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# The resource, waste and climate conundrum: essays on the economics of recycling

## Etienne Lorang

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### ► To cite this version:

Etienne Lorang. The resource, waste and climate conundrum: essays on the economics of recycling. Economics and Finance. Université de Lorraine, 2022. English. NNT: 2022LORR0016. tel-03736518

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https://hal.univ-lorraine.fr/tel-03736518

Submitted on 22 Jul 2022

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Distributed under a Creative Commons Attribution 4.0 International License
AVERTISSEMENT

Ce document est le fruit d'un long travail approuvé par le jury de soutenance et mis à disposition de l'ensemble de la communauté universitaire élargie.

Il est soumis à la propriété intellectuelle de l'auteur. Ceci implique une obligation de citation et de référencement lors de l'utilisation de ce document.

D'autre part, toute contrefaçon, plagiat, reproduction illicite
encourt une poursuite pénale.

Contact bibliothèque : ddoc-theses-contact@univ-lorraine.fr

*(Cette adresse ne permet pas de contacter les auteurs)*

LIENS

Code de la Propriété Intellectuelle. articles L 122. 4

Code de la Propriété Intellectuelle. articles L 335.2- L 335.10

http://www.cfcopies.com/V2/leg/leg_droi.php

http://www.culture.gouv.fr/culture/infos-pratiques/droits/protection.htm
# The resource, waste and climate conundrum:
## Essays on the economics of recycling

### Thèse de doctorat de l'université de Lorraine

École Doctorale : Sciences Juridiques, Politiques, Économiques et de Gestion (SJPEG)

Laboratoire : Bureau d'Économie Théorique et Appliquée (BETA)

Mention : Sciences Économiques

Thèse présentée et soutenue à Nancy, le 18 Mars 2022, par

### Etienne LORANG

#### Composition du Jury

**Sophie BERNARD**
Professeure, Polytechnique Montréal

Examinatrice

**Mouez FODHA**
Professeur, Université Paris 1

Rapporteur

**Ankinée KIRAKOZIAN**
Professeure, Université de Lorraine

Présidente du jury

**Aude POMMERET**
Professeure, Université Savoie Mont Blanc

Rapporteuse

**Philippe DELACOTE**
Directeur de recherche, INRAE

Directeur de thèse

**Gilles LAFFORGUE**
Professeur, Toulouse Business School

Directeur de thèse
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THE RESOURCE, WASTE AND CLIMATE CONUNDRUM:
ESSAYS ON THE ECONOMICS OF RECYCLING
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Résumé en français

Dans cette thèse, je propose une analyse de l'économie du recyclage lorsque trois types d'interactions entre les activités anthropiques et les milieux naturels sont considérés : i) la production/consommation de biens et la disponibilité en matières premières vierges, ii) la production/consommation et l'accumulation de déchets, et, iii) la production/consommation et les émissions carbonées associées. Au cours de quatre chapitres, j'explore le potentiel que représente le recyclage pour le développement d'un modèle économique durable en évaluant ses impacts environnementaux, en analysant les objectifs de politiques publiques et en remettant en question le cadre d'analyse concernant la durabilité.

Recyclage sous contraintes environnementales, climatiques et de
ressources

Ce chapitre étudie les possibilités de recyclage d'un secteur industriel limité par les capacités en matière de ressources, de climat et de déchets. Avec les difficultés de développement de marchés et d'industries spécifiques, les intrants recyclés luttent pour concurrencer les intrants ordinaires dans la production, en particulier lorsque des taux de recyclage élevés entraînent des coûts importants. Cependant, j'analyse comment la prise en compte des coûts sociaux liés à l'extraction des ressources, à leur utilisation et à leur fin de vie, pourrait modifier l'arbitrage entre extraction et recyclage. Cette recherche contribue à l'analyse du recyclage à travers les outils de la littérature sur l'économie des ressources.

Pour ce faire, je modélise le cycle de vie complet d'un bien afin de prendre en compte
les déchets et les émissions de gaz à effet de serre (GES) provenant à la fois de sa produc-
tion – à partir de matériaux vierges ou recyclés – et de sa consommation. Le planificateur
social choisit les quantités de ressources (vierges ou recyclées) et la part de déchets à val-
oriser. Dans une première itération, seule l'accumulation des déchets est prise en compte
par l'introduction d'une fonction de dommage aux déchets. Dans une seconde itération, les
émissions de GES sont introduites à travers un budget carbone.

J'identifie d'abord les trajectoires optimales d'utilisation des ressources, principalement en fonction de la rareté relative des ressources et de leurs émissions lorsqu'elles sont incluses. Bien que le recyclage soit généralement, et à juste titre, considéré comme une opportunité de réduire l'impact de la consommation sur les ressources primaires et les déchets, je considère également les éventuelles conséquences environnementales négatives du recyclage et je discute des arbitrages qui en découlent. Le recyclage est en effet également une source d'émissions de GES, bien que dans la plupart des cas, elles soient plus faibles que pour les ressources extraites. La structure du modèle (coûts privés linéaires et substitution parfaite) donne des solutions en coin concernant le choix de la ressource. La ressource vierge est exclusivement utilisée pendant une première phase de production. Ensuite, la ressource recyclée peut être utilisée si elle devient socialement plus rentable, soit parce que la rareté de la ressource vierge prévaut, soit parce que les coûts privés favorisent l'utilisation du recyclage, et, soit parce que la différence d'émissions entre les deux ressources est suffisamment grande. Je montre également qu'il peut y avoir une phase de rattrapage de la consommation à la fin du programme du planificateur social.

Je caractérise aussi la stratégie optimale de recyclage et je montre que dans certains cas, le taux de recyclage à travers le temps est en forme de U inversé. Le recyclage est bénéfique à deux égards : il allège le coût de l'accumulation des déchets et permet de produire à partir d'un intrant recyclé. Cela montre la dualité de l'activité de tri des déchets : un objectif pratique de réduction des coûts et un objectif spéculatif d'utilisation potentielle d'intrants recyclés à l'avenir. Lorsque le recyclage est trop coûteux, une phase où le taux de recyclage est nul se produit (bien qu'il soit toujours possible de produire à partir des matériaux recyclés déjà stockés).

Enfin, je discute des implications politiques du modèle en identifiant et en analysant l'ensemble des schémas optimaux de taxes-subventions, et je mets en évidence l'existence d'externalités environnementales standards ainsi qu'une externalité positive liée à l'absence
d'un marché des déchets. Pour implémenter cette solution dit first-best, il faut introduire
dans les deux secteurs une taxation basée sur les émissions de GES, pondérée par l'intensité
carbone de la branche. Elle s'ajoute à un système de taxation-subvention basé sur les déchets
et la reconstitution des stocks, reflétant l'externalité positive.

Impacts sectoriels, sur les ressources et le carbone, de l'augmentation
du recyclage du papier et du carton

Dans ce chapitre, j'évalue le rôle du recyclage du papier et du carton dans le secteur forestier, tant du point de vue économique que de celui du climat. La question de savoir si le recyclage du papier et du carton a un impact positif ou négatif sur le carbone est cruciale car le secteur forestier mondial joue un rôle central dans l'atténuation du changement climatique : le secteur du recyclage du papier et du carton est bien développé en France, avec 66% de la production provenant de déchets récupérés. Des impacts complexes en cascade sur la ressource et les autres produits bois suggèrent des impacts économiques et environnementaux incertains. Historiquement, les facteurs de déplacement des GES du recyclage du papier ont été étudiés avec des hypothèses économiques fortes. Cela a conduit à des résultats contradictoires entre les études, en raison des différences dans les méthodes de comptabilisation des émissions, et en raison des limites organisationnelles, spatiales et temporelles du secteur considéré.

Pour cette raison, je modélise l'industrie du recyclage du papier et du carton dans le
cadre d'un modèle d'évaluation intégrée de la forêt française, afin de la relier aux autres
produits bois et à la ressource forestière. Sur la base de la situation actuelle du secteur
forestier français, j'examine l'impact d'une augmentation du recyclage. Pour simuler une
augmentation du recyclage, je mets en œuvre une réduction des coûts dans le secteur. Un
module économique d'équilibre partiel des produits bois et une structure récursive de la
dynamique forestière étendent une analyse du cycle de vie standard avec à la fois des impacts
économiques sur d'autres secteurs de produits bois et des variations dans la substitution et
la séquestration du carbone.

En explicitant le modèle, il est clair que les résultats dépendent fortement des hypothèses concernant la substituabilité de la pâte à papier recyclée et de la pâte à papier vierge, alors que les nombreuses études existantes ne tranchent pas sur la question de savoir si c'est la substitution ou la complémentarité qui prévaut (certaines soutiennent qu'avec le temps, la complémentarité prévaut). Empiriquement, les élasticités semblent être faibles et dépendent de l'utilisation finale du produit en papier ou en carton. Les résultats qualitatifs attendus concernant la complémentarité ou la substitution des deux produits sont trouvés, avec respectivement un impact positif ou négatif sur la demande lorsque le recyclage augmente. Les émissions de GES du secteur de la pâte à papier sont en augmentation dans les deux cas, ce qui soulève la question de leur éventuelle atténuation par le reste du secteur forestier. Les effets sur les autres produits bois sont faibles, en raison des faibles élasticités-prix des produits en concurrence pour les ressources en bois et des faibles élasticités croisées entre vierge et recyclé.

Dans l'ensemble, l'impact sur la séquestration nette totale est : positif lorsque les produits à base de pâte sont des substituts forts ; très faible lorsque les produits à base de pâte sont des substituts faibles ; et négatif lorsque les produits à base de pâte sont des compléments. Les effets relatifs sont faibles (entre 0,4 % et 0,8 % d'augmentation du bilan carbone du secteur en cas de substitution forte, un scénario très optimiste selon la littérature économétrique). Comme la faible complémentarité entre le papier vierge et le papier recyclé semble être l'hypothèse la plus valable, l'idée suggérant que le recyclage et l'économie circulaire sont une bonne solution contre le réchauffement climatique doit être prise avec précaution : dans cette situation, une augmentation de l'utilisation du recyclage ne s'accompagne pas d'une économie des ressources en bois.
Une évaluation de la réglementation européenne sur le recyclage
des batteries des véhicules électriques

Ce chapitre étudie une récente proposition de réglementation européenne visant à favoriser l'émergence d'une industrie du recyclage des batteries en Europe. La mobilité électrique est jugée nécessaire pour réduire les émissions de CO2 dans le secteur des transports. De plus, les batteries lithium-ion (Li-ion) - les composants centraux des véhicules électriques (VE) - ont fait l'objet de progrès technologiques remarquables ces dernières années. En conséquence, leur coût est passé de 1100 USD par kilowattheure (kWh) en 2010 à 156 USD/kWh en 2020. Cependant, les impacts industriels et environnementaux de la fabrication des batteries lithium-ion sont controversés. Parmi eux, la fin de vie des batteries peut être particulièrement dangereuse. Ajouté à une potentielle rareté de certains métaux (lithium, mais aussi terres rares), le recyclage des batteries usagées apparaît comme une solution séduisante. Pour faire face à ces problématiques, l'Union Européenne (UE) a développé plusieurs initiatives, telles que la European Battery Alliance et la Raw Materials Initiative. La nouvelle proposition de la Commission européenne vise à ajouter un cadre législatif à ces initiatives.

Le cadre réglementaire actuel de l'UE sur les piles et batteries est fixé par une directive de la Commission européenne de 2006. Cette directive vise à fournir des règles communes à travers le marché européen qui limiteraient les impacts environnementaux des déchets de batteries. Cependant, le bouleversement provoqué par l'arrivée des batteries Li-ion, alimenté par le développement de l'électromobilité, a rendu cette ancienne directive obsolète. Une récente proposition réglementaire de la Commission européenne introduit l'obligation d'atteindre une série de seuils minimaux de matériaux recyclés pour les nouvelles batteries qui seront fabriquées après 2030. Les matériaux concernés pour les batteries Li-ion sont le nickel, le cobalt et le lithium. Je discute dans ce chapitre des conditions requises pour que cette obligation soit remplie.

Tout d'abord, je développe un cadre d'analyse qualitative pour discuter des principales
tendances caractéristiques du secteur des batteries et de la manière dont elles devraient affecter l'approvisionnement en matériaux recyclés. Ce cadre d'analyse permet de séparer les différents mécanismes qui déterminent la quantité de matériaux recyclés pouvant être incorporés dans les nouvelles batteries : l'évolution de la demande en batteries, la composition des matériaux liée aux changements technologiques, l'efficacité du recyclage et la probabilité de fin de vie.

Deuxièmement, je calibre un modèle de flux de matières qui effectue des projections de flux de déchets de batteries et leur potentiel de recyclage. En utilisant des données provenant des rapports économiques de l'Agence internationale de l'énergie et de la littérature académique, le modèle indique que la faisabilité des seuils actuellement proposés n'est pas très sensible aux changements d'intensité des matériaux liés aux changements technologiques des batteries, aux efficacités de recyclage ou à l'accélération de la demande. Au contraire, la durée de vie des batteries est le paramètre le plus crucial pour le potentiel de recyclage et le respect des seuils proposés. En effet, la plupart des objectifs fixés par la Commission européenne ne sont pas atteints pour des durées de vie moyennes des batteries supérieures à 10 ans. Une telle hypothèse peut être considérée comme irréaliste compte tenu des progrès technologiques réalisés dans le secteur des batteries, avec des durées de vie plus longues. En outre, un scénario lors duquel la seconde vie des batteries serait développée réduirait considérablement le potentiel de contenu recyclé. Dans ce cas, les batteries des VEs sont remises à neuf lorsqu'elles ne répondent pas aux normes de puissance des VEs et sont utilisées pour le stockage stationnaire.

Les recommandations de politiques publiques se déclinent en deux volets. Premièrement, je recommande d'abaisser les seuils pour améliorer la crédibilité de la réglementation. Bien que les objectifs puissent être modifiés à l'avenir, les acteurs industriels du secteur (fabricants de batteries et de voitures, industrie du recyclage) auraient besoin d'une réglementation lisible et crédible aujourd'hui pour faire les bons investissements et les bons choix technologiques. Deuxièmement, la réglementation devrait intégrer d'autres objectifs qui con-
cernent la durée de vie des batteries. En effet, le développement d'un secteur durable des batteries ne doit pas reposer uniquement sur des objectifs de recyclage et de contenu recyclé. De plus, le fait de ne fixer que des objectifs de contenu élevés pourrait inciter à réduire la durée de vie des batteries, car une durée de vie plus courte permettrait une circulation plus rapide des matériaux dans l'économie, augmentant ainsi l'utilisation potentielle de matériaux recyclés.

## Le recyclage dans un cadre durable

Dans ce dernier chapitre, j'étudie la question du recyclage en macroéconomie dans le cadre de la durabilité. En effet, l'étude du recyclage en macroéconomie vise à évaluer comment les boucles fermées de matériaux sont capables d'améliorer la croissance à long terme avec un épuisement optimal des stocks de ressources. Cette analyse peut également prendre en compte la réduction des externalités environnementales grâce au recyclage. En ce sens, le recyclage, et plus généralement le concept d'économie circulaire, sont censés être un moyen de suivre une voie de développement durable. Cependant, l'économie de la durabilité est également liée aux questions d'équité en termes de consommation et d'impacts environnementaux. A ce sujet, la littérature sur le recyclage ne s'appuie pas sur un critère durable ; dans sa grande majorité, elle s'appuie sur l'utilité escomptée, un critère ne favorisant pas l'équité.

Pour explorer le sujet du recyclage dans un cadre de modélisation durable, je développe un modèle de ressources et de déchets résolu à l'aide du critère du maximin. La valeur maximin est définie comme le plus haut niveau d'utilité qui peut être maintenu à l'infini. J'utilise un modèle basé sur le modèle de Dasgupta-Heal-Solow, pour lequel la littérature indique qu'un chemin maximin existe dans le cas de la durabilité faible. Cela correspond au cas où la fonction de production ne dépend pas fortement de la ressource naturelle (pour une fonction de Cobb Douglas, lorsque l'élasticité de la ressource dans la production est inférieure à celle du capital).
Ce modèle simple complexifie l'analyse d'un niveau minimal de consommation durable. Je suis en mesure de caractériser un chemin optimal et ses valeurs fictives, et de discuter de certaines trajectoires à long terme. Des règles plus complexes de gestion des ressources apparaissent via une nouvelle règle d'investissement de Hartwick qui dépend maintenant de l'accumulation des déchets et de l'utilisation du recyclage dans la production. Dans le cas spécifique d'un stock de déchets vidé en temps fini, il est possible de discuter des chemins de consommation explicites len spécifiant les fonctions de production. Dans le cas d'une fonction de production Cobb-Douglas, l'hypothèse d'une élasticité du capital supérieure à l'élasticité de la ressource dans la production est cruciale pour qu'une trajectoire durable existe.

Ce cas spécifique fragilise l'hypothèse que le recyclage puisse permettre des trajectoires durables dans le cas de la durabilité forte. Ce modèle est un premier pas vers l'étude du recyclage à l'aide des outils de l'économie de la durabilité. À ma connaissance, c'est en effet le premier modèle qui étudie le recyclage avec le critère du maximin, et qui met en évidence comment les règles d'investissement optimales sont modifiées.
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REMERCIEMENTS

L'aboutissement de ce travail de plus de trois ans n'a pu avoir lieu sans les interventions et l'aide de nombreuses personnes que je souhaite très chaleureusement remercier.

Je remercie tout d'abord mes directeurs de thèse, Philippe Delacote et Gilles Lafforgue. Ils m'ont accordé leur totale confiance, leurs conseils ainsi que leur grande disponibilité qui ont été indispensables pour mener à bien mes travaux. Je ne pouvais pas espérer de meilleurs encadrants.

Je remercie également Sophie Bernard, Mouez Fodha, Ankinee Kirakozian et Aude Pom-meret d'avoir accepté de faire partie de mon jury et de lire mon manuscrit. J'espère qu'il les intéressera.

Cette thèse a été réalisée grâce à l'accueil et au soutien financier de la Chaire Économie du Climat. Je remercie son fondateur Christian de Perthuis pour son accueil, et Anna Creti qui en a repris la direction avec brio. Merci pour la confiance qu'ils m'ont accordé. J'espère avoir pu apporter ma petite pierre au développement de ce beau projet.

Mon laboratoire pendant ces trois années, le BETA à Nancy, a également été indispens-able à l'accomplissement de ce projet. Je remercie chaleureusement Serge Garcia de m'avoir accueilli, malgré la distance et mes trop brefs passages.

Mon travail n'aurait pu être effectué sans la collaboration déterminante de mes co-
auteurs. Je tiens déjà à remercier à nouveau Philippe et Gilles, qui en plus d'être d'incroyables encadrants m'ont permis de beaucoup apprendre en travaillant à leurs côtés. Je remercie Antonello Lobianco, qui, avec patience et longueur de temps, a su m'introduire aux questions de modélisation forestière. Je remercie également Quentin Hoarau, qui a formé un duo d'excellents partenaires de coworking avec Maxime Ollier, et dont le savoir faire technique et les connaissances m'ont beaucoup apporté.

