%. For the other three sectors, the value added ratios are 45%, 20% and 40% respectively. Finally, we obtain:

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

$$p = \tilde{\mathcal{L}}v = \mathbf{1}_4$$

We check that the prices in a monetary input-output table are normalized to one dollar. In this basic economy, the total final demand $y^\top p$ is equal to $12050$, which is equal to the total value added $x^\top v$. Suppose we have a variation in the labour/capital costs. From Equation (8.31), we obtain $\Delta p = \tilde{\mathcal{L}}\Delta v$. For example, a 10% increase in costs in the energy sector means that the price of energy increases by 11.88%, the price of materials by 3.89%, and so on:

$$\Delta p = \tilde{\mathcal{L}}\Delta v = \begin{pmatrix} 1.1881 & 0.1678 & 0.1430 & 0.0715 \\ 0.3894 & 1.2552 & 0.4110 & 0.1718 \\ 0.4919 & 0.4336 & 1.6303 & 0.2993 \\ 0.2884 & 0.1891 & 0.3044 & 1.6087 \end{pmatrix} \begin{pmatrix} 0.10 \\ 0.00 \\ 0.00 \\ 0.00 \end{pmatrix} = \begin{pmatrix} 11.88\% \\ 3.89\% \\ 4.92\% \\ 2.88\% \end{pmatrix}$$

The definition of a price index is:

$$\mathcal{PI} = \sum_{j=1}^{n} \alpha_{j} p_{j} = \alpha^{\top} p$$

where $\alpha = (\alpha_1, \dots, \alpha_n)$ is the weights of the basket of items. We deduce that the inflation rate between two dates $t_0$ and $t_1$ is:

$$\pi = \frac{\mathcal{PI}(t_1) - \mathcal{PI}(t_0)}{\mathcal{PI}(t_0)} = \frac{\alpha^\top (I_n - A^\top)^{-1} \Delta v(t_0, t_1)}{\alpha^\top (I_n - A^\top)^{-1} v(t_0)}$$

We can simplify this formula because $p(t_0) = (I_n - A^\top)^{-1} v(t_0) = \mathbf{1}_n$ and $\mathbf{1}_n^\top \alpha = 1$. Finally, we have:

$$\pi = \alpha^{\top} (I_n - A^{\top})^{-1} \Delta v \qquad (8.32)$$

In general, we define two price indexes: the producer price index (PPI) where the basket weights are proportional to the output ($\alpha_j \propto x_j$) and the consumer price index (CPI) where the basket weights are proportional to the final demand ($\alpha_j \propto y_j$). In the latter case we obtain:

$$\text{CPI} = \frac{y^{\top} (I_n - A^{\top})^{-1} \Delta v}{\mathbf{1}_n^{\top} y} = \frac{\Delta v^{\top} (I_n - A)^{-1} y}{\mathbf{1}_n^{\top} y} = \frac{\Delta v^{\top} x}{\mathbf{1}_n^{\top} y} = \frac{\sum_{j=1}^{n} x_j \Delta v_j}{\sum_{j=1}^{n} y_j}$$

Looking at the previous example, a 10% increase in energy costs will cause the producer price index to rise by 5.10% and the consumer price index by 4.15%.

## Mathematical properties

Since we have $(I_n - A^\top)^{-1} = ((I_n - A)^\top)^{-1} = ((I_n - A)^{-1})^\top$, we deduce that $\tilde{\mathcal{L}} = \mathcal{L}^\top$. So we can study $\mathcal{L}$ or $\tilde{\mathcal{L}}$ indifferently, because they share the same properties. The matrix $\mathcal{L}$ admits the following Neumann series:

$$\begin{aligned} \mathcal{L} &= (I_n - A)^{-1} \\ &= I_n + A + A^2 + A^3 + \dots \\ &= \sum_{k=0}^{\infty} A^k \end{aligned} \qquad (8.33)$$
Box 8.19: Neumann series

A Neumann series is a mathematical series of the form $S := \sum_{k=0}^{\infty} T^k$ where $T$ is a bounded linear operator and $T^k = T^{k-1} \circ T = T \circ T^{k-1}$. If the Neumann series converges in the operator norm, then $\mathrm{Id} - T$ is invertible and its inverse is the Neumann series:

$$ (\mathrm{Id} - T)^{-1} = S = \sum_{k=0}^{\infty} T^k $$

where $\mathrm{Id}$ is the identity operator. A sufficient condition is that the spectral radius of $T$ is less than one. If $A$ is an invertible matrix, we conclude that:

$$ (I_n - A)^{-1} = \sum_{k=0}^{\infty} A^k $$

and $\lim_{k \to 0} \|A^k\| = 0$. This result generalizes the geometric series:

$$ \frac{1}{1-x} = 1 + x + x^2 + \dots \quad \text{where } |x| < 1 $$

Since $A$ is a doubly substochastic matrix$^{135}$, the eigendecomposition of $A$ is $A = V\Lambda V^{-1}$ where $V$ is the matrix of eigenvectors, $\Lambda = \mathrm{diag}(\lambda_1, \dots, \lambda_n)$ and $|\lambda_i| \le 1$. If we assume that $A$ is irreducible, the Perron-Frobenius theorem states that the spectral radius $\rho(A)$ is strictly lower than 1 and corresponds to the largest positive eigenvalue $\lambda_1$. This implies that all the eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$ satisfies $|\lambda_i| < 1$. As we have $A^k = V\Lambda^k V^{-1}$, we deduce that $\lim_{k \to \infty} A^k = V\Lambda^k V^{-1} = \mathbf{0}_{n,n}$. The Neumann series $\sum_{k=0}^{\infty} A^k$ converges then to a finite matrix, which implies that the multiplier matrix $\mathcal{L}$ is nonsingular.

