# Network-based allocation of responsibility for GHG emissions

Rosa van den Ende, Antoine Mandel, Agnieszka Rusinowska

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Rosa van den Ende, Antoine Mandel, Agnieszka Rusinowska. Network-based allocation of responsibility for GHG emissions. 2023. halshs-04188365

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CES

Centre d'Économie de la Sorbonne
UMR 8174

Network-based allocation of responsibility
for GHG emissions

Rosa VAN DEN ENDE, Antoine MANDEL, Agnieszka RUSINOWSKA

2023.13
Network-based allocation of responsibility for GHG emissions *

Rosa van den Ende †‡

Antoine Mandel §

Agnieszka Rusinowska ¶

August 3, 2023

**Abstract**

We provide an axiomatic approach to the allocation of responsibility for GHG emissions in supply chains. Considering a set of axioms standardly used in networks and decision theory, and consistent with legal principles underlying responsibility, we show that responsibility measures shall be based on exponential discounting of upstream and downstream emissions. From a network theory perspective, the proposed responsibility measure corresponds to a convex combination of the Bonacich centralities for the upstream and downstream weighted adjacency matrices. Scope 1 emissions, consumption-based accounting and income-based accounting are obtained as particular cases of our approach, which also gives a precise meaning to scope 3 emissions while avoiding double-counting. We apply our approach to the assessment of country-level responsibility for global GHG emissions and to sector-level responsibility in the USA. We examine how the responsibility of sectors/countries varies with the discounting of indirect emissions. We identify three groups of countries/sectors: producers of emissions whose responsibility decreases with the discounting factor, consumers of emissions whose responsibility increases with the discounting factor, and an intermediary group whose responsibility mostly depends on the network position and varies non-monotonically with the discounting factor. Overall, our axiomatic approach provides strong normative foundations for the definition of reporting requirements for indirect emissions and for the allocation of responsibility in claims for climate-related loss and damage.

**Keywords:** upstream and downstream emission responsibilities, supply chains and networks, responsibility measure, axiomatization, Bonacich centrality

<footnote>*This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 956107, "Economic Policy in Complex Environments (EPOC)".</footnote>

<footnote>†Corresponding author</footnote>

<footnote>‡Université Paris 1 Panthéon-Sorbonne, Centre d'Économie de la Sorbonne, Université Bielefeld, rosa.van-denne@univ-paris1.fr</footnote>

<footnote>§Université Paris 1 Panthéon-Sorbonne, Centre d'Économie de la Sorbonne, Paris School of Economics, antoine.mandel@univ-paris1.fr</footnote>

<footnote>¶CNRS, Université Paris 1 Panthéon-Sorbonne, Centre d'Économie de la Sorbonne, Paris School of Economics, agnieszka.rusinowska@univ-paris1.fr</footnote>
# 1 Introduction

With the emergence of anthropogenic climate change as a core socio-environmental challenge at the global scale [Rockström et al., 2009], greenhouse gas (GHG) emissions have become a key metric of environmental performance at the individual, company, or country level. Embodied GHG emissions are defining the individual carbon footprint [Peters, 2010] and they are the core indicator for climate-related financial disclosures at the company level [TCFD, 2017, U.S. Securities and Exchange Commission, 2022]. They form the basis for the carbon border adjustment mechanism [Kortum and Weisbach, 2017] and more generally for carbon pricing and taxation at the product level. At the country level, emission targets are the central tenet in international environmental negotiations [Falkner, 2016, Schleussner et al., 2016]. Because of its centrality in monitoring performance, the measure of emissions, i.e. the definition of the scope of GHG emissions for which an actor carries responsibility, is crucial to provide the appropriate incentives to reduce these emissions [Gopalakrishnan et al., 2021a,b]. A clear definition of responsibility is made all the more necessary by the recent rise in climate litigation [Burger et al., 2020]. Notably, a Dutch court has recently ordered Shell to reduce its total emissions by 45 per cent in 2030, relative to their 2019 levels<fnref n="1" />.

Against this background, there exists a range of carbon accounting standards and principles. The GHG protocol [WRI and WBCSD, 2011], a widely used standard for corporate reporting, distinguishes between three emission scopes: direct emissions from sources controlled by the company (scope 1), emissions from the generation of purchased electricity consumed by the company (scope 2), all other indirect emissions (scope 3). The definition of scope 3 emissions is extremely broad. It can lead to inconsistencies at the aggregate level because of double-counting [Caro et al., 2013, Gopalakrishnan, 2022] and, in actual company reports, is often found missing [Depoers et al., 2016] or inconsistent [Busch et al., 2022]. At the country level, the UNFCCC<fnref n="2" /> assigns to a country responsibility for the emissions that occur within its borders [see e.g. Harrison, 2015]. There have been substantial criticisms of this “production-based” approach in particular in terms of fairness and of the incentives it provides for carbon leakage [see e.g. Peters, 2008, Steininger et al., 2014, and references therein]. An alternative “consumption-based” approach has accordingly been put forward [Lenzen et al., 2007, Davis and Caldeira, 2010]. Overall, several problems remain to be solved in order to provide a consistent carbon accounting framework: a precise definition of indirect emissions is lacking; the requirements for a consistent measurement of direct and indirect emissions at the individual and at the aggregate levels are unspecified; the relationships between direct emissions, indirect emissions and consumption-based emissions are unclear and last but not least, responsibility (indirect emissions) measures lack normative foundations if they ought to be used in

<footnote n="1">¹The Hague District Court, 26th of May 2021, ECLI:NL:RBDHA:2021:5339</footnote>

<footnote n="2">²United Nations Framework Convention on Climate Change</footnote>
a legal context.

In this paper, we propose a network-based approach to the measure of GHG emissions that addresses these issues both from theoretical and empirical points of view. From the theoretical point of view, we provide axiomatic foundations for a class of measures that define responsibility for GHG emissions in a supply chain as a convex combination of the discounted sums of upstream and downstream emissions. From a network perspective, these correspond to a convex combination of the Bonacich centralities of adjacency matrices normalized from an upstream and a downstream perspective respectively and weighted according to individual emissions. We further show that this responsibility measure is consistent with both the production-based/direct emission approach, which corresponds to the case where the discount factor tends towards zero, and the consumption-based and income-based approach, which corresponds to the case where the discount factor tends towards one. Furthermore, we show that in the no-discounting case, an agent's responsibility depends only on its network position and not on its individual emissions. From the empirical point of view, we build on the world input-output (WIOD) database [Timmer et al., 2015] and its satellite environmental accounts [Corsatea et al., 2016] to characterize, at the country level and at the sectoral level for the USA, how responsibility for GHG emissions evolves when one shifts from a production-based to a consumption-based approach by accounting progressively for indirect emissions and network effects. We find three broad classes of countries: (i) the most developed and service-oriented economies whose share of responsibility grows as one increasingly accounts for emissions (upstream or downstream) in the supply chains, (ii) manufacturing-oriented economies whose share of responsibility decreases as one increasingly accounts for emissions in the supply chains and (iii) an intermediary class of economies whose responsibility can exhibit non-monotonic relationships with respect to the discounting of indirect emissions because they have a relatively high emission intensity but are more closely connected to the most developed service-oriented economies than to the manufacturing-oriented economies. At the sectoral level, we find that the few sectors that are responsible for most GHG emissions (e.g. energy utilities and transport) see their responsibility decrease very rapidly when indirect emissions are accounted for. The service sector accordingly sees its share of responsibility increase rapidly. The responsibility of the mining and quarrying sector (which embeds fossil-fuel extraction) has an interesting non-monotonic (increasing and decreasing) behaviour: it is closely connected downstream to the most emitting sectors but is not very central in the overall economic system.

From the methodological point of view, our axiomatization builds on the characterization of centrality measures by Bloch et al. [2023]. The sequence of indirect emissions (with increasing network distance, upstream and downstream) can be interpreted as a nodal statistic and the discounted emissions measure is then characterized by a series of six axioms. First, anonymity states that the
responsibility of a player is not influenced by their identity. Second, monotonicity sets forth that responsibility is monotonic with respect to the natural order on emission sequences. Additivity amounts to considering that total responsibility is invariant when an agent is split into subunits or when two agents are merged. Tail-convergence ensures consistent measurement of infinite emission sequences. Recursivity states that the responsibility ratio between two emission patterns is independent of their level of upstreamness/downstreamness. This axiom is closely connected to the notion of stationarity in the axiomatization of discounted expected utility by Koopmans [1960] and induces the exponential discounting aspect of the responsibility measure. Finally, consistency ensures that the responsibility allocates exactly the total amount of emissions in the system and thus prevents double-counting. All these axioms are consistent with legal principles underlying responsibility and can contribute to sustainable operation and carbon accounting management.

**Related Literature** This paper contributes to different strands of literature. First, it relates to studies in supply chain management, in particular, supply chain emission accounting and apportioning of shared emissions. In particular, Sunar and Plambeck [2016] investigate the allocation of emission among co-products. Gopalakrishnan et al. [2021b][2021a] focus on the allocation of responsibility for upstream GHG emissions in supply chains using a cooperative game-theoretic approach. They put forward firm-level emission measures using solution concepts such as the Shapley [1953] value and the nucleolus [Schmeidler, 1969]. Cooperative game theoretic approaches have been also applied more broadly to supply chain management [see e.g. Nagarajan and Sošič, 2008, for a survey], to cost-sharing problems in the context of river management [see e.g. Ni and Wang, 2007, Dong et al., 2012, Gómez-Rúa, 2013, Alcalde-Unzu et al., 2015, van den Brink et al., 2018], to loss allocation in energy transmission networks [Bergañtinos et al., 2017, Bergantiños et al., 2019], to allocate the global carbon budget [Giménez-Gómez et al., 2016] or carbon emission permits [Moretti and Trabelsi, 2021] and to responsibility in the context of hazardous transportation problems [Teker, 2022]. These approaches often focus on specific network structures (e.g. trees) and their axiomatization (when available) can hardly be applied in the context of GHG emission allocation. In the present paper, a complementary network-based and normative approach is adopted. We propose a responsibility measure that allows for both upstream and downstream responsibilities and applies to a generic network structure. The axiomatization of this measure is tailored to the problem of GHG emission allocations. Our approach is also much less computationally intensive than cooperative game solution concepts as it only involves the computation of network centrality measures via matrix multiplication.

A second strand of literature focuses on the allocation of responsibility of GHG emissions across countries using input-output analysis [see e.g. Gallego and Lenzen, 2005, Lenzen et al., 2007, Benjaa-
far et al., 2013, Caro et al., 2013]. This literature has notably introduced the notion of consumption-
based [Eder and Narodoslawsky, 1999, Munksgaard and Pedersen, 2001] and income-based [Marques
et al., 2012] emissions. However, it has mostly focused on country-level analysis and generally lacks
axiomatic foundations. Our approach subsumes the different notions of responsibility introduced in
this literature and clarifies the relationships they entertain. It also provides axiomatic foundations
that were missing or incomplete.

From a more theoretical perspective, our approach relates to the network theory literature that
aims to provide axiomatic characterizations of centrality measures [see e.g. Jackson, 2008, Newman,
2010, for surveys]. Garg [2009] axiomatizes the degree, decay and closeness centralities. Palacios-
Huerta and Volij [2004], Slutzki and Volij [2006], Dequiedt and Zenou [2017], Kitti [2016] provide
characterizations of some prestige and eigenvector-related centrality measures. Boldi and Vigna
[2014] axiomatize a variation of closeness centrality. Bloch et al. [2023] characterize the standard
centrality measures within a unified framework. They introduce the concept of nodal statistics
which captures a node's position in a network and show that each of these centrality measures
can be viewed as an additively separable weighted average of some nodal statistic. A network
and centrality based approach is also adopted by King et al. [2019] in the context of carbon tax
reforms. They show that taxing specific sectors based on their position in the production network
can lead to a larger reduction of aggregate emissions than when targeting sectors based on only their
direct emissions. In order to assign responsibility for direct and indirect emissions, we investigate
and characterize axiomatically a more general centrality measure which corresponds to a convex
combination of the Bonacich centralities for the upstream and downstream weighted adjacency
matrices.

**Structure of the Paper** The remaining part of the paper is organized as follows. In section 2, we introduce our model and the responsibility measure, and illustrate the new setting by a simple numerical example. Section 3 provides an axiomatic characterization of the responsibility measure. Section 4 investigates relationships with existing measures. In section 5, we apply our approach to the allocation of responsibility for global emissions across countries and to the allocation of sectoral emissions in the USA. Section 6 provides concluding remarks. The paper contains three appendices. In Appendix A, we recall some basic concepts in network theory that are used in our study. Appendix B presents all proofs. Appendix C contains supplementary empirical results.
# 2 The Model

## 2.1 Responsibility Measure

We consider a production network formed by economic agents that are linked by supply relationships and whose productive activities generate GHG emissions. Formally, we denote by $N \subset \mathbb{N}$ the set of agents, by $F := (f_i)_{i \in N} \in \mathbb{R}_+^N$ the vector of GHG emissions where $f_i$ is the quantity of direct emissions from agent $i$, and by $A = (a_{i,j})_{i,j \in N} \in \mathbb{R}_+^{N \times N}$ the matrix of supply relationships where $a_{i,j}$ is the (nominal) value of the output of agent $i$ used as input by agent $j$. We refer to such a triple $(N, F, A)$ as an emission system. The relative strength of supply relationships might be of greater concern than the absolute one in our setting. Therefore, we associate to the matrix of supply relationships $A$, two auxiliary adjacency matrices normalized from the upstream and downstream perspective. Namely, we denote by $\phi_{i,j} := \frac{a_{j,i}}{\sum_{k \in N} a_{j,k}}$ the share of output from agent $j$ used as input by agent $i$. The matrix $\Phi := (\phi_{i,j})_{i,j \in N}$ is the adjacency matrix of the production network normalized from an upstream perspective: the weighted sum of links upstream towards a supplier equals 1, i.e., $\sum_{i \in N} \phi_{i,j} = 1$, and $\Phi$ is column-stochastic. Conversely, we denote by $\psi_{i,j} := \frac{a_{i,j}}{\sum_{k \in N} a_{k,j}}$ the share of agent $i$ in the input basket of agent $j$. The matrix $\Psi := (\psi_{i,j})_{i,j \in N}$ is the adjacency matrix of the production network normalized from a downstream perspective: the weighted sum of links downstream towards agent $j$ equals 1, i.e., $\sum_{i \in N} \psi_{i,j} = 1$ and $\Psi$ is column-stochastic. With a slight abuse of terminology, we shall refer to $(N, F, \Phi, \Psi)$ as an emission system interchangeably with $(N, F, A)$. We denote by $\mathcal{E}$ the set of all emission systems.

**Example 1** Let us consider a three-firm production network with the set of nodes $N = \{a, b, c\}$ and supply relationships given by the matrix $A$ such that

$$A = \begin{pmatrix} 0 & 1 & 2 \\ 4 & 0 & 3 \\ 0 & 0 & 5 \end{pmatrix}$$

For any vector of emissions $F = (f_a, f_b, f_c)^T \in \mathbb{R}_+^3$, the triple $(N, F, A)$ defines an emission system. A graphical representation of the emission system is provided in Figure 1.