Merci également aux nombreux relecteurs de mes travaux, qui ont su apporter leurs réflexions et corrections sans lesquelles ce manuscrit ne serait pas ce qu'il est: les déjà cités Maxime et Quentin bien sûr, mais également Jérôme Deyris, Eugénie Joltreau, Vincent Martinet, Olivier Massol, Yannick Perez et Simon Quemin. Je remercie tout particulièrement Kevin Basek sans lequel la langue anglaise dans ce manuscrit aurait été fortement amochée. Merci à Arthur, Aude, Gaspard, Marc-Daniel, Marie, Romain et Arthur dont les relectures ont permis d'éviter quelques coquilles malvenues.

Cette thèse doit beaucoup à l'environnement chaleureux dans lequel elle a été réalisée. Merci au doctorant.e.s et stagiaires du Palais Brongniart, et notamment Marie et Mathias que j'ai eu le plaisir de coencadrer. Merci également aux chercheur.e.s et doctorant.e.s du BETA pour leur accueil en terre nancéienne. Merci aux multiples chercheur.e.s rencontré.e.s aux conférences de la FAERE, de l'EAERE ou de la WCNRM. Merci à Mehdi Senouci de m'avoir permis d'enseigner son cours exigeant. Merci beaucoup à Francesco Ricci de m'avoir proposé de participer à son projet ANR ScarCyclET. Merci également à Nicolas Humez et Cédric L'Elchat pour leur disponibilité et l'intérêt qu'ils ont porté à mes travaux. Merci à Claire Bérenger et Évelyne Lenel pour leur assistance au cours de ces années.

Enfin, je remercie mes parents, frère, grand-parents, tantes, cousin.e.s et ami.e.s dont la patience et le soutien sans faille ont été si précieux ces dernières années.
* * *

Où s'arrête le jeu, où commence l'art [...]
la récup' ménagère est entrée dans le marché de l'art
et quand je dis marché, je ne dis pas bon marché.

Les glaneurs et la glaneuse - Agnes Varda (2000)

* * *
CONTENTS

<table><tr><td>List of Figures</td><td>xxiii</td></tr><tr><td>List of Tables</td><td>xxv</td></tr><tr><td>Introduction</td><td>3</td></tr><tr><td>An economy of material resources</td><td>3</td></tr><tr><td>An historical perspective on a material economy</td><td>3</td></tr><tr><td>Material resources in the economy</td><td>5</td></tr><tr><td>Local pollution and social impacts</td><td>7</td></tr><tr><td>Climate change</td><td>9</td></tr><tr><td>Waste accumulation</td><td>10</td></tr><tr><td>The recycling challenge</td><td>12</td></tr><tr><td>Historical perspective</td><td>12</td></tr><tr><td>Policy framework</td><td>14</td></tr><tr><td>Trade-off between resource, waste and climate</td><td>17</td></tr><tr><td>Contributions of this dissertation</td><td>18</td></tr><tr><td>Scope of the dissertation</td><td>18</td></tr></table>
<table>
  <tbody>
    <tr>
      <td></td>
      <td>Description of the chapters</td>
      <td>18</td>
    </tr>
    <tr>
      <td>1</td>
      <td>Recycling under environmental, climate and resource constraints</td>
      <td>27</td>
    </tr>
    <tr>
      <td>1.1</td>
      <td>Introduction</td>
      <td>28</td>
    </tr>
    <tr>
      <td>1.2</td>
      <td>Related literature</td>
      <td>31</td>
    </tr>
    <tr>
      <td>1.3</td>
      <td>A basic optimal recycling model</td>
      <td>33</td>
    </tr>
    <tr>
      <td>1.3.1</td>
      <td>Setup of the model</td>
      <td>33</td>
    </tr>
    <tr>
      <td>1.3.2</td>
      <td>Central planner program and optimal conditions</td>
      <td>36</td>
    </tr>
    <tr>
      <td>1.3.3</td>
      <td>Arbitration on resources use</td>
      <td>39</td>
    </tr>
    <tr>
      <td>1.3.4</td>
      <td>Recovering waste</td>
      <td>42</td>
    </tr>
    <tr>
      <td>1.3.5</td>
      <td>Dynamics of resource flows</td>
      <td>45</td>
    </tr>
    <tr>
      <td>1.4</td>
      <td>Accounting for climate change impacts</td>
      <td>48</td>
    </tr>
    <tr>
      <td>1.4.1</td>
      <td>Optimal program with greenhouse gas emissions</td>
      <td>48</td>
    </tr>
    <tr>
      <td>1.4.2</td>
      <td>Relative scarcity of the stocks and budget</td>
      <td>50</td>
    </tr>
    <tr>
      <td>1.4.3</td>
      <td>Arbitration on resources use</td>
      <td>52</td>
    </tr>
    <tr>
      <td>1.4.4</td>
      <td>Dynamics of resource flows</td>
      <td>53</td>
    </tr>
    <tr>
      <td>1.5</td>
      <td>Decentralization and policy implications</td>
      <td>55</td>
    </tr>
    <tr>
      <td>1.5.1</td>
      <td>Equilibrium in the absence of a waste market</td>
      <td>56</td>
    </tr>
    <tr>
      <td>1.5.2</td>
      <td>Existence of a waste market</td>
      <td>61</td>
    </tr>
    <tr>
      <td>1.6</td>
      <td>Conclusion</td>
      <td>62</td>
    </tr>
    <tr>
      <td></td>
      <td>Appendix</td>
      <td>64</td>
    </tr>
    <tr>
      <td>1.A</td>
      <td>Appendix</td>
      <td>64</td>
    </tr>
    <tr>
      <td>1.A.1</td>
      <td>Scrap value for waste</td>
      <td>64</td>
    </tr>
    <tr>
      <td>1.A.2</td>
      <td>Decoupling emissions of the recycling branch</td>
      <td>66</td>
    </tr>
    <tr>
      <td>1.A.3</td>
      <td>Limiting the capacity of the recycling sector</td>
      <td>68</td>
    </tr>
    <tr>
      <td>2</td>
      <td>Sectoral, resource and carbon impacts of increased paper and cardboard recycling</td>
      <td>73</td>
    </tr>
  </tbody>
</table>
<table><tr><td>2.1</td><td>Introduction</td><td>74</td></tr><tr><td>2.1.1</td><td>Context and motivations</td><td>74</td></tr><tr><td>2.1.2</td><td>Related literature</td><td>75</td></tr><tr><td>2.2</td><td>Adding a paper recycling loop to a forest sector model</td><td>79</td></tr><tr><td>2.2.1</td><td>Modular structure of FFSM</td><td>79</td></tr><tr><td>2.2.2</td><td>Dynamics of pulp recycling</td><td>81</td></tr><tr><td>2.2.3</td><td>Calibration and simulations strategy</td><td>83</td></tr><tr><td>2.3</td><td>Impact of paper recycling on the carbon balance</td><td>85</td></tr><tr><td>2.3.1</td><td>Impacts on the pulp industry</td><td>85</td></tr><tr><td>2.3.2</td><td>Impacts on other forest products</td><td>87</td></tr><tr><td>2.3.3</td><td>Impacts on the forest resource</td><td>88</td></tr><tr><td>2.3.4</td><td>Global carbon impact on the forest sector</td><td>90</td></tr><tr><td>2.4</td><td>Conclusion</td><td>91</td></tr></table>

3 An assessment of the European regulation on battery recycling for electric vehicles 95

<table><tr><td>3.1</td><td>Introduction</td><td>96</td></tr><tr><td>3.2</td><td>Proposal of the European Commission</td><td>99</td></tr><tr><td>3.3</td><td>Qualitative Insights</td><td>101</td></tr><tr><td>3.4</td><td>Quantitative Analysis: Methods</td><td>103</td></tr><tr><td>3.4.1</td><td>Model</td><td>103</td></tr><tr><td>3.4.2</td><td>Data</td><td>105</td></tr><tr><td>3.5</td><td>Quantitative Analysis: Results</td><td>109</td></tr><tr><td>3.5.1</td><td>Evolution of RMD Ratios and Effect of Demand Growth</td><td>109</td></tr><tr><td>3.5.2</td><td>Effect of the Battery Lifespan</td><td>110</td></tr><tr><td>3.5.3</td><td>Effect of Technological Change</td><td>111</td></tr><tr><td>3.5.4</td><td>Effect of Recycling Efficiencies</td><td>112</td></tr><tr><td>3.6</td><td>Conclusion and Policy Implications</td><td>113</td></tr></table>
<table><tr><td>4 Recycling in a sustainable framework</td><td>117</td></tr><tr><td>4.1 Context and motivations</td><td>118</td></tr><tr><td>4.2 Recycling and sustainability</td><td>121</td></tr><tr><td>4.3 A maximin model of recycling</td><td>124</td></tr><tr><td>4.3.1 Setup of the model</td><td>124</td></tr><tr><td>4.3.2 Characterization of the maximin path</td><td>129</td></tr><tr><td>4.3.3 Hartwick rule and sustainability</td><td>133</td></tr><tr><td>4.4 Conclusion</td><td>139</td></tr><tr><td>General conclusion</td><td>143</td></tr><tr><td>Transversal and policy considerations</td><td>144</td></tr><tr><td>An holistic approach of recycling</td><td>144</td></tr><tr><td>Can recycling be bad?</td><td>145</td></tr><tr><td>Substitution and complementarity</td><td>145</td></tr><tr><td>Recycling and circular economy as a goal?</td><td>146</td></tr><tr><td>Future work</td><td>146</td></tr><tr><td>Recycling and sustainability accounting</td><td>146</td></tr><tr><td>Oligopolistic market structures for critical raw materials</td><td>147</td></tr><tr><td>Substituting virgin and recycled resources</td><td>148</td></tr><tr><td>Bibliography</td><td>151</td></tr></table>
LIST OF FIGURES

<table><tr><td>1.1</td><td>(a) Saturation of β and (b) Maximum time T̂<sub>β</sub></td><td>45</td></tr><tr><td>2.1</td><td>Recursive structure of FFSM</td><td>80</td></tr><tr><td>2.2</td><td>Products structure of FFSM</td><td>81</td></tr><tr><td>2.3</td><td>Changes in demand for pulp between baseline and med for substitutability (η &lt; 0) and complementarity (η &gt; 0) in 2030</td><td>86</td></tr><tr><td>2.4</td><td>Relative changes in demand for other wood products between baseline and med in 2030</td><td>87</td></tr><tr><td>3.1</td><td>Projected EV batteries growth for two demand scenarios (extrapolated from IEA)</td><td>106</td></tr><tr><td>3.2</td><td>Ratios for Li, Ni and Co (policy objectives in dashed lines)</td><td>109</td></tr></table>
LIST OF TABLES

<table><tbody><tr><td>1.1 Examples of emissions rates in France (in kg CO<sub>2</sub>e/t)</td><td>29</td></tr><tr><td>1.2 Possible shapes of the gross marginal surplus and conditions of occurrence</td><td>48</td></tr><tr><td>1.3 Examples of policy-mix</td><td>60</td></tr><tr><td>2.1 Tested recycling scenarios</td><td>85</td></tr><tr><td>2.2 Changes in emissions linked to transformation in the pulp industry (compared with scenario baseline) - yearly average for 2013-2080</td><td>86</td></tr><tr><td>2.3 Changes in carbon substitution for other wood products (compared with scenario baseline) - yearly average for 2013-2080</td><td>88</td></tr><tr><td>2.4 Changes in carbon stocks (compared with scenario baseline) - yearly average for 2013-2080</td><td>89</td></tr><tr><td>2.5 Changes in forest volumes and areas between baseline and med in 2070</td><td>89</td></tr><tr><td>2.6 Changes in global GHG balance (compared with scenario baseline) - yearly average for 2013-2080</td><td>90</td></tr><tr><td>3.1 Targets on battery recycling of the 2020 EC proposal</td><td>100</td></tr><tr><td>3.2 Indexes of the model</td><td>104</td></tr><tr><td>3.3 Battery Composition for each battery type. Values are given in kg/kWh.</td><td>107</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td>3.4</td>
      <td>Description technological scenarios with their respective market shares (in %).</td>
      <td>107</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Description of scenarios on lifespans and other relevant parameter for the Weibull function of batteries (in years)</td>
      <td>108</td>
    </tr>
    <tr>
      <td>3.6</td>
      <td>Ratios of recycled materials over demand (RMD ratios) for alternative battery lifetimes with the STEPS demand scenario. Values of RMD ratios above the respective EC minimum threshold are written in bold.</td>
      <td>110</td>
    </tr>
    <tr>
      <td>3.7</td>
      <td>Ratios of recycled materials over demand (RMD ratios) for alternative technological scenarios with the STEPS demand scenario. Values of RMD ratios above the respective EC minimum threshold are written in bold.</td>
      <td>111</td>
    </tr>
    <tr>
      <td>3.8</td>
      <td>Ratios of recycled materials over demand (RMD ratios) for an alternative recycling efficiency scenarios with the STEPS demand scenario. Values of RMD ratios above the respective EC minimum threshold are written in bold.</td>
      <td>112</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Share of reinvested production θ(t)</td>
      <td>135</td>
    </tr>
  </tbody>
</table>
# INTRODUCTION

## An economy of material resources

### An historical perspective on a material economy

Global primary material use is expected to double to support world gross domestic product (GDP) growth in the next 40 years (OECD, 2019). These figures remind us that economic development remains widely based on material resources. It may seem outdated in the 21st century, with an ever-increasing presence of a so-called dematerialized economy and a reduction of the share of agriculture and industry in the production apparatus of developed countries.

Historically, economic development prior to the industrial revolution was strongly linked to the exploitation of natural resources. Availability of resources was central in the economic thought, with what is called an "organic regime of growth". The focus on resource availability goes from the agricultural structuration at the core of physiocracy to manufacturing and commercial exchanges that structure the emerging classical economy with Adam Smith (Charbonnier, 2021). However with the introduction of coal in the 18th-19th centuries (the start of the industrial revolution), a transformation of the industrial apparatus allowed for an apparent freeing from the limitations of the organic regime. This freedom came from intensive growth via a significant increase in productivity, as well as an extensive growth in
access to natural resources.

The fatal relationship between economic growth and the finite resource (coal) was already questioned by William Stanley Jevons who anticipated a limit to the techno-industrial superiority of Great-Britain (Jevons, 1865). Jevons introduced a now widely known paradox: gains in productivity which lead to a more sparing use of coal, are offset by more extensive use. Even knowing the paradox, this concern is put aside with the use of coal's modern counterpart, oil, due to the "freedom" from domestic resource constraints for western economies, with an apparent abundance. It took until the second half of the 20th century for concerns regarding natural resources availability to be revived. Those concerns are most famously expressed by the Club of Rome, whose report on the limits to growth showed that physical limits of natural stocks lead to a dead-end (Meadows et al., 1972). The report considers the economic system as a global one that is built on natural capital. This approach is contemporary of Georgescu-Roegen's work on the integration of physical concepts (mainly thermodynamics) into economics in order to reconcile economic sciences with physics laws that rule the indispensable use of energy and material resources in production (Georgescu-Roegen, 1970). These considerations are added to an ever-growing ecological conscience regarding risks to natural capital: local pollution, disruption of ecosystems and climate change. This unfolding of economic thought lead us to the current observation of a prominent role for natural resources in the economy, from biological to energy to mining resources.

At the same time, certain neoclassical economists have also integrated resource considerations in their models and analysis for years. Solow's classical growth model was enriched by the inclusion of natural resources alongside capital and labour as factors of production. However, limitations of natural resource supply were overcome by the assumption of unbounded material productivity. Unbounded productivity relies on two concepts: substitution and technical progress, both of which are derived from the hypothesis of neoclassical growth models (Daly, 1997; Couix, 2019). This controversy between neoclassical resource
economics and ecological economics gained in interest in the 21st century, as concepts of dematerialized and circular economy became more broadly discussed. The main question is whether a transition towards an economy decorrelated from material constraints can be achieved. In the same way, the history of energy sources is put in terms of transitions while some argue that additions of energy sources is more accurate (Fressoz, 2020).

Material resources in the economy

Material flows in the economy are now accompanied by a broad topic of concerns. The significant growth rate of resource use for metals and non-metallic minerals motivates the monitoring of material flows in the economy. Closer monitoring has revealed a growth of demand driven by demographic and economic growth. The nature of the demand is also changing, with variations across countries and across time as new technologies emerge. A so-called dematerialization of the economy puts away, for instance, material considerations in production, especially in developed countries. However, the material dependency of the economy is still subject to scarcity issues that come from several factors.

First, the risk of physical scarcity of resource stocks has been put forward since the Meadows report of 1972. However, the notion of scarcity bears several limits as there is a difference between the "existing resource" and the "available stock". The size of the available stock is used to define the size of the reserve and can be described as the currently exploitable resource. The level of exploitable resource is conditioned on the known size of the stock and the current price of the resource that would make extraction economically feasible. This economic dimension allows to define a second stock that includes the geologically available resources that are not economically exploitable. With this first form of criticality lies a trade-off in resource use, which is choosing the specific stock to extract it from, based on the underlying cost. To these stocks should be added potential stocks of resources that are either identified but unexplored, or unidentified but potentially existing. For instance, in Central and South America, estimations of available copper resources are three times higher
than the current reserves, which already represents half of the world's supply (Seck et al., 2020).

Second, the criticality of material resources also relies on their uneven geographical distribution. This situation leads to geostrategic risks when the resource access can be limited. For instance, lithium reserves are concentrated in South America (Bolivia, Argentina and Chile) while this metal is central to the development of new technologies for the energy transition. Another example is the concentration of rare earth elements; after world supply was dominated by the US who extracted from the single Mountain Pass Mine until the 1990s, China has since become the major supplier with over 90% of the production. The strategical reduction of their extraction through quotas in years 2000s lead to a dramatic increase in prices and subsequent rulings by the World Trade Organization in 2014 (Wan and Wen, 2017). On top of this geopolitical framework, the oligopolistic structure of these resources gives certain actors control of a majority of the resource supply. For instance, Glencore produced 27% of cobalt and Albemarle produced 24% of lithium in 2019 (IEA, 2021b).

Third, environmental degradation can also lead to degraded access to natural resources, resulting in lower quality or higher prices. This is the case for the forest resource which is vulnerable to climate change and subject to over exploitation (Lobianco, Delacote, Caurla and Barkaoui, 2016).

With these considerations, countries and organizations have the incentive to list and monitor critical raw materials (CRMs). To this end, the European Union (EU) established the Raw Materials Initiative to assess the criticality of materials based on the economic importance and the existing risk for their supply. <fnref n="1" /> However, this kind of listing often remains arbitrary, as criticality can be defined according to different criteria, methodologies or industrial and strategical priorities being taken into account (Hayes and McCullough, 2018).

<footnote n="1">¹https://ec.europa.eu/growth/sectors/raw-materials/policy-and-strategy-raw-materials_fr</footnote>
Resource economics play a role in assessing those risks of scarcity. It involves arbitrating in two directions. First, between resource types, as increasing prices can lead to turning away from one supplier of a mineral to another, or even to a different resource substitute. Second, trade-offs across time, as a limited resource supply has to be allocated between generation of consumers. One famous economic result is Hotelling's rule which defines the price of the resource when the economic surplus is maximized across time (Hotelling, 1931). It isolates a scarcity rent, called the Hotelling rate, that increases at the discount rate. Economics also give three indicators to measure resource scarcity. First, the marginal cost of extraction: it takes into account technological change but omits other costs (costs to access to the market, exploration costs,...). Second, the market price: it includes all costs and the scarcity rents, but, it also takes into account other distortions such as economic policies and geopolitical contexts. Third, the scarcity rent, or opportunity cost: theoretically, the most accurate indicator, but, difficult to observe. However, as we will discuss in the following, this trade-off in resource use can be extended by considering externalities linked to their extraction and use (Watari et al., 2020).