From Equation (8.33), we deduce that $\mathcal{L} \succeq I_n$ because $A^k \succeq \mathbf{0}_{n,n}$ for $k \ge 1$ (Property NN4 on page 1148). We also get the following decomposition:

$$ \begin{aligned} x &= \sum_{k=0}^{\infty} A^k y \\ &= y + Ay + A^2 y + \dots \\ &= \sum_{k=0}^{\infty} y_{(k)} \end{aligned} $$

where $y_{(0)} = y$ is the final demand (or zero-tier intermediate demand), $y_{(1)} = Ay$ is the first-tier intermediate demand, $y_{(2)} = A^2 y$ is the second-tier intermediate demand, and $y_{(k)} = A^k y$ is the $k^{\mathrm{th}}$-tier intermediate demand. Furthermore, we have:

$$ \frac{\partial x}{\partial y} = (I_n - A)^{-1} \equiv \mathcal{L} \succeq I_n $$

We can better understand why the matrix $\mathcal{L}$ is also called the multiplier matrix because it is an analogy to Keynesian consumption theory and the effect of a change in aggregate demand on the output.

<footnote n="135">135 See Appendix A.1.4 on page 1148.</footnote>
<table><caption>Table 8.42: Tier decomposition of the output (Example 29)</caption><thead><tr><th>k</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td rowspan="4">y(k)</td><td>Energy</td><td>850.0</td><td>1 622.5</td><td>1 066.0</td><td>630.8</td><td>362.1</td><td>205.2</td></tr><tr><td>Materials</td><td>875.0</td><td>1 183.8</td><td>805.1</td><td>486.3</td><td>282.1</td><td>160.7</td></tr><tr><td>Industrials</td><td>3 300.0</td><td>1 910.0</td><td>1 175.8</td><td>695.1</td><td>400.0</td><td>227.1</td></tr><tr><td>Services</td><td>7 025.0</td><td>2 849.5</td><td>1 280.0</td><td>627.1</td><td>325.9</td><td>175.4</td></tr><tr><td rowspan="4">y(0:k)</td><td>Energy</td><td>850.0</td><td>2 472.5</td><td>3 538.5</td><td>4 169.2</td><td>4 531.3</td><td>4 736.5</td></tr><tr><td>Materials</td><td>875.0</td><td>2 058.8</td><td>2 863.9</td><td>3 350.1</td><td>3 632.2</td><td>3 792.9</td></tr><tr><td>Industrials</td><td>3 300.0</td><td>5 210.0</td><td>6 385.8</td><td>7 080.9</td><td>7 480.9</td><td>7 708.0</td></tr><tr><td>Services</td><td>7 025.0</td><td>9 874.5</td><td>11 154.5</td><td>11 781.6</td><td>12 107.5</td><td>12 283.0</td></tr></tbody></table>

In Tables 8.42 and 8.43, we report the values of intermediate demand $y_{(k)}$ in the case of Example 29. It can be seen that the first-tier intermediate demand is larger than final demand when we look at the energy and materials sector. For instance, we have $y_{(1),1} = 1622.5$ but $y_{(0),1} = y_1 = 850$. In the case of industrials and services, we obtain the opposite effect. We also check that the k-tier intermediate demand converges to zero. We have also reported the cumulative demand of the first k tiers: $y_{(0:k)} = \sum_{h=0}^{k} y_{(k)}$. This illustrates that $\lim_{k \to \infty} y_{(0:k)} = x$.

<table><caption>Table 8.43: Tier decomposition of the output (Example 29)</caption><thead><tr><th>k</th><th>10</th><th>20</th><th>25</th></tr></thead><tbody><tr><td rowspan="4">y(k)</td><td>Energy</td><td>11.45</td><td>0.03</td><td>0.00</td></tr><tr><td>Materials</td><td>9.01</td><td>0.03</td><td>0.00</td></tr><tr><td>Industrials</td><td>12.69</td><td>0.04</td><td>0.00</td></tr><tr><td>Services</td><td>9.29</td><td>0.03</td><td>0.00</td></tr><tr><td rowspan="4">y(0:k)</td><td>Energy</td><td>4 985.42</td><td>4 999.96</td><td>5 000.00</td></tr><tr><td>Materials</td><td>3 988.52</td><td>3 999.97</td><td>4 000.00</td></tr><tr><td>Industrials</td><td>7 983.83</td><td>7 999.95</td><td>8 000.00</td></tr><tr><td>Services</td><td>12 488.18</td><td>12 499.96</td><td>12 500.00</td></tr></tbody></table>

Since $y_{(k)}$ converges to $\mathbf{0}_n$, we can ask whether this convergence is monotone. In particular, do we check that $y_{(k)} \succeq y_{(k+1)}$? The answer to this question is no. In fact, we can easily find counterexamples. The reason is that $A^k \not\succeq A^{k+1}$. For instance, we have seen that $y_{(1),1} \ge y_{(0),1}$ in Example 29. Nevertheless, we observe empirically that the relation $y_{(k)} \succeq y_{(k+1)}$ is satisfied for $k \ge k^*$. This means that for sufficiently large $k$, the contribution of the $k^{th}$ tier decreases with respect to $k$. A sufficient (but not necessary) condition is that $A^k \succeq A^{k+1}$ for $k = k^*$. Let us assume that $A^k \succeq A^{k+1}$ holds for $k = k^*$. Let us note $B = A^k$, $C = A^{k+1}$ and $D = A$. Property NN1 (Appendix A.1.4 on page 1148) implies that $BD \succeq CD$. This means that if $A^k \succeq A^{k+1}$, then $A^{k+1} \succeq A^{k+2}$. We conclude that if $A^k \succeq A^{k+1}$ for $k = k^*$, then $A^k \succeq A^{k+1}$ for $k \ge k^*$ and the relation $y_{(k)} \succeq y_{(k+1)}$ is satisfied for $k \ge k^*$ if we assume that $y \succeq \mathbf{0}_n$.