In the setting of Example 1, the upstream adjacency matrix $\Phi$ is obtained by transposing links and ensuring that the sum of incoming links equals to 1 while the downstream adjacency matrix $\Psi$ is obtained by normalizing links such that the sum of incoming links towards each node equals 1. The
<figure><img src="image_2.png" /><figcaption>Figure 1: Graphical representation of the emission system (N, F, A)</figcaption></figure>

corresponding matrices are given below and represented graphically in Figure 2.

$$ \Phi = \begin{pmatrix} 0 & \frac{4}{7} & 0 \\ \frac{1}{3} & 0 & 0 \\ \frac{2}{3} & \frac{3}{7} & 1 \end{pmatrix} \qquad \Psi = \begin{pmatrix} 0 & 1 & \frac{2}{10} \\ 1 & 0 & \frac{3}{10} \\ 0 & 0 & \frac{5}{10} \end{pmatrix} $$

<figure><img src="image_3.png" /><figcaption>Figure 2: Upstream (left) and Downstream (right) adjacency matrix for the emission system (N, F, A)</figcaption></figure>

In an emission system, agent i uses a vector of input $a_{.,i} \in \mathbb{R}_+^N$ to produce a quantity of output $\sum_{j \in N} a_{i,j}$ that generates, as a by-product, a quantity $f_i$ of emissions. If all inputs are assumed to be necessary for production, the suppliers of agent i are, at least indirectly, necessary for the emissions $f_i$ to occur. Accordingly, if the existence of a demand is a necessary condition for production, customers of agent i are, at least indirectly, necessary for the emissions $f_i$ to occur. These causal relationships suggest that agents upstream and downstream from i in the production network shall carry a responsibility for the emissions $f_i$. In the following, we are concerned with the measure of this indirect responsibility, i.e., we aim to define a measure that associates with each node/agent in an emission system, the amount of emissions it shall be considered responsible for and/or that
it shall declare in its annual report statement, given its position in the production network and the distribution of emissions. More formally, we define an emission responsibility measure as follows.

**Definition 1** An emission responsibility measure is a mapping $\rho$ that associates to every emission system $(N, F, A) \in \mathcal{E}$, a vector $\rho(N, F, A) \in \mathbb{R}^N$ of responsibilities such that, for all $i \in N$, $\rho_i(N, F, A)$ is the amount of emissions for which agent $i$ is deemed responsible.

From a network theory perspective, defining the responsibility of agent $i$ amounts to define the centrality of agent $i$ in the network $A$ considering that each node $i$ is weighted by its emissions $f_i$. A standard approach in this context is to define centrality in terms of discounted sum of walks in the network [see e.g. Jackson, 2008]. In our setting, this amounts to define the responsibility for GHG emissions in a supply chain through the discounted sum of indirect emissions. Namely, we shall associate to agent $i$ in the emission system $E = (N, F, \Phi, \Psi)$, a measure of responsibility of the form:

$$ \bar{\rho}_i(E, \lambda, \gamma, \delta) = \lambda(1-\gamma) \sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \Phi_{i,j}^{\nu} f_j + (1-\lambda)(1-\delta) \sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \Psi_{i,j}^{\nu} f_j \quad (1) $$

where $\gamma, \delta \in [0, 1)$ are discount factors, $\lambda \in [0, 1]$ measures the relative share of responsibility for upstream and downstream emissions, $\Phi_{i,j}^{\nu}$ is the sum of walks of length $\nu$ upstream from $i$ to $j$ (given by the $(i, j)$ coefficient of the $\nu$th power of $\Phi$) and $\Psi_{i,j}^{\nu}$ is the sum of walks of length $\nu$ downstream from $i$ to $j$ (given by the $(i, j)$ coefficient of the $\nu$th power of $\Psi$). Hence, $\bar{\rho}_i(E, \lambda, \gamma, \delta)$ associates to agent $i$ a share of responsibility in the emissions generated by its upstream and downstream connections in the network that is proportional to the strength of their supply/demand relationships, as measured by the sum of walks, and discounted according to the network distance. Formally, the measure is similar to discounted intertemporal expected utility: the sum of weighted walks of length $\nu$ between $i$ and $j$ corresponds to the probability to reach node $j$ from node $i$ in $\nu$ steps. It is also closely related to Bonacich centrality [Bonacich, 1987]. Equation (1) can indeed be written as:

$$ \bar{\rho}(E, \lambda, \gamma, \delta) = \lambda(1-\gamma)(I-\gamma\Phi)^{-1}F + (1-\lambda)(1-\delta)(I-\delta\Psi)^{-1}F \quad (2) $$

where $\bar{\rho} := (\bar{\rho}_i)_{i \in N}$. Hence, $\bar{\rho}$ corresponds to a convex combination of the Bonacich centralities for the upstream and downstream adjacency matrices weighted by the vector of emissions.

**Example 1 (continued)** In the case of Example 1 considered above, Equation (2) implies that the
responsibility measure for firms a, b and c is equal to:

$$ \bar{\rho}_a(E, \lambda, \gamma, \delta) = \lambda \left[ \frac{21(1-\gamma)}{21-4\gamma^2} f_a + \frac{12\gamma(1-\gamma)}{21-4\gamma^2} f_b \right] \\ + (1-\lambda) \left[ \frac{1}{1+\delta} f_a + \frac{\delta}{1+\delta} f_b + \frac{\delta(3\delta+2)}{5(2-\delta)(1+\delta)} f_c \right] $$

$$ \bar{\rho}_b(E, \lambda, \gamma, \delta) = \lambda \left[ \frac{7\gamma(1-\gamma)}{21-4\gamma^2} f_a + \frac{21(1-\gamma)}{21-4\gamma^2} f_b \right] \\ + (1-\lambda) \left[ \frac{\delta}{1+\delta} f_a + \frac{1}{1+\delta} f_b + \frac{\delta(2\delta+3)}{5(2-\delta)(1+\delta)} f_c \right] $$

$$ \bar{\rho}_c(E, \lambda, \gamma, \delta) = \lambda \left[ \frac{\gamma(3\gamma+14)}{21-4\gamma^2} f_a + \frac{\gamma(8\gamma+9)}{21-4\gamma^2} f_b + f_c \right] \\ + (1-\lambda) \frac{2(1-\delta)}{2-\delta} f_c $$

Note that if $\gamma = \delta = 0$, then $\bar{\rho}(E, \lambda, \gamma, \delta) = \begin{pmatrix} f_a \\ f_b \\ f_c \end{pmatrix} = F$.

Moreover, if $\gamma \to 1$ and $\delta \to 1$, then

$$ \bar{\rho}(E, \lambda, \gamma, \delta) \to \lambda \begin{pmatrix} 0 \\ 0 \\ f_a + f_b + f_c \end{pmatrix} + (1-\lambda) \begin{pmatrix} \frac{1}{2}(f_a + f_b + f_c) \\ \frac{1}{2}(f_a + f_b + f_c) \\ 0 \end{pmatrix} $$

These last two results correspond to the general results for the particular cases ($\gamma = \delta = 0$, $\gamma \to 1$ and $\delta \to 1$), as will be discussed in Section 4.

## 3 Axiomatic Foundations of the Responsibility Measure

Normative foundations are crucial if the discounted emission responsibility measure is to be used in a legal context or included in standardised corporate disclosures. In order to provide such normative foundations, we shall show that $\bar{\rho}$ is characterized by a set of axioms that are consistent with legal principles underlying responsibility [Burger et al., 2020]. These axioms are closely related to these used for the axiomatization of network centrality measures [in particular Bloch et al., 2023] and discounted expected utility.
In this respect, a first standard requirement is to assume that the responsibility measure is anonymous, i.e., that the responsibility of an agent is not influenced by its identity. Namely, one has:

**Axiom 1 (Anonymity)** The responsibility measure $\rho$ is anonymous if for all emission systems $E = (N, F, \Phi, \Psi) \in \mathcal{E}$, for all permutations $\pi$ of $N$ and for all $i \in N$ one has:

$$ \rho_i(E, \lambda, \gamma, \delta) = \rho_{\pi(i)}(E_{\pi(i)}, \lambda, \gamma, \delta) $$

where $E_{\pi} := (N, F_{\pi}, \Phi_{\pi}, \Psi_{\pi})$, $F_{\pi} := (F_{\pi(i)})_{i \in N}$, $\Phi_{\pi} := (\Phi_{\pi(i),\pi(j)})_{i,j \in N}$ and $\Psi_{\pi} := (\Psi_{\pi(i),\pi(j)})_{i,j \in N}$.

From a more fundamental perspective, our second axiom defines the drivers of responsibility. It shall increase with (direct and indirect) emissions and with the strength of relationships with upstream/downstream emitters. Formally, let us associate to agent $i$ in the emission system $(N, F, \Phi, \Psi)$, the (generalized) nodal statistic $(u_i(F, \Phi), v_i(F, \Psi)) \in \mathbb{R}_+^N \times \mathbb{R}_+^N$ defined by:

$$ u_i(F, \Phi) := \left( \sum_{j \in N} \Phi_{i,j}^{\nu} f_j \right)_{\nu \in \mathbb{N}} = (f_i, \sum_{j \in N} \Phi_{i,j} f_j, \dots, \sum_{j \in N} \Phi_{i,j}^{\nu} f_j, \dots) $$

$$ v_i(F, \Psi) := \left( \sum_{j \in N} \Psi_{i,j}^{\nu} f_j \right)_{\nu \in \mathbb{N}} = (f_i, \sum_{j \in N} \Psi_{i,j} f_j, \dots, \sum_{j \in N} \Psi_{i,j}^{\nu} f_j, \dots) $$

The statistic $u_i(F, \Phi)$ corresponds to the sequence of upstream emissions for agent $i$, i.e., for each $\nu \in \mathbb{N}$, the sum of emissions occurring $\nu$ steps upstream from $i$ weighted by the (indirect) share of $i$ in the consumption of the corresponding output. Respectively, $v_i(F, \Psi)$ corresponds to the sequence of downstream emissions for agent $i$, i.e., for each $\nu \in \mathbb{N}$, the sum of emissions occurring $\nu$ steps downstream from $i$ weighted by the (indirect) share of $i$ in the corresponding input mix. Our second axiom assumes that the responsibility measure is monotonic with respect to the Euclidian partial order on the sequences of upstream and downstream emissions, namely<fnref n="3" />:

**Axiom 2 (Monotonicity)** The responsibility measure $\rho$ is monotonic if for all $i \in N$, and all $(F, \Phi, \Psi)$ and $(\tilde{F}, \tilde{\Phi}, \tilde{\Psi})$ such that for all $\nu \in \mathbb{N}$:

$$ \sum_{j \in N} \Phi_{i,j}^{\nu} f_j \geq \sum_{j \in N} \tilde{\Phi}_{i,j}^{\nu} \tilde{f}_j \quad \text{and} \quad \sum_{j \in N} \Psi_{i,j}^{\nu} f_j \geq \sum_{j \in N} \tilde{\Psi}_{i,j}^{\nu} \tilde{f}_j, $$

one has:

$$ \rho_i(E, \lambda, \gamma, \delta) \geq \rho_i(\tilde{E}, \lambda, \gamma, \delta), $$

where $E = (N, F, \Phi, \Psi)$ and $\tilde{E} = (N, \tilde{F}, \tilde{\Phi}, \tilde{\Psi})$.

<footnote n="3">³We denote by $\geq$ the partial Euclidian order on $\mathbb{R}^N$ so that $u \geq v$ if and only if $u_n \geq v_n$ for all $n \in \mathbb{N}$.</footnote>
Hence, $\rho$ satisfies monotonicity if the responsibility of an agent (weakly) increases when its weighted share in upstream and downstream emissions increases. In particular, ceteris paribus, the responsibility of an agent (weakly) increases (i) if emissions increase upstream or downstream in its supply chain or (ii) if the strengths of its supply or demand relationships increase.

Following [Bloch et al., 2023], anonymity and monotonicity imply that the responsibility measure is a function of the sequence of upstream and downstream emissions. Namely, one has:

**Lemma 1** If the responsibility measure $\rho$ satisfies anonymity and monotonicity, then there exists a function $\sigma : \mathbb{R}_+^N \times \mathbb{R}_+^N \to \mathbb{R}$ such that for all emission systems $E = (N, F, \Phi, \Psi)$, for all $i \in N$, one has

$$ \rho_i(E, \lambda, \gamma, \delta) = \sigma(u_i(F, \Phi), v_i(F, \Psi)). $$

Furthermore, $\sigma$ is monotonic with respect to the Euclidian partial order on $\mathbb{R}_+^N \times \mathbb{R}_+^N$.

In view of Lemma 1, assuming in the following that anonymity and monotonicity hold, further axioms can equivalently be stated in terms of $\rho$ or $\sigma$. In particular, our third axiom assumes the additivity of the responsibility measure.

**Axiom 3 (Additivity)** A responsibility measure that is monotonic and anonymous is additive if for all $(u, v), (u', v') \in \mathbb{R}_+^N \times \mathbb{R}_+^N$, one has

$$ \sigma((u, v) + (u', v')) = \sigma(u, v) + \sigma(u', v'). $$

Additivity plays in our setting a similar role to the independence axiom in the Von Neumann-Morgenstern expected utility theory. It assumes that the change in responsibility induced by additional upstream or downstream emissions is independent of the preexisting level of responsibility. It can also be interpreted as a form of invariance with respect to corporate structure: total responsibility shall not be modified if an agent is split into subunits or if two agents are merged. The opposite could lead to strategic behaviour in the choice of organisational/corporate structure to reduce total responsibility. Additivity implies in particular that one can separate upstream and downstream responsibility of an agent. Namely:

**Lemma 2** If the responsibility measure $\rho$ satisfies anonymity, monotonicity and additivity, then there exist functions $\alpha : \mathbb{R}_+^N \to \mathbb{R}$ and $\beta : \mathbb{R}_+^N \to \mathbb{R}$ such that for all emission systems $E = (N, F, \Phi, \Psi)$ and for all $i \in N$, one has

$$ \rho_i(E, \lambda, \gamma, \delta) = \alpha(u_i(F, \Phi)) + \beta(v_i(F, \Psi)). $$

Furthermore, $\alpha$ and $\beta$ are additive and monotone.
In view of Lemma 2, assuming that anonymity, monotonicity and additivity hold, further axioms can be stated in terms of the functions $\alpha$ and $\beta$. In this setting, our fourth axiom characterizes the tail behaviour of the responsibility measure.