## Local pollution and social impacts

Relationships between natural resources and the environment are numerous and intervene at all stages of their use in the economy: extraction, transformation and final use.

First, many externalities relate to the extraction of the resource. Metal mining often consumes large volumes of water and can therefore contribute to groundwater pollution. The case of lithium in South America is a characteristic one, as it has been reported that lithium extraction in Chile's Atacama region uses 65% of the region's water tables. Extracting one ton of lithium uses 2,000 tons of water,<fnref n="2" /> seriously disrupting local ecosystems and releasing toxic chemicals. Regarding biological resources, exploitation of the forest resource can harm ecosystems via the use of chemical intrants and a lack of diversity in planted species. These

<footnote n="2">²Katwala, A. The spiralling environmental cost of our lithium battery addiction. Wired https://www.wired.co.uk/article/lithium-batteries-environment-impact (2018)</footnote>
things cause soil and water pollution as well as a reduction in ecosystem services provided by the forest (Gamfeldt et al., 2013). Additionally, these extraction impacts can last long after the activity, thus posing intertemporal threats to deal with.

Second, transformation of resources can rely on heavy industries, with important environmental impacts of their own. In the copper industry, tailing resources have to be stored on site and pose threats in terms of chemicals infiltration in groundwater (acid and heavy metals), as well as a risk of dams breaking. Moreover, metal smelting produces large amounts of metal fumes (arsenic and lead) and health-threatening gases ($SO_2$) (Li et al., 2017).

As mentioned previously, resource exploitation activities are often geographically localized. As such, polluting industries often relocate to countries where environmental constraints are less stringent (Bernard, 2015). This increases the risk of negative environmental impacts of resource extraction and transformation, as industries are less incentivized to reduce their externalities. This localization also adds up to social impacts of mining activities. A weaker institutional context as well as lesser capacities to enforce law lead to resource management practices with many health and safety issues as well as child labor concerns. A well documented example is the case of cobalt extraction in the Democratic Republic of Congo (DRC) where 70% of the world production came from in 2019 (Seck et al., 2022). Part of these social risks comes from the fact that around 20% of DRC mines are artisanal with workers and local populations confronted to more severe and unregulated environmental pollution. Furthermore, the DRC is subject to political instability leading to regular surges of violence in the region, affecting mining industries and adding to the scarcity risks mentioned before. While the sources of those conflicts come from various causes, presence of resource extraction activities has been highlighted as one of the factors (Fourati et al., 2021; Mukwege, 2021).

These considerations add up to define the trade-offs for resource use. Given the various environmental externalities and negative social impacts of resource extraction and transformation, the analytical framework of economists cannot be limited considering only to the
cost of extracting a resource. Analytical frameworks must also take into account various explicit or implicit costs associated with resource extraction. Additionally, long term climate impacts and end-of-life management of the resource have to be taken into account to properly choose between the different kinds of resources available for production in the economy.

## Climate change

In addition to local externalities due to effluents of the production and transformation of natural resources, virgin material consumption has an important impact on climate change. In the case of plastics, the industry contributes to fossil fuel exploitation as oil and gas are crucial for their manufacturing. Up to 6% of EU's consumption of oil and gas is dedicated to the production of plastics (Plastics Europe, 2017). Metal production is also a huge emitter of greenhouse gases (GHG): it has been estimated that around 7 to 8% of the world's energy supply is dedicated to mining and refining metals (UNEP, 2011). Furthermore, 23% of global GHG emissions (2015) are related to some form of material harvesting (UNEP, 2019). For the specific case of aluminum, harvesting it accounts for 1% of global GHG emissions (Gautam et al., 2018). The relevant local energy framework plays an important role in these results, as we can find significant variations across countries with different energy mix. It can vary from 8,000 kg CO2e per ton of produced aluminum for France to 40,000 kg CO2e/t for China.

In a different way, the forest resource is also strongly connected to GHG emissions re-
duction as it is one of the main carbon sinks, alongside with the oceans, related to many impacts of climate change (Cook-Patton et al., 2020). Harvesting the forest for the produc-
tion of wood products gives the opportunity for a renew of the forest cover, thus playing an important role in GHG removal. In France, the carbon sequestration potential is estimated between 66.2 and 125.3 MtCO2 per year (15 to 28% of CO2 equivalent emissions in France in 2019) (Lobianco, Caurla, Delacote and Barkaoui, 2016).
Climate change and resource extraction are also linked through investments needed for the energy transition. Considerable amounts of minerals are required in order to develop the necessary infrastructure to abate GHG emissions, especially in the energy sector (IEA, 2021b). In the future decades, some mineral extraction sectors are expected to be increasingly dedicated to these, for instance lithium, nickel and cobalt for the electrification of vehicles. This observation is also an incentive for countries and international organizations to carefully monitor material supply chains as well as the underlying environmental and social impacts. The previously presented Raw Material Initiative in the EU is one of them.

## Waste accumulation

A central concern of material use in the economy is the accumulation of waste. Waste is defined as material discarded after use, or any economic process including extraction and transformation of resources (industrial waste). For instance, we account for the previously mentioned copper tailing and production scraps into the broad definition of waste: materials that are disposed of when they have been used and they do not bear any value for the consumer (solid waste). In that sense, ownership of waste is ambiguous as it tests the boundaries of what is defined by ownership: as defined by Locke, waste land does not have value or an owner from the moment when it is not exploited (Lupton, 2011). However, the definition can be challenged by thinking of waste (in the literal sense) as something that has both an economic value and a destined use. Additionally, waste has a host of negative externalities to consider and characterize. We will first focus on the latter in this section.

Environmental damage from waste mainly comes from the different management methods. First, landfilling can be the source of groundwater pollution, soil contamination, fire hazards and olfactory nuisances. In addition to these externalities, landfilling can be the source of high costs due the large land areas allocated to waste deposits.<fnref n="3" /> Second, waste

<footnote n="3">³Talpin, J. Saturée de déchets, La Réunion veut stopper le tout enfouissement. Le Monde https://www.lemonde.fr/planete/article/2019/12/28/saturee-de-dechets-la-reunion-veut-stopper-le-tout-enfouissement_6024253_3244.html (2019)</footnote>
incineration generates air pollution and slags that are highly polluting (especially because the presence of heavy metals) while also requiring waste storage capacities. on top of the externalities relative to waste management, we must also add externalities linked to waste outside of the waste management sector. More specifically, we must consider waste that is illegally disposed of, waste that ends up in the environment by mistake or carelessness, or that is produced through the consumption of products that are dispersed in the environment. An iconic example is the case of marine plastic pollution. Driven by the growth of its production (from 0.5 Mt in 2050 to 260 Mt in 2009 and 320 Mt in 2015), plastic now accounts for a stockpile of 8.3 Gt worldwide (McNicholas and Cotton, 2019). It is estimated that now 40% of plastic production is not managed by the time it becomes waste, thus ending up in terrestrial or marine ecosystems. Macroscopic and microscopic waste (after decomposition) are persistent pollutants with sometimes irreversible consequences on the environment. Macro plastics induce impacts on biodiversity and on human activity through thing like lethal ingestion by animals, a beaches clean-up cost, and disruptions to fishing and tourism. The impacts of microplastics are subject to more nuance and uncertainties. For example, microplastics can be ingested by fish and then by humans through fish consumption. As such there are multidimensional health threats. While marine plastic pollution is rapidly increasing in geographically located areas, such as the Great Pacific Garbage Patch (Lebreton et al., 2018), it is hard to assess damages for waste, as the impacts are often scattered across several environments with complex behaviors and synergies. In studying marine ecosystem services, economists have estimated costs ranging from $3,300 to $33,000 per ton of marine plastic (Beaumont et al., 2019).

As for environmental impacts of resource management, we observe a concentration of waste damage in developing countries where environmental regulations are weaker and the cost of labor is cheaper (Bernard, 2015). For instance, South East Asia is now a major importer of waste worldwide. There have been consecutive geopolitical crisis when China strictly restricted conditions for waste importation in 2018. A good example is the ship
dismantling industry, which is concentrated in Asia due to limited regulatory environments (Devaux and Nicolaï, 2020). Social and environmental consequences are numerous: unsafe working conditions, high amounts of toxic effluents damaging the environment, and innumerable health hazards for local populations.

## The recycling challenge

Confronted with the joint problems of increasing amounts of waste to manage and a growing demand in primary material, recycling gained economical and political momentum in recent years as part of the concept of circular economy. As opposed to a linear economy, recycling it is part of a wider reflection on how to transform the economy from a unidirectional model based on the "extract-consume-throw away" tryptic to a more sustainable one. As the industrial process that enables to "close the material loop" in the economy, recycling is a key element of the circular economy. However, the circular economy is a broader vision of a societal change that supplements recycling with changes in supply and demand of products (ADEME, n.d.).

## Historical perspective

Concerns regarding a linear model first arose from the nuisances related to production residuals, especially in urban territories. These concerns were widely documented as early as the Middle-Age (de Silguy, 1996). Managing them is then a matter of public health and results in the progressive implication of governments: waste collection, paved streets, sewers... However residuals are rapidly considered as useful resources. Starting in the industrial revolution and then picking up in the 19th century, waste becomes the subject of an intense economic activity. An prime example is the paper industry.<fnref n="4" /> To meet the demand for paper rags were initially used. Ragpickers form an important social body, greatly influencing au-

<footnote n="4">⁴We can also cite animal bones collection for many manufactured objects and manure for agricultural fertilization.</footnote>
thors and painters during the 19th century. Rag workers represented up to 100,000 workers in France (Compagnon, 2017).

The modern conception of waste, especially solid waste, was built around the turn of the 20th century. Near that time, the profession of ragpickers disappears as the paper industry turns towards vegetable cellulose.<fnref n="5" /> The modern concept of waste focused on eliminating it from the public (and private) space, with landfilling becoming the common practice, especially from a purely economic sense (Monsaingeon, 2020).

A new transformation of relationships to waste occured in the second half of the 20th century, with environmental issues carving out a path in the public debate. Saturation of landfills and related pollution drove governments of western countries to legislate the waste management sector. Additionally, the development of incineration and recycling spawned out of an attempt to combat increased consumption, and the use of plastic packaging. In France, waste sorting emerged locally in the 1970s, with glass container collection, and then generalized in the 1990s to include plastic containers. In 1975, laws and regulations on waste management required local authorities to put in place solid waste collection services. However, elimination remained the first alternative to landfilling, with almost 40% of solid waste being incinerated in 1989 and still 52% ending up in landfills (Lupton, 2011). It wasn't until the development of the concept of a circular economy that recycling gained prominence (in the late 20th - early 21st century).

The circular economy first emerged in the 1990s and it is still widely discussed and contested today. It first became popular in non-academic circles, promoted by non-governmental organisations, and later by businesses and governments. The *Ellen McArthur Foundation* has for instance played an important role in its popularization. One key approach is to consider the use of resource *from cradle to cradle* as formalized by Braungart and McDonough (2002). The idea is to rethink the economic system so that all resources can be recovered after use and reintegrated in the economy or the environment, without any damage. Be-

<footnote n="5">⁵Bones collection is also replaced by synthetic resins and fertilization now uses chemical fertilizers.</footnote>
yond the purely economical aspect, the concept now also stands as part of a new societal vision embedded in a sustainable economic model (Arnsperger and Bourg, 2017). One could consider the circular economy as a mobilizing concept or an essentially contested concept, as it binds many topics of concerns with the objective of creating a new sustainable paradigm everyone agrees on, but without a concrete idea of its feasibility or the path to achieving it (Gallie, 1956; Korhonen, Nuur, Feldmann and Birkie, 2018). While it has been linked to the topic of sustainability, the wide range of research looking into the circular economy can lead academics to consider it not only as an economic theory, but as a new holistic approach for ecological and economical concerns. Moreover, its apparent objective of breaking free from ecological limits through circularity appears to be an attempt to reconcile physical constraints developed by ecological economics and sustainable growth advocated by neoclassical economics. With the increasing involvement of academic research in the topic, some limits have been pointed out in that sense (Korhonen, Honkasalo and Seppälä, 2018). For instance, it still bears thermodynamic challenges regarding resource and energy uses. From these considerations, developing a circular economy appears more as a tool for sustainable development than a goal in itself (Millar et al., 2019).

## Policy framework

Reducing the impact of waste accumulation and promoting recycling is a central political question. An important feature is that it affects a large variety of environmental policies, from the ones specifically dedicated to waste management, to more general ones like carbon pricing. This holistic dimension comes from the several impacts of resource use in the economy, as described before. It gives policy makers a wide range of possibilities when regulating, but it involves complex impact studies when multiple instruments are involved. We will take here the example of France. Since 1975, France has deployed a large range of policies aiming at supporting recycling and lowering the impact of resource use from extraction to waste.
Extended Producer Responsibility (EPR) is the core principle for waste management. In that sense, an economic actor that produces a good is held responsible for the environmental impacts during its entire lifecycle. The objective of EPR is to incite the producer to adopt more environmental friendly usages. In practice, producers organize to form Producer Responsibility Organisations (PRO). PROs subsidize recycling by collecting fees from upstream producers that are set to match recycling costs. Financing waste management can also be completed by taxation by local or central authorities.<fnref n="6" /> These policies are complemented by adjustments on the value-added tax (VAT) for collection and sorting services. These solutions are not the textbook pigovian tax advocated by economists, and their efficiency can be criticized. In particular, subsidizing recycling activities appears to be a less efficient policy. A better alternative is putting in place a deposit-refund program. Under such a policy, the consumer bears a supplementary cost only if the good is thrown away (Palmer et al., 1997; Ino, 2011; Acuff and Kaffine, 2013). In France, the approach has recently regained interest as demonstrated by the deposit refund system for plastic bottles which will be implemented under the *anti-gaspillage pour une économie circulaire (AGEC)* law (February 2020).<fnref n="7" />

Although economists have long advocated for direct environmental taxation due to its efficiency regarding the internalization of the social cost of externalities, it has rarely been implemented. As mineral and material industries are carbon intensive sectors from extraction to waste, taxation on GHG emissions can have significant reduction impacts (Acuff and Kaffine, 2013). At the European level, a trading scheme for CO₂ emissions has been implemented, with an forthcoming border adjustment mechanism to avoid carbon leakage outside the EU. Resource transformation is covered by this European Trading Scheme (ETS). However, end-of-life management of resources is subject to many exemptions. In particular, recycling and incineration are not subject to the ETS. On the other hand, the cement clinker

<footnote n="6">⁶In France, local authorities have different options for financing waste collection and curbside recycling, either independently or related to the provided service https://www.ecologie.gouv.fr/fiscalite-des-dechets.</footnote>

<footnote n="7">⁷LOI n° 2020-105 du 10 février 2020 relative à la lutte contre le gaspillage et à l'économie circulaire https://www.legifrance.gouv.fr/jorf/id/JORFTEXT000041553759/</footnote>
industry, which is an important part of waste incineration, is included, while benefitting an
large share of free carbon allowances.

Taxation directly addressing pollution (other than GHG) also exists with the *Taxe Générale sur les Activités Polluantes* (TGAP). Created in 1999, it can be broken down into several components with each one targeting a different effluent. In particular, it includes a tax on waste flowing into landfilling and incineration industries, with adjustments depending on environmental performances (for instance, incineration with co-production of electricity). This TGAP is complemented by a disposal fee for solid waste. Disposal fees are still implemented on a voluntary basis from local authorities. In France in 2021, 6 millions inhabitants are subject to disposal taxation. Based on volume or weight, its goal is to discourage the disposal of non-recyclable goods (Bueno and Valente, 2019). However, economists have pointed out some limitations such as an increase of illicit waste burning or dumping (Fullerton and Kinnaman, 1995), ex-ante opposition to the taxation (Carattini et al., 2018) and the influence of private values and social norms when confronted with individual waste management practices (Viscusi et al., 2011; Czajkowski et al., 2019).

Another widely used type of policy instrument is the use of regulation to impose norms
on producers of goods. For instance, with the *loi AGEC*, France introduced a ban on
microplastics and a progressive ban of single use plastic products. Given the variety of effects
on the environment and their persistence through time, there are difficulties in assessing the
cost of plastic pollution and taxing it accordingly. Complementary to EPR mechanisms
used to incentivize the production of greener products, norms on their composition are
also enforced. New regulations aim at imposing a minimum share of recycled material in
new products. Some initiatives have voluntarily originated from private actors at first,
and later will become mandatory. For example, in France, the production of PET plastic
bottles will require a minimum amount of recycled material after 2025. In addition to
the upstream impact, it also forces a demand for recycled material. However this kind
of measure can be problematic at an international level. Asymmetries between countries,
due to varying abilities of recycling waste, can lead to large increases of trade instead of
changes in environmental performances (Huhtala and Samakovlis, 2002). Finally, policies
also target information available for the consumer of the product. It includes reparability
indexes for electronic products, sorting instructions and environmental labels. The large
variety of existing labels, both from public and private incentives, is a limit regarding their
efficiency and drove lawmakers to start standardizing them. Furthermore, many discussions
exist regarding the efficiency of social norm nudging, especially when norms are very high
to meet (Czajkowski et al., 2019).

Trade-off between resource, waste and climate

Historically, recycling has either been considered as a solution for a waste problem or a re-source availability problem. However, recycling cannot separated from a wider analysis that includes the full lifecycle of materials. As detailed before, recycling is part of a path towards sustainability, as long as it is studied under an holistic approach. Resource availability, local pollution, social impacts, climate change and waste accumulation are all topics related to the use of resources in the economy. Therefore the decision to recycle goes from a single trade-off problem on resource stocks and pollutions levels to a multi criteria trade-off.

This multidimensional trade-off problem is at the core of this dissertation. The starting
point is a standard stock and flows analysis, where recycling alleviates resource scarcity and
allows for less damage from waste accumulation. However, as it has been noted before,
welfare losses and gains have to include underlying effects on climate change. In that sense,
we have to consider GHG emissions as an element of the trade-off. The objective is to assess
the eventual synergies or antagonisms between the fight against climate change and waste,
and resource regulations.
# Contributions of this dissertation

## Scope of the dissertation

In this dissertation, we propose an analysis of the economics of recycling when three types of interactions between anthropogenic activities and natural environments are considered: i) the production/consumption of goods and the availability of virgin raw materials, ii) the production/consumption and accumulation of waste, and, iii) the production/consumption and associated carbon emissions. Over four chapters, we explore the potential that recycling holds for the development of a sustainable economic model by assessing its environmental impacts, analyzing policy objectives and questioning the framework of analysis regarding sustainability.

## Description of the chapters

### Recycling under environmental, climate and resource constraints<fnref n="8" />

This chapter investigates the recycling opportunities of an industrial sector constrained by resource, climate, and waste capacities. With development difficulties of specific markets and industries, recycled inputs struggle to compete with regular inputs in production, especially when high recycling rates bear significant costs. However, we analyze how accounting for social costs from resource extraction, use and end-of-life, could change the trade-off between extraction and recycling. This research contributes to the analysis of recycling with the instruments of the literature on resource economics.