**Remark 89** Theoretically, the partial monotonicity property does not hold if some elements of final demand are negative. Empirically, we observe that the property is always satisfied because $y_{(0:k)}$ converges to $\mathcal{L}y$ and the production is always positive: $\mathcal{L}y \equiv x \succeq \mathbf{0}_n$.

Looking again at our numerical example, we check that the monotonicity property $y_{(k)} \succeq y_{(k+1)}$ is satisfied for $k \ge 1$.
Multi-regional input-output analysis

While an input-output table (IOT) usually covers only one region and tracks economic flows between sectors within that region, a multi-regional input-output table (MRIO) involves several regions. It includes national harmonized input-output tables and export and import data calculated within the same sectoral structure. The two best known MRIO databases are GTAP (global trade analysis project) and WIOD (world input-output database). They can be downloaded from www.gtap.agecon.purdue.edu and www.rug.nl/ggdc/valuechain/wiod respectively. The advantage of WIOD is that it is free, which is not the case with GTAP. However, the latest version is from November 2014. An alternative is to use the input-output tables provided by the OECD. The data can be found at https://stats.oecd.org/Index.aspx?DataSetCode=IOTS. Below we describe the WIOD table in order to understand the basic structure of an MRIO database.

We download the zipped file WIOTS_in_EXCEL.zip with the button WIOT tables Excel from www.rug.nl/ggdc/valuechain/wiod/wiod-2016-release and use the Excel binary spreadsheet WIOT2014_Nov16_ROW.xlsx, which is shown in Figure 8.132. This file contains the 2014 input-output table for 44 regions (28 EU countries, 15 other major countries and a global region corresponding to the rest-of-the-world aggregate) and 56 sectors. The list of countries and their ISO codes can be found in Table 8.45. ROW is the ISO code for the rest of the world. We also report the list of industries in Table 8.44. A comprehensive description of database can be found in Dietzenbacher et al. (2023) and Timmer et al. (2015). The structure of the database is:

<table><thead><tr><th></th><th>Z<br>(56 × 44) × (56 × 44)</th><th>y<br>(56 × 44) × (5 × 44)</th><th>x<br>(56 × 44) × 1</th></tr></thead><tbody><tr><th>Sum</th><td>1 × (56 × 44)</td><td>1 × (5 × 44)</td><td><b>0</b><sub>1,1</sub></td></tr><tr><th>Value<br>added</th><td>V<br>6 × (56 × 44)</td><td>6 × (5 × 44)</td><td><b>0</b><sub>6,1</sub></td></tr><tr><th>Output</th><td>w<br>1 × (56 × 44)</td><td><b>0</b><sub>6,5×44</sub></td><td><b>0</b><sub>1,1</sub></td></tr></tbody></table>

We have five main blocks of matrices: Z, y, x, V and w. First we have the matrix Z with 2464 rows and 2464 columns. It shows the flows between all sectors for all countries. For instance, the first 56 rows correspond to the 56 sectors of Australia (AUS), the first 56 columns correspond to the 56 sectors of Australia (AUS), the next 56 columns correspond to the 56 sectors of Austria (AUT). The structure of Z is then:

<table><thead><tr><th></th><th></th><th>AUS</th><th>AUT</th><th>...</th><th>USA</th><th>ROW</th></tr></thead><tbody><tr><th rowspan="3">From</th><th>AUS</th><td>AUS → AUS</td><td>AUS → AUT</td><td>...</td><td>AUS → USA</td><td>AUS → ROW</td></tr><tr><th>AUT</th><td>AUT → AUS</td><td>AUT → AUT</td><td>...</td><td>AUT → USA</td><td>AUT → ROW</td></tr><tr><th>⋮</th><td>⋮</td><td>⋮</td><td>...</td><td>⋮</td><td>⋮</td></tr><tr><th>ROW</th><td></td><td>ROW → AUS</td><td>ROW → AUT</td><td>...</td><td>ROW → USA</td><td>ROW → ROW</td></tr></tbody></table>