**Axiom 4 (Tail-convergence)** A responsibility measure that is monotonic, anonymous and additive is tail-convergent if for every $(u, v) \in \mathbb{R}_+^N \times \mathbb{R}_+^N$, one has

$$ \lim_{\nu \to +\infty} \alpha(u_1, \dots, u_\nu, 0, \dots) = \alpha(u) \quad \text{and} \quad \lim_{\nu \to +\infty} \beta(v_1, \dots, v_\nu, 0, \dots) = \beta(v). $$

The tail-convergence axiom implies that distant upstream and downstream emissions eventually become negligible so that the responsibility for an emission sequence can be approximated by truncated emission sequences of sufficient length. Tail-convergence ensures continuity and consistent measurement of infinite emission sequences. Our fifth axiom assumes the recursivity of the responsibility measure in the sense of Bloch et al. [2023]. Namely:

**Axiom 5 (Recursivity)** A responsibility measure that is monotonic, anonymous and additive is recursive if for all $(u, v), (u', v') \in \mathbb{R}_+^N \times \mathbb{R}_+^N$ and all $\nu \in \mathbb{N}$, one has

$$ \frac{\alpha(0, \dots, 0, u_{\nu+1}, u_{\nu+2}, \dots)}{\alpha(0, \dots, 0, u'_{\nu+1}, u'_{\nu+2}, \dots)} = \frac{\alpha(u_{\nu+1}, u_{\nu+2}, \dots)}{\alpha(u'_{\nu+1}, u'_{\nu+2}, \dots)} $$

and

$$ \frac{\beta(0, \dots, 0, v_{\nu+1}, v_{\nu+2}, \dots)}{\beta(0, \dots, 0, v'_{\nu+1}, v'_{\nu+2}, \dots)} = \frac{\beta(v_{\nu+1}, v_{\nu+2}, \dots)}{\beta(v'_{\nu+1}, v'_{\nu+2}, \dots)}. $$

The recursivity axiom is closely related to the stationarity axioms introduced by [Koopmans, 1960] in its axiomatization of discounting. In our setting, the axiom implies that the responsibility ratio between two emission patterns is independent of their level of upstreamness or downstreamness. In other words, the relative importance of an emission pattern does not depend on its level of upstreamness/downstreamness. Consistently with Koopmans [1960] and Bloch et al. [2023], recursivity implies that the responsibility measure is given by the discounted sum of emissions. Namely:

**Lemma 3** If the responsibility measure $\rho$ satisfies anonymity, monotonicity, additivity, tail-convergence and recursivity, then there exist $\gamma, \delta \in [0, 1]$ and $a, b \in \mathbb{R}_+$ such that for all emission systems $E = (N, F, \Phi, \Psi)$ and for all $i \in \mathbb{N}$, one has

$$ \rho_i(E, \lambda, \gamma, \delta) = a\left(\sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \Phi_{i,j}^{\nu} f_j\right) + b\left(\sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \Psi_{i,j}^{\nu} f_j\right). $$
Our final axiom ensures the consistency of the responsibility measure by requiring that it covers all emissions in the system. Namely:

**Axiom 6 (Consistency)** The responsibility measure $\rho$ is consistent if for all emission systems $E = (N, F, \Phi, \Psi) \in \mathcal{E}$, one has:

$$ \sum_{i \in N} \rho_i(E, \lambda, \gamma, \delta) = \sum_{i \in N} f_i. $$

Consistency ensures that responsibility is allocated for all emissions in the system and that there is no double-counting of emissions. In other words, the responsibility for every unit of emission is allocated once and only once. This property is required for a range of empirical applications, in particular, to define liability in an accounting or legal setting. It also allows us to compare the responsibility measure with direct emissions for each agent. Formally, the consistency axiom leads to a normalization of the responsibility measure and its characterization. Namely, one has:

**Theorem 1** A responsibility measure satisfies anonymity, monotonicity, additivity, tail-convergence, recursivity and consistency if and only if, for every emission system $E = (N, F, \Phi, \Psi)$, it is of the form

$$ \bar{\rho}_i(E, \lambda, \gamma, \delta) = \lambda(1-\gamma) \sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \Phi_{i,j}^{\nu} f_j + (1-\lambda)(1-\delta) \sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \Psi_{i,j}^{\nu} f_j $$

with $\gamma, \delta \in [0, 1)$ and $\lambda \in [0, 1]$.

Theorem 1 thus provides the desired normative foundations for the discounted emission responsibility measures.

# 4 Relation with Existing Measures

In this section, we shall show that a number of responsibility measures currently in use or put forward in the literature are in fact particular cases of the discounted emission metric $\bar{\rho}$.

## 4.1 Upstream vs Downstream Responsibility and Direct Emissions

First, if $\lambda = 1$, the responsibility measure is of the form

$$ \bar{\rho}_i(E, \lambda, \gamma, \delta) = (1 - \gamma) \sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \Phi_{i,j}^{\nu} f_j $$
and thus only accounts for upstream emissions. Conversely, if $\lambda = 0$, the responsibility measure is of the form

$$ \bar{\rho}_i(E, \lambda, \gamma, \delta) = (1 - \delta) \sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \Psi_{i,j}^{\nu} f_j $$

and thus only accounts for downstream emissions.

Second, if $\gamma = \delta = 0$, one has $\bar{\rho}_i(E, \lambda, \gamma, \delta) = f_i$ and the responsibility measure only accounts for direct emissions. It thus coincides with scope 1 emissions in the sense of the GHG protocol [WRI and WBCSD, 2011]. In such a setting, discounting is “infinitely strong” and indirect emissions are discarded.

## 4.2 No-discounting Limit and Independence From Own Emissions

We then consider the case when $\gamma$ and $\delta$ tend towards 1 so that the effect of discounting becomes negligible and indirect emissions become as important as direct ones. The limit of the responsibility measure can be characterized through the eigenvector centrality of the adjacency matrices. More precisely, let $Eig(M)$ denote a matrix for which all rows are equal to the normalized vector of eigenvector centralities $eig(M)$ in $M$ with the sum of coordinates equal to 1. Let $Eig^T(M)$ be the transpose of $Eig(M)$ with its $i$-row denoted by $Eig_i^T(M)$. Then one has:

**Proposition 1** If the adjacency matrix $A$ is aperiodic and irreducible, then for every $i \in N$:

$$ \lim_{(\gamma, \delta) \to (1,1)} \lambda(1-\gamma) \sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \Phi_{i,j}^{\nu} f_j + (1-\lambda)(1-\delta) \sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \Psi_{i,j}^{\nu} f_j = \\ \lambda Eig_i^T(\Phi) F + (1-\lambda) Eig_i^T(\Psi) F. $$

Hence, in the “no-discounting” limit, the responsibility of an agent is independent of its level of direct or indirect emissions but only depends on its network position. The more central an agent, the larger its responsibility for GHG emissions.

**Example 2** Applying Proposition 1 to the emission system introduced in Example 1, one has

$$ \mathrm{Eig}^T(\Phi) = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{pmatrix} \qquad \mathrm{Eig}^T(\Psi) = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 0 \end{pmatrix} $$

consistently with the computations of $\bar{\rho}(E, \lambda, \gamma, \delta)$ performed in Example 1.
## 4.3 Consumption-based accounting

Consumption-based accounting corresponds to a particular case of the responsibility where only upstream responsibility is taken into account (i.e., $\lambda = 1$) and one considers the no-discounting limit ($\gamma \to 1$ as considered in Proposition 1). Indeed, in its standard formulation [see Yamano and Guilhoto, 2020, Rodrigues and Domingos, 2008, Gallego and Lenzen, 2005], consumption-based accounting considers a set of $L$ countries and $M$ industries per country. Let us denote by $\mathcal{L}$ the set of countries, by $\mathcal{M}$ the set of industries and by $\mathcal{J} = \mathcal{L} \times \mathcal{M}$ the set of country-industry pairs. In the following, let $i, j \in \mathcal{J}$ and $k \in \mathcal{L}$. Production relationships are defined by an intermediary consumption matrix $C = (c_{i,j}) \in \mathbb{R}_+^{\mathcal{J} \times \mathcal{J}}$ where $c_{i,j}$ corresponds to the amount of input that country-industry pair $j$ requires from $i$ in order to produce one unit of output. The final demand to each country-industry pair from each country is given by a matrix $D = (d_{i,k}) \in \mathbb{R}_+^{\mathcal{J} \times \mathcal{L}}$ where $d_{i,k}$ is the final demand from country $k$ to the country-industry pair $i$. Emission intensities are given by a vector $g \in \mathbb{R}_+^{\mathcal{J}}$ where $g_i$ is the amount of GHG emissions per unit of production for the country-industry pair $i$.

In this setting, the vector of country-level consumption-based emissions $\chi \in \mathbb{R}^\mathcal{L}$ is defined as

$$ \chi = e^T \mathrm{diag}(g) (I - C)^{-1} D = \left( \sum_{j \in \mathcal{J}} \sum_{i \in \mathcal{J}} g_j (I - C)^{-1}_{j,i} D_{i,k} \right)_{k \in \mathcal{L}} $$

where $e = (1, \dots, 1) \in \mathbb{R}^\mathcal{J}$ is the column vector of all ones, diag($g$) denotes the diagonal matrix whose diagonal is given by $g$. Moreover, $(I - C)^{-1} D$ is the $\mathcal{J} \times \mathcal{L}$ matrix whose coefficients correspond to the total production in each country-sector pair required to fulfil the final demand. Namely, $(I - C)^{-1}$ is the Leontief inverse [Leontief, 1970] that we will denote by $Q = (q_{i,j})_{i,j \in \mathcal{J}}$, an element $q_{i,j}$ of which represents the output from $i$ that is directly and indirectly required to meet the demand of $j$. We denote by $q_i = \sum_{j \in \mathcal{J}} q_{i,j}$ the total output of country-industry pair $i \in \mathcal{J}$. Accordingly, the total direct emissions $f_i$ of country-industry pair $i$ are given by $f_i = g_i q_i$ and we denote by $v_{i,j} = c_{i,j} q_j$ the total amount of output of $i$ used by $j$.

We then consider the network with nodes in $\mathcal{N} := \mathcal{J} \cup \mathcal{L}$, i.e., one node per country-industry pair and one node to represent final demand in each country. As emphasized above, we focus on the upstream adjacency matrix. Hence, for $i, j \in \mathcal{J}$, the value of the adjacency coefficient is given by the share of output from $j$ used by $i$ as input, $\phi_{i,j} = \frac{v_{j,i}}{q_j}$. For $k \in \mathcal{L}$ and $j \in \mathcal{J}$, the value of the adjacency coefficient is given by the share of output from $j$ supplied to $k$ as final demand $\phi_{k,j} = \frac{D_{j,k}}{q_j}$. Finally, by convention, we consider that $k \in \mathcal{L}$ only has supply relationship towards itself so that $\phi_{k,k} = 1$ and $\phi_{j,k} = 0$ for $j \neq k$; see Figure 3. Hence, the upstream adjacency matrix $\Phi \in \mathbb{R}^{\mathcal{N} \times \mathcal{N}}$ is column-stochastic and consumption-based accounting corresponds to the upstream
<figure><img src="image_4.png" /><figcaption>Figure 3: Consumption-based accounting. Six country-industry pairs are represented by the circle nodes and the final demand of the countries is represented by the square nodes. The solid arrows represent the supply from one country-industry pair to another, whereas the dashed arrows represent the final demand from a country to a country-industry pair.</figcaption></figure>

responsibility in the network in the no-discounting limit. Namely, one has

**Proposition 2** For the emission system defined above, one has for all $k \in \mathcal{L}$:

$$ \chi_k = \sum_{j \in \mathcal{J}} \sum_{i \in \mathcal{I}} g_j (I - C)_{j,i}^{-1} D_{i,k} = \lim_{\gamma \to 1} \sum_{j \in \mathcal{J}} (1 - \gamma)(I - \gamma\Phi)_{k,j}^{-1} f_j. $$

## 4.4 Income-based accounting

Besides considering the emissions embodied in the final demand, one can also consider emissions embodied in the primary inputs. This approach is referred to as income-based or revenue-based accounting in the literature [Marques et al., 2012, Liang et al., 2017, Rodrigues et al., 2006, Pinero et al., 2019]. It corresponds to the particular case of the discounted emission metric where we only take downstream responsibility into account ($\lambda = 0$) and consider the no-discounting limit ($\delta \to 1$).

Using the same notation as in Section ?? unless stated otherwise, we let $\bar{C} \in \mathbb{R}_{+}^{\mathcal{J} \times \mathcal{J}}$ be the intermediary output matrix, an element $\bar{c}_{i,j}$ of which is the share of output of country-industry pair $i$ that is used by country-industry pair $j$. That is, $\bar{c}_{i,j} = \frac{v_{i,j}}{\bar{q}_i}$ with $\bar{q}_i$ the total output of $i$ from
its direct and indirect inputs. We denote the primary input from every country $k \in \mathcal{L}$ to every country-industry pair $i \in \mathcal{J}$ by the matrix $P = (p_{k,i}) \in \mathbb{R}_+^{\mathcal{L} \times \mathcal{J}}$. Then, the vector of country-level income-based emissions $\xi \in \mathbb{R}^\mathcal{L}$ is given by

$$ \xi = P(I - \bar{C})^{-1}\mathrm{diag}(g)e = \left( \sum_{j \in \mathcal{J}} \sum_{i \in \mathcal{J}} P_{k,i} (I - \bar{C})^{-1}_{i,j} g_j \right)_{k \in \mathcal{L}} $$

where $(I - \bar{C})^{-1} = (\bar{q}_{i,j})_{i,j \in \mathcal{J}}$ is the Ghosh inverse [Ghosh, 1958] with $\bar{q}_{i,j}$ representing the value of production in country-industry $j$ per unit of input in country-industry pair $i$. Accordingly, $P(I - \bar{C})^{-1} \in \mathbb{R}^{\mathcal{L} \times \mathcal{J}}$ represents the production in every country-industry pair that is required for the primary inputs to be attainable. Again, we consider the network with nodes $\mathcal{N} := \mathcal{L} \cup \mathcal{J}$ where the nodes $k \in \mathcal{L}$ now represent the primary input in country $k$. Considering the downstream matrix, we obtain for $k \in \mathcal{L}$ and $i, j \in \mathcal{J}$ the following values of the adjacency coefficients: $\psi_{k,j} = \frac{P_{k,j}}{\bar{q}_j}$, and $\psi_{i,j} = \frac{v_{i,j}}{\bar{q}_j}$. As before, we have by convention $\psi_{k,k} = 1$ and $\psi_{j,k} = 0$. This leads to column-stochastic matrix $\Psi \in \mathbb{R}_+^{\mathcal{N} \times \mathcal{N}}$, for which we then have:

**Proposition 3** For the emission system defined above, one has for all $k \in \mathcal{L}$:

$$ \xi_k = \sum_{j \in \mathcal{J}} \sum_{i \in \mathcal{J}} P_{k,i} (I - B)^{-1}_{i,j} g_j = \lim_{\delta \to 1} \sum_{j \in \mathcal{J}} (1 - \delta)(I - \delta\Psi)^{-1}_{k,j} f_j. $$

# 5 Applications

In this section we illustrate the application of the discounted emission responsibility measure to the allocation of global emissions across countries and to the allocation of sectoral responsibility in the USA economy.