To do this, we model the full lifecycle of a good to consider the waste and GHG emissions coming from both its production — from virgin or recycled materials — and consumption. The social planner chooses resource quantities (virgin or recycled) and the share of waste to be recovered. In a first iteration, only waste accumulation is taken into account through

<footnote n="8">⁸This chapter was written in collaboration with Gilles Lafforgue. Published in *Resource and Energy Economics* (2022) https://doi.org/10.1016/j.reseneeco.2021.101278</footnote>
the introduction of a waste damage function. In a second iteration, GHG emissions are introduced through a carbon budget.

We first identify the optimal trajectories of resources use, mainly depending on the relative scarcity of the resources and on their emissions when included. Although recycling is usually, and correctly, noted as an opportunity to reduce the impact of consumption on primary resources and waste, we also consider the possible negative environmental consequences of recycling and we discuss the resulting trade-offs. Recycling is indeed also a source of GHG emissions, although in most cases they are lower than for extracted resources. The structure of the model (linear private costs and perfect substitution) gives us corner solutions regarding the choice of the resource. Virgin resource is exclusively used during a first phase of production. Then, the recycled resource can be used if it becomes more socially profitable, either because the scarcity of the virgin resource prevails, because private costs favor the use of recycling, and, or because, the emissions difference between both resources is sufficiently large. We also show that there can be a catch-up phase of consumption at the end of the social planner program.

We also characterize the optimal recycling strategy and we show that in some cases, the recycling rate through time is an inverted U-shape. Recycling is beneficial in two ways: it alleviates the cost of waste accumulation and allows for production from a recycled input. This shows the duality of the activity of sorting waste: a practical goal being the reduction of costs and a speculative goal of potentially using recycled input in the future. When recycling is too costly, a phase where the recycling rate is zero occurs (although it is still possible to produce from the already stockpiled recycled material).

Finally, we discuss the policy implications of our model by identifying and analyzing the set of optimal tax-subsidy schemes, and we highlight the existence of standard environmental externalities as well as a positive externality linked to the absence of a market for waste. To do this first best solution, a taxation based on GHG emissions must be introduced in both sectors, weighed by the carbon intensity of the branch. This comes in addition to a tax-
subsidy system based on waste and stock replenishment, reflecting the positive externality.

Sectoral, resource and carbon impacts of increased paper and cardboard recy-
cling<fnref n="9" />

In this chapter, we assess the role of paper and cardboard recycling on the forest sector, both from an economic and carbon perspective. The question of whether paper and cardboard recycling has a positive or negative carbon impact is crucial as the global forest sector plays a central role in climate change mitigation: the paper and cardboard recycling sector is well developed in France, with 66% of the production coming from recovered waste. Complex cascading impacts on the resource and other wood products suggest unclear economic and environmental impacts. Historically, GHG displacement factors of paper recycling have been studied with strong economic assumptions. This has led to contradictory results between studies, due to differences in accounting methods of emissions, and due to the organisational, spatial and temporal boundaries of the sector considered.

For that reason, we model the paper and cardboard recycling industry within an existing integrated assessment model of the French forest, in order to relate it to other wood products and forest resource. Based on the present situation of the French forest sector, we examine the impact of increased recycling. To simulate increased recycling, we implement a cost reduction in the sector. A partial equilibrium module of wood products and a recursive structure of the forest dynamics expands a standard Life Cycle Analysis (LCA) with both economic impacts on other wood-product sectors and variations in carbon substitution and sequestration.

In formulating the model, it is clear that results are highly dependant on assumptions regarding the substitutability of recycled and virgin pulp, while many existing studies do not settle on whether substitution or complementarity prevails (some argue that with time, complementarity prevails). Empirically, elasticities appear to be low and depend on the

<footnote n="9">⁹This chapter was written in collaboration with Philippe Delacote and Antonello Lobianco. FAERE working paper (2021) and submitted to *Environmental Modeling and Assessment* (2021)</footnote>
final use of the paper or cardboard product. Expected qualitative results regarding the complementarity or substitution of both products are found, with respectively a positive or negative impact on demand when recycling increases. GHG emissions of the pulp sector are always increasing, thus raising the question of their possible mitigation through the rest of the forest sector. Effects on other wood products are small, due to small price-elasticities of products competing for timber resources and small cross-elasticities between virgin and recycled.

Overall, the impact on total net sequestration is: positive when pulp products are strong substitutes; very small when pulp products are weak substitutes; and, negative when pulp products are complements. Relative effects are small (between 0.4% and 0.8% in the additional carbon balance of the sector when strong substitutes, a very optimistic scenario according to the econometric literature). As weak complementarity between virgin and recycled paper seems to be more accurate, the idea suggesting that recycling and circular economy is a good solution against global warming should be taken cautiously: in this situation, an increase in the use of recycling does not come along with savings of wood resources.

An assessment of the European regulation on battery recycling for electric vehicles<fnref n="10" />

This chapter investigates the design of a recent regulatory proposal aimed at favoring the emergence of a battery recycling industry in Europe. Electric mobility is deemed necessary to cut CO₂ emissions in the transport sector. Moreover, Lithium-ion (Li-ion) batteries - the central components of electric vehicles (EVs) - have been subject to remarkable technological progress in the recent years. As a result, their costs have fallen from USD 1100 per kilowatt-hour (kWh) in 2010 to USD 156/kWh in 2020. However, the industrial and environmental impacts of lithium-ion battery manufacturing are controversial. Among them, the end-

<footnote n="10">¹⁰This chapter was written in collaboration with Quentin Hoarau. Published in *Energy Policy* (2022)
https://doi.org/10.1016/j.enpol.2021.112770</footnote>
of-life of batteries may be particularly hazardous. Added to a potential scarcity of some
metals (lithium, but also rare earth materials), recycling of used batteries appears as an
appealing solution. To address these issues, the EU has developed several initiatives, such
as the European Battery Alliance and the Raw Materials Initiative. The new proposal of
the European Commission intends to bring a legislative framework to these initiatives.

The current EU regulatory framework on batteries is currently established by a directive from the European Commission in 2006 (Council of European Union, 2006). This directive aims at providing common rules across the EU market that would limit environmental impacts of battery wastes. However, the disruption of Li-ion batteries fueled by the development of electromobility has rendered the former directive outdated. A recent regulatory proposal from the European Commission introduces the obligation to attain a series of minimum thresholds of recycled materials for the new batteries to be manufactured after 2030. The relevant materials for Li-ion batteries are nickel, cobalt and lithium. We discuss the conditions required for that obligation to be fulfilled.

First, we develop a framework to qualitatively discuss the main characteristic trends of
the battery sector and how these should affect sourcing from recycled materials. It allows
us to separate the different mechanisms driving the amount of recycled material that can be
incorporated in new batteries: the change in battery demand, material composition linked
to technological changes, recycling efficiency, and end-of-life probability.

Second, we calibrate a material flow model that projects battery wastes and their recy-
cling potential. Using data from economic reports of the International Energy Agency and
academic literature, the model indicates that the feasibility of currently proposed thresholds
is not very sensitive to changes of material intensities from battery technology shifts, recy-
cling efficiencies, or the faster uptake of demand. On the contrary, battery lifetimes are the
most crucial parameters for recycling potential and meeting proposed thresholds. Indeed,
most of the targets given by the European Commission are not met for average battery
lifetimes beyond 10 years. Such an hypothesis can be considered as unrealistic given the
technological progress in the battery sector, with longer lifetimes. Moreover, a scenario with
a developed second life of batteries would dramatically lower the potential for a recycled
content. In this case, EV batteries are being refurbished when they do not meet EV power
standards and are used for stationary storage.

Our policy recommendations are therefore twofold. First, we recommend lowering the thresholds to improve the credibility of the regulation. Although targets might be changed in the future, industrial actors of the sectors (battery and cars manufacturers, recycling industry) would need readable and credible regulations today to make the right investments and technology choices. Second, the regulation should integrate other objectives that address the lifetime of batteries. Indeed, developing a sustainable battery sector should not rely on recycling and recycled content targets only. Moreover, only setting high content targets could give incentive to decreasing battery lifetimes, as shorter lifetimes would allow a faster circulation of material in the economy, thus increasing the potential use of recycled material.

Recycling in a sustainable framework <fnref n="11" />

In this last chapter, we study the question of recycling in macroeconomics with the scope
of sustainability. Indeed, the macroeconomics of recycling aim at assessing how closed ma-
terial loops are able to improve long term growth with optimal depletion of resource stocks.
Such analysis can also take into account the reduction of environmental externalities through
recycling. In that sense, recycling, and more generally the concept of a circular economy,
are expected to be a means for following a sustainable development path. However, sustain-
ability economics also relates to equity issues in terms of consumption and environmental
impacts. On this matter, literature on recycling does not rely on a sustainable criterion; in
its vast majority, it relies on discounted utility, a criterion that does not favour equity and
is thus unsustainable.

To explore the topic of recycling in a sustainable modeling framework, we develop a
resource and waste model to be solved using the maximin criterion. The maximin value is

<footnote n="11">¹¹Ongoing working paper</footnote>
defined as the highest level of utility that can be sustained forever. We use a model based
on the Dasgupta-Heal-Solow model, for which literature states that a maximin path exists
in the case of weak sustainability. It corresponds to the case when the production function
is not strongly dependent on the natural resource (for a Cobb Douglas function, when the
elasticity of the resource in production is lower than the one of capital).

This simple model adds complexity to the analysis of a minimum sustainable consumption
level. We are able to characterize an optimal path and its shadow values, and discuss
some long-run analysis. More complex resource management rules are reflected by a new
Hartwick investment rule which now depends on waste accumulation and the use of recycling
in production. In the specific case of a waste stock emptied in finite time, it is possible to
discuss explicit consumption paths when specifying production functions. In the case of a
Cobb-Douglas production function, the assumption of an elasticity of capital higher than an
elasticity of resources in production is crucial for a sustainable path to exist.

This specific case weakens the idea of recycling being able to allow sustainable paths in
the case of strong sustainability. This model is a first step towards the study of recycling
using the tools of sustainability economics. To our knowledge, it is indeed the first model
that studies recycling with the maximin criterion, and highlights how optimal investment
rules are modified.
CHAPTER 1

RECYCLING UNDER ENVIRONMENTAL,
CLIMATE AND RESOURCE
CONSTRAINTS

***

With development difficulties of specific markets and industries, recycled inputs struggle to compete with regular inputs in production, especially when high recycling rates bear significant costs. However, accounting for social costs from resource extraction, use and end-of-life of products, could change the trade-off between extraction and recycling. Although recycling is usually, and correctly, noted as an opportunity to reduce the impact of consumption on primary resources and waste, we also consider in this chapter the possible negative environmental consequences of recycling and we discuss the resulting arbitrations. We discuss the policy implications, and we highlight the existence of standard environmental externalities as well as a positive externality linked to the absence of a market for waste.

***
## 1.1 Introduction

The early economic studies on secondary materials long focused on recycling's ability to save extracted resources (UNEP, 2019). More recently, however, economists have begun to consider recycling's ability to mitigate waste pollution and greenhouse gas (GHG) emissions. With this new view, the concept of a circular economy, in which recycling is one of the cornerstones, appeared as a solution for a more sustainable economical model, as formalized by Braungart and McDonough (2002) in *Cradle to Cradle*. This concept has generated a significant amount of "grey literature" through many non-governmental organizations like the Ellen McArthur Foundation. It can be noted that a large concentration of the grey literature, as well as of the academic literature, occurs in Europe and Asia. Both of these regions have many academic and institutional initiatives in this field (de Jesus and Mendonça, 2018).

In this chapter, we contribute to the literature on circular economy by examining the impact of recycling in an industrial sector facing environmental, resource and climate constraints. In this context, we must consider three different balances: a material balance in order to examine the saving of natural resources and the reduction of waste accumulation; a carbon balance for the topic of climate change; and, an economic balance for the evolution of consumption. Hence, we see environmental objectives of recycling going in three different directions: the saving of natural resources while its shortage could lead to economic difficulties; the reduction of waste accumulation that is costly to manage for both public and private entities, and poses a threat to the environment; the fight against climate change and especially the reduction of GHG emissions. A good illustration of these features can be found in the metal sector: scarcity of critical metals raises the question of their sustainable use and end of life management, as well as the emissions rates of their lifecycle (UNEP, 2011).

One intended specificity of this chapter is to extend the study of recycling by considering
climate change as an additional externality. Recycling can indeed harm the environment, in particular because of the extra energy required by this activity and its resulting potential GHG emissions. This additional externality leads to new arbitrations we describe in the model: in most cases, recycling is a way of reducing the use of resources with a high carbon footprint (ADEME and FEDEREC, 2017), but recycling is still the source of GHG emissions. It has already been highlighted that circularity and environmental issues are connected in an industrial sector, with for instance used tires (Lonca et al., 2018). In France, studies of ADEME and FEDEREC (2017), ADEME and Bio by Deloitte (2017) and ADEME (2019) focused on quantifying different impacts of recycling in terms of GHG emissions, showing that industrial processes are often highly carbon intensive compared to recycling industries (see Table 1). However, in the end, recycling does not appear to be the ideal clean substitute to regular production: recycling produces its own emissions; there is a need in initial production from a regular source; and, recycling comes with a cost (UNEP, 2011). These environmental considerations contribute to the current policy framework, with many countries implementing emissions reduction targets that affect polluting industries.

<table><caption>Table 1.1: Examples of emissions rates in France (in kg CO₂e/t)</caption><thead><tr><th></th><th>PET Plastics</th><th>Aluminum</th><th>Glass</th><th>Copper</th><th>Scrap/Steel</th></tr></thead><tbody><tr><td>Virgin material</td><td>3 270</td><td>7 803</td><td>923</td><td>1 445</td><td>2 211</td></tr><tr><td>Recycled material</td><td>202</td><td>562</td><td>409</td><td>1 304</td><td>938</td></tr></tbody></table>

Note: Values obtained from ADEME and FEDEREC (2017) and from bilans-ges.ademe.fr

We do not pretend to calibrate policies to implement, but we try to analytically highlight the cross-effects and eventual synergies between climate change mitigation, preservation of resources, and the limitation of waste disposal. To do that, we develop a dynamic model of resource use where a final manufactured good can be produced from a virgin exhaustible resource or from recycled materials, these two inputs being taken as perfect substitutes.
In order to clearly understand their various effects, we successively introduce two negative
environmental externalities. First, we examine waste accumulation that harms the economy
through a specific damage function, and then we add cumulative GHG emissions that are
constrained by an exogenous carbon budget. Consequently, we observe that a third source
of (positive) externality must also be considered as long as there is no waste market, as
consumption of the final good provided a waste stream which can be reused thanks to an
endogenous effort of recycling.

In this framework, we characterize the main properties of the optimal trajectories of the
model. In particular, we discuss the merit order in using each type of resource, depending
on the relative scarcity of resources and their emissions rates. We analyze the optimized
recycling rate curve through time and we show that, under some conditions, it can be an
inverted U-shape. We also discuss the respective dynamics of resource use, which sometimes
can result in a catch-up phase of consumption at the end of the program. Last, we show
that in a competitive market economy, this optimal outcome can be implemented by a set
of tax-subsidy schemes and we discuss their policy implications depending on the identity
of the tax payer or the subsidy beneficiary. The introduction of a waste market allows to
account for the positive externality linked to production that relaxes resources restriction
thanks to recycling. This externality comes from the dynamic framework of the model and
has not been considered by the static economic literature on recycling.

The rest of this chapter is organized as follows. Section 1.2 provides a brief review of the literature on recycling. Section 1.3 characterizes the basic model and describes the different possible scenarios of consumption and recycling. Section 1.4 considers the introduction of GHG emissions through an emissions ceiling.<fnref n="1" /> Section 1.5 studies the decentralized equilibrium outcome and discusses policy implications. Section 1.6 concludes.

<footnote n="1">¹We also propose three other extensions in appendix: accounting for a scrap value for waste, decoupling emissions between collection and transformation in the recycling branch, limiting the capacity of the recycling sector.</footnote>
## 1.2 Related literature

Our work can be related to three strands of literature on recycling: resource scarcity, waste management, and environmental policies.

Early studies from the 70s-80s already tackled resource scarcity. For instance, Smith (1972) puts forward social costs linked to waste accumulation and stock diminution. He focuses on the dynamics of waste when recycling is under consideration, and he shows that there is a trade-off between private costs (labor, material) and social costs (waste accumulation, resource depletion). Such dynamic models informed the first economic guidelines motivating recycling. Waste accumulation issues were added to the topic through various models intending to find the optimal level of pollution in an economy (see for instance Plourde (1972), Forster (1973), or Hoel (1978)). Later, the work of Chakravorty et al. (2006) and Chakravorty et al. (2008) focused on the order of resource extraction and gave many insights on situations where resource depletion induces pollution, that changes the extraction order. However, these studies do not include recycling in their model.

An important part of the relevant literature was later developed around the topic of using green policies to promote recycling. Palmer and Walls (1997) use a static micro-economic model to analyze the effects of diverse economic incentives such as subsidies, waste tax and deposit-refunds. This approach gives many policy insights but only takes into account waste and recycling activities. The work started by Palmer and Walls (1997) was then expanded to include environmental effects associated with recycling and resource extraction (Fullerton and Kinnaman, 1995; Palmer et al., 1997). Going further in this type of analysis, Walls and Palmer (2001) integrate life-cycle aspects of production and consumption and discuss optimal policy instruments. However, they only include an effluent linked to a third, non-material, input in production. Thus they do not differentiate between recycling and extraction. In a later work, Acuff and Kaffine (2013) add carbon emissions to the model with a direct link to the input choice and show that the objective of reducing GHG is also a
strong incentive to increase recycling, and that green policies can be implemented with this
goal.

These articles add a significant contribution regarding public intervention linked to recy-
cling activities. However this kind of static analysis omits the dynamic aspects of resource
stocks mentioned above. A further analysis is needed to examine the arbitration between
environmental externalities and resource depletion. For instance, extending the Acuff and
Kaffine (2013) model (initially being an extension of the Palmer and Walls (1997) model)
to a dynamic system.

An example of this can be found in Huhtala (1999). This is one of the few to analyze
the optimal use of an exhaustible resource while considering issues of waste accumulation,
resource depletion, and pollutant emissions at the same time. She describes the best arbi-
tration of labor between recycling and primary production, and designs a fitting tax-subsidy
scheme to achieve it under a balanced budget. This work is complemented by different
recent studies on resource economics, with for instance Pittel et al. (2010) who model a
decentralized economy with a recycling activity and highlight the market failure resulting
from the absence of a market for waste, despite their economical value. They provide an
optimum by setting up a market for waste and subsidizing recycling activities.