The dimension of each submatrix $C_i \to C_j$ is equal to $56 \times 56$. All the items are expressed in millions of dollars. For example, we read in Figure 8.132 that $Z_{1,1} = \$12,924 mn$ and $Z_{1,2} = \$112 mn$. This means that the sector *crop and animal production, hunting and related service activities* in Australia sells \$12.924 bn and \$112 mn of goods and services to this sector and the sector *forestry and logging* in Australia. Second, the matrix *y* consists of five final demand items: (a) final consumption expenditure by households; (b) final consumption expenditure by non-profit organisations serving households (NPISH); (c) final consumption expenditure by government; (d)
<table><caption>Figure 8.132: World input-output database (November 2014)</caption><thead><tr><th></th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th><th>F</th><th>G</th><th>H</th><th>I</th><th>J</th><th>K</th><th>L</th><th>M</th><th>N</th><th>O</th><th>P</th><th>Q</th><th>R</th><th>S</th></tr></thead><tbody><tr><td>1</td><td colspan="20">Intercountry Input-Output Table</td></tr><tr><td>2</td><td colspan="20">43 countries, in current prices</td></tr><tr><td>3</td><td colspan="20">(industry-by-industry)</td></tr><tr><td>4</td><td colspan="20">(millions of US$)</td></tr><tr><td>5</td><td colspan="20"></td></tr><tr><td>6</td><td colspan="20"></td></tr><tr><td>7</td><td>A01</td><td colspan="19">addition, hunting and related service activities</td></tr><tr><td>8</td><td>A02</td><td colspan="19">Forestry and logging</td></tr><tr><td>9</td><td>A03</td><td colspan="19">Mining and aquaculture</td></tr><tr><td>10</td><td>B</td><td colspan="19">Mining and quarrying</td></tr><tr><td>11</td><td>C10-C12</td><td colspan="19">al products, beverages and tobacco products</td></tr><tr><td>12</td><td>C13-C15</td><td colspan="19">textiles, wearing apparel and leather products</td></tr><tr><td>13</td><td>C16</td><td colspan="19">al products of articles of straw and plaiting materials</td></tr><tr><td>14</td><td>C17</td><td colspan="19">Manufacture of paper and paper products</td></tr><tr><td>15</td><td>C18</td><td colspan="19">Printing and reproduction of recorded media</td></tr><tr><td>16</td><td>C19</td><td colspan="19">al products of coke and refined petroleum products</td></tr><tr><td>17</td><td>C20</td><td colspan="19">ufacture of chemicals and chemical products</td></tr><tr><td>18</td><td>C21</td><td colspan="19">al products and pharmaceutical preparations</td></tr><tr><td>19</td><td>C22</td><td colspan="19">Manufacture of rubber and plastic products</td></tr><tr><td>20</td><td>C23</td><td colspan="19">acture of other non-metallic mineral products</td></tr><tr><td>21</td><td>C24</td><td colspan="19">Manufacture of basic metals</td></tr><tr><td>22</td><td>C25</td><td colspan="19">al products, except machinery and equipment</td></tr><tr><td>23</td><td>C26</td><td colspan="19">re of computer, electronic and optical products</td></tr><tr><td>24</td><td>C27</td><td colspan="19">Manufacture of electrical equipment</td></tr><tr><td>25</td><td>C28</td><td colspan="19">anufacture of machinery and equipment in e.c.</td></tr><tr><td>26</td><td>C29</td><td colspan="19">ure of motor vehicles, trailers and semi-trailers</td></tr><tr><td>27</td><td>C30</td><td colspan="19">Manufacture of other transport equipment</td></tr><tr><td>28</td><td>C31-C32</td><td colspan="19">Manufacture of furniture, other manufacturing</td></tr><tr><td>29</td><td>C33</td><td colspan="19">r and installation of machinery and equipment</td></tr><tr><td>30</td><td>D05</td><td colspan="19">strictly, gas, steam and air conditioning supply</td></tr><tr><td>31</td><td>E36</td><td colspan="19">Water collection, treatment and supply</td></tr><tr><td>32</td><td>E37-E39</td><td colspan="19">activities and other waste management services</td></tr><tr><td>33</td><td>F</td><td colspan="19">Construction</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

Source: Dietzenbacher et al. (2023); Timmer et al. (2015) & www.rug.nl/ggdc/valuechain/wiod.
<table><caption>Table 8.44: List of industries/sectors in the 2014 WIOD table</caption><thead><tr><th>No.</th><th>Name</th></tr></thead><tbody><tr><td>S1</td><td>Accommodation and food service activities</td></tr><tr><td>S2</td><td>Activities auxiliary to financial services and insurance activities</td></tr><tr><td>S3</td><td>Activities of extraterritorial organizations and bodies</td></tr><tr><td>S4</td><td>Activities of households as employers; undifferentiated goods- and services-producing activities of households for own use</td></tr><tr><td>S5</td><td>Administrative and support service activities</td></tr><tr><td>S6</td><td>Advertising and market research</td></tr><tr><td>S7</td><td>Air transport</td></tr><tr><td>S8</td><td>Architectural and engineering activities; technical testing and analysis</td></tr><tr><td>S9</td><td>Computer programming, consultancy and related activities; information service activities</td></tr><tr><td>S10</td><td>Construction</td></tr><tr><td>S11</td><td>Crop and animal production, hunting and related service activities</td></tr><tr><td>S12</td><td>Education</td></tr><tr><td>S13</td><td>Electricity, gas, steam and air conditioning supply</td></tr><tr><td>S14</td><td>Financial service activities, except insurance and pension funding</td></tr><tr><td>S15</td><td>Fishing and aquaculture</td></tr><tr><td>S16</td><td>Forestry and logging</td></tr><tr><td>S17</td><td>Human health and social work activities</td></tr><tr><td>S18</td><td>Insurance, reinsurance and pension funding, except compulsory social security</td></tr><tr><td>S19</td><td>Land transport and transport via pipelines</td></tr><tr><td>S20</td><td>Legal and accounting activities; activities of head offices; management consultancy activities</td></tr><tr><td>S21</td><td>Manufacture of basic metals</td></tr><tr><td>S22</td><td>Manufacture of basic pharmaceutical products and pharmaceutical preparations</td></tr><tr><td>S23</td><td>Manufacture of chemicals and chemical products</td></tr><tr><td>S24</td><td>Manufacture of coke and refined petroleum products</td></tr><tr><td>S25</td><td>Manufacture of computer, electronic and optical products</td></tr><tr><td>S26</td><td>Manufacture of electrical equipment</td></tr><tr><td>S27</td><td>Manufacture of fabricated metal products, except machinery and equipment</td></tr><tr><td>S28</td><td>Manufacture of food products, beverages and tobacco products</td></tr><tr><td>S29</td><td>Manufacture of furniture; other manufacturing</td></tr><tr><td>S30</td><td>Manufacture of machinery and equipment n.e.c.</td></tr><tr><td>S31</td><td>Manufacture of motor vehicles, trailers and semi-trailers</td></tr><tr><td>S32</td><td>Manufacture of other non-metallic mineral products</td></tr><tr><td>S33</td><td>Manufacture of other transport equipment</td></tr><tr><td>S34</td><td>Manufacture of paper and paper products</td></tr><tr><td>S35</td><td>Manufacture of rubber and plastic products</td></tr><tr><td>S36</td><td>Manufacture of textiles, wearing apparel and leather products</td></tr><tr><td>S37</td><td>Manufacture of wood and of products of wood and cork, except furniture; manufacture of articles of straw and plaiting materials</td></tr><tr><td>S38</td><td>Mining and quarrying</td></tr><tr><td>S39</td><td>Motion picture, video and television programme production, sound recording and music publishing activities; programming and broadcasting activities</td></tr><tr><td>S40</td><td>Other professional, scientific and technical activities; veterinary activities</td></tr><tr><td>S41</td><td>Other service activities</td></tr><tr><td>S42</td><td>Postal and courier activities</td></tr><tr><td>S43</td><td>Printing and reproduction of recorded media</td></tr><tr><td>S44</td><td>Public administration and defence; compulsory social security</td></tr><tr><td>S45</td><td>Publishing activities</td></tr><tr><td>S46</td><td>Real estate activities</td></tr><tr><td>S47</td><td>Repair and installation of machinery and equipment</td></tr><tr><td>S48</td><td>Retail trade, except of motor vehicles and motorcycles</td></tr><tr><td>S49</td><td>Scientific research and development</td></tr><tr><td>S50</td><td>Sewerage; waste collection, treatment and disposal activities; materials recovery; remediation activities and other waste management services</td></tr><tr><td>S51</td><td>Telecommunications</td></tr><tr><td>S52</td><td>Warehousing and support activities for transportation</td></tr><tr><td>S53</td><td>Water collection, treatment and supply</td></tr><tr><td>S54</td><td>Water transport</td></tr><tr><td>S55</td><td>Wholesale and retail trade and repair of motor vehicles and motorcycles</td></tr><tr><td>S56</td><td>Wholesale trade, except of motor vehicles and motorcycles</td></tr></tbody></table>
gross fixed capital formation; (e) changes in inventories and valuables. The sum of these five items is then the aggregate final demand. Third, the vector $x$ corresponds to the output. For example, $x_1 = \$70 292 mn$ is the total production of the sector *crop and animal production, hunting and related service activities* in Australia. The fourth block matrix is $V$ and has six rows: (a) taxes less subsidies on products; (b) cif/fob adjustments on exports; (c) direct purchases abroad by residents; (d) purchases on the domestic territory by non-residents; (e) value added at basic prices; (f) international transport margins. Again, we can aggregate these six items to calculate a total value added. Finally, the last block is the row vector $w$. By construction, we have $w_j = x_j$. For example, we check that $w_1$ is equal to \$70 292 mn with the following decomposition: the total intermediate consumption is equal to \$39 039 mn while the total value added is equal to $502 + 0 + 0 + 0 + 30 489 + 262 = \$31 253 mn.