## 5.1 Data

We use the world input-output (WIOD) [2016 release, see Timmer et al., 2015] and the associated Environmental Accounts [Corsatea et al., 2016]. A world input-output table provides information about all the transactions between industries and final users in the global economy. The data concern the year 2014 and contain the trade of all 56 sectors of all EU-countries plus seventeen large economies. With 56 sectors and 44 countries, we thus obtain a matrix with 2464 rows and 2464 columns. The columns regard the production processes, the corresponding products of which
can be either used as intermediates for other sectors, or as final products by firms, governments or households. The rows provide information on how the output of industries is distributed over those user categories. Denote by A the world input-output matrix, then, an element $a_{i,j}$ is thus the value of the output of agent i used as input by agent j. The Environmental Account data consists of the carbon emissions of all sectors of these countries.<fnref n="4" />

## 5.2 Allocation of responsibility for global emissions

<table><caption>Table 1: Responsibility for Switzerland (CHE), China (CHN), France (FRA), United States of America (USA), Russia (RUS) with $\gamma = 0.7$ and $\delta = 0.7$, in kton $CO_2$.</caption><thead><tr><th>Country</th><th>Emissions</th><th>λ = 1 (upstream)</th><th>λ = 0 (downstream)</th><th>λ = 1/2</th></tr></thead><tbody><tr><td>CHE</td><td>44,067</td><td>99,565</td><td>145,574</td><td>122,570</td></tr><tr><td>CHN</td><td>10,528,550</td><td>10,516,086</td><td>10,234,564</td><td>10,375,325</td></tr><tr><td>FRA</td><td>345,586</td><td>504,557</td><td>478,519</td><td>491,538</td></tr><tr><td>USA</td><td>5,257,452</td><td>5,433,719</td><td>5,198,942</td><td>5,3163,31</td></tr><tr><td>RUS</td><td>1,723,086</td><td>1,208,205</td><td>1,777,928</td><td>1,493,067</td></tr></tbody></table>

In Table 1, we can see some examples of assigned responsibility in kton $CO_2$ for Switzerland (CHE), China (CHN), France (FRA), United States of America (USA) and Russia (RUS) when we set both $\gamma = 0.7$ and $\delta = 0.7$. All the upstream and downstream responsibilities for all countries and for various levels of discounting are given in Table 2 and Table 3 in Appendix C. In Figure 4 we see both the upstream and downstream responsibility for Brazil, Canada, Indonesia, South Korea and France, with $\nu_{max} = 1000$, for various values of $\gamma$ and $\delta$.

There are a few interesting remarks we can make based on this responsibility data. First, we observe that the responsibility of the most developed and service-oriented economies increases rapidly when one accounts for indirect emissions upstream or downstream in the supply chain. For those countries, the value that $\gamma$ and $\delta$ take makes a large difference in what responsibility they are assigned. For example, for Switzerland, when the discounting factor $\delta$ is high, for example $\delta = 0.9$ and $\lambda = 0$, its responsibility is almost 6 times its direct emissions. The same effect occurs when we take $\lambda = 1$ and evaluate the upstream responsibility. In this case, for $\gamma = 0.9$ the responsibility is approximately 3.5 times as large as the direct emissions. We can see this in Figure 5. Other examples of such countries are The Netherlands and Luxembourg.

Secondly, we can distinguish a class of countries with manufacturing-oriented economies that have high direct emissions for whom the share of responsibility decreases as one increasingly accounts for upstream or downstream emissions in the supply chain. A clear example of such a country is

<footnote n="4">⁴The data set does not include the carbon emissions of the Netherlands, which have therefore been added manually with the data from [Olivier et al., 2015].</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 4: Upstream (left) and downstream responsibility for Brazil, Canada, Indonesia, Korea and France in kton CO₂, with $v_{max}$ = 1000, for $\gamma$ and $\delta$ ranging from 0.0 to 0.99.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Upstream and downstream responsibility for Switzerland in kton CO₂, with $v_{max}$ = 1000, for various values of $\gamma$ and $\delta$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6: Upstream and downstream responsibility for China in kton CO₂, with $v_{max}$ = 1000, for various values of $\gamma$ and $\delta$.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: Upstream and downstream responsibility for Poland, $\nu_{max}$ = 1000, for various values of $\gamma$ and $\delta$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8: Upstream and downstream responsibility for Norway, $\nu_{max}$ = 1000, for various values of $\gamma$ and $\delta$.</figcaption></figure>

China, which has a high input as well as output, but due to its high direct emissions, the direct responsibility is very large compared to the indirect responsibility. Accounting from a production-based point of view yields a high responsibility for China, while moving towards a consumption-based point of view by increasingly accounting for the emissions in the supply chain results in a lower responsibility, as we can see in Figure 6. Other countries in this class are for example India and Indonesia. These patterns indeed confirm that in the “no-discounting” limit, the responsibility of an agent depends only on its network position and not on its direct and indirect emissions.

Thirdly, we find an intermediary class of economies for whom the responsibility does not always increase or decrease monotonically when increasingly accounting for emissions in the supply chain. An example of this is given by the downstream responsibility of Poland, whose responsibility decreases when $\delta$ increases until $\delta \approx 0.85$, but increases when $\delta$ continues to increase, as depicted in Figure 7. From the downstream perspective, we see similar behaviour for example for Hungary and Romania. The same behaviour but from the upstream perspective is exhibited for example by Mexico, Turkey or Canada. The countries in this class have a relatively high emission intensity but are more closely connected to service-oriented countries than to emission-intensive manufacturing countries. Then, as we can interpret $\delta$ and $\gamma$ as a factor that determines to what extent we take the walks that are sufficiently far away into account, the centrality of the agent in the network can differ for various values of $\delta$ and $\gamma$ which results in different responsibilities and non-monotonic behaviour with respect to the discounting factor.

Remark that for most countries, the upstream responsibility and downstream responsibility show the same kind of behaviour. However, when a country either exports or imports more, the upstream
responsibility can show different behaviour. Norway, known for its export of emission-intensive petroleum gases and oils, is an insightful example: its downstream responsibility is substantially higher than its upstream responsibility, as depicted in Figure 8.

## 5.3 Sectoral allocation of responsibility for US emissions

In order to allocate responsibility within one country, we assign responsibility on a sectoral level of the USA. Instead of aggregating over all sectors with the data of Timmer et al. [2015] and Corsatea et al. [2016] as in the previous section, we now just evaluate the input and output for every sector of the USA and evaluate the responsibility. For the 56 sectors, we use the NACE classification codes, which can be found in Appendix C. The total amount of emissions of the USA that we want to allocate is 4.343.481 kton CO2. In Appendix C, one can find the responsibility of every (sub-)sector for various values of γ and δ. In the following, we evaluate the responsibility of a few sectors.

According to the Environmental Accounts [Corsatea et al., 2016], about 48% of the emissions of the USA in 2014 comes from the sector *D35: Electricity, gas, steam and air conditioning supply*. For this sector, there is a large discrepancy between the share of emissions of this sector and the share of total input and output, which are respectively only 0.9% and 1.6%. As we increasingly account for indirect emissions, the responsibility of this sector decreases rapidly, as can be seen in Figure 9. The same behaviour is exhibited by for example the Transport sector *H*, which consists of the subsectors *H49 Land transport and transport via pipelines*; *H50 Water transport*; *H51 Air transport*; *H52 Warehousing and support activities for transportation* and *H53 Postal and courier activities*, and that emits 29% of the total USA emissions. For this sector, the responsibility also decreases very quickly when the discount factors γ or δ increase.

The sector *B: Mining and Quarrying*, which embeds fossil fuel extraction, exhibits non-monotonic behaviour. We see that the responsibility decreases when we increasingly account for the upstream emissions in the supply chain. On the other hand, accounting for downstream emissions in the supply chain, results in a non-monotonic increase in responsibility, see Figure 10. From the downstream perspective, this sector is closely connected to the emission-intensive sectors such as the energy utilities and transport sector, but in the overall economic system, this sector of fossil fuel extraction does not have a central position.

Lastly, the manufacturing sector *C*, which consists of nineteen subsectors amongst which the manufacture of food and beverages, textiles, and machinery, emits 15% of the total emissions of the USA, and has a 25% share of output and a share of input of 27%. The upstream and downstream responsibility of this sector as a whole are depicted in Figure 11. Within this sector, though, there can be large differences. For example, in the upstream perspective, the manufacture of food products, beverages and tobacco products (C10-C12), as well as the manufacture of motor vehicles
<figure><img src="image_10.png" /></figure>

Figure 9: Upstream and downstream responsibility for sector D35: Electricity, gas, steam and air conditioning supply in kton CO₂, with $v_{max}$ = 1000, for $\gamma$ and $\delta$ ranging from 0.0 to 0.99.

<figure><img src="image_11.png" /></figure>

Figure 10: Upstream and downstream responsibility for the Mining and Quarrying sector B in kton CO₂, with $v_{max}$ = 1000, for $\gamma$ and $\delta$ ranging from 0.0 to 0.99.

and trailers (C29) has a high centrality and therefore a high responsibility compared to its direct emissions in the non-discounting limit, whereas the manufacture of chemicals (C20) has a fairly low centrality and therefore low responsibility in this limit. We can see these differences in the upstream responsibility in Figure 12.

# 6 Concluding Remarks

In this paper, we consider the problem of allocating responsibility for emissions of greenhouse gases in production networks. We thus introduce a class of responsibility measures based on the discounted emissions across paths in the production network. We show that this measure has a strong normative foundation as it is characterized by a set of axioms consistent with legal principles underlying responsibility.

A policy maker can adjust the measure to its use case (e.g. allocation of responsibility across emitters in a system, definition of individual liability in a legal context) by adjusting the discount factors and relative share of responsibility for upstream and downstream emissions. In particular, when indirect emissions are infinitely discounted, the responsibility of each agent coincides with its direct emissions. Oppositely, in the “no-discounting” limit, the responsibility of an agent depends entirely on its position in the network and is independent of its level of emissions. Furthermore, a number of responsibility measures currently in use or put forward in the literature, including
<figure><img src="image_12.png" /><figcaption>Figure 11: Upstream and downstream responsibility for the Manufacturing sector *C*, in kton CO₂, with $v_{max}$ = 1000, for $\gamma$ and $\delta$ ranging from 0.0 to 0.99</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 12: Upstream responsibility for all sub-sectors of the Manufacturing sector *C*, in kton CO₂, with $v_{max}$ = 1000, for $\gamma$ ranging from 0.0 to 0.99</figcaption></figure>

consumption-based and income-based accounting, are obtained as a particular case of the discounted emission measure.

The responsibility measure can have far-reaching policy implications. As this measure is shown to be consistent, we overcome the issue of double counting, which could facilitate setting and meeting GHG emission reduction goals. Alternatively, the responsibility measure could also serve as a basis for carbon taxing when agents pay taxes over the amount of emissions they are responsible for, or the responsibility measure could guide climate litigation cases.

## Acknowledgement

This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 956107, "Economic Policy in Complex Environments (EPOC)".

## Appendix A – Basic Concepts in Network Theory

In this appendix, we recall some basic concepts in network theory (see, e.g., Jackson [2008]) that are used in the main part of the paper.

A *directed network* or *directed graph* (*N*, *g*) is an ordered pair of a set *N* = {1, ..., *n*} of nodes also called agents or players and a set of edges *g* ⊆ {*i*j ∈ *N* × *N*}, where *i*j denotes the directed link
between node *i* and *j*. If *ij* ∈ *g* for *i* = *j*, the network is said to have *loops*. Equivalently, a directed network is represented by a set of nodes *N* = {1,...,*n*} and the *adjacency matrix G* = [*g*<sub>*ij*</sub>] being a real-valued *n* × *n* matrix with *g*<sub>*ij*</sub> ∈ [0,1], where *ij* ∈ *g* if and only if *g*<sub>*ij*</sub> > 0. If *g*<sub>*ij*</sub> ∈ {0,1} for all *i*,*j* ∈ *N*, then the network is referred to as an *unweighed* network, otherwise, i.e., if the entries of **G** can take on more than two values 0 or 1, the network is *weighted*. When *N* is fixed or given, one usually refers to *g* as being a network.

A *walk* in a network *g* between *i* and *j* is a sequence of links *i*₁*i*₂, ..., *i*ₖ₋₁*i*ₖ such that *i*ₖ*i*ₖ₊₁ ∈ *g* for each *k* ∈ {1,...,K-1}, where *i*₁ = *i* and *i*ₖ = *j*. Note that for a walk, the nodes or the links in the sequence *i*₁, ..., *i*ₖ may appear more than once. A *path* is a walk in which all nodes and links are distinct. A *cycle* is a walk with at least 3 nodes, in which all links are distinct and the initial and end nodes are the same. A network is *connected* if there exists a path between any pair of nodes *i*, *j* ∈ *N*. A *tree* is a connected network that has no cycles.

Let $\mathbf{G}^k = [g_{ij}^k]$ denote the $k$th power of the adjacency matrix $\mathbf{G} = [g_{ij}]$ with $g_{ij} \in \{0,1\}$, where $\mathbf{G}^0$ is the $n \times n$ identity matrix $I$. Then $g_{ij}^k$ is equal to the number of walks of length $k$ that exist between $i$ and $j$ in the network $g$.

The issue of the most important or central nodes in a network has been widely studied in the literature since the late 1940's. As a consequence, a variety of centrality measures have been developed to capture different aspects of prominence or centrality of a node. One of the well-known and widely used centrality measures is *Bonacich centrality* [Bonacich, 1987] defined in matrix terms by:

$$
\text{Bon}(\mathbf{G}, c, d) = (I - c\mathbf{G})^{-1} d\mathbf{G}1 \quad (3)
$$

where 0 < *c* < 1 is an attenuation parameter adjusting the measure for the lower 'effectiveness'
of longer walks in a network, *d* > 0 is a parameter capturing the base value on each node and is
sufficiently small so that the measure (3) is well defined, and **1** is the *n* × 1 vector of 1's. The
Bonacich centrality of a node is therefore equal to a weighted sum of the walks that emanate from
it, and a walk of length *k* is of worth *c*<sup>*k*</sup>.