This dynamic approach is also the perspective of Di Vita (2001) and Di Vita (2007) who
assess the possibility of an economic and welfare growth under a material constraint, thanks
to investments in recycling. Sorensen (2017) also considers a recycling technology in a Ram-
sey model that alleviates externalities due to resource extraction and consumption. These
articles share the use of optimal control theory, but propose different models to represent a
circular economy. Additionally, Di Vita (2001) and Di Vita (2007) propose models that do
not respect a material balance in the economy in contrast to the physical reality of the use
of secondary materials. With these propositions, they arrive to the conclusion that recycling
allows a stationary growth path. An alternative modeling approach is given by Boyce (2012)
who chooses to specify a recycling stock separate from accumulated waste. He examines the
dynamic of this stock when there is perfect substitution between virgin and recycled material
and he describes economic trade-offs between the two material to manufacture a final good.

The results of all these studies are in part driven by the substitutability characteristics
of the production function. Models often use perfect substitution for virgin and recycled
uses (Boyce, 2012; Hollander and Lasserre, 1988; Hoogmartens et al., 2018). While being a
strong hypothesis, it remains relevant in our case of metal recycling with efficient processes
(Villalba et al., 2002, 2004). While metal recycling has been a motivated topic for a long
time (Ayres, 1997; Sigman, 1995), more recent concerns connected to technical changes and
the energy transition have begun to be included. The essential place in modern technological
applications taken by some materials (such as rare earth or lithium) also raises the question
of how to bare the storage costs inherent to the industry as they are not evenly allocated
worldwide (Rosendahl and Rubiano, 2019; Ba et al., 2020). As such, one could consider
in-use material as a stock of resource (Batteries Europe, 2020).

1.3 A basic optimal recycling model

1.3.1 Setup of the model

We consider an industry that is managing a natural resource stock from which a final con-
sumption good can be produced. We set an exogenous time-limit $T$ to the management
program, corresponding to the horizon of exploitation of the resource, before a shut down
of the mine (Lappi and Ollikainen, 2019).

The industry can produce a quantity $q$ of final good from two different inputs: the
virgin resource and a recycled material, of relative quantities $v$ and $r$. We assume perfect
substitution between these two inputs, involving that $q(t) = v(t) + r(t)$ for any time $t$. We
also pose that waste is a one-to-one co-product of the final consumption good. Thus, $q(t)$
also denotes the instantaneous flow of waste, before any potential recycling process.<fnref n="2" />

Consuming $q$ units of final good provides a gross surplus $u(q)$ to the final user, where $u(.)$ follows the standard hypotheses of utility functions: of class $C^2$, increasing ($u' > 0$), concave ($u'' < 0$) and verifying the Inada conditions, i.e. $\lim_{q \to 0^+} u'(q) = +\infty$ and $\lim_{q \to \infty} u'(q) = 0$. To account for the previously stated hypothesis of perfect substitution in the production function, we assume the utility drawn from the consumption of a good made out of virgin material is the same as that taken from the consumption of a good made out of recycled material.<fnref n="3" />

Input flows $v$ and $r$ come from two different primary sources, a virgin stock and a recycled stock. We denote the average delivery costs of these two resources as $c_v$ and $c_r$, respectively. These average delivery costs include the extraction, production and transportation costs, all of which are assumed to be constant through time.

The industry is initially endowed with a stock $V_0 > 0$ of virgin resource. We suppose that recycling has never happened by the past, meaning that the initial stock of recycled material is zero: $R_0 = 0$. At any time $t$, the current levels of virgin and recycled stocks $V(t)$ and $R(t)$ are governed, respectively, by the following extraction processes:

$$ \dot{V}(t) = -v(t) \quad , \quad V(0) = V_0 \qquad (1.1) $$

$$ \dot{R}(t) = -r(t) + \beta(t)q(t) \quad , \quad R(0) = 0, \qquad (1.2) $$

where $\beta, \beta \in [0, 1]$, represents the endogenous recycling intensity (i.e. the share of production which is collected, sorted and incorporated into the recycled stock after use).<fnref n="4" /> However,

<footnote n="2">²An alternative would be to consider that a constant share of the final good is destroyed by consumption, and then unrecoverable. In that case, consuming a quantity $q$ of good would release a flow $\gamma q$ of waste, with $\gamma < 1$ being the waste content rate of the manufactured good. This would not change significantly the analysis but does not respect the physical law of material balance.</footnote>

<footnote n="3">³Note that when focusing on the use of different resources, perfect substitution is a common simplifying assumption in the literature (Amigues et al., 2011; Boyce, 2012; Chakravorty et al., 2006, 2008).</footnote>

<footnote n="4">⁴While $\beta$ does not exactly fit the common definition, we will also refer to it as "recycling rate" throughout the rest of the chapter. Moreover, as we model recycled resource and waste as two different stocks, $\beta$ must be a control variable in order to have an endogenous recycling. A similar approach is used by Boyce (2012)</footnote>
note that recovering waste from the flow $q$ and using it in production are two different actions that are only correlated by the cost arbitration. As a matter of fact, it can be optimal to use a recycled input in production while deciding not to redirect waste into the recycled stock ($\beta = 0$), as long as this stock is not empty.

Recovering material from waste is costly and this cost is added to the other private costs related to the recycled input. The recycling cost per unit of waste is given by the function $f(\beta)$, such that $f(0) = 0$, $f'(\beta) > 0$ with $\beta \in [0, 1[$, $f'(1) = +\infty$ and $f''(\beta) > 0$. Recycling the share $\beta$ of the waste flow $q$ thus costs $qf(\beta)$. The convexity of $f(.)$ means decreasing returns in the recycling technology, which reflects the technical difficulties to retrieve all material produced and consumed. This assumption, combined with $\lim_{\beta \to 1} f'(\beta) = +\infty$, implies that full recycling is not attainable. There should always be a part of the waste that cannot be recovered, at any cost. It can be due to fatal transformation of the original resource during the manufacturing process or due to a very dispersive use of the resource which renders it technologically impossible to retrieve all of the resource after consumption.<fnref n="5" />

The remaining share of final good which could not be collected and recycled yields a waste stream $(1 - \beta)q$ that accumulates.<fnref n="6" /> We denote $W(t)$ as the resulting stock of waste at time $t$, with $W_0$ as the initial stock. This waste stock is reduced by a bio-decomposition, or another natural resorption mechanism, at a constant and exogenous rate $\alpha$, $\alpha \in (0, 1)$. The dynamics of $W$ are then given by:

$$ \dot{W}(t) = (1 - \beta(t))q(t) - \alpha W(t), \quad W(0) = W_0. \qquad (1.3) $$

<footnote>as he considers a sorting cost and assigns to the social planner the choice of sorting waste or not (a binary decision). This choice of modeling two stocks allows to represent the material balance constraint in our economy.</footnote>

<footnote n="5">⁵This hypothesis reflects the dispersive use of many materials as well as complex applications at mass production scales. In fact, thermodynamic limitations to recycling can involve important economic costs, or even impracticability, thus forbidding perfect recycling (Ayres, 1999; UNEP, 2011).</footnote>

<footnote n="6">⁶Note that the share $(1 - \beta)$ of the waste stream that is not redirected to the recycled stock at instant $t$ cannot be recycled later in the program.</footnote>
Finally, waste accumulation generates local environmental degradation that harms the economy through a damage function $D(W)$, with $D'(W) > 0$ and $D''(W) \ge 0$. In order to simplify the analysis, we will consider a linear damage function, with a constant marginal damage denoted by $c_W > 0$.

### 1.3.2 Central planner program and optimal conditions

The objective of the central planner is to determine the resource extraction and recycling paths that maximize the discounted sum of net surpluses over the planning horizon.<fnref n="7" /> Formally, denoting the time discount rate as $\rho > 0$, we want to solve the following problem:<fnref n="8" />

$$ \max_{\{v,r,\beta\}} \int_0^T [u(v+r) - c_v v - c_r r - qf(\beta) - D(W)] e^{-\rho t} dt, \quad (1.4) $$

subject to the dynamic constraints (1.1)-(1.3) and to the non-negativity constraints on $v$, $r$ and $\beta$. The Hamiltonian can be written as:

$$ \mathcal{H} = u(v+r) - c_v v - c_r r - qf(\beta) - D(W) + \lambda_V(-v) + \lambda_R[\beta(v+r) - r] \\ - \lambda_W[(1-\beta)(v+r) - \alpha W], \quad (1.5) $$

where $\lambda_V$, $\lambda_R$ and $-\lambda_W$ are the co-state variables attached to the virgin stock, the recycled stock and the waste stock, respectively.<fnref n="9" />

<footnote n="7">⁷Remind that, here, the ending time $T$ is exogenous and finite. An alternative would be to let $T$ free. However, as the industry evolves in a finite world where none renewable resource is available and because of Inada conditions, it would be optimal to postpone $T$ as much as possible: $T \to \infty$.</footnote>

<footnote n="8">⁸In order to simplify notations, we will hide the time subscript whenever it is convenient and clear from the context.</footnote>

<footnote n="9">⁹As waste accumulation generates a negative externality through a damage function, its shadow value is non-positive. For the purpose of simplifying the notations, we introduced a positive shadow cost by considering formally $-\lambda_W$ as co-state variable.</footnote>
The optimal solution must satisfy the following first-order conditions:

$$u' \leq c_v + f(\beta) + \lambda_V - \beta\lambda_R + (1 - \beta)\lambda_W, \quad (= \text{if } v > 0) \quad (1.6a)$$

$$u' \leq c_r + f(\beta) + \lambda_R - \beta\lambda_R + (1 - \beta)\lambda_W, \quad (= \text{if } r > 0) \quad (1.6b)$$

$$qf'(\beta) \geq (\lambda_R + \lambda_W)q, \quad (= \text{if } \beta > 0) \quad (1.6c)$$

$$\dot{\lambda}_V = \rho\lambda_V \quad (1.6d)$$

$$\dot{\lambda}_R = \rho\lambda_R \quad (1.6e)$$

$$\dot{\lambda}_W = (\rho + \alpha)\lambda_W - D'(W), \quad (1.6f)$$

completed by the transversality conditions:

$$\lambda_V(T)V(T)e^{-\rho T} = \lambda_R(T)R(T)e^{-\rho T} = \lambda_W(T)W(T)e^{-\rho T} = 0. \quad (1.7)$$

Conditions (1.6a) and (1.6b) compare the gross marginal surplus of using each type of resource with its full marginal cost and state that it is optimal to use the resource when both are equal. The full marginal cost (FMC) of each input $v$ or $r$ is composed of: the delivery cost ($c_v$ or $c_r$) of the input; the unitary cost $f(\beta)$ of recovering material from the waste stream; the scarcity rent ($\lambda_V$ or $\lambda_R$) of the resource stock; the social marginal cost $-\beta\lambda_R$ of replenishing the recycled stock (a negative cost, this stock being a good for society); and, the shadow marginal cost $(1 - \beta)\lambda_W$ of waste accumulation.

We can already determine the motivation of recycling as implemented in this resource model: recycling relaxes the resource constraint by replenishing the recycled stock and reducing waste accumulation. However, the choice of using recycled inputs in production here only relaxes pressure on virgin stock. Damage due to waste accumulation is not impacted, as both inputs have the same impact on waste after consumption. Waste reduction is controlled by the choice of the recycling rate $\beta$.

The optimal path for the recycling behavior is obtained from (1.6c) by comparing the
marginal cost $f'(\beta)$ per unit of final good with the marginal social gain ($\lambda_R + \lambda_W$) of recycling. Indeed, for a given material stream $q$, increasing the share of recycled material by $\Delta\beta$ allows to increase the stock of recycled resource, whose marginal shadow value is $\lambda_R$, by quantity $\Delta\beta q$, and to reduce waste generation by this same unit, thus saving the marginal shadow cost $\lambda_W$ of the waste stock.

Equations (1.6d) to (1.6f) rule the dynamics of the system. (1.6d) and (1.6e) illustrate, in particular, the Hotelling rule which state that the scarcity rents of the two resources grow at a rate equal to the discount rate $\rho$. Hence we can write: $\lambda_V(t) = \lambda_{V0}e^{\rho t}$ and $\lambda_R(t) = \lambda_{R0}e^{\rho t}$, with $\lambda_{V0} \equiv \lambda_V(0)$ and $\lambda_{R0} \equiv \lambda_R(0)$.

In (1.7), we can replace $\lambda_V$ and $\lambda_R$ by the above expressions, and $V$ and $R$ by the solutions of the differential equations (1.1) and (1.2), respectively. Assuming that neither resource stock is abundant, i.e. $\lambda_{V0} > 0$ and $\lambda_{R0} > 0$, the transversality condition implies that both stocks must be fully depleted at the terminal time $T$:

$$ \int_{0}^{T} v(t)dt = V_{0} \qquad (1.8) $$

$$ \int_{0}^{T} [\beta(t)v(t) - (1 - \beta(t))r(t)]dt = 0. \qquad (1.9) $$

Equation (1.6f), combined with the transversality condition (1.7), gives us the trajectory of the shadow cost of waste accumulation:<fnref n="10" />

$$ \lambda_{W}(t) = \int_{t}^{T} D'(W)e^{-(\rho+\alpha)(s-t)}ds. \qquad (1.10) $$

We see here that the shadow cost of waste accumulation is equal to the intertemporal sum of the flows of marginal damages, discounted at rate $(\rho + \alpha)$ since waste is not only a flow, but

<footnote n="10">¹⁰Solutions of equations (1.3) and (1.6f) are, respectively: $W = e^{-\alpha t}[W_0 + \int_0^t (1 - \beta)qe^{\alpha s}ds]$ and $\lambda_W = e^{(\rho+\alpha)t}[\lambda_{W0} - \int_0^t D'(W)e^{-(\rho+\alpha)s}ds]$. Replacing $W$ and $\lambda_W$ by these expressions into the transversality condition (1.7), we get: $\lambda_{W0} = \int_0^T D'(W)e^{-(\rho+\alpha)t}dt$. The initial value of the waste social cost is independently given by conditions on waste, only taking into account the dynamic characteristics $\alpha$ of the stock and its cost $D(W)$ for society.</footnote>
also a stock that shrinks at rate $\alpha$. With a constant marginal damage $c_W$, this expression
can be simplified as follows:

$$ \lambda_W(t) = \frac{c_W}{\rho + \alpha} \left[ 1 - e^{-(\rho+\alpha)(T-t)} \right]. \qquad (1.11) $$

This implies that $\lambda_W$ is decreasing through time: $\dot{\lambda}_W(t) = -c_W e^{-(\rho+\alpha)(T-t)} < 0$. In fact, this social cost accounts for the future damage due to waste. As the program approaches its end, the constraint has less impact.<fnref n="11" />

From first-order conditions (1.6a) and (1.6b), we can infer that the optimal solution is one or more consecutive phases of production from one input or the other, with, in parallel, a specific path for rate $\beta$ ruled by equation (1.6c). Perfect substitution and fixed private costs do not allow simultaneous use of each input. Following the display of the model, we will be studying the switch from one input to the other, the needed initial conditions, and the optimal choice regarding the redirection of the waste flow to the recycled stock.

### 1.3.3 Arbitration on resources use

In order to get the optimal consumption path, we have to compare the full marginal costs of using each specific input ($FMC_i$, with $i \in \{v; r\}$), given by the right-hand side of (1.6a) and (1.6b). We define $\Delta FMC$ as their difference:

$$ \Delta FMC \equiv FMC_v - FMC_r = c_v - c_r + (\lambda_{V0} - \lambda_{R0})e^{\rho t}. \qquad (1.12) $$

Examining the sign of this expression gives us arbitration at stakes when applying Herfindahl least-cost principle analysis (Herfindahl, 1967). It can be noted that neither recycling nor the waste stock come into play in the trade-off. In fact, products made out of

<footnote n="11">¹¹We discuss in the appendix the case where the damage caused by the left-over waste stock $W(T)$ still harms the economy after the terminal date $T$.</footnote>
recycled or virgin materials have the same impact on waste accumulation and the activity of waste recovery does not depend on the origin of the input flow. In this basic version of the model, the relative competitiveness of each type of resource is simply driven by the differential of resource extraction costs and by their relative scarcity. The relative magnitude of these two effects can lead to a switch between inputs, mathematically materialized by the change in sign of $\Delta FMC$ before the end of the program. Formally, if $\Delta FMC < 0$ then we produce from the virgin input, and if $\Delta FMC > 0$ then we produce from the recycled input.

From this analysis, Proposition 1 below describes the main characteristics of the optimal sequence of input use.

**Proposition 1.** Regardless of the recycling rate $\beta(t)$, the optimal path is such that:

1. It must start with a phase of production from extraction of virgin resource in all scenarios.

2. There can be at most one switch of input during the program. Let us call this time of switch $\tilde{T}$, whose existence within $[0; T]$ is not guaranteed. Then, if it exists within $[0; T]$, $\exists! \tilde{T} > 0$ such that $\Delta FMC(t) < 0$ for any $t < \tilde{T}$ and $\Delta FMC(\tilde{T}) = 0$.

3. When such a switching time $\tilde{T}$ exists, it is defined by:

$$e^{\rho \tilde{T}} = \frac{c_r - c_v}{\lambda_{V0} - \lambda_{R0}}. \qquad (1.13)$$

*Proof.* (1.) For any $t \in [0, \tilde{T}]$, if $v(t) = 0$ then, from (1.2), $R(\tilde{T}) = -\int_0^{\tilde{T}} (1 - \beta)rds \le 0$ which is not possible. (2.) Differentiating (1.12) with respect to time yields $\Delta FMC = \rho(\lambda_{V0} - \lambda_{R0})e^{\rho t}$. As this expression is constant in sign, $\Delta FMC$ is monotonic and there can be at most one switch of inputs during the optimal program of the model. (3.) This is simply the solution of $\Delta FMC(t) = 0$. $\square$

The first result highlights the need for initial conventional production: recycling can only be achieved when there has been enough extracted input consumed, and then collected
from the waste stream. Also, by the assumption of the recycled stock being empty at the beginning ($R_0 = 0$), there must always be a first phase of virgin production to fill the recycled stock before being able to use it. Thus, $\Delta FMC$ must be initially negative, which requires $\lambda_{V0} - \lambda_{R0} < c_r - c_v$. The second result implies that, if it exists, the time of the switch $\tilde{T}$ is the solution of the following equation $c_v + \lambda_V(\tilde{T}) = c_r + \lambda_R(\tilde{T})$. A single instant solution to $\Delta FMC = 0$ as given by (1.13) also means that a simultaneous use of virgin and recycled resource cannot occur. Different scenarios can occur, depending on the sign of ($c_r - c_v$). Proposition 2 below summarizes them.

**Proposition 2.** Regardless of the recycling rate path $\beta(t)$, we can summarize the different optimal scenarios to the following ones:

1. If extracting input from the recycled stock is costlier than from the virgin stock ($c_r > c_v$):

(a) We only use the virgin resource if this resource is relatively more abundant than the recycled one ($\lambda_{V0} < \lambda_{R0}$);

(b) We use the virgin resource up to time $\tilde{T}$, with $\tilde{T} \le T$, and then the recycled resource if the later is relatively more abundant than the former ($\lambda_{V0} > \lambda_{R0}$);

2. If extracting input from the virgin stock is costlier than from the recycled stock ($c_r < c_v$), we use the virgin resource up to time $\tilde{T}$ and then the recycled resource. In this case, the virgin resource must also be relatively more abundant than the recycled one ($\lambda_{V0} < \lambda_{R0}$).