<table><caption>Table 8.45: List of countries/regions in the 2014 WIOD table</caption><thead><tr><th>No.</th><th>ISO</th><th>Name</th><th>No.</th><th>ISO</th><th>Name</th></tr></thead><tbody><tr><td>C<sub>1</sub></td><td>AUS</td><td>Australia</td><td>C<sub>2</sub></td><td>AUT</td><td>Austria</td></tr><tr><td>C<sub>3</sub></td><td>BEL</td><td>Belgium</td><td>C<sub>4</sub></td><td>BGR</td><td>Bulgaria</td></tr><tr><td>C<sub>5</sub></td><td>BRA</td><td>Brazil</td><td>C<sub>6</sub></td><td>CAN</td><td>Canada</td></tr><tr><td>C<sub>7</sub></td><td>CHE</td><td>Switzerland</td><td>C<sub>8</sub></td><td>CHN</td><td>China</td></tr><tr><td>C<sub>9</sub></td><td>CYP</td><td>Cyprus</td><td>C<sub>10</sub></td><td>CZE</td><td>Czech Republic</td></tr><tr><td>C<sub>11</sub></td><td>DEU</td><td>Germany</td><td>C<sub>12</sub></td><td>DNK</td><td>Denmark</td></tr><tr><td>C<sub>13</sub></td><td>ESP</td><td>Spain</td><td>C<sub>14</sub></td><td>EST</td><td>Estonia</td></tr><tr><td>C<sub>15</sub></td><td>FIN</td><td>Finland</td><td>C<sub>16</sub></td><td>FRA</td><td>France</td></tr><tr><td>C<sub>17</sub></td><td>GBR</td><td>United Kingdom</td><td>C<sub>18</sub></td><td>GRC</td><td>Greece</td></tr><tr><td>C<sub>19</sub></td><td>HRV</td><td>Croatia</td><td>C<sub>20</sub></td><td>HUN</td><td>Hungary</td></tr><tr><td>C&#773;<sub>21</sub></td><td>IDN</td><td>Indonesia</td><td>C&#773;<sub>22</sub></td><td>IDN</td><td>India</td></tr><tr><td>C<sub>23</sub></td><td>IRL</td><td>Ireland</td><td>C<sub>24</sub></td><td>ITA</td><td>Italy</td></tr><tr><td>C<sub>25</sub></td><td>JPN</td><td>Japan</td><td>C<sub>26</sub></td><td>KOR</td><td>Republic of Korea</td></tr><tr><td>C<sub>27</sub></td><td>LTU</td><td>Lithuania</td><td>C<sub>28</sub></td><td>LUX</td><td>Luxembourg</td></tr><tr><td>C<sub>29</sub></td><td>LVA</td><td>Latvia</td><td>C<sub>30</sub></td><td>MEX</td><td>Mexico</td></tr><tr><td>C<sub>31</sub></td><td>MLT</td><td>Malta</td><td>C<sub>32</sub></td><td>NLD</td><td>Netherlands</td></tr><tr><td>C<sub>33</sub></td><td>NOR</td><td>Norway</td><td>C<sub>34</sub></td><td>POL</td><td>Poland</td></tr><tr><td>C<sub>35</sub></td><td>PRT</td><td>Portugal</td><td>C<sub>36</sub></td><td>ROU</td><td>Romania</td></tr><tr><td>C<sub>37</sub></td><td>RUS</td><td>Russian Federation</td><td>C<sub>38</sub></td><td>SVK</td><td>Slovakia</td></tr><tr><td>C<sub>39</sub></td><td>SVN</td><td>Slovenia</td><td>C<sub>40</sub></td><td>SWE</td><td>Sweden</td></tr><tr><td>C&#773;<sub>41</sub></td><td>TUR</td><td>Turkey</td><td>C&#773;<sub>42</sub></td><td>TWN</td><td>Taiwan</td></tr><tr><td>C<sub>43</sub></td><td>USA</td><td>United States of America</td><td>C<sub>44</sub></td><td>ROW</td><td>Rest-of-the-world</td></tr></tbody></table>