Another prominent and widely used centrality measure is eigenvector centrality [Bonacich, 1972]
defined in matrix terms by:

$$
\tilde{\lambda}\operatorname{eig}(\mathbf{G}) = \mathbf{G}\operatorname{eig}(\mathbf{G}) \quad (4)
$$

and thus eig(**G**) is an eigenvector of **G** with $\tilde{\lambda}$ being the corresponding largest eigenvalue of matrix **G**. The eigenvector centrality of a node is therefore proportional to the sum of the centrality of its neighbors.
# Appendix B – Proofs

**Proof of Lemma 1.** Assume $E = (N, F, \Phi, \Psi)$ and $E' = (N, F', \Phi', \Psi')$ are such that $(u_i(F, \Phi), v_i(F, \Psi)) = (u_i(F', \Phi'), v_i(F', \Psi'))$. By reflexivity of the Euclidian partial order, one has $(u_i(F, \Phi), v_i(F, \Psi)) \ge (u_i(F', \Phi'), v_i(F', \Psi'))$ and $(u_i(F', \Phi'), v_i(F', \Psi')) \ge (u_i(F, \Phi), v_i(F, \Psi))$. Thus monotonicity implies that $\rho_i(E, \lambda, \gamma, \delta) \ge \rho_i(E', \lambda, \gamma, \delta)$ and $\rho_i(E', \lambda, \gamma, \delta) \ge \rho_i(E, \lambda, \gamma, \delta)$. Hence, $\rho_i(E, \lambda, \gamma, \delta) = \rho_i(E', \lambda, \gamma, \delta)$ and there exists a function $\sigma_i: \mathbb{R}_+^N \times \mathbb{R}_+^N \to \mathbb{R}$ such that for all $E = (N, F, \Phi, \Psi) \in \mathcal{E}$, $\rho_i(E, \lambda, \gamma, \delta) = \sigma_i(u_i(F, \Phi), v_i(F, \Psi))$. The domain of $\sigma_i$ is $\mathbb{R}_+^N \times \mathbb{R}_+^N$ as this is the range of $(u_i, v_i)$ on $\mathcal{E}$. The fact that $\sigma_i$ is independent of $i$ is a direct consequence of anonymity and the monotonicity of $\sigma$ is a direct consequence of the monotonicity axiom. ■

**Proof of Lemma 2.** Following Lemma 1, one has for all emission systems $E = (N, F, \Phi, \Psi)$ and for all $i \in N$:

$$ \rho_i(E, \lambda, \gamma, \delta) = \sigma(u_i(F, \Phi), v_i(F, \Psi)). $$

Additivity then implies that

$$ \rho_i(E, \lambda, \gamma, \delta) = \sigma(u_i(F, \Phi), 0) + \sigma(0, v_i(F, \Psi)). $$

Letting for all $(u, v) \in \mathbb{R}_+^N \times \mathbb{R}_+^N$, $\alpha(u) = \sigma(u, 0)$ and $\beta(v) = \sigma(0, v)$ yields the required decomposition. The additivity and the monotonicity of $\alpha$ and $\beta$ then are direct consequences of the additivity and monotonicity of $\sigma$. ■

**Proof of Lemma 3.** Following Lemma 2, it is straightforward that for all emission systems $E = (N, F, \Phi, \Psi)$ and for all $i \in N$, one has:

$$ \rho_i(E, \lambda, \gamma, \delta) = \alpha(u_i(F, \Phi)) + \beta(v_i(F, \Psi)). \qquad (5) $$

Then it suffices to show that there exist $\gamma \ge 0$ and $a \ge 0$ such that for all $u \in \mathbb{R}_+^N$:

$$ \alpha(u) = a \sum_{\nu=0}^{+\infty} \gamma^{\nu} u_{\nu}. $$

Indeed, the proof that there exist $\delta \ge 0$ and $b \ge 0$ such that for all $v \in \mathbb{R}_+^N$:

$$ \beta(v) = b \sum_{\nu=0}^{+\infty} \delta^{\nu} v_{\nu} $$

follows by a symmetric argument. Let us first remark that additivity implies that $\alpha(0) = 0$. Let
us then consider $u \in \mathbb{R}_+^N$ and $\nu \in \mathbb{N}$. Considering the truncation of $u$ at rank $\nu$, one has using additivity that:

$$ \alpha(u_i, \dots, u_\nu, 0, \dots) = \sum_{\tau=1}^{\nu} \alpha(0, \dots, 0, u_\tau, 0, \dots) $$

which can equivalently be written as

$$ \alpha(u_i, \dots, u_\nu, 0, \dots) = \sum_{\tau=1}^{\nu} \alpha_\tau(u_\tau) \qquad (6) $$

where for $x \in \mathbb{R}_+$, one sets $\alpha_\tau(x) := \alpha(0, \dots, 0, x, 0, \dots)$.

Using recursivity, one has that for all $x \ge 0$, there exists $\delta(x) \ge 0$ such that for all $\tau \le \nu$, one has:

$$ \frac{\alpha_{\tau+1}(x)}{\alpha_{\tau}(x)} = \frac{\alpha_{2}(x)}{\alpha_{1}(x)} = \gamma(x). $$

Recursivity also implies that for all $x, x' \ge 0$:

$$ \frac{\alpha_1(x')}{\alpha_1(x)} = \frac{\alpha_2(x')}{\alpha_2(x)}, $$

so that

$$ \frac{\alpha_2(x)}{\alpha_1(x)} = \frac{\alpha_2(x')}{\alpha_1(x')} $$

and $\gamma(x)$ is in fact independent of $x$ and can be denoted by $\gamma \ge 0$. Thus Equation (6) can in fact be written as

$$ \alpha(u_1, \dots, u_\nu, 0, \dots) = \sum_{\tau=1}^{\nu} \gamma^{\tau-1} \alpha_1(u_\tau). $$

The additivity of $\alpha$ implies the additivity of $\alpha_1$, $\alpha_1$ is thus a group morphism of $\mathbb{R}$ and consequently linear according to standard arguments. This implies that there exists $a \ge 0$ (non-negativity is implied by monotonicity) such that

$$ \alpha(u_1, \dots, u_\nu, 0, \dots) = a \sum_{\tau=1}^{\nu} \gamma^{\tau-1} u_\tau. $$

As this holds for all $\nu \ge 0$, tail-convergence eventually yields:

$$ \alpha(u) = a \sum_{\tau=1}^{+\infty} \gamma^{\tau-1} u_{\tau}. \qquad (7) $$

As mentioned above, a symmetric argument yields that there exist $\delta \ge 0$ and $b \ge 0$ such that for
all $v \in \mathbb{R}_+^N$

$$ \beta(v) = b \sum_{\tau=1}^{+\infty} \gamma^{\tau-1} v_{\tau}. \qquad (8) $$

Substituting Equations (7) and (8) into Equation (5) eventually yields

$$ \rho_i(E, \lambda, \gamma, \delta) = a\left(\sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \Phi_{i,j}^{\nu} f_j\right) + b\left(\sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \Psi_{i,j}^{\nu} f_j\right). $$

<figure><img src="image_14.png" /></figure>

**Proof of Theorem 1.** It is straightforward to show that if $\rho_i$ is of the required form, it satisfies all the axioms. We thus focus on the converse.

Using Lemma 3, one knows that $\bar{\rho}_i$ is of the form

$$ \bar{\rho}_i(E, \lambda, \gamma, \delta) = a\left(\sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \Phi_{i,j}^{\nu} f_j\right) + b\left(\sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \Psi_{i,j}^{\nu} f_j\right). \qquad (9) $$

Tail-convergence further implies that $\gamma, \delta \in [0, 1)$.

Summing over $i \in N$, Equation (9) yields:

$$ \sum_{i \in N} \bar{\rho}_i(E, \lambda, \gamma, \delta) = a\left(\sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \sum_{i \in N} \Phi_{i,j}^{\nu} f_j\right) + b\left(\sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \sum_{i \in N} \Psi_{i,j}^{\nu} f_j\right). $$

Using consistency, this yields

$$ \sum_{i \in N} f_i = a\left(\sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \sum_{i \in N} \Phi_{i,j}^{\nu} f_j\right) + b\left(\sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \sum_{i \in N} \Psi_{i,j}^{\nu} f_j\right). $$

Now, $\Phi$ and $\Psi$ are column-stochastic. This implies that for all $\nu \in \mathbb{N}$, $\Phi^{\nu}$ and $\Psi^{\nu}$ are also column-stochastic so that

$$ \sum_{i \in N} f_i = a\left(\sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} f_j\right) + b\left(\sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} f_j\right) $$

and thus:

$$ 1 = a \sum_{\nu=0}^{+\infty} \gamma^{\nu} + b \sum_{\nu=0}^{+\infty} \delta^{\nu}. $$

As $\gamma, \delta \in [0, 1)$, this yields

$$ 1 = \frac{a}{1 - \gamma} + \frac{b}{1 - \delta}. $$
Letting $\lambda = \frac{a}{1-\gamma}$, one obtains $\lambda \in [0, 1]$, $a = \lambda(1-\gamma)$ and $b = (1-\lambda)(1-\delta)$, which eventually yields:

$$ \bar{\rho}_i(E, \lambda, \gamma, \delta) = (1-\gamma)\lambda\left(\sum_{\nu=0}^{+\infty} \gamma^{\nu} \sum_{j \in N} \Phi_{i,j}^{\nu} f_j\right) + (1-\delta)(1-\lambda)\left(\sum_{\nu=0}^{+\infty} \delta^{\nu} \sum_{j \in N} \Psi_{i,j}^{\nu} f_j\right). $$

**Proof of Proposition 1.** If $A$ is aperiodic and irreducible, so are $\Phi$ and $\Psi$ since they describe the same graph. Then it is a consequence of the linearity of the limit and the Perron-Frobenius theorem that for a column-stochastic matrix $M$, as $\delta$ tends towards 1, the matrix $(I - \delta M)^{-1}$ converges to $Eig^T(M)$. The proof follows. ■

**Proof of Proposition 2.** In the following, given a walk $w := (i_1, \dots, i_\tau) \in \mathcal{N}^\tau$, we denote, with a slight abuse of notation its weight as $\phi_w := \prod_{t=1}^{\tau-1} \phi_{i_t, i_{t+1}}$. We shall also refer more specifically to the set of walks of length $\tau$ between $i$ and $j$: $W_{i,j}^\tau := \{(i_1, \dots, i_\tau) \in \mathcal{N}^\tau \mid i_1 = i \wedge i_\tau = j\}$, and the set of direct walks of length $\tau$ between $i$ and $j$: $D_{i,j}^\tau := \{(i_1, \dots, i_\tau) \in W_{i,j}^\tau \mid \forall t < \tau \mid i_t \neq j\}$. Let us then consider a walk of length $\tau$, $(i_1, \dots, i_\tau)$. One has

$$ g_{i_1} \prod_{t=1}^{\tau-1} c_{i_t, i_{t+1}} = \frac{f_{i_1}}{q_{i_1}} \prod_{t=1}^{\tau-1} \frac{v_{i_t, i_{t+1}}}{q_{i_{t+1}}} = f_{i_1} \left( \prod_{t=1}^{\tau-1} \frac{v_{i_t, i_{t+1}}}{q_{i_t}} \right) \frac{1}{q_{i_\tau}} = f_{i_1} \left( \prod_{t=1}^{\tau-1} \phi_{i_{t+1}, i_t} \right) \frac{1}{q_{i_\tau}}. \quad (10) $$

As for $j \neq k$, one has $\phi_{j,k} = 0$, there is no walk between $i \in \mathcal{J}$ and $j \in \mathcal{J}$ that can go through $k \in \mathcal{L}$ and thus one has for all $i, j \in \mathcal{J}$ and $k \in \mathcal{L}$:

$$ g_j C_{j,i}^\tau D_{i,k} = \frac{D_{i,k}}{q_i} \left( \sum_{w \in W_{i,j}^\tau} \phi_w \right) f_j. \quad (11) $$

This yields

$$ g_j C_{j,i}^\tau D_{i,k} = \phi_{k,i} \left( \sum_{w \in W_{i,j}^\tau} \phi_w \right) f_j. \quad (12) $$

Now, a walk of the form $(k, w)$ with $w \in W_{i,j}^\tau$ and $i, j \in \mathcal{J}$ is a direct walk of length $\tau + 1$ from $k$ to $j$. Furthermore all direct walks are of this form because no walk from $k$ to $j$ can go through $\ell \in \mathcal{L} / \{k\}$. Thus one has:

$$ \sum_{i=1}^{ML} g_j C_{j,i}^\tau D_{i,k} = \left( \sum_{w \in D_{k,j}^{\tau+1}} \phi_w \right) f_j. \quad (13) $$
Accordingly, for all $\gamma \in [0, 1]$, one has:

$$ \gamma \sum_{i=1}^{ML} g_j \gamma^\tau C_{j,i}^\tau D_{i,k} = \left( \sum_{w \in \mathcal{D}_{k,j}^{\tau+1}} \gamma^{\tau+1} \phi_w \right) f_j. \quad (14) $$

Summing over $\tau$ this yields:

$$ \gamma \sum_{i=1}^{ML} g_j (I - \gamma C)_{j,i}^{-1} D_{i,k} = \gamma \sum_{\tau=0}^{+\infty} \sum_{i=1}^{ML} g_j \gamma^\tau C_{j,i}^\tau D_{i,k} = \left( \sum_{\tau=0}^{+\infty} \sum_{w \in \mathcal{D}_{k,j}^{\tau+1}} \gamma^{\tau+1} \phi_w \right) f_j. \quad (15) $$

Now $(\sum_{\tau=0}^{+\infty} \sum_{w \in \mathcal{D}_{k,j}^{\tau+1}} \gamma^{\tau+1} \phi_w)$ is the discounted sum of all direct walks from $k$ to $j$. Any walk from $k$ to $j$ consists in an arbitrary number of cycles around $k$ followed by a direct walk from $k$ to $j$. In our setting, all cycles around $k$ consist in the self-loop $\phi_{k,k} = 1$ and thus a walk from $k$ to $j$ consists in an arbitrary number of self-loops around $k$ followed by a direct walk from $k$ to $j$. Therefrom, one gets that

$$ \sum_{\tau=0}^{+\infty} \sum_{w \in \mathcal{W}_{k,j}^{\tau+1}} \gamma^{\tau+1} \phi_w = \sum_{k \in \mathbb{N}} \gamma^k \sum_{\tau=0}^{+\infty} \sum_{w \in \mathcal{D}_{k,j}^{\tau+1}} \gamma^{\tau+1} \phi_w = \frac{1}{1-\gamma} \sum_{\tau=0}^{+\infty} \sum_{w \in \mathcal{D}_{k,j}^{\tau+1}} \gamma^{\tau+1} \phi_w. \quad (16) $$

In turn this yields that for all $\tau \ge 0$, for all $k \in \mathcal{L}$ and all $j \in \mathcal{N}$, one gets:

$$ (1 - \gamma)(\gamma\Phi)_{k,j}^{\tau+1} = \sum_{\tau=0}^{+\infty} \sum_{w \in \mathcal{D}_{k,j}^{\tau+1}} \gamma^{\tau+1} \phi_w. \quad (17) $$

Substituting Equation (17) into Equation (15), one gets

$$ \gamma \sum_{i \in \mathcal{J}} g_j (I - \gamma C)_{j,i}^{-1} D_{i,k} = \sum_{\tau=0}^{+\infty} (1 - \gamma)(\gamma\Phi)_{k,j}^{\tau+1} f_j = (1 - \gamma)(I - \gamma\Phi)_{k,j}^{-1} f_j \quad (18) $$

and then

$$ \gamma \sum_{j \in \mathcal{J}} \sum_{i \in \mathcal{J}} g_j (I - \gamma C)_{j,i}^{-1} D_{i,k} = \sum_{j \in \mathcal{J}} (1 - \gamma)(I - \gamma\Phi)_{k,j}^{-1} f_j. \quad (19) $$

So that, taking the limit as $\gamma$ tends towards 1, one gets

$$ \sum_{j \in \mathcal{J}} \sum_{i \in \mathcal{J}} g_j (I - C)_{j,i}^{-1} D_{i,k} = \lim_{\gamma \to 1} \sum_{j \in \mathcal{J}} (1 - \gamma)(I - \gamma\Phi)_{k,j}^{-1} f_j. \quad (20) $$

<figure><img src="image_15.png" /></figure>
**Proof of Proposition 3.** Proof follows from the proof of Proposition 2. ■