*Proof.* The different optimal paths can be drawn from the positivity conditions for $\Delta FMC$. Moreover, in case 1(b), as the transversality condition guarantees that the virgin resource must be exhausted at the terminal date $T$, the switch to the recycled resource occurs no later than $T: \tilde{T} \le T$. $\square$

In scenario 1(a), using the recycled resource does not show any interest as the virgin resource is both less expensive and relatively more abundant than the recycled one. In
scenario 1(b), the virgin stock exhibits relative scarcity that can be relaxed by using the recycled resource, despite its higher extraction cost. Last, scenario 2 illustrates the case where the recycled resource is cheaper than the virgin one. In this case, the only way to justify an initial phase of virgin resource use, which is required to provide recycled waste, is to assume that the virgin stock is relatively abundant.

As we separate marginal delivery cost $c_r$ and waste recovery cost $f(\beta)$, we do not assume at first any relationship between delivery costs $c_i$. However, the difference in maturity between these two materials can be observed as the recycling branch is usually not favored when social costs are not internalized, leading to the reasonable hypothesis $c_r + f(\beta) > c_v$<fnref n="12" />. However we do not assume anything more as differences between sectors could lead to recycling costs relying on delivery ($c_r \gg f(\beta)$ thus $c_r > c_v$) or on collection ($f(\beta) \gg c_r$ thus potentially $c_v > c_r$).

At the end, the optimal path of resource consumption is characterized by the triplet of endogenous elements {$\lambda_{V0}, \lambda_{R0}, \tilde{T}$}, which is determined as the solution of the three equations-system (1.8), (1.9) and (1.13). For the remaining of the section, we will consider the most interesting case where both resources are successively used, which means that the switching time $\tilde{T}$ exists within $]0, T[$. Hence, $\lambda_{V0}$ and $\lambda_{R0}$ must satisfy the following additional condition:

$$1 < \frac{c_r - c_v}{\lambda_{V0} - \lambda_{R0}} < e^{\rho T}. \qquad (1.14)$$

### 1.3.4 Recovering waste

The optimal path of the recycling rate $\beta$ is determined from equation (1.6c). In order to simplify the analysis, we define the function $\Phi_{\beta} \equiv \lambda_{R} + \lambda_{W}$ as the marginal social gain of the recycling effort (both in avoiding accumulation of waste and replenishing the stock of

<footnote n="12">¹²Di Vita (2007) precises that an industry of secondary materials with the same properties as virgin inputs would bear very high cost, not sustainable with current market prices. With the exemple of lithium, Rosendahl and Rubiano (2019) use higher initial costs for extraction than recycling.</footnote>
recycled resource). Recycling intensity results from the arbitration between its marginal
cost and this marginal gain. Formally, $\beta = 0$ when $\Phi_{\beta} \le f'(0)$, and $\beta \in ]0; 1[$ such that
$\Phi_{\beta} = f'(\beta)$ when $\Phi_{\beta} > f'(0)$. Remark that there can be recycling $\beta > 0$ while the virgin
resource is not used: recycling only alleviates the pressure on waste accumulation in this
case.

The optimal recycling program can be described by having three material moments with characteristics detailed in proposition 3. The principle material moment is time $\hat{T}_{\beta}$ which represents the time of maximal recycling rate $\beta(t)$. This moment is accompanied by two other material moments $\underline{T}_{\beta}$ and $\bar{T}_{\beta}$ which represent the start and finish of recycling, respectively. Formally we specify $\underline{T}_{\beta}$ and $\bar{T}_{\beta}$ such that $\beta(t) = 0$ if $t < \underline{T}_{\beta}$ or $t > \bar{T}_{\beta}$. Note that we restrained our analysis to the case where both V and R are extracted, meaning that there must be a phase of recycling during the program.

**Proposition 3.** *Regardless of the resource extraction regime:*

1. There can be zero, one or two phases where the recycling rate $\beta$ saturates and is zero.
These phases can either be:

(a) No saturation phase (we always recycle);

(b) One phase at the beginning or the end of the program, such that $\beta(t) = 0$ when $t \in [0; \underline{T}_{\beta}]$ or $t \in [\bar{T}_{\beta}; T]$, where the values of times $\underline{T}_{\beta}$ or $\bar{T}_{\beta}$ are the solution in $[0; T]$ of equation $\Phi_{\beta}(t) = f'(0)$;

(c) Two phases at the beginning and the end of the program, such that $\beta(t) = 0$ when $t \in [0; \underline{T}_{\beta}]$ and $t \in [\bar{T}_{\beta}; T]$, where the values of times $\underline{T}_{\beta}$ and $\bar{T}_{\beta}$ are the solutions in $[0; T]$ of equation $\Phi_{\beta}(t) = f'(0)$ with $0 < \underline{T}_{\beta} < \bar{T}_{\beta} < T$.

2. When the recycling rate $\beta(t)$ is not always zero, it reaches a maximum at the time $\hat{T}_{\beta}$
defined as follows:

$$ \hat{T}_{\beta} \left\{ \begin{array}{ll} = 0 & , \text{ if } e^{(\rho+\alpha)T} \le \frac{c_W}{\rho\lambda_{R0}} \\ = T & , \text{ if } \frac{c_W}{\rho\lambda_{R0}} \le e^{\rho T} \\ \in ]0, T[ \text{ and s.t. } e^{\alpha\hat{T}_{\beta}} = \frac{\rho\lambda_{R0}}{c_W} e^{(\rho+\alpha)T} & , \text{ if } e^{\rho T} < \frac{c_W}{\rho\lambda_{R0}} < e^{(\rho+\alpha)T}. \end{array} \right. \qquad (1.15) $$

*Proof.* As the cost function associated to the recycling rate $f(\beta)$ is increasing and convex: from (1.6c), the proof of proposition 3 comes from the study of the shape of function $\Phi_{\beta}$, illustrated by Figure 1.1(a). We can study the dynamics of $\beta$ by studying the evolution of $\Phi_{\beta}$:<fnref n="13" />

$$ \Phi_{\beta}(t) = \lambda_{R0}e^{\rho t} + \frac{c_W}{\rho + \alpha} (1 - e^{-(\rho+\alpha)(T-t)}). $$

Function $\Phi_{\beta}(t)$ defined on $\mathbb{R}^{+}$ is proved to be increasing from $\Phi_{\beta}(0) = \lambda_{R0} + \frac{c_W}{(\rho+\alpha)}[1 - e^{-(\rho+\alpha)T}]$ to its maximum $\Phi_{\beta}(\hat{T}_{\beta})$, and then declining, with $\lim_{t \to \infty} \Phi_{\beta}(t) = -\infty$.

(1.) We can see that the equation $\Phi_{\beta}(t) = f'(0)$ has zero solution when $\forall t \in \mathbb{R}^{+}$, $\Phi_{\beta}(t) < f'(0)$, one solution when $f'(0) \le \Phi_{\beta}(0)$ and two solutions otherwise.<fnref n="14" /> We can determine $\underline{T}_{\beta}$ and $\overline{T}_{\beta}$ as the results of this equation, when they exist in $[0; T]$.

(2.) The existence of $\hat{T}_{\beta}$ in the boundaries of the program $[0; T]$, as illustrated by Figure 1.1(b), directly follows the study of the equation $\Phi'_{\beta}(t) = 0$ and gives the results of the second item of Proposition 3. □

It follows from this analysis that as the optimal path of $\beta(t)$ is defined by the path of function $\Phi_{\beta}(t)$, the optimal recycling rate is inverted U-shaped in $t$ when positive. Also, the permanent arbitration with the marginal social benefit of recycling can induce phases with

<footnote n="13">¹³For the purpose of this analysis, we extend the definition of $\Phi_{\beta}(t)$ to $\mathbb{R}^{+}$, and then discuss the existence of solutions in $[0; T]$.</footnote>

<footnote n="14">¹⁴We intentionally omit the specific case when there is a tangential solution for the equation, such as $\max_{\mathbb{R}^{+}} \Phi_{\beta} = f'(0)$.</footnote>
<figure><img src="image_8.png" /><figcaption>Figure 1.1: (a) Saturation of $\beta$ and (b) Maximum time $\hat{T}_{\beta}$</figcaption></figure>

$\beta = 0$. When this phase comes first, it means that the resource constraint on stock $R$ is not high enough to start recycling. The social planner only initiates it after $T_{\beta}$. On the other hand, there can also be a final phase without recycling as the shrinking social cost of waste accumulation makes it no more optimal. This is visible from the evolution of maximum time exposed in (1.15). Given the inverted U-shape of $\beta(t)$, a maximum instant $\hat{T}_{\beta}$ is given by the comparison between the sum of discounted marginal waste damage $\frac{c_W}{\rho} = \int_0^{\infty} c_W e^{-\rho t} dt$, and the initial scarcity of the recycled resource $\lambda_{R0}$. For high damage wastes (or a low constraint on the recycled resource), the maximum of recycling occurs at the beginning.

In section 1.3.5, we will consider the most interesting case where the recycling rate $\beta(t)$ reaches a maximum at instant $\hat{T}_{\beta}$, corresponding to the third case of (1.15), where $0 < \hat{T}_{\beta} < T$.

### 1.3.5 Dynamics of resource flows

The objective is now to superimpose the respective dynamics of resource use and recycling that have been previously analyzed separately. For that we consider a phase during which $v > 0$ (and $r = 0$) or $r > 0$ (and $v = 0$). We look at the dynamics of $v(t)$ and $r(t)$ by examining the time derivatives of conditions (1.6a) and (1.6b). Using (1.11), for any value
of $\beta$, we obtain:

$$FM\dot{C}_v = [\rho(\lambda_{V0} - \beta\lambda_{R0}) - (1-\beta)c_W e^{-(\rho+\alpha)T}e^{\alpha t}]e^{\rho t}, \quad (1.16)$$

$$FM\dot{C}_r = (1-\beta)[\rho\lambda_{R0} - c_W e^{-(\rho+\alpha)T}e^{\alpha t}]e^{\rho t}. \quad (1.17)$$

Remark that, defining $\Phi_v(t) = [1-\beta(t)]e^{\alpha t} + \beta(t)e^{\alpha \hat{T}_\beta}$, with $\hat{T}_\beta$ such that $e^{\alpha \hat{T}_\beta} = \rho \lambda_{R0} e^{(\rho+\alpha)T}/c_W$, we can rewrite (1.16) as follows:

$$FM\dot{C}_v = \left[ \frac{\rho \lambda_{V0}}{c_W} e^{(\rho+\alpha)T} - \Phi_v \right] c_W e^{\rho t} e^{-(\rho+\alpha)T}. \quad (1.18)$$

**Proposition 4.** The full marginal cost $FMC_v$ (resp. $FMC_r$) of producing with the virgin resource (resp. recycled) can reach a maximum at time $\hat{T}_v$ (resp. $\hat{T}_r$).

1. A maximum for $FMC_v$ exists if $\Phi_v(0) < \frac{\rho\lambda_{V0}}{c_W}e^{(\rho+\alpha)T} < \Phi_v(T)$. It is determined by:

$$\Phi_v(\hat{T}_v) = \frac{\rho\lambda_{V0}}{c_W}e^{(\rho+\alpha)T}.$$

2. If $FMC_r$ is non-monotonous, it reaches a maximum at time $\hat{T}_r = \hat{T}_\beta$, with $\hat{T}_\beta$ being characterized by (1.15).

3. If the $\hat{T}$'s exist, then $\hat{T}_v < \hat{T}_r$ (resp. $>$) if $\lambda_{V0} < \lambda_{R0}$ (resp. $>$).

*Proof.* (1.) The full marginal cost $FMC_v$ will reach an extremum when $FM\dot{C}_v$ as given by (1.18) equals 0. In addition, $\Phi_v$ is strictly increasing: $\Phi'_v(t) = (e^{\alpha \hat{T}_\beta} - e^{\alpha t})\dot{\beta} + (1-\beta)e^{\alpha t} > 0$ for any $t$, as $\hat{T}_\beta$ is the maximum instant of $\beta$. Then, if $FM\dot{C}_v = 0$, the extremum of $FMC_v$ is a maximum. (2.) From (1.17), and given that $\beta(t) < 1 \ \forall t$, it comes immediately that $FM\dot{C}_r(t) = 0 \Leftrightarrow t = \hat{T}_\beta$. Since $\rho\lambda_{R0} - c_W e^{-(\rho+\alpha)T}e^{\alpha t}$ is decreasing in $t$, the extremum is a maximum. (3.) From the characterization of $\hat{T}_v$ given by the first part of the proof, we can
write: $e^{\alpha \hat{T}_v} - e^{\alpha \hat{T}_\beta} = \left(\frac{1}{1-\beta(\hat{T}_v)}\right) \left[ \frac{\rho \lambda_{V0}}{c_w} e^{(\rho+\alpha)T} - \beta(\hat{T}_v) e^{\alpha \hat{T}_\beta} \right] - e^{\alpha \hat{T}_\beta} = \left(\frac{1}{1-\beta(\hat{T}_v)}\right) \frac{\rho(\lambda_{V0}-\lambda_{R0})}{c_w} e^{(\rho+\alpha)T}$. 
Hence, $\mathrm{sign}(\hat{T}_v - \hat{T}_\beta) = \mathrm{sign}(\lambda_{V0} - \lambda_{R0})$. $\square$

Analyzing possible outcomes for FMC curves and considering their characteristics described in Propositions 1, 2 and 4 allows us to determine the typical trajectories of the gross marginal surplus $u'(q)$. From (1.6a) and (1.6b), we know that $u'(q) = FMC_v(t) \le FMC_r(t)$ for $t \le \tilde{T}$ and $u'(q) = FMC_r(t) < FMC_v(t)$ for $t > \tilde{T}$. Given the possible shapes of the FMC's as described in Proposition 4, the time dependent marginal gross surplus curve can thus be either always increasing, or always declining, or an inverted U-shape. In this last case, the maximum of $u'(q)$ can be attained before the switching time $\tilde{T}$, after it, or precisely at that time.

Table 2 summarizes all these possible cases depending on whether the recycled material is more expensive (second column) or cheaper (third column) than the virgin resource, and characterizes the conditions on the dates $\hat{T}_v$ and $\hat{T}_r$ for their occurrence. Notice that, as we focus on the scenario of a successive use of both resources, i.e. such that $\tilde{T} \in (0, T)$, if $c_r - c_v > 0$ (resp. $<$), then we must have $\lambda_{V0} - \lambda_{R0} > 0$ (resp. $<$) from condition (1.14), which implies $\hat{T}_v > \hat{T}_r$ (resp. $<$) from Proposition 4.3. This last result allows for reducing the number of possible trajectories.

By concavity of $u$, the dynamics of resource extraction $q$, either from the virgin stock or from the recycled stock, is the inverse of those of the marginal surplus as characterized in Table 2. When the gross marginal surplus curve is increasing, the Hotelling effect prevails due to the scarcity of both resources (i.e. $\lambda_{V0}$ and/or $\lambda_{R0}$ large), and the extraction path must be decreasing. However, as previously explained, this scarcity effect on the resource stocks can be counterbalanced by a recycling effect aiming at increasing the resource use in order to provide raw materials to be recycled, which corresponds to phases where $u'(q)$ is decreasing through time.
<table><caption>Table 1.2: Possible shapes of the gross marginal surplus and conditions of occurrence</caption><thead><tr><th>Shape of u'(q)</th><th>If c<sub>v</sub> &lt; c<sub>r</sub> (and T&#x0302;<sub>r</sub> &lt; T&#x0302;<sub>v</sub>)</th><th>If c<sub>r</sub> &lt; c<sub>v</sub> (and T&#x0302;<sub>v</sub> &lt; T&#x0302;<sub>r</sub>)</th></tr></thead><tbody><tr><td>Always increasing</td><td>T &le; T&#x0302;<sub>r</sub></td><td>T&#x0302; &le; T&#x0302;<sub>v</sub> and T &le; T&#x0302;<sub>r</sub></td></tr><tr><td>Increasing and then decreasing with a maximum attained after T&#x0302;</td><td>T&#x0302; &lt; T&#x0302;<sub>r</sub> &lt; T</td><td>T&#x0302; &lt; T&#x0302;<sub>v</sub> &lt; T&#x0302;<sub>r</sub> &lt; T</td></tr><tr><td>Increasing and then decreasing with a maximum attained at T&#x0302;</td><td>T&#x0302;<sub>r</sub> &le; T&#x0302; &le; T&#x0302;<sub>v</sub></td><td>impossible</td></tr><tr><td>Increasing and then decreasing with a maximum attained before T&#x0302;</td><td>0 &lt; T&#x0302;<sub>v</sub> &lt; T&#x0302;</td><td>0 &lt; T&#x0302;<sub>v</sub> &lt; T&#x0302;<sub>r</sub> &lt; T&#x0302;</td></tr><tr><td>Always decreasing</td><td>T&#x0302;<sub>v</sub> &le; 0</td><td>T&#x0302;<sub>v</sub> &le; 0 and T&#x0302;<sub>r</sub> &lt; T&#x0302;</td></tr></tbody></table>

As the trajectory of $u'$ is possibly inverted U-shaped, the total flow of material can increase at the end of the program (after time $T_i$). This constitutes a catch-up phase: the lower pressure on waste (as the end of the program approaches) and a higher appreciation of the replenishment of the recycled stock counterbalance depletion of the virgin stock. The constraint $T_β = T_r < T_v$ follows when $T_β$ comes before the end of the program and the scarcity of V is more prominent than R. A catch-up phase with an increasing production comes after the start of a phase when the interest in recovering waste is lowered $\dot{\beta} < 0$, meaning that the social cost of waste accumulation declines more than the social cost of the resource.

# 1.4 Accounting for climate change impacts

## 1.4.1 Optimal program with greenhouse gas emissions

We now take into account another environmental externality that comes from GHG emissions due to both virgin and recycled resource uses. We assume that input flows $v$ and $r$
respectively contribute at rates $\delta_v$ and $\delta_r$ to these emissions.<fnref n="15" /> Then, the cumulative GHG emissions $E(t)$ at any time $t$ evolves as:

$$ \dot{E}(t) = \delta_v v(t) + \delta_r r(t), \quad E(0) = 0. \qquad (1.19) $$

Contrary to waste accumulation which creates local damages, GHG emissions are a global externality which do not directly harm the industry. However, for this last externality to be binding in our model, we define a GHG stabilization cap beyond which damages are supposed to be too high to be supported on a global scale. Such cumulative emission targets are set by international environmental efforts in order to curb global warming. We suppose in our model that this global target has been divided between different countries, leading to national emissions objectives (or budgets) that can be dispatched between industries. We set this sectoral carbon budget to $\bar{E}$ so that, at any time $t$, the cumulative emissions cannot be higher:

$$ E(t) \leq \bar{E}, \quad \forall t. \qquad (1.20) $$

As we consider neither carbon-free alternative resources nor processes of reducing cumulative emissions, we can no longer use inputs, virgin or recycled, once the cap is reached. With the Inada conditions, it forces $q = 0$ as late as possible, simplifying hypothesis (1.20) to a final state condition. Hence, the exogenous time-limit $T$ of the model also corresponds to climate objectives of the industrial sector and its associated emissions ceiling, involving the constraint:

$$ E(T) \leq \bar{E}. \qquad (1.21) $$

<footnote n="15">¹⁵We suppose for the moment that the contribution of the recycling activity represented by rate $\beta$ to total emissions is included in rate $\delta_r$ given for the use of the recycled input. In appendix, we show how relaxing this assumption modify the results.</footnote>
Compared to the set of optimal conditions obtained in the basic model without GHG emissions, only (1.6a) and (1.6b) are modified as shown in (1.22a) and (1.22b) and an additional condition (1.22c) on GHG dynamics is introduced. In this condition, $-\lambda_E$ is the co-state variable associated to the dynamics of GHG emissions:

$$u' \le c_v + f(\beta) + \lambda_V + \delta_v \lambda_E - \beta \lambda_R + (1-\beta) \lambda_W, \quad (= \text{if } v > 0) \quad (1.22a)$$

$$u' \le c_r + f(\beta) + \lambda_R + \delta_r \lambda_E - \beta \lambda_R + (1-\beta) \lambda_W, \quad (= \text{if } r > 0) \quad (1.22b)$$

$$\dot{\lambda}_E = \rho \lambda_E \Leftrightarrow \lambda_E = \lambda_{E0} e^{\rho t}, \quad (1.22c)$$

Conditions (1.6c)-(1.6f) are unchanged. The transversality condition (1.7) also remains true but it must be expanded to include:

$$\lambda_E(T)(\bar{E} - E(T))e^{-\rho T} = 0. \quad (1.23)$$

The new FMC expression for each input now also includes the social marginal cost $\lambda_E$ of cumulative emissions weighted by the carbon intensity of each input, $\delta_v$ and $\delta_r$. We can see that beyond relaxing the resource constraint and reducing waste accumulation, recycling can also be motivated by potentially reducing GHG emissions when the virgin input is more polluting than the recycled one ($\delta_v > \delta_r$).