In Figure 8.133 we plot the sparsity pattern of the input-output matrix, and only the values of $A_{i,j}$ greater than 5% are colored. We see that the density of the matrix is mainly within the country submatrices. Outside of these intra-country matrices, the input-output table is sparse except for a few countries: China, Germany, Russia, USA and the rest-of-the-world region.

Let us now analyze the Leontief matrix $\mathcal{L} = (I - A)^{-1}$. In Figure 8.134, we perform the eigen-decomposition $A = V\Lambda V^{-1}$ and plot the spectrum of $A$. Figure 8.135 shows the Frobenius norm of the matrix $A^k$ for $k = 0, \dots, 10$ and the Leontief matrix $\mathcal{L}$. We find that the convergence of the Leontief matrix is achieved very fast since the Frobenius norm of $A^k$ is less than 1 after the third tier and 0.1 after the seventh tier.
<figure><img src="image_414.png" /><figcaption>Figure 8.134: Spectrum of the matrix A</figcaption></figure>

<figure><img src="image_415.png" /></figure>
<figure><img src="image_416.png" /><figcaption>Figure 8.135: Frobenius norm of the matrix $A^k$</figcaption></figure>

## 8.4.2 Application to environmental problems

In the late 1960s, several authors proposed linking economic and ecological systems using generalized input-output models. For example, Daly (1968) suggested extending the technical coefficients with additional rows/columns to reflect non-human sectors such as animals, plants, and bacteria, and non-living sectors such as the atmosphere, hydrosphere, and lithosphere. Leontief (1970) himself explained how externalities such as pollution could be incorporated into a basic input-output model. Since these first contributions, input-output analysis has been extended to many environmental problems<fnref n="136" />.

### Production-based vs. consumption-based inventory

To understand how input-output analysis can be used to measure carbon emissions, consider the mathematical problem of calculating the contribution of carbon emissions per product. Following Miller and Blair (2009), we denote by $C^{(x)} = (C_{g,j}^{(x)})$ the pollution output matrix where $C_{g,j}^{(x)}$ is the total amount of the $g^{th}$ pollutant generated by the output of the $j^{th}$ sector. Similarly, we define $D^{(y)} = C^{(x)} \text{diag}(x)^{-1} = (D_{g,j}^{(y)})$ the matrix of direct impact coefficients where $D_{g,j}^{(y)} = c_{g,j}^{(y)}/x_j$ is the amount of the $g^{th}$ pollutant generated by 1$ of the output of the $j^{th}$ sector. Let $\bar{\omega} = (\bar{\omega}_1, \dots, \bar{\omega}_m)$ be the vector of pollution level. We have:

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

where $D^{(y)} = D^{(x)}(I_n - A)^{-1}$ is the pollutant multiplier matrix with respect to the final demand $y$. $D^{(y)}$ also measures the product carbon footprint (PCF). Since we have the following identity

<footnote n="136">¹³⁶See Chapters 9 and 10 of Miller and Blair (2009).</footnote>
$\bar{\omega}_g = (D^{(y)}y)_g = \sum_{j=1}^{n} D_{g,j}^{(y)} y_j$, we deduce that the total contribution of sector $j$ to the $g^{th}$ pollutant is equal to:

$$C_{g,j}^{(x)} = \frac{\partial \bar{\omega}_g}{\partial y_j} y_j = D_{g,j}^{(y)} y_j$$

Again, we can decompose the pollutant level according to the $k^{th}$ tier. We have:

$$\bar{\omega} = D^{(y)}y = \sum_{k=0}^{\infty} D^{(x)}A^k y = \sum_{k=0}^{\infty} \bar{\omega}_{(k)}$$

where $\bar{\omega}_{(0)} = D^{(x)}y$ is the pollutant level due to the final demand (or the zero-tier pollutant level), $\bar{\omega}_{(1)} = D^{(x)}Ay$ is the pollutant level due to the first-tier supply chain, and $\bar{\omega}_{(k)} = D^{(x)}A^k y$ is the $k^{th}$-tier pollutant level. The matrix $D_{(k)}^{(y)} = D^{(x)}A^k$ is called the $k^{th}$-tier multiplier matrix and satisfies the identity $D^{(y)} = \sum_{k=0}^{\infty} D_{(k)}^{(y)}$.