# Appendix C – Empirical Results
<table><caption>Table 2: Upstream responsibility ($\lambda = 1$, $\nu_{max} = 1000$) for various values of $\gamma$, in kton CO$_2$.</caption><thead><tr><th>Country</th><th>γ = 0.0</th><th>γ = 0.1</th><th>γ = 0.2</th><th>...</th><th>γ = 0.7</th><th>γ = 0.8</th><th>γ = 0.9</th><th>γ = 0.99</th></tr></thead><tbody><tr><td>AUS</td><td>404,864</td><td>405,276</td><td>405,749</td><td>...</td><td>409,283</td><td>409,854</td><td>408,961</td><td>402,119</td></tr><tr><td>AUT</td><td>57,324</td><td>59,466</td><td>61,956</td><td>...</td><td>85,015</td><td>94,475</td><td>108,549</td><td>131,216</td></tr><tr><td>BEL</td><td>97,560</td><td>101,616</td><td>106,305</td><td>...</td><td>148,168</td><td>164,595</td><td>188,352</td><td>225,096</td></tr><tr><td>BGR</td><td>45,142</td><td>44,577</td><td>43,920</td><td>...</td><td>37,876</td><td>35,500</td><td>32,257</td><td>28,445</td></tr><tr><td>BRA</td><td>572,372</td><td>576,092</td><td>580,588</td><td>...</td><td>632,584</td><td>660,570</td><td>710,911</td><td>813,335</td></tr><tr><td>CAN</td><td>576,979</td><td>576,458</td><td>575,878</td><td>...</td><td>572,901</td><td>573,842</td><td>579,583</td><td>607,279</td></tr><tr><td>CHE</td><td>44,067</td><td>47,977</td><td>52,615</td><td>...</td><td>99,565</td><td>120,607</td><td>153,222</td><td>206,574</td></tr><tr><td>CHN</td><td>10,528,550</td><td>10,529,692</td><td>10,530,744</td><td>...</td><td>10,516,086</td><td>10,485,845</td><td>10,383,828</td><td>9,917,066</td></tr><tr><td>CYP</td><td>7,029</td><td>6,982</td><td>6,927</td><td>...</td><td>6,358</td><td>6,095</td><td>5,690</td><td>5,125</td></tr><tr><td>CZE</td><td>95,800</td><td>96,071</td><td>96,386</td><td>...</td><td>99,343</td><td>100,637</td><td>102,852</td><td>108,062</td></tr><tr><td>DEU</td><td>855,312</td><td>857,768</td><td>861,000</td><td>...</td><td>910,545</td><td>941,553</td><td>999,772</td><td>1,123,097</td></tr><tr><td>DNK</td><td>71,497</td><td>72,369</td><td>73,409</td><td>...</td><td>84,238</td><td>89,247</td><td>97,177</td><td>110,740</td></tr><tr><td>ESP</td><td>270,385</td><td>274,773</td><td>280,081</td><td>...</td><td>341,534</td><td>374,529</td><td>433,571</td><td>551,804</td></tr><tr><td>EST</td><td>19,565</td><td>19,138</td><td>18,649</td><td>...</td><td>14,563</td><td>13,107</td><td>11,193</td><td>8,896</td></tr><tr><td>FIN</td><td>51,307</td><td>52,434</td><td>53,749</td><td>...</td><td>65,952</td><td>70,850</td><td>77,846</td><td>88,227</td></tr><tr><td>FRA</td><td>345,586</td><td>355,340</td><td>367,164</td><td>...</td><td>504,557</td><td>577,692</td><td>706,210</td><td>953,274</td></tr><tr><td>GBR</td><td>488,332</td><td>493,499</td><td>499,836</td><td>...</td><td>577,171</td><td>620,063</td><td>697,299</td><td>851,647</td></tr><tr><td>GRC</td><td>78,655</td><td>78,349</td><td>77,981</td><td>...</td><td>73,919</td><td>71,902</td><td>68,651</td><td>63,887</td></tr><tr><td>HRV</td><td>17,850</td><td>17,774</td><td>17,683</td><td>...</td><td>16,817</td><td>16,468</td><td>16,014</td><td>15,707</td></tr><tr><td>HUN</td><td>47,587</td><td>47,919</td><td>48,312</td><td>...</td><td>52,238</td><td>54,022</td><td>56,909</td><td>62,284</td></tr><tr><td>IDN</td><td>542,875</td><td>539,667</td><td>535,808</td><td>...</td><td>492,612</td><td>470,398</td><td>432,212</td><td>362,284</td></tr><tr><td>IND</td><td>2,195,307</td><td>2,182,361</td><td>2,166,548</td><td>...</td><td>1,971,004</td><td>1,856,618</td><td>1,636,829</td><td>1,155,133</td></tr><tr><td>IRL</td><td>44,079</td><td>47,364</td><td>50,986</td><td>...</td><td>77,138</td><td>85,297</td><td>95,895</td><td>111,164</td></tr><tr><td>ITA</td><td>356,596</td><td>363,367</td><td>371,574</td><td>...</td><td>467,310</td><td>518,909</td><td>611,226</td><td>795,770</td></tr><tr><td>JPN</td><td>1,251,980</td><td>1,267,237</td><td>1,285,496</td><td>...</td><td>1,483,183</td><td>1,580,098</td><td>1,739,147</td><td>2,006,186</td></tr><tr><td>KOB</td><td>670,005</td><td>676,433</td><td>683,978</td><td>...</td><td>755,965</td><td>785,231</td><td>825,470</td><td>872,898</td></tr><tr><td>LTU</td><td>17,832</td><td>17,810</td><td>17,766</td><td>...</td><td>16,489</td><td>15,590</td><td>13,997</td><td>11,376</td></tr><tr><td>LUX</td><td>8,270</td><td>10,307</td><td>12,587</td><td>...</td><td>30,406</td><td>36,611</td><td>45,228</td><td>58,297</td></tr><tr><td>LVA</td><td>8,695</td><td>8,840</td><td>9,002</td><td>...</td><td>10,133</td><td>10,343</td><td>10,342</td><td>9,749</td></tr><tr><td>MEX</td><td>488,677</td><td>487,749</td><td>486,694</td><td>...</td><td>479,055</td><td>477,828</td><td>479,692</td><td>497,593</td></tr><tr><td>MLT</td><td>3,634</td><td>3,728</td><td>3,843</td><td>...</td><td>5,139</td><td>5,738</td><td>6,665</td><td>8,190</td></tr><tr><td>NLD</td><td>160,000</td><td>163,472</td><td>167,494</td><td>...</td><td>203,835</td><td>218,337</td><td>239,540</td><td>272,755</td></tr><tr><td>NOR</td><td>51,116</td><td>51,430</td><td>51,826</td><td>...</td><td>57,037</td><td>59,957</td><td>65,079</td><td>74,947</td></tr><tr><td>POL</td><td>315,600</td><td>312,842</td><td>309,552</td><td>...</td><td>275,111</td><td>259,321</td><td>235,524</td><td>203,693</td></tr><tr><td>PRT</td><td>49,364</td><td>49,780</td><td>50,291</td><td>...</td><td>56,632</td><td>60,265</td><td>67,029</td><td>81,271</td></tr><tr><td>ROU</td><td>80,216</td><td>80,220</td><td>80,217</td><td>...</td><td>79,727</td><td>79,322</td><td>78,691</td><td>78,792</td></tr><tr><td>RUS</td><td>1,723,086</td><td>1,686,183</td><td>1,642,597</td><td>...</td><td>1,208,205</td><td>1,017,945</td><td>733,706</td><td>324,820</td></tr><tr><td>SVK</td><td>33,564</td><td>34,265</td><td>35,068</td><td>...</td><td>41,889</td><td>44,370</td><td>47,763</td><td>52,824</td></tr><tr><td>SVN</td><td>14,409</td><td>14,368</td><td>14,325</td><td>...</td><td>14,220</td><td>14,345</td><td>14,733</td><td>15,894</td></tr><tr><td>SWE</td><td>51,175</td><td>53,690</td><td>56,641</td><td>...</td><td>85,041</td><td>97,081</td><td>115,053</td><td>142,976</td></tr><tr><td>TUR</td><td>357,560</td><td>357,881</td><td>358,221</td><td>...</td><td>358,945</td><td>357,509</td><td>352,969</td><td>341,016</td></tr><tr><td>TWN</td><td>314,702</td><td>312,608</td><td>310,224</td><td>...</td><td>290,580</td><td>283,623</td><td>274,228</td><td>261,170</td></tr><tr><td>USA</td><td>5,257,452</td><td>5,267,728</td><td>5,280,252</td><td>...</td><td>5,433,719</td><td>5,524,109</td><td>5,705,225</td><td>6,181,470</td></tr><tr><td>ROW</td><td>7,835,123</td><td>7,804,173</td><td>7,767,151</td><td>...</td><td>7,364,989</td><td>7,166,778</td><td>6,841,984</td><td>6,311,360</td></tr></tbody></table>
<table><caption>Table 3: Downstream responsibility ($\lambda = 0, \nu_{max} = 1000$) for various values of $\delta$, in kton CO$_2$.</caption><thead><tr><th>Country</th><th>δ = 0.0</th><th>δ = 0.1</th><th>δ = 0.2</th><th>...</th><th>δ = 0.7</th><th>δ = 0.8</th><th>δ = 0.9</th><th>δ = 0.99</th></tr></thead><tbody><tr><td>AUS</td><td>404,864</td><td>411,880</td><td>420,335</td><td>...</td><td>515,477</td><td>563,960</td><td>644,076</td><td>762,315</td></tr><tr><td>AUT</td><td>57,324</td><td>59,962</td><td>63,093</td><td>...</td><td>96,276</td><td>112,855</td><td>142,036</td><td>202,937</td></tr><tr><td>BEL</td><td>97,560</td><td>101,232</td><td>105,560</td><td>...</td><td>149,328</td><td>169,997</td><td>205,077</td><td>275,319</td></tr><tr><td>BGR</td><td>45,142</td><td>44,502</td><td>43,759</td><td>...</td><td>37,159</td><td>34,740</td><td>31,771</td><td>29,634</td></tr><tr><td>BRA</td><td>572,372</td><td>575,045</td><td>578,240</td><td>...</td><td>612,501</td><td>629,017</td><td>655,493</td><td>696,607</td></tr><tr><td>CAN</td><td>576,979</td><td>580,520</td><td>584,676</td><td>...</td><td>624,490</td><td>641,041</td><td>665,166</td><td>705,285</td></tr><tr><td>CHE</td><td>44,067</td><td>50,873</td><td>58,998</td><td>...</td><td>145,574</td><td>187,465</td><td>256,963</td><td>384,610</td></tr><tr><td>CHN</td><td>10,528,550</td><td>10,513,650</td><td>10,495,100</td><td>...</td><td>10,234,564</td><td>10,055,027</td><td>9,647,863</td><td>8,379,388</td></tr><tr><td>CYP</td><td>7,029</td><td>7,070</td><td>7,118</td><td>...</td><td>7,573</td><td>7,792</td><td>8,207</td><td>9,280</td></tr><tr><td>CZE</td><td>95,800</td><td>95,447</td><td>95,097</td><td>...</td><td>95,819</td><td>98,905</td><td>107,811</td><td>136,042</td></tr><tr><td>DEU</td><td>855,312</td><td>868,318</td><td>884,102</td><td>...</td><td>1,071,930</td><td>1,177,611</td><td>1,377,584</td><td>1,831,071</td></tr><tr><td>DNK</td><td>71,497</td><td>72,323</td><td>73,319</td><td>...</td><td>84,843</td><td>91,170</td><td>102,942</td><td>128,855</td></tr><tr><td>ESP</td><td>270,385</td><td>272,622</td><td>275,336</td><td>...</td><td>308,032</td><td>326,998</td><td>364,305</td><td>455,507</td></tr><tr><td>EST</td><td>19,565</td><td>19,230</td><td>18,855</td><td>...</td><td>16,276</td><td>15,775</td><td>15,784</td><td>17,989</td></tr><tr><td>FIN</td><td>51,307</td><td>52,442</td><td>53,817</td><td>...</td><td>69,895</td><td>78,734</td><td>95,072</td><td>130,534</td></tr><tr><td>FRA</td><td>345,586</td><td>353,539</td><td>363,207</td><td>...</td><td>478,519</td><td>542,944</td><td>663,246</td><td>928,458</td></tr><tr><td>GBR</td><td>488,332</td><td>497,181</td><td>508,044</td><td>...</td><td>641,555</td><td>716,906</td><td>856,633</td><td>1,157,654</td></tr><tr><td>GRC</td><td>78,655</td><td>78,464</td><td>78,221</td><td>...</td><td>74,901</td><td>72,974</td><td>69,687</td><td>64,922</td></tr><tr><td>HRV</td><td>17,850</td><td>17,842</td><td>17,833</td><td>...</td><td>17,874</td><td>18,021</td><td>18,524</td><td>20,503</td></tr><tr><td>HUN</td><td>47,587</td><td>47,175</td><td>46,752</td><td>...</td><td>45,604</td><td>46,656</td><td>50,168</td><td>61,590</td></tr><tr><td>IDN</td><td>542,875</td><td>540,128</td><td>536,796</td><td>...</td><td>497,582</td><td>476,110</td><td>437,241</td><td>359,385</td></tr><tr><td>IND</td><td>2,195,307</td><td>2,167,262</td><td>2,133,501</td><td>...</td><td>1,754,716</td><td>1,560,511</td><td>1,230,246</td><td>647,479</td></tr><tr><td>IRL</td><td>44,079</td><td>46,820</td><td>49,849</td><td>...</td><td>72,393</td><td>79,982</td><td>90,686</td><td>108,435</td></tr><tr><td>ITA</td><td>356,596</td><td>361,738</td><td>368,060</td><td>...</td><td>449,635</td><td>500,063</td><td>602,617</td><td>859,185</td></tr><tr><td>JPN</td><td>1,251,980</td><td>1,256,531</td><td>1,261,963</td><td>...</td><td>1,318,327</td><td>1,343,077</td><td>1,376,584</td><td>1,397,754</td></tr><tr><td>KOR</td><td>670,005</td><td>676,793</td><td>684,718</td><td>...</td><td>757,123</td><td>783,928</td><td>815,592</td><td>830,156</td></tr><tr><td>LTU</td><td>17,832</td><td>17,894</td><td>17,969</td><td>...</td><td>18,855</td><td>19,414</td><td>20,621</td><td>23,953</td></tr><tr><td>LUX</td><td>8,270</td><td>10,713</td><td>13,403</td><td>...</td><td>32,839</td><td>39,000</td><td>47,224</td><td>59,717</td></tr><tr><td>LVA</td><td>8,695</td><td>8,875</td><td>9,091</td><td>...