### 1.4.2 Relative scarcity of the stocks and budget

Transversality conditions (1.7) and (1.23) state that at the final time $T$, stock $V$ (resp. $R$ and $\bar{E} - E$) is either a scarce resource, with $V(T) = 0$ (resp. $R(T) = 0$ and $E(T) = \bar{E}$), or it is an abundant resource and its shadow value is always zero, and thus $\lambda_{V0} = 0$ (resp. $\lambda_{R0} = 0$ and $\lambda_{E0} = 0$). From (1.1), (1.2) and (1.19), final stocks are linked with the following
equation:

$$E(T) = \delta_v(V_0 - V(T)) + \delta_r \left( \int_0^T \beta(s)q(s)ds - R(T) \right). \quad (1.24)$$

Therefore, a full exhaustion of all stocks means that the exogenous decision on a carbon budget perfectly meets the emissions potential of both resources and accounts for the optimal trajectory of recycling. This politically-hard-to-design scenario can then be ruled out.

For the problem to be meaningful, we place our model in a situation where the carbon budget is set to answer a pressing climate constraint, where there should be significant changes of behaviours in the economy. In other words, we assume that the carbon budget $\bar{E}$ to be small enough, as compared with the initial endowment in carbon-emitting resources, so that it will be exhausted no later than time $T$. As there is no carbon-free option available in this economy, it is then optimal to postpone as much as possible the exhaustion of the budget, $E(T) = \bar{E}$, which implies a strictly positive shadow cost of emissions: $\forall t, \lambda_E(t) = \lambda_{E0}e^{\rho t} > 0$.

However, exhausting the virgin resource would mean $\delta_v V_0 \le \bar{E}$ from equations (1.1) and (1.19). This situation is similar to what we would call a "business as usual" scenario, where imposing a carbon budget has no impact on resource exhaustion on a finite period. For this reason we should consider the virgin resource to be relatively abundant, meaning that $V(T) > 0$ and, from (1.7), that $\lambda_V(t) = 0 \forall t$. These observations lead us to the following assumption:

**Assumption 1.** The terminal state of the model with climate change is such that the carbon budget is saturated $E(T) = \bar{E}$ and the virgin resource is abundant $V(T) > 0$.
### 1.4.3 Arbitration on resources use

FMCs and their difference are now expressed with the social cost of GHG and without the scarcity rent on the virgin resource:

$$ FMC_v = c_v - \beta\lambda_R + \delta_v\lambda_E + (1 - \beta)\lambda_W + f(\beta) \quad (1.25) $$

$$ FMC_r = c_r + (1 - \beta)\lambda_R + \delta_r\lambda_E + (1 - \beta)\lambda_W + f(\beta) \quad (1.26) $$

$$ \Delta FMC = FMC_v - FMC_r = -(c_r - c_v) + [(\delta_v - \delta_r)\lambda_{E0} - \lambda_{R0}]e^{\rho t}. \quad (1.27) $$

The change of inputs now also depends on the GHG emissions difference. Proposition 1 remains overall the same, except for a new eventual time of input switch which is now defined by:

$$ e^{\rho \tilde{T}} = \frac{c_r - c_v}{(\delta_v - \delta_r)\lambda_{E0} - \lambda_{R0}} \quad (1.28) $$

Different scenarios depending on the costs and emissions rates of resources exist and are summarized with Proposition 5:

**Proposition 5.** Regardless of the recycling rate path $\beta(t)$, we can summarize the different optimal scenarios under a carbon budget to the following:

1. If extracting input from the recycled stock is costlier than from the virgin stock ($c_r > c_v$):

(a) If the use of the virgin resource emits less than the recycled one ($\delta_v < \delta_r$), or if the scarcity of the recycled resource weighs more than the emissions difference, we use only the virgin resource;

(b) If the virgin resource is more polluting ($\delta_v > \delta_r$), at a level enabling $\tilde{T}$ to exist as defined in (1.28) (i.e. ($\delta_v - \delta_r)\lambda_{E0} > \lambda_{R0}$) and provided that $\tilde{T} \le T$, we use the virgin resource up to time $\tilde{T}$;

2. If extracting input from the virgin stock is costlier than from the recycled stock ($c_r < c_v$),
we use the virgin resource up to time $\tilde{T}$ and then the recycled resource. In this case, the emissions difference must also weigh less than the scarcity of the recycled resource $((\delta_v - \delta_r)\lambda_{E0} < \lambda_{R0})$.

*Proof.* The different optimal paths can be drawn from the positivity conditions for $\Delta FMC$.

□

The arbitration on resources use is significantly influenced by the carbon constraint, as the scarcity of the GHG budget also leads to a switch in inputs, while the virgin resource remains abundant. However, the reasoning on recycling rate $\beta$ developed earlier remains valid, as equation (1.6c) does not include the social cost of carbon. Last, as previously mentioned, we restrict the analysis to the case of an interior switch of inputs, i.e. $\tilde{T} \in (0; T)$, and when recycling is non-monotonous, i.e. $0 < \hat{T}_{\beta} < T$.

### 1.4.4 Dynamics of resource flows

While no closed-form expression for the consumption trajectories of resources can be determined analytically in this program, first-order conditions can give us some insights on the different phases of production (as seen previously) and the fluctuations of the material flows. For that we consider a phase during which $v > 0$ (and $r = 0$) or $r > 0$ (and $v = 0$). We look at the dynamics of $v(t)$ and $r(t)$ by examining the time derivatives of conditions (1.6a) and (1.6b). Using (1.11), we obtain for any value of $\beta(t)$:

$$FM\dot{C}_v = [\rho\delta_v\lambda_{E0} - \rho\beta\lambda_{R0} - (1-\beta)c_W e^{-(\rho+\alpha)T}e^{\alpha t}]e^{\rho t}, \quad (1.29)$$

$$FM\dot{C}_r = [\rho\delta_r\lambda_{E0} + \rho(1-\beta)\lambda_{R0} - (1-\beta)c_W e^{-(\rho+\alpha)T}e^{\alpha t}]e^{\rho t}. \quad (1.30)$$
For this we define functions $\Phi_v$ and $\Phi_r$ as follow:

$$ \Phi_v(t) \equiv [1 - \beta(t)]e^{\alpha t} + \beta(t)e^{\alpha \hat{T}_\beta}, \qquad (1.31) $$

$$ \Phi_r(t) \equiv [1 - \beta(t)]e^{\alpha t} - [1 - \beta(t)]e^{\alpha \hat{T}_\beta}, \qquad (1.32) $$

where, from (1.15), $\hat{T}_\beta$ is such that $e^{\alpha \hat{T}_\beta} = \frac{\rho \lambda_{R0} e^{(\rho+\alpha)T}}{c_W}$ (which is, if it exists in $]0; T[$, the instant when recycling reaches its maximum).

**Proposition 6.** The full marginal cost $FMC_v$ (resp. $FMC_r$) of producing with the virgin resource (resp. recycled) can reach a maximum at time $\hat{T}_v$ (resp. $\hat{T}_r$).

1. A maximum exists if $\Phi_v(0) < \frac{\rho \delta_v \lambda_{E0}}{c_W} e^{(\rho+\alpha)T} < \Phi_v(T)$ (resp. $\Phi_r(0) < \frac{\rho \delta_r \lambda_{E0}}{c_W} e^{(\rho+\alpha)T} < \Phi_r(T)$). These instants correspond to maxima of the FMCs and are determined by:

$$ \Phi_v(\hat{T}_v) = \frac{\rho \delta_v \lambda_{E0}}{c_W} e^{(\rho+\alpha)T} \quad \text{and} \quad \Phi_r(\hat{T}_r) = \frac{\rho \delta_r \lambda_{E0}}{c_W} e^{(\rho+\alpha)T}. $$

2. If true, we have:

$$ \hat{T}_{\beta} < \hat{T}_{r} < \hat{T}_{v} \text{ if } c_{v} < c_{r}, \\ \hat{T}_{\beta} < \hat{T}_{r} \text{ and } \hat{T}_{v} < \hat{T}_{r} \text{ if } c_{r} < c_{v}. $$

*Proof.* (1.) The full marginal cost $FMC_v$ will reach an extremum when $FMC_v = 0$. Using (1.29) and (1.31), this condition becomes: $FMC_v = \left[ \frac{\rho \delta_v \lambda_{E0}}{c_W} e^{(\rho+\alpha)T} - \Phi_v \right] c_W e^{\rho t} e^{-(\rho+\alpha)T} = 0$. In addition, $\Phi_v$ is strictly increasing: $\Phi'_v(t) = (e^{\alpha \hat{T}_\beta} - e^{\alpha t}) \dot{\beta} + (1-\beta)e^{\alpha t} > 0$ for any $t$, as $\hat{T}_\beta$ is the maximum instant of $\beta$. Then, if $FMC_v = 0$, the extremum of $FMC_v$ is a maximum. The same reasoning is applied to $FMC_r$, with $FMC_r = \left[ \frac{\rho \delta_r \lambda_{E0}}{c_W} e^{(\rho+\alpha)T} - \Phi_r \right] c_W e^{\rho t} e^{-(\rho+\alpha)T}$.

(2.) First, from the expression of $\Phi_r$, condition $\Phi_r(\hat{T}_r) = \frac{\rho \delta_r \lambda_{E0}}{c_W} e^{(\rho+\alpha)T}$ implies that $\hat{T}_\beta < \hat{T}_r$.
Next, when $c_v < c_r$, from (1.27), we can write $\Delta F\dot{M}C = \rho[\Delta FMC + (c_r - c_v)]$. As $\tilde{T}$ exists, we know that there is a second phase during which the recycled input is used, meaning $\Delta FMC > 0$ for $t > \tilde{T}$, which involves $\Delta F\dot{M}C > 0$. This result implies $[(\delta_v - \delta_r)\lambda_{E0} - \lambda_{R0}] > 0$. As it is independent of time, we can generalize with $\Delta F\dot{M}C > 0$, $\forall t$. Hence $F\dot{M}C_v > F\dot{M}C_r$. By rewriting expressions of $F\dot{M}C_i$ with $\Phi_i (i \in \{v; r\})$, which are proved to be increasing functions of time, thus $F\dot{M}C_i > 0$ before $\hat{T}_i$ and $F\dot{M}C_i < 0$ after. This gives us that $\hat{T}_r < \hat{T}_v$.

Else, when $c_r < c_v$, expression (1.28) gives us that $[(\delta_v - \delta_r)\lambda_{E0} - \lambda_{R0}] < 0$. When used in the expression of $\Delta F\dot{M}C$, it leads to $F\dot{M}C_r > F\dot{M}C_v$, hence $\hat{T}_v < \hat{T}_r$. $\square$

From Proposition 6, we can induce the same type of results on the dynamics of the gross marginal surplus and the dynamics of the resource extraction as those contained in Table 2. Whatever the relative cost of each input, we obtain a time trajectory of the resource extraction that is either strictly increasing, or strictly decreasing or U-shaped. Hence, as in the basic model, a catch-up phase can occur at the end of the program, during which the total flow of material increases. However, this phase is not driven anymore by the relative scarcity of each resource stocks (the virgin stock is assumed to be abundant), but rather by the relative carbon footprint of each resource and by the carbon budget constraint on cumulative emissions with respect to the environmental damage due to waste accumulation.

## 1.5 Decentralization and policy implications

In order to analyze environmental policy tools used to promote recycling and the reduction of carbon emissions, there is a need to discuss a decentralized version of our model. To decentralize the model, transfer functions for the different actors in the economy are introduced. These functions correct for the externalities previously developed in this chapter.
The basis of the decentralized model is the extended model developed in Section 1.4 that includes GHG emissions, as it offers a richer discussion on combined externalities and allows for the observation of different mixes of policies.

As a basic structure, we assume that the economy is composed of four agents: the final consumer, the producer of the manufactured good, the virgin resource sector, and the recycled resource sector which manages both the recycling activity and the exploitation of the recycled stock.<fnref n="16" /> The property rights of each resource stocks are correctly defined so that each extracting sector is owner of its reserves. These agents can take actions on four markets, assumed to be perfectly competitive: the virgin resource market (price $p_v$), the recycled resource market (price $p_r$), the manufactured good market (price $p_q$), and the waste market (price $p_w$), provided that such a market exists. Initially, we omit the waste market and we introduce it in a second iteration of the decentralized model.<fnref n="17" />

The policy-maker can influence private decisions on each market thanks to a set of monetary transfer functions, denoted by $T_C(\cdot)$, $T_P(\cdot)$, $T_V(\cdot)$ and $T_R(\cdot)$, to the consumer, the producer, the virgin resource sector and the recycling sector, respectively.

### 1.5.1 Equilibrium in the absence of a waste market

The consumer determines the quantity of final good $q$ they will consume in order to maximize her instantaneous surplus function $S_C \equiv [u(q) - p_q q + T_C(q)]$.<fnref n="18" /> The producer of the manufactured good chooses the quantity of inputs $v$ and $r$ in order to maximize instantaneous profits $S_P \equiv [p_q q - p_v v - p_r r + T_P(v, r)]$ subject to the technological constraint $q = v + r$. The virgin resource owner chooses the extraction rate $v$ maximizing profits

<footnote n="16">¹⁶We can obtain similar results with a less developed model, where the production sector is omitted. In this case, the final consumer directly consumes the two types of resources which are perfect substitutes for each other.</footnote>

<footnote n="17">¹⁷Note that by considering this market economy, we only apply a partial decentralization process, in the sense that we do not develop explicitly the financial market. We assume that private agents discount their monetary flows at the rate $\rho$. Moreover, we do not include transaction and transportation costs here.</footnote>

<footnote n="18">¹⁸There is no dynamic budget constraint, as we do not develop the financial market.</footnote>
during an exogenous finite time $T$, expressed by $S_V \equiv \int_0^T [p_v v - c_v v + T_V(v)]e^{-\rho t}dt$ subject to (1.1). Last, the objective of the recycling sector is to determine both the share of final consumption good to be recycled $\beta$, and the recycled resource extraction $r$ that maximize $S_R \equiv \int_0^T [p_r r - c_r r - qf(\beta) + T_R(r, \beta)]e^{-\rho t}dt$ subject to (1.2). Note that we directly focus here on interior values of $\beta$ (the conditions for corner solution has been discussed previously in the central planner problem).

We introduce notations $\lambda_i^*$ and $\tilde{\lambda}_i$ ($i \in \{R; E; W\}$) that refer respectively to the co-state variables of the central planner problem and the co-state variables of the decentralized problem.

**Proposition 7.** Given $\lambda_i^* i \in \{E; W\}$, the shadow costs for an optimal trajectory, the set of policy instruments that restores optimality in the absence of a market for waste must satisfy the following conditions:<fnref n="19" />

$$ \frac{1}{q} \frac{\partial T_R(r, \beta)}{\partial \beta} = \lambda_W^* \qquad (1.33) $$

$$ T'_{V}(v) + \frac{\partial T_{P}(v, r)}{\partial v} + T'_{C}(q) = \pi(\beta) - \delta_{v}\lambda_{E}^{*} - \lambda_{W}^{*} \qquad (1.34) $$

$$ \frac{\partial T_R(r, \beta)}{\partial r} + \frac{\partial T_P(v, r)}{\partial r} + T'_C(q) = \pi(\beta) - \delta_r \lambda_E^* - \lambda_W^*, \qquad (1.35) $$

with $\pi(\beta) \equiv \beta f'(\beta) - f(\beta) \ge 0$ for any $\beta$.

*Proof.* Maximizing surplus functions of the consumer, the producer, the resource owner and

<footnote n="19">¹⁹Here, as we follow a partial equilibrium approach, we do not consider the budget balance equation of the policy-maker. In particular, there is no reason at all to suppose that the net sum of all these monetary transfers must be equal to zero.</footnote>
the recycling sector, we get the following first-order conditions:

$$u'(q) \leq p_q - T'_C(q), \quad (= \text{if } q > 0) \qquad (1.36a)$$

$$p_q \leq p_v - \frac{\partial T_P}{\partial v} \quad \text{and} \quad p_q \leq p_r - \frac{\partial T_P}{\partial r}, \quad (= \text{resp. if } v > 0 \text{ or } r > 0) \qquad (1.36b)$$

$$p_v \leq c_v + \tilde{\lambda}_V - T'_V(v), \quad (= \text{if } v > 0), \quad \text{with } \tilde{\lambda}_V = \tilde{\lambda}_{V0}e^{\rho t} \qquad (1.36c)$$

$$p_r \leq c_r + \tilde{\lambda}_R - \frac{\partial T_R}{\partial r}, \quad (= \text{if } r > 0), \quad \text{with } \tilde{\lambda}_R = \tilde{\lambda}_{R0}e^{\rho t} \qquad (1.36d)$$

$$f'(\beta) \geq \tilde{\lambda}_R + \frac{1}{q} \frac{\partial T_R}{\partial \beta}, \quad (= \text{if } \beta > 0). \qquad (1.36e)$$

Given Eq. (1.36a)-(1.36d), the market-clearing conditions that characterize an equilibrium are:

$$u'(q) = c_v + \tilde{\lambda}_V - T'_V(v) - \frac{\partial T_P}{\partial v} - T'_C(q) \qquad (1.37a)$$

$$u'(q) = c_r + \tilde{\lambda}_R - \frac{\partial T_R}{\partial r} - \frac{\partial T_P}{\partial r} - T'_C(q), \qquad (1.37b)$$

Additionally, we include condition (1.36e) to characterize the equilibrium. With these conditions, we can then compare them to the first-order conditions for optimal interior solutions in the central planner model (conditions (1.22a), (1.22b) and (1.6c)) to get equations (1.33) to (1.35). □

Simple economic policies can be illustrated when considering linear and additive separable transfer functions, i.e. tax-subsidy schemes: $T_C(q) = T_C \times q$, $T_P(v, r) = T_{Pv} \times v + T_{Pr} \times r$, $T_V(v) = T_V \times v$ and $T_R(r, \beta) = T_{Rr} \times r + T_{R\beta} \times \beta q$. In this case, first-best implementation requires four policy instruments.