**Example 31** We consider three products, whose input-output table is given below:

<table><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="3">To</th><th>Final demand</th><th>Total output</th></tr><tr><th>P<sub>1</sub></th><th>P<sub>2</sub></th><th>P<sub>3</sub></th><th>y</th><th>x</th></tr></thead><tbody><tr><td rowspan="3">From</td><td>P<sub>1</sub></td><td>100</td><td>300</td><td>100</td><td>500</td><td>1 000</td></tr><tr><td>P<sub>2</sub></td><td>250</td><td>150</td><td>200</td><td>1 600</td><td>2 000</td></tr><tr><td>P<sub>3</sub></td><td>25</td><td>200</td><td>75</td><td>200</td><td>500</td></tr><tr><td colspan="2">Value added</td><td>625</td><td>1 350</td><td>125</td><td></td><td></td></tr><tr><td colspan="2">Total outlays</td><td>1 000</td><td>2 000</td><td>500</td><td></td><td></td></tr><tr><td rowspan="2">GHG</td><td>CO<sub>2</sub></td><td>50</td><td>20</td><td>5</td><td>75</td><td></td></tr><tr><td>CH<sub>4</sub></td><td>3</td><td>1</td><td>0</td><td>4</td><td></td></tr></tbody></table>

Intermediate production of $100 of P_1$, $300 of P_2$, and $100 of P_3$ is required to produce $500 of P_1$. This environmentally-extended input-output table has two additional rows corresponding to the GHG emissions. For instance, the production of $P_1$ causes 50 kgCO$_2$ and 3 kgCH$_4$.

The matrix of technical coefficients is equal to:

$$A = Z \text{diag}(x)^{-1} = \begin{pmatrix} 10.0\% & 15.0\% & 20.0\% \\ 25.0\% & 7.5\% & 40.0\% \\ 2.5\% & 10.0\% & 15.0\% \end{pmatrix}$$

It follows that the matrix of multipliers is equal to:

$$\mathcal{L} = (I_3 - A)^{-1} = \begin{pmatrix} 1.1871 & 0.2346 & 0.3897 \\ 0.3539 & 1.2090 & 0.6522 \\ 0.0766 & 0.1491 & 1.2647 \end{pmatrix}$$

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

$$D^{(x)} = \begin{pmatrix} 50/1000 & 20/2000 & 5/500 \\ 3/1000 & 1/2000 & 0/500 \end{pmatrix} = \begin{pmatrix} 0.05 & 0.01 & 0.01 \\ 0.003 & 0.0005 & 0 \end{pmatrix}$$
The unit of $D^{(x)}$ is expressed in kilogram of the gas per dollar. For instance, the GHG intensities of the product $P_1$ are equal to 0.05 kgCO$_2$/ and 0.003 kgCH$_4$/ . Finally, we obtain:

$$
D^{(y)} = D^{(x)} \mathcal{L} = \begin{pmatrix} 0.0637 & 0.0253 & 0.0387 \\ 0.0037 & 0.0013 & 0.0015 \end{pmatrix}
$$

While $D^{(x)}$ corresponds to the production-based inventory, $D^{(y)}$ measures the carbon footprint from the perspective of the consumption-based inventory (Kitzes, 2013). This gives us the following decomposition:

$$
C^{(y)} = \begin{pmatrix} 31.83 & 35.44 & 7.73 \\ 1.87 & 1.83 & 0.30 \end{pmatrix} \neq \begin{pmatrix} 50 & 20 & 5 \\ 3 & 1 & 0 \end{pmatrix} = C^{(x)}
$$

We notice that the two contribution matrices are different. For instance, while the production of $P_1$ is responsible of 50 kgCO$_2$, the final consumption of $P_1$ is responsible of only 31.83 kgCO$_2$, meaning that 18.17 kgCO$_2$ are emitted by $P_1$ for the other two products. The difference between $C^{(x)}$ and $C^{(y)}$ depends on the structure of the matrix $A$ and the vector $y$. Let us apply a perturbation to the previous example. We assume that there is no final demand of $P_1$: $y_1 = 0$. For consistency, we also assume that the GHG emissions of Product $P_1$ are divided by two because the production is reduced by 50%. We find that:

$$
C^{(y)} = \begin{pmatrix} 0 & 40.96 & 9.04 \\ 0 & 2.13 & 0.37 \end{pmatrix} \neq \begin{pmatrix} 25 & 20 & 5 \\ 1.5 & 1 & 0 \end{pmatrix} = C^{(x)}
$$

The consumption-based contribution of $P_1$ is null because $P_1$ is manufactured for the other two products.

**Remark 90** We can show that $C^{(x)} = C^{(y)}$ implies that $A$ is a diagonal matrix. We conclude that the supply chain and interconnectedness between sectors can lead to a misperception of the sectoral carbon footprint.

The previous framework can be applied to many problems involving the calculation of carbon footprints. Miller and Blair (2009) examined three categories of EEIO analysis: generalized input-output, economic-ecological, and commodity-by-industry models. An overview of generalized input-output models can be found in Minx et al. (2009) and Wiedmann (2009). These models are commonly used to calculate the carbon footprint of nations, sectors, supply chains, etc., and to analyze the impact of foreign trade. The use of economic-ecological models is less popular, since it involves building an input-output table for ecological sectors (species, plants, etc.). Commodity-by-industry models are more studied because it is easier to collect data for the commodity sector.

EEIO databases

The use of environmentally-extended input-output models requires credible database. Two EEIO databases dominate the market: Eora and Exiobase. According to https://worldmrio.com, the Eora global supply chain database uses more than 15 000 sectors across 190 countries, and contains about 2 700 environmental indicators covering GHG emissions, air pollution, energy use, water demand, land use, etc. Exiobase is a multi-regional environmentally-extended supply-use and input-output model with 44 countries, 163 industries, 200 products and 417 emission categories (www.exiobase.eu).