</td><td>11,523</td><td>12,840</td><td>15,307</td><td>20,908</td></tr><tr><td>MEX</td><td>488,677</td><td>481,923</td><td>474,025</td><td>...</td><td>399,657</td><td>369,316</td><td>326,338</td><td>271,565</td></tr><tr><td>MLT</td><td>3,634</td><td>3,602</td><td>3,572</td><td>...</td><td>3,621</td><td>3,797</td><td>4,231</td><td>5,391</td></tr><tr><td>NLD</td><td>160,000</td><td>166,658</td><td>174,569</td><td>...</td><td>257,964</td><td>299,012</td><td>370,172</td><td>515,257</td></tr><tr><td>NOR</td><td>51,116</td><td>55,572</td><td>60,937</td><td>...</td><td>121,659</td><td>153,661</td><td>210,731</td><td>327,186</td></tr><tr><td>POL</td><td>315,600</td><td>312,021</td><td>307,910</td><td>...</td><td>275,098</td><td>266,693</td><td>264,160</td><td>297,402</td></tr><tr><td>PRT</td><td>49,364</td><td>49,827</td><td>50,380</td><td>...</td><td>56,407</td><td>59,530</td><td>65,232</td><td>78,076</td></tr><tr><td>ROU</td><td>80,216</td><td>80,038</td><td>79,842</td><td>...</td><td>79,021</td><td>79,624</td><td>82,445</td><td>94,673</td></tr><tr><td>RUS</td><td>1,723,086</td><td>1,726,585</td><td>17,30,773</td><td>...</td><td>1,777,928</td><td>1,804,321</td><td>1,858,641</td><td>2,024,597</td></tr><tr><td>SVK</td><td>33,564</td><td>33,752</td><td>33,981</td><td>...</td><td>37,032</td><td>39,073</td><td>43,497</td><td>55,418</td></tr><tr><td>SVN</td><td>14,409</td><td>14,499</td><td>14,612</td><td>...</td><td>16,181</td><td>17,211</td><td>19,370</td><td>24,940</td></tr><tr><td>SWE</td><td>51,175</td><td>55,027</td><td>59,661</td><td>...</td><td>111,974</td><td>139,461</td><td>188,434</td><td>288,571</td></tr><tr><td>TUR</td><td>357,560</td><td>355,676</td><td>353,429</td><td>...</td><td>330,224</td><td>320,049</td><td>306,116</td><td>296,115</td></tr><tr><td>TWN</td><td>314,702</td><td>317,454</td><td>320,561</td><td>...</td><td>343,663</td><td>349,507</td><td>352,997</td><td>341,826</td></tr><tr><td>USA</td><td>5,257,452</td><td>5,253,461</td><td>5,248,737</td><td>...</td><td>5,198,942</td><td>5,175,710</td><td>5,144,894</td><td>5,173,991</td></tr><tr><td>ROW</td><td>7,835,123</td><td>7,794,910</td><td>7,747,274</td><td>...</td><td>7,256,524</td><td>7,026,595</td><td>6,655,291</td><td>6,025,021</td></tr></tbody></table>
<table><caption>Table 4: Sectoral upstream responsibility ($\lambda = 1, \nu_{max} = 1000$) in kton CO$_2$. Note that sector U, activities of extraterritorial organizations and bodies, does emissions nor nonzero supply relationships, and carries therefore zero responsibility. Neither does sector T, Activities of households as employers; undifferentiated goods- and services-producing activities of households for own use have any emissions, but the nonzero supply relationships of this sector result in a nonzero responsibility.</caption><thead><tr><th>Sector</th><th>γ = 0.0</th><th>γ = 0.1</th><th>γ = 0.2</th><th>...</th><th>γ = 0.7</th><th>γ = 0.8</th><th>γ = 0.9</th><th>γ = 0.99</th></tr></thead><tbody><tr><td>A01</td><td>53,153</td><td>56,532</td><td>59,653</td><td>...</td><td>69,931</td><td>70,385</td><td>69,955</td><td>68,553</td></tr><tr><td>A02</td><td>2,756</td><td>2,659</td><td>2,569</td><td>...</td><td>2,211</td><td>2,145</td><td>2,073</td><td>2,000</td></tr><tr><td>A03</td><td>1,099</td><td>1,098</td><td>1,102</td><td>...</td><td>1,176</td><td>1,195</td><td>1,211</td><td>1,220</td></tr><tr><td>B</td><td>130,773</td><td>125,099</td><td>118,818</td><td>...</td><td>77,741</td><td>67,377</td><td>56,165</td><td>45,269</td></tr><tr><td>C10 - C12</td><td>59,146</td><td>75,239</td><td>91,830</td><td>...</td><td>179,980</td><td>197,122</td><td>212,765</td><td>224,402</td></tr><tr><td>C13 - C15</td><td>4,523</td><td>5,898</td><td>7,269</td><td>...</td><td>13,833</td><td>14,989</td><td>16,028</td><td>16,820</td></tr><tr><td>C16</td><td>5,907</td><td>7,454</td><td>8,883</td><td>...</td><td>13,920</td><td>14,399</td><td>14,644</td><td>14,628</td></tr><tr><td>C17</td><td>32,447</td><td>37,028</td><td>40,879</td><td>...</td><td>46,135</td><td>43,525</td><td>39,273</td><td>33,812</td></tr><tr><td>C18</td><td>2,949</td><td>4,252</td><td>5,546</td><td>...</td><td>11,445</td><td>12,357</td><td>13,092</td><td>13,549</td></tr><tr><td>C19</td><td>133,686</td><td>136,925</td><td>138,458</td><td>...</td><td>115,358</td><td>103,336</td><td>88,304</td><td>71,928</td></tr><tr><td>C20</td><td>160,432</td><td>159,329</td><td>157,107</td><td>...</td><td>123,244</td><td>110,130</td><td>93,981</td><td>76,327</td></tr><tr><td>C21</td><td>1,003</td><td>6,242</td><td>11,082</td><td>...</td><td>27,148</td><td>28,095</td><td>27,958</td><td>26,719</td></tr><tr><td>C22</td><td>30,125</td><td>33,102</td><td>35,784</td><td>...</td><td>41,675</td><td>40,402</td><td>37,819</td><td>34,092</td></tr><tr><td>C23</td><td>114,851</td><td>108,649</td><td>101,711</td><td>...</td><td>54,819</td><td>42,706</td><td>29,554</td><td>16,770</td></tr><tr><td>C24</td><td>113,632</td><td>113,760</td><td>112,719</td><td>...</td><td>85,823</td><td>74,987</td><td>61,823</td><td>47,728</td></tr><tr><td>C25</td><td>9,751</td><td>16,893</td><td>23,667</td><td>...</td><td>47,610</td><td>49,142</td><td>48,967</td><td>47,024</td></tr><tr><td>C26</td><td>4,900</td><td>6,271</td><td>7,714</td><td>...</td><td>15,644</td><td>17,225</td><td>18,716</td><td>19,928</td></tr><tr><td>C27</td><td>4,072</td><td>5,667</td><td>7,236</td><td>...</td><td>13,399</td><td>13,918</td><td>14,011</td><td>13,630</td></tr><tr><td>C28</td><td>23,151</td><td>26,690</td><td>30,432</td><td>...</td><td>50,494</td><td>54,081</td><td>57,090</td><td>59,017</td></tr><tr><td>C29</td><td>5,498</td><td>12,511</td><td>20,547</td><td>...</td><td>81,579</td><td>99,180</td><td>118,905</td><td>138,412</td></tr><tr><td>C30</td><td>10,213</td><td>13,052</td><td>16,205</td><td>...</td><td>38,083</td><td>43,932</td><td>50,291</td><td>56,383</td></tr><tr><td>C31, C32</td><td>9,171</td><td>11,594</td><td>14,083</td><td>...</td><td>26,393</td><td>28,433</td><td>30,110</td><td>31,175</td></tr><tr><td>C33</td><td>2,858</td><td>2,977</td><td>3,133</td><td>...</td><td>4,375</td><td>4,695</td><td>5,032</td><td>5,346</td></tr><tr><td>D35</td><td>2,094,483</td><td>1,890,303</td><td>1,685,676</td><td>...</td><td>655,035</td><td>447,187</td><td>238,660</td><td>50,348</td></tr><tr><td>E36</td><td>18,548</td><td>16,873</td><td>15,183</td><td>...</td><td>6,477</td><td>4,676</td><td>2,853</td><td>1,190</td></tr><tr><td>E37 - E39</td><td>7,867</td><td>8,269</td><td>8,738</td><td>...</td><td>12,194</td><td>13,122</td><td>14,127</td><td>15,093</td></tr><tr><td>F</td><td>62,943</td><td>72,908</td><td>83,811</td><td>...</td><td>148,804</td><td>162,144</td><td>174,574</td><td>184,398</td></tr><tr><td>G45</td><td>12,770</td><td>14,728</td><td>16,960</td><td>...</td><td>31,928</td><td>35,664</td><td>39,656</td><td>43,470</td></tr><tr><td>G46</td><td>26,891</td><td>35,622</td><td>45,328</td><td>...</td><td>109,818</td><td>126,474</td><td>144,744</td><td>162,798</td></tr><tr><td>G47</td><td>61,415</td><td>70,059</td><td>79,491</td><td>...</td><td>132,332</td><td>142,814</td><td>152,865</td><td>161,389</td></tr><tr><td>H49</td><td>246,868</td><td>229,431</td><td>212,205</td><td>...</td><td>128,027</td><td>111,347</td><td>94,661</td><td>79,637</td></tr><tr><td>H50</td><td>64,157</td><td>59,595</td><td>54,928</td><td>...</td><td>29,714</td><td>24,234</td><td>18,591</td><td>13,372</td></tr><tr><td>H51</td><td>194,809</td><td>177,368</td><td>160,062</td><td>...</td><td>74,006</td><td>56,521</td><td>38,793</td><td>22,563</td></tr><tr><td>H52</td><td>21,217</td><td>22,814</td><td>24,332</td><td>...</td><td>30,049</td><td>30,684</td><td>31,102</td><td>31,277</td></tr><tr><td>H53</td><td>7,109</td><td>8,167</td><td>9,259</td><td>...</td><td>15,138</td><td>16,382</td><td>17,643</td><td>18,790</td></tr><tr><td>I</td><td>61,085</td><td>68,220</td><td>75,910</td><td>...</td><td>122,050</td><td>132,410</td><td>142,760</td><td>151,700</td></tr><tr><td>J58</td><td>52</td><td>1,361</td><td>2,930</td><td>...</td><td>16,618</td><td>21,098</td><td>26,443</td><td>32,146</td></tr><tr><td>J59, J60</td><td>174</td><td>1,819</td><td>3,739</td><td>...</td><td>19,093</td><td>23,770</td><td>29,193</td><td>34,814</td></tr><tr><td>J61</td><td>3,673</td><td>7,814</td><td>12,610</td><td>...</td><td>50,293</td><td>61,634</td><td>74,730</td><td>88,244</td></tr><tr><td>J62, J63</td><td>14,379</td><td>17,591</td><td>21,169</td><td>...</td><td>45,145</td><td>51,319</td><td>58,023</td><td>64,538</td></tr><tr><td>K64</td><td>11,505</td><td>14,052</td><td>17,004</td><td>...</td><td>41,832</td><td>49,934</td><td>59,634</td><td>70,029</td></tr><tr><td>K65</td><td>10,217</td><td>12,630</td><td>16,039</td><td>...</td><td>63,148</td><td>84,453</td><td>113,625</td><td>149,435</td></tr><tr><td>K66</td><td>4,377</td><td>7,427</td><td>11,129</td><td>...</td><td>42,018</td><td>51,565</td><td>62,740</td><td>74,498</td></tr><tr><td>L68</td><td>12,913</td><td>73,164</td><td>126,572</td><td>...</td><td>287,080</td><td>296,706</td><td>298,188</td><td>292,178</td></tr></tbody></table>
<table><caption>Table 5: Sectoral upstream responsibility ($\lambda = 1, \nu_{max} = 1000$) in kton CO$_2$ (continued)</caption><thead><tr><th>Sector</th><th>γ = 0.0</th><th>γ = 0.1</th><th>γ = 0.2</th><th>...</th><th>γ = 0.7</th><th>γ = 0.8</th><th>γ = 0.9</th><th>γ = 0.99</th></tr></thead><tbody><tr><td>M69-M70</td><td>20,294</td><td>28,298</td><td>36,927</td><td>...</td><td>91,087</td><td>104,670</td><td>119,465</td><td>133,977</td></tr><tr><td>M71</td><td>12,801</td><td>15,647</td><td>18,848</td><td>...</td><td>40,298</td><td>45,734</td><td>51,584</td><td>57,217</td></tr><tr><td>M72</td><td>6,897</td><td>8,430</td><td>10,155</td><td>...</td><td>21,712</td><td>24,641</td><td>27,793</td><td>30,828</td></tr><tr><td>M73</td><td>6,945</td><td>8,489</td><td>10,226</td><td>...</td><td>21,864</td><td>24,813</td><td>27,987</td><td>31,043</td></tr><tr><td>M74,M75</td><td>2,267</td><td>2,771</td><td>3,338</td><td>...</td><td>7,135</td><td>8,097</td><td>9,131</td><td>10,127</td></tr><tr><td>N</td><td>59,695</td><td>62,498</td><td>66,025</td><td>...</td><td>96,635</td><td>105,946</td><td>116,608</td><td>127,505</td></tr><tr><td>O84</td><td>242,418</td><td>267,882</td><td>294,056</td><td>...</td><td>448,490</td><td>487,028</td><td>529,282</td><td>570,996</td></tr><tr><td>P85</td><td>21,931</td><td>30,851</td><td>38,665</td><td>...</td><td>58,181</td><td>57,511</td><td>55,038</td><td>51,123</td></tr><tr><td>Q</td><td>84,405</td><td>101,134</td><td>120,701</td><td>...</td><td>269,046</td><td>311,567</td><td>360,051</td><td>409,919</td></tr><tr><td>R,S</td><td>32,280</td><td>38,096</td><td>44,545</td><td>...</td><td>84,256</td><td>93,285</td><td>102,531</td><td>110,966</td></tr><tr><td>T</td><td>0</td><td>222</td><td>465</td><td>...</td><td>1,934</td><td>2,267</td><td>2,609</td><td>2,924</td></tr><tr><td>U</td><td>0</td><td>0</td><td>0</td><td>...</td><td>0</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