We consider the Exiobase data for the year 2022. To do this, we download the MRIO archive in industry by industry format (IOT_2022_ixi.zip) and the MRIO archive in product by product format (IOT_2022_pxp.zip) from the website: https://zenodo.org/record/5589597. Both archives
have the same structure. The economic input-output data are stored in the root of the archive,
which contains four data files:

1. The *Z* matrix (*Z*.txt — flow/transactions matrix)

2. The *A* matrix (*A*.txt — direct requirements matrix)

3. The *y* matrix (*y*.txt — final demand)

4. The *x* vector (*x*.txt — gross/total output)

and four description files (*finaldemand.txt*, *industries.txt*, *products.txt* and *units.txt*). Exiobase provides also two set of extension data stored in the sub-folders *satellite* (uncharacterized stressors data — e.g., CO₂ emissions, land use per category, etc.) and *impacts* (characterized stressors data — e.g., total GWP100, total land use, etc.). The structure of the two sub-folders is the following:

• Factors of productions/stressors/impacts: *F*.txt

• Stressors/impacts of the final demand: *F_Y*.txt

• Direct stressor/impact coefficients: *S*.txt

• Stressor/impact coefficients of the final demand: *S_Y*.txt

• MRIO extension multipliers (total requirement factors of consumption): *M*.txt

• Consumption based accounts per sector: *D_cba*.txt

• Production based accounts per sector: *D_pba*.txt

• Consumption based accounts per region: *D_cba_reg*.txt

• Production based accounts per region: *D_pba_reg*.txt

• Import accounts per region: *D_imp_reg*.txt

• Export accounts per region: *D_exp_reg*.txt

• Absolute units of the stressor and impacts: *unit*.txt

Previously, we have given the detailed sectors in the WIOD database, which is a 56 × 56 industry by industry IOT. Exiobase is both a 163 × 163 industry by industry IOT and a 200 × 200 product by product IOT. The first fifteen products are listed here: paddy rice, wheat, cereal grains nec, vegetables, fruit, nuts, oil seeds, sugar cane and beet, plant-based fibers, crops nec, cattle, pigs, poultry, meat animals nec, animal products nec, raw milk, wool and silk-worm cocoons. In the impact sub-directory, we find measurement that concerns employment (in hour), GHG emissions (in GWP100), water consumption (in Mm³), Nitrogen (in kg), land-use crop, forest, pasture (in km²), etc. In the satellite sub-directory, we have similar information but in a disaggregated form. For instance, the employment item has seven categories: low-skilled male, low-skilled female, medium-skilled male, medium-skilled female, high-skilled male, high-skilled female, vulnerable.
### 8.4.3 Estimation of first-tier and indirect emissions

#### Basic formula

We assume that the carbon footprint is evaluated in CO₂e, which means that the input-output analysis considers only one pollutant, with all greenhouse gases being converted to the carbon based on their global warming potential. In this case, $D^{(x)}$ is a row vector of dimension $n$, and $D_j^{(x)}$ measures the direct emission intensity of sector $j$. We reiterate that the total emission intensities are equal to $D^{(y)} = D^{(x)}\mathcal{L} = D^{(x)}(I_n - A)^{-1}$. $D^{(y)}$ is a row vector of dimension $n$, and $D_j^{(y)}$ measures the direct and indirect emission intensity of sector $j$. Using the usual notation $\mathcal{CI}$ for the carbon intensity, we have<fnref n="137" />:

$$ 
\begin{aligned}
\mathcal{CI}_{\text{total}} &= \mathcal{L}^{\top}\mathcal{CI}_{1} \\
&= (I_n - A^{\top})^{-1}\mathcal{CI}_{1} \\
&= \tilde{\mathcal{L}}\mathcal{CI}_{1}
\end{aligned}
\qquad (8.34) $$

where $\mathcal{CI}_1 = \mathcal{CI}_{\text{direct}}$ is the vector of direct carbon intensities and $\mathcal{CI}_{\text{total}}$ is the vector of direct plus indirect carbon intensities. It follows that the indirect carbon intensities are given by:

$$ 
\begin{aligned}
\mathcal{CI}_{\text{indirect}} &= \mathcal{CI}_{\text{total}} - \mathcal{CI}_{1} \\
&= \left( (I_n - A^{\top})^{-1} - I_n \right) \mathcal{CI}_{\text{direct}}
\end{aligned}
\qquad (8.35) $$

In particular, we can decompose $\mathcal{CI}_{\text{indirect}}$ using the Neumann series:

$$ \mathcal{CI}_{\text{indirect}} = \underbrace{A^{\top}\mathcal{CI}_{1}}_{\text{First-tier}} + \underbrace{(A^{\top})^2\mathcal{CI}_{1}}_{\text{Second-tier}} + \dots + \underbrace{(A^{\top})^k\mathcal{CI}_{1}}_{k^{\text{th-tier}}} + \dots \qquad (8.36) $$

and we have:

$$ \mathcal{CI}_{\text{total}} = \underbrace{\mathcal{CI}_{1}}_{\text{Scope 1}} + \underbrace{A^{\top}\mathcal{CI}_{1}}_{\text{First-tier}} + \underbrace{(A^{\top})^2\mathcal{CI}_{1}}_{\text{Second-tier}} + \dots + \underbrace{(A^{\top})^k\mathcal{CI}_{1}}_{k^{\text{th-tier}}} + \dots \qquad (8.37) $$

Equations (8.34–8.37) are the core formulas of the consumption-based inventory approach and the calculation of indirect carbon intensities.

#### Illustration

Continuing with Example 30, let us assume that the carbon emissions, expressed in ktCO₂e, are as follows: 500 for the energy sector, 200 for the materials sector, 200 for the industrials sector and 125 for the services sector. We deduce that the vector of Scope 1 carbon intensities is equal to:

$$ \mathcal{CI}_1 = \text{diag}(x)^{-1} \mathcal{CE}_1 = \begin{pmatrix} 500/5\,000 \\ 200/4\,000 \\ 200/8\,000 \\ 125/12\,500 \end{pmatrix} \times 10^3 = \begin{pmatrix} 100 \\ 50 \\ 25 \\ 10 \end{pmatrix} $$

<footnote n="137">¹³⁷Because $D^{(x)} = \mathcal{CI}_1^{\top}$ and $D^{(y)} = \mathcal{CI}_{\text{total}}^{\top}$.</footnote>