<table><caption>Table 6: Sectoral downstream responsibility ($\lambda = 0, \nu_{max} = 1000$) in kton CO$_2$. Note that sector U, activities of extraterritorial organizations and bodies, does emissions nor nonzero supply relationships, and carries therefore zero responsibility. Neither does sector T, Activities of households as employers; undifferentiated goods- and services-producing activities of households for own use have any emissions, but the nonzero supply relationships of this sector result in a nonzero responsibility.</caption><thead><tr><th>Sector</th><th>δ = 0.0</th><th>δ = 0.1</th><th>δ = 0.2</th><th>...</th><th>δ = 0.7</th><th>δ = 0.8</th><th>δ = 0.9</th><th>δ = 0.99</th></tr></thead><tbody><tr><td>A01</td><td>53,153</td><td>52,094</td><td>51,002</td><td>...</td><td>44,980</td><td>43,906</td><td>43,181</td><td>43,123</td></tr><tr><td>A02</td><td>2,756</td><td>3,000</td><td>3,264</td><td>...</td><td>5,263</td><td>5,952</td><td>6,847</td><td>7,908</td></tr><tr><td>A03</td><td>1,099</td><td>1,298</td><td>1,509</td><td>...</td><td>2,968</td><td>3,433</td><td>4,019</td><td>4,700</td></tr><tr><td>B</td><td>130,773</td><td>182,410</td><td>230,805</td><td>...</td><td>343,865</td><td>317,582</td><td>263,509</td><td>184,740</td></tr><tr><td>C10 - C12</td><td>59,146</td><td>59,151</td><td>59,033</td><td>...</td><td>57,758</td><td>58,043</td><td>59,024</td><td>60,882</td></tr><tr><td>C13 - C15</td><td>4,523</td><td>4,765</td><td>5,053</td><td>...</td><td>8,061</td><td>9,279</td><td>10,870</td><td>12,717</td></tr><tr><td>C16</td><td>5,907</td><td>6,620</td><td>7,513</td><td>...</td><td>15,640</td><td>18,332</td><td>21,565</td><td>25,043</td></tr><tr><td>C17</td><td>32,447</td><td>32,105</td><td>31,937</td><td>...</td><td>35,847</td><td>38,273</td><td>41,598</td><td>45,531</td></tr><tr><td>C18</td><td>2,949</td><td>3,734</td><td>4,716</td><td>...</td><td>14,254</td><td>17,613</td><td>21,741</td><td>26,283</td></tr><tr><td>C19</td><td>133,686</td><td>173,060</td><td>205,530</td><td>...</td><td>245,175</td><td>222,898</td><td>187,625</td><td>143,069</td></tr><tr><td>C20</td><td>160,432</td><td>154,144</td><td>147,943</td><td>...</td><td>117,812</td><td>111,296</td><td>103,943</td><td>96,001</td></tr><tr><td>C21</td><td>1,003</td><td>5,814</td><td>10,658</td><td>...</td><td>35,120</td><td>39,707</td><td>43,843</td><td>46,864</td></tr><tr><td>C22</td><td>30,125</td><td>30,049</td><td>30,314</td><td>...</td><td>37,115</td><td>39,515</td><td>42,145</td><td>44,579</td></tr><tr><td>C23</td><td>114,851</td><td>106,710</td><td>98,463</td><td>...</td><td>55,039</td><td>45,718</td><td>36,060</td><td>26,987</td></tr><tr><td>C24</td><td>113,632</td><td>108,149</td><td>103,064</td><td>...</td><td>88,643</td><td>88,708</td><td>89,550</td><td>90,395</td></tr><tr><td>C25</td><td>9,751</td><td>14,557</td><td>20,059</td><td>...</td><td>59,490</td><td>69,679</td><td>80,281</td><td>89,775</td></tr><tr><td>C26</td><td>4,900</td><td>6,686</td><td>8,849</td><td>...</td><td>29,127</td><td>36,212</td><td>44,869</td><td>54,293</td></tr><tr><td>C27</td><td>4,072</td><td>4,669</td><td>5,424</td><td>...</td><td>12,361</td><td>14,568</td><td>17,100</td><td>19,653</td></tr><tr><td>C28</td><td>23,151</td><td>24,878</td><td>27,075</td><td>...</td><td>41,636</td><td>43,557</td><td>43,996</td><td>42,353</td></tr><tr><td>C29</td><td>5,498</td><td>7,281</td><td>9,374</td><td>...</td><td>25,080</td><td>29,281</td><td>33,735</td><td>37,833</td></tr><tr><td>C30</td><td>10,213</td><td>10,974</td><td>11,763</td><td>...</td><td>16,140</td><td>17,067</td><td>17,968</td><td>18,717</td></tr><tr><td>C31,C32</td><td>9,171</td><td>9,514</td><td>9,916</td><td>...</td><td>13,346</td><td>14,456</td><td>15,759</td><td>17,110</td></tr><tr><td>C33</td><td>2,858</td><td>2,793</td><td>2,757</td><td>...</td><td>3,160</td><td>3,404</td><td>3,728</td><td>4,101</td></tr></tbody></table>
<table><caption>Table 7: Sectoral downstream responsibility ($\lambda = 0, \nu_{max} = 1000$) in kton CO$_2$ (continued)</caption><thead><tr><th>Sector</th><th>δ = 0.0</th><th>δ = 0.1</th><th>δ = 0.2</th><th>...</th><th>δ = 0.7</th><th>δ = 0.8</th><th>δ = 0.9</th><th>δ = 0.99</th></tr></thead><tbody><tr><td>D35</td><td>2,094,483</td><td>1,892,898</td><td>1,690,987</td><td>...</td><td>676,866</td><td>473,170</td><td>269,160</td><td>85,230</td></tr><tr><td>E36</td><td>18,548</td><td>16,962</td><td>15,365</td><td>...</td><td>7,223</td><td>5,564</td><td>3,894</td><td>2,380</td></tr><tr><td>E37 - E39</td><td>7,867</td><td>10,340</td><td>12,831</td><td>...</td><td>24,944</td><td>27,030</td><td>28,803</td><td>29,963</td></tr><tr><td>F</td><td>62,943</td><td>67,608</td><td>71,566</td><td>...</td><td>80,543</td><td>80,168</td><td>79,117</td><td>77,647</td></tr><tr><td>G45</td><td>12,770</td><td>13,181</td><td>13,772</td><td>...</td><td>20,353</td><td>22,688</td><td>25,521</td><td>28,582</td></tr><tr><td>G46</td><td>26,891</td><td>47,796</td><td>68,543</td><td>...</td><td>167,618</td><td>185,485</td><td>201,921</td><td>214,831</td></tr><tr><td>G47</td><td>61,415</td><td>58,932</td><td>56,537</td><td>...</td><td>45,074</td><td>42,815</td><td>40,593</td><td>38,666</td></tr><tr><td>H49</td><td>246,868</td><td>248,125</td><td>246,664</td><td>...</td><td>189,845</td><td>165,572</td><td>135,273</td><td>101,772</td></tr><tr><td>H50</td><td>64,157</td><td>58,906</td><td>53,602</td><td>...</td><td>26,129</td><td>20,376</td><td>14,487</td><td>9,038</td></tr><tr><td>H51</td><td>194,809</td><td>177,243</td><td>159,746</td><td>...</td><td>74,029</td><td>57,487</td><td>41,278</td><td>27,049</td></tr><tr><td>H52</td><td>21,217</td><td>30,885</td><td>39,917</td><td>...</td><td>70,357</td><td>72,037</td><td>71,515</td><td>68,738</td></tr><tr><td>H53</td><td>7,109</td><td>16,032</td><td>24,129</td><td>...</td><td>48,276</td><td>48,763</td><td>47,279</td><td>43,966</td></tr><tr><td>I</td><td>61,085</td><td>61,765</td><td>62,546</td><td>...</td><td>72,600</td><td>77,290</td><td>83,643</td><td>91,229</td></tr><tr><td>J58</td><td>52</td><td>2,141</td><td>4,571</td><td>...</td><td>25,508</td><td>32,610</td><td>41,293</td><td>50,806</td></tr><tr><td>J59,J60</td><td>174</td><td>1,630</td><td>3,537</td><td>...</td><td>26,209</td><td>35,652</td><td>48,103</td><td>62,765</td></tr><tr><td>J61</td><td>3,673</td><td>7,996</td><td>13,211</td><td>...</td><td>61,736</td><td>78,939</td><td>100,347</td><td>124,255</td></tr><tr><td>J62,J63</td><td>14,379</td><td>21,155</td><td>28,661</td><td>...</td><td>81,912</td><td>96,747</td><td>113,373</td><td>130,043</td></tr><tr><td>K64</td><td>11,505</td><td>26,618</td><td>41,763</td><td>...</td><td>126,389</td><td>148,202</td><td>173,587</td><td>200,813</td></tr><tr><td>K65</td><td>10,217</td><td>14,704</td><td>20,550</td><td>...</td><td>84,994</td><td>110,596</td><td>144,260</td><td>184,332</td></tr><tr><td>K66</td><td>4,377</td><td>8,732</td><td>13,953</td><td>...</td><td>61,247</td><td>78,143</td><td>99,699</td><td>124,725</td></tr><tr><td>L68</td><td>12,913</td><td>26,752</td><td>41,918</td><td>...</td><td>152,510</td><td>186,364</td><td>226,722</td><td>270,116</td></tr><tr><td>M69-M70</td><td>20,294</td><td>43,682</td><td>69,764</td><td>...</td><td>238,138</td><td>277,578</td><td>317,542</td><td>352,940</td></tr><tr><td>M71</td><td>12,801</td><td>19,309</td><td>26,211</td><td>...</td><td>71,879</td><td>84,698</td><td>99,429</td><td>114,676</td></tr><tr><td>M72</td><td>6,897</td><td>10,516</td><td>14,352</td><td>...</td><td>39,707</td><td>46,816</td><td>54,983</td><td>63,433</td></tr><tr><td>M73</td><td>6,945</td><td>10,588</td><td>14,450</td><td>...</td><td>39,980</td><td>47,138</td><td>55,362</td><td>63,872</td></tr><tr><td>M74,M75</td><td>2,267</td><td>3,433</td><td>4,670</td><td>...</td><td>12,849</td><td>15,144</td><td>17,780</td><td>20,509</td></tr><tr><td>N</td><td>59,695</td><td>80,675</td><td>103,200</td><td>...</td><td>252,017</td><td>292,759</td><td>339,010</td><td>386,346</td></tr><tr><td>O84</td><td>242,418</td><td>229,014</td><td>215,821</td><td>...</td><td>153,721</td><td>142,373</td><td>131,570</td><td>122,434</td></tr><tr><td>P85</td><td>21,931</td><td>20,980</td><td>20,011</td><td>...</td><td>15,441</td><td>14,741</td><td>14,193</td><td>13,874</td></tr><tr><td>Q</td><td>84,405</td><td>77,136</td><td>69,784</td><td>...</td><td>32,024</td><td>24,341</td><td>16,649</td><td>9,740</td></tr><tr><td>R,S</td><td>32,280</td><td>33,141</td><td>34,541</td><td>...</td><td>54,193</td><td>62,136</td><td>72,225</td><td>83,614</td></tr><tr><td>T</td><td>0</td><td>121</td><td>259</td><td>...</td><td>1,261</td><td>1,551</td><td>1,884</td><td>2,227</td></tr><tr><td>U</td><td>0</td><td>0</td><td>0</td><td>...</td><td>0</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>
<table><caption>Table 8: Sector codes</caption><thead><tr><td>Code</td><td>Sector</td></tr></thead><tbody><tr><td>A01</td><td>Crop and animal production, hunting and related service activities</td></tr><tr><td>A02</td><td>Forestry and logging</td></tr><tr><td>A03</td><td>Fishing and aquaculture</td></tr><tr><td>B</td><td>Mining and quarrying</td></tr><tr><td>C10-C12</td><td>Manufacture of food products, beverages and tobacco products</td></tr><tr><td>C13-C15</td><td>Manufacture of textiles, wearing apparel and leather products</td></tr><tr><td>C16</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>C17</td><td>Manufacture of paper and paper products</td></tr><tr><td>C18</td><td>Printing and reproduction of recorded media</td></tr><tr><td>C19</td><td>Manufacture of coke and refined petroleum products</td></tr><tr><td>C20</td><td>Manufacture of chemicals and chemical products</td></tr><tr><td>C21</td><td>Manufacture of basic pharmaceutical products and pharmaceutical preparations</td></tr><tr><td>C22</td><td>Manufacture of rubber and plastic products</td></tr><tr><td>C23</td><td>Manufacture of other non-metallic mineral products</td></tr><tr><td>C24</td><td>Manufacture of basic metals</td></tr><tr><td>C25</td><td>Manufacture of fabricated metal products, except machinery and equipment</td></tr><tr><td>C26</td><td>Manufacture of computer, electronic and optical products</td></tr><tr><td>C27</td><td>Manufacture of electrical equipment</td></tr><tr><td>C28</td><td>Manufacture of machinery and equipment n.e.c.</td></tr><tr><td>C29</td><td>Manufacture of motor vehicles, trailers and semi-trailers</td></tr><tr><td>C30</td><td>Manufacture of other transport equipment</td></tr><tr><td>C31-C32</td><td>Manufacture of furniture; other manufacturing</td></tr><tr><td>C33</td><td>Repair and installation of machinery and equipment</td></tr><tr><td>D35</td><td>Electricity, gas, steam and air conditioning supply</td></tr><tr><td>E36</td><td>Water collection, treatment and supply</td></tr><tr><td>E37-E39</td><td>Sewerage; waste collection, treatment and disposal activities; materials recovery; remediation activities and other waste management services</td></tr><tr><td>F</td><td>Construction</td></tr><tr><td>G45</td><td>Wholesale and retail trade and repair of motor vehicles and motorcycles</td></tr><tr><td>G46</td><td>Wholesale trade, except of motor vehicles and motorcycles</td></tr><tr><td>G47</td><td>Retail trade, except of motor vehicles and motorcycles</td></tr><tr><td>H49</td><td>Land transport and transport via pipelines</td></tr><tr><td>H50</td><td>Water transport</td></tr><tr><td>H51</td><td>Air transport</td></tr><tr><td>H52</td><td>Warehousing and support activities for transportation</td></tr><tr><td>H53</td><td>Postal and courier activities</td></tr><tr><td>I</td><td>Accommodation and food service activities</td></tr><tr><td>J58</td><td>Publishing activities</td></tr><tr><td>J59-J60</td><td>Motion picture, video and television programme production, sound recording and music publishing activities; programming and broadcasting activities</td></tr><tr><td>J61</td><td>Telecommunications</td></tr><tr><td>J62-J63</td><td>Computer programming, consultancy and related activities; information service activities</td></tr><tr><td>K64</td><td>Financial service activities, except insurance and pension funding</td></tr><tr><td>K65</td><td>Insurance, reinsurance and pension funding, except compulsory social security</td></tr><tr><td>K66</td><td>Activities auxiliary to financial services and insurance activities</td></tr><tr><td>L68</td><td>Real estate activities</td></tr></tbody></table>
<table><caption>Table 9: Sector codes (continued)</caption><thead><tr><td>Code</td><td>Sector</td></tr></thead><tbody><tr><td>M69-M70</td><td>Legal and accounting activities; activities of head offices; management consultancy activities</td></tr><tr><td>M71</td><td>Architectural and engineering activities; technical testing and analysis</td></tr><tr><td>M72</td><td>Scientific research and development</td></tr><tr><td>M73</td><td>Advertising and market research</td></tr><tr><td>M74-M75</td><td>Other professional, scientific and technical activities; veterinary activities</td></tr><tr><td>N</td><td>Administrative and support service activities</td></tr><tr><td>O84</td><td>Public administration and defence; compulsory social security</td></tr><tr><td>P85</td><td>Education</td></tr><tr><td>Q</td><td>Human health and social work activities</td></tr><tr><td>R-S</td><td>Other service activities</td></tr><tr><td>T</td><td>Activities of households as employers; undifferentiated goods- and services-producing activities of households for own use</td></tr><tr><td>U</td><td>Activities of extraterritorial organizations and bodies</td></tr></tbody></table>

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