The first instrument is a unit subsidy $\pi(\beta)$ (always non-negative given the properties of function $f(.)$) on the flow of produced/consumed goods. For the case of an interior solution ($\beta > 0$), from (1.6c), this subsidy can be rewritten as follows: $\pi(\beta) = [\beta q\Phi_{\beta} - qf(\beta)]/q$. This expression reads as the average net social gain of recycling the share $\beta$ of the flow $q$ of final good (remind that $\Phi_{\beta}$ is the marginal social benefit of the recycling effort and $f(\beta)$ is
the recycling cost per unit of output). The policy-maker must implement such a subsidy in order to correct for the positive externality generated by the waste production for free in the absence of waste market. It stems from the absence of a market for waste and the dynamic framework of the model. Recycled input accumulation is indeed not accounted for in the static modeling literature, thus not finding this externality. To the best of our knowledge, only the dynamic work of Pittel et al. (2010) highlights this market failure.

The second instrument is a unit subsidy of value $\lambda_W^*$ on the flow of recycled good $\beta q$. It is required due to the avoided waste accumulation. This subsidy is always attributed to the recycling industry.

The third and fourth instruments are unit taxes corresponding to the two environmental negative externalities: GHG emissions and waste accumulation.

To integrate the GHG emissions externality in the model, we define the first unit tax using (1.34) and (1.35) to write: $T_{Rr} + T_{Pr} - T_V - T_{Pv} = (\delta_v - \delta_r)\lambda_E^*$. In this expression, $\lambda_E^*$ is a uniform carbon tax that is applied to carbon content rate of each primary material. This tax can be imposed either to the producer of the final good or to the extraction sectors.

To integrate the waste accumulation externality in the model, we define the second unit tax of value $\lambda_W^*$ that focuses on the potential waste generation from production/consumption of the final good $q$. This tax can be imposed either to the producer of the final good or to the consumer.

Observing the two monetary transfers associated with waste accumulation (instruments two and four), we can notice that when aggregated together, they represent a single tax of value $\lambda_W^*$ on the effective waste accumulation $(1 - \beta)q$.

Table 1.3 illustrates various possibilities of who carries the burden or receives the benefit of these instruments.<fnref n="20" /> Note that all these options are revenue-equivalent for the policy-maker.

<footnote n="20">²⁰As usual, the transfer of the tax burden between producers and consumers may depend on the price elasticity of demand and supply functions. It is not the purpose here to develop this aspect.</footnote>
<table><caption>Table 1.3: Examples of policy-mix</caption><thead><tr><th>Transfers to:</th><th>Example 1</th><th>Example 2</th><th>Example 3</th></tr></thead><tbody><tr><td>Final consumer (T<sub>C</sub>)</td><td>π(β)q</td><td>(π(β) - λ<sub>W</sub><sup>*</sup>)q</td><td>-λ<sub>W</sub><sup>*</sup>q</td></tr><tr><td>Producer (T<sub>P</sub>)</td><td>-λ<sub>W</sub><sup>*</sup>(v + r)</td><td>-λ<sub>E</sub><sup>*</sup>(δ<sub>v</sub>v + δ<sub>r</sub>r)</td><td>-λ<sub>E</sub><sup>*</sup>(δ<sub>v</sub>v + δ<sub>r</sub>r)</td></tr><tr><td>Virgin resource (T<sub>V</sub>)</td><td>-λ<sub>E</sub><sup>*</sup>δ<sub>v</sub>v</td><td>0</td><td>π(β)v</td></tr><tr><td>Recycling sector (T<sub>R</sub>)</td><td>λ<sub>W</sub><sup>*</sup>βq - λ<sub>E</sub><sup>*</sup>δ<sub>r</sub>r</td><td>λ<sub>W</sub><sup>*</sup>βq</td><td>π(β)r + λ<sub>W</sub><sup>*</sup>βq</td></tr></tbody></table>

These examples show how the carbon tax should be distributed between the producer ($T_P$) and resource managers ($T_V$ and $T_R$). The implementation of this tax is the subject of a wide range of literature, and is not the focus of this chapter. While existing policies are often developed to favor the recycling industry, our model acknowledges the level of the GHG emissions of the sector, and advocates for an appropriate carbon tax on this industry. The tax does not replace the existing support for the recycling industry. These tax-subsidy support schemes remain as they address other issues unrelated to GHG emissions.

It is interesting to note that whatever the transfer structure, there is a subsidy for the recovered flow of material for the recycling sector. The combination of this subsidy with a taxation on products is often promoted in economic literature as a deposit-refund scheme (Fullerton and Kinnaman, 1995; Choe and Fraser, 1999). Additionally, this combination accounts for the specific characteristics of the industrial sector like emissions rates, private and social costs of waste management (Acuff and Kaffine, 2013). Although the system suggested by our model is still hardly implemented, there are several subsidy schemes for recycling industries and several taxation schemes on waste.

In the examples of Table 1.3, taxation relies on the global flow of material subject to potential waste damage. Extended Producer Responsibility (EPR) systems are an example where producers have to assume financial responsibility for the potential environmental damage of their products during their whole life-cycle (through an eco-contribution). In
example 2 and 3, the cost of potential waste damage is assumed by the final consumer. We
can note that even if the tax is applied to the producer, it will eventually be reflected on
the price of the product and the burden of the tax will be assumed by the consumer. In
existing examples, EPR instruments are implemented by industrial sector, in varying sizes.
In France in 2020, Citeo (an agency organizing EPR for packaging industries) fixes a baseline
level for eco-contributions of 16,53 ct€/kg for paper and cardboard, and 28,88 ct€/kg for
light PET (colourless polyethylene terephthalate).<fnref n="21" /> Note that almost 80% of these amounts
are redistributed to the recycling sector (collecting, sorting and resource production), thus
covering the subsidy part of our monetary transfers (Citeo, 2019).

In the case of example 3, note that subsidy $\pi(\beta)$ is applied to resource producers instead of
the final consumer. This reflects the prospective activity of recycling, as consumed resources
have the potential to be recycled resources or waste.

1.5.2 Existence of a waste market

Let us now assume that the final consumer can sell its flow of waste $q$ at price $p_w$ on a
specific market. The consumer's surplus must now include the revenue from the waste sale
$p_w q$: $S_C = [u(q) + (p_w - p_q)q + T_C(q)]$. The new expression of the intertemporal surplus
of the recycling sector is $S_R = \int_0^T [p_r r - c_r r - qf(\beta) - p_w q + T_R(r, \beta, q)]e^{-\rho t} dt$, which must
be maximized subject to (1.2). The private marginal cost of recycling now also includes
purchasing waste. Behaviors of the other agents are unchanged.

**Proposition 8.** Given $\lambda_i^* \ i \in \{E; W\}$, the shadow costs for an optimal trajectory, the set of policy instruments that restores optimality with a market for waste must satisfy the following

<footnote n="21"><sup>21</sup>This amount is completed by a unit based amount and a bonus-penalty system depending on the eco-design of the product.</footnote>
conditions:

$$ \frac{1}{q} \frac{\partial T_R}{\partial \beta} = \lambda_W^* \qquad (1.38) $$

$$ T'_{V}(v) + \frac{\partial T_{R}}{\partial q} + \frac{\partial T_{P}}{\partial v} + T'_{C}(q) = -\delta_{v}\lambda_{E}^{*} - (1 - \beta)\lambda_{W}^{*} \qquad (1.39) $$

$$ \frac{\partial T_R}{\partial r} + \frac{\partial T_R}{\partial q} + \frac{\partial T_P}{\partial r} + T'_C(q) = -\delta_r \lambda_E^* - (1 - \beta) \lambda_W^* \qquad (1.40) $$

*Proof.* Programs of the final good producer and of the virgin resource extracting sector are not affected by the existence of a waste market. Thus, Eq. (1.36b) and (1.36c) still hold. Additionally, the first-order conditions (1.36d) and (1.36e) are unchanged for the recycling sector, but another condition appears for this sector (1.41a) and the condition for the consumer is changed (1.41b):

$$ u'(q) + p_w = p_q - T'_{C}(q), \qquad (1.41a) $$

$$ f(\beta) + p_w = \beta \tilde{\lambda}_R + \frac{\partial T_R}{\partial q}. \qquad (1.41b) $$

When comparing these equations with the first-order conditions for optimal interior solutions in the central planner model given by (1.22a), (1.22b) and (1.6c), we get equations (1.38) to (1.40). □

As expected, the waste market allows to fully internalize the positive externality associated with the waste generation by the final consumer.

## 1.6 Conclusion

In this chapter, we provided a discussion on optimal use of resources when climate change and waste are both additional constraints for the social planner. Depending on initial characteristics of the industrial sector, the optimal path can be divided into phases of virgin and recycled production, with a potential switch of inputs if recycling becomes more socially
profitable. This change in the sector intervenes when the scarcity of the virgin resource prevails and/or private costs also favour the use of recycling. When introducing a climate constraint and a difference between emissions rates, a change of inputs can also occur when this difference is sufficiently large, especially compared to the fixed difference between private delivery costs of inputs. In parallel, the social planner will develop a recycling (or waste recovering) strategy, such as it alleviates the cost of waste accumulation and allows for production from a recycled input. This shows the duality of the activity of sorting waste: a practical goal being the reduction of costs and a speculative goal as the use of recycled input can potentially occur in the future. This leads to inverse U-shaped recycling programs, where it is optimal to increase recycling at first and then decrease it. Moreover, when the marginal cost of recycling is too high, it leads to phases where the recycling rate is zero (although it is still possible to produce from the existing stock of recycled input at the same time). In parallel, under certain conditions, production shows catch-up phases at the end of the program, as full marginal costs start decreasing (involving an increase in production). Our model gives a better understanding of the complementary nature of developing climate change objectives and addressing resource scarcity issues. The model addresses the fact that recycled inputs, while often a cleaner option for production also delaying scarcity, still come with GHG emissions.

While the central planner program gives many insights regarding optimal recycling activities, it does not give suitable results for policy making. The decentralized model of the sector allows to specify the need for public intervention. This new model highlights a possible tax/subsidy scheme in order to implement a first-best solution. To do this first best solution, a taxation based on GHG emissions must be introduced in both sectors, weighed by the carbon intensity of the branch (the carbon intensity is higher for virgin extraction in most cases). This comes in addition to a tax-subsidy system based on waste and stock replenishment which are already demonstrated in Extended Producer Responsibility programs currently implemented for certain products.
Our model is mainly focused on the case of metal industries, where strong GHG emissions assymetries and perfect substitution are relevant hypothesis. In the appendix we address other constraints on recycling: the existence of a scrap value for waste, a decoupling of GHG emissions between collection of waste and transformation into an input, and a limited capacity for the stock of recycled input. Extensions toward other industries would require relaxing some hypothesis used here. For instance, loss of quality is in fact an important topic for the academic research on circular economy, especially when the efficiency of a process is studied (Figge et al., 2018). This is, for instance, the case in the paper and cardboard industry, when each recycling loop lowers the quality of the pulp and only allows a limited amount of cycles depending on the needed quality (usually 7 and 8 cycles are technically possible, but fiber is rather used 3.5 times on average in Europe (ERPC, 2015)). Lafforgue and Rouge (2019) provide the basis for understanding the optimal production path when technological progress in recycling allows for higher quality in recycled inputs.

# 1.A Appendix

Our model can be extended to account more precisely for existing environmental externalities. First, we assume the existence of a scrap value for the waste stock after the program. Second, we decouple the GHG emissions of the recycling process into those related to the collection of waste and to those related to production of the recycled input. To end, we consider the limited capacity of the recycling sector by imposing a cap on the cumulative flows of recycled materials.

## 1.A.1 Scrap value for waste

Here, we assume that waste accumulation is no longer damaging from time $T$ onward: $D(W(t)) = 0, \forall t \ge T$. From (1.3), given that extraction/consumption flows are zero, the remaining stock of wastes evolves as: $W(t) = W(T)e^{-\alpha(t-T)}$, for $t \ge T$. A more realistic
approach consists of attaching a scrap value $\Omega$ to $W(T)$ in case where the damage caused by the left-over waste stock remains after the end of the exploitation period of time:

$$ \Omega(W(T)) = \int_{T}^{\infty} D(W(T)e^{-\alpha(t-T)}) e^{-\rho t} dt. \qquad (1.42) $$

This scrap value must be added to the value function (1.4) of the social planner program, which slightly modifies the transversality condition relative to stock $W$:

$$ [\lambda_{W}(T) - \Omega'(W(T))] W(T) e^{-\rho T} = 0. \qquad (1.43) $$

Using (1.6f), (1.42) and (1.43), the shadow cost of the waste stock at any point in time can be expressed as:

$$ \begin{aligned} \lambda_{W}(t) &= e^{(\rho+\alpha)t} \left[ \Omega'(W(T))e^{-(\rho+\alpha)T} + \int_{t}^{T} D'(W)e^{-(\rho+\alpha)s} ds \right] \\ &= \int_{t}^{T} D'(W)e^{-(\rho+\alpha)(s-t)} ds + e^{-\rho T} \int_{T}^{\infty} D'(W)e^{-(\rho+\alpha)(s-t)} ds. \end{aligned} \qquad (1.44) $$

At any time $t$, the shadow marginal cost of waste contains two components: the sum from $t$ to $T$ of the marginal damage, discounted at rate $(\rho + \alpha)$ to reflect the marginal absorption process of the stock of waste by the environment, and the value at time $T$ of the sum from $T$ onward of the discounted marginal damage, i.e. the scrap value term. With constant marginal damage, this shadow cost becomes:

$$ \lambda_{W}(t) = \frac{\theta}{(\rho + \alpha)} \left(1 - (1 - e^{-\rho T})e^{-(\rho + \alpha)(T-t)}\right), \qquad (1.45) $$

and we still have $\dot{\lambda}_W < 0$. The scrap value intervenes by adding an exogenous factor $1 - e^{-\rho T} \in [0, 1[$ to the decreasing term of the shadow cost. The remaining shadow cost for waste at the end of the program is the marginal scrap value $\frac{c_W}{\rho+\alpha}e^{-\rho T} = \Omega'(W(T))$, instead of zero for the model previously discussed.
This additional assumption does not change the change the arbitration in the model, as the first-order conditions (1.6a)-(1.6f) remain the same. However, the optimal recycling rate follows a different time path, according to the following proposition.

**Proposition 9.** When the recycling rate $\beta(t)$ is not always zero and considering a scrap value for the waste stock damage, it reaches a maximum at time $\hat{T}_{\beta}$. This instant can take the following values:

$$ \hat{T}_{\beta} = \begin{cases} 0, & \text{if } \frac{e^{(\rho+\alpha)T}}{(1-e^{-\rho T})} \le \frac{c_W}{\rho\lambda_{R0}} \text{ or } \lambda_{R0} = 0 \\ T, & \text{if } \frac{c_W}{\rho\lambda_{R0}} \le \frac{e^{\rho T}}{(1-e^{-\rho T})} \\ \frac{1}{\alpha} \left[ \ln \left( \frac{\rho\lambda_{R0}}{c_W(1-e^{-\rho T})} \right) + (\rho+\alpha)T \right] \in [0, T], & \text{if } \frac{e^{\rho T}}{(1-e^{-\rho T})} < \frac{c_W}{\rho\lambda_{R0}} < \frac{e^{(\rho+\alpha)T}}{(1-e^{-\rho T})}. \end{cases} \quad (1.46) $$

*Proof.* Now the marginal benefit of recycling is:

$$ \Phi_{\beta}(t) = \lambda_{R0}e^{\rho t} + \frac{c_W}{\rho + \alpha} \left(1 - \left(1 - e^{-\rho T}\right) e^{-(\rho+\alpha)(T-t)}\right) $$

Following the same analysis as the general case, we get Proposition (9). □

### 1.A.2 Decoupling emissions of the recycling branch

Another possible extension of the model is to consider a decoupling of the emissions related to the recycling process. In that case, parameter $\delta_r$ only concerns the transformation and use of the recycled input, while we add a constant rate $\delta_{\beta}$ for the activity of collecting and sorting from the waste flow: $\delta_{\beta}\beta(t)q(t)$. This leads to the following new dynamic equation for emissions accumulation:

$$ \dot{E}(t) = [\delta_v + \beta(t)\delta_{\beta}]v(t) + [\delta_r + \beta(t)\delta_{\beta}]r(t), \quad E(0) = 0. \quad (1.47) $$
Practically speaking, this modification in the model does not change the arbitration between the two resources. This is because the term relative to this type of emissions disappears with the Full Marginal Costs difference, the same way it does for terms relative to waste accumulation, recycled stock replenishment and the cost of recycling. However, there is an impact on the first-order condition relative to recycling, with a new function $\Phi_{\beta}$:

$$ \Phi_{\beta} = \lambda_{R} + \lambda_{W} - \delta_{\beta}\lambda_{E} \qquad (1.48) $$

This more complex expression of the marginal social gain of recycling can be easily reinterpreted: we now recycle to replenish stock R and alleviate the cost of stock W but it costs emissions to the industrial sector, at rate $\delta_{\beta}$. Developing this expression, we obtain: $\Phi_{\beta} = (\lambda_{R0} - \delta_{\beta}\lambda_{E0})e^{\rho t} + \frac{c_{W}}{\rho+\alpha}[1 - e^{-(\rho+\alpha)(T-t)}]$. The same analysis as before can be done for the evolution of the recycling rate. However we can also add the following proposition:

**Proposition 10.** It is never optimal to start recycling under the two following conditions:

* There is a higher initial social cost for emissions than the depletion of the recycled stock, i.e. $\lambda_{R0} \le \delta_{\beta}\lambda_{E0}$;

* Damage of waste accumulation is such that $\frac{c_{W}}{\rho+\alpha} \le f'(0)$.

In this case, the recycled resource is never used, $\lambda_{R0} = 0$.

*Proof.* Given the new expression of $\Phi_{\beta}$, we see that it is decreasing when $\lambda_{R0} \le \delta_{\beta}\lambda_{E0}$. As $\lim_{-\infty} \Phi_{\beta} = \frac{c_{W}}{\rho+\alpha}$, equation $\Phi_{\beta} = f'(\beta)$ does not have a solution when $\frac{c_{W}}{\rho+\alpha} \le f'(0)$, proving the necessary condition of the proposition. As there is no recycled stock initially, we never use the recycled resource. Note that a more constraining condition for the absence of recycling is $\Phi_{\beta}(0) \le f'(0)$. $\square$

By adding an emission term for recycling activities, we can highlight a simple condition on the parameters of the model for which recycling is never optimal. If the social cost of
GHG emissions relating to the recycling activity is too high, or if the avoided cost of waste accumulation is too low, recycling is never optimal. Finally, note that we also observe the catch-up phase for the use of inputs with this extension of the model.

### 1.A.3 Limiting the capacity of the recycling sector

#### Optimal program with a cap on $R(t)$

In this final iteration of the model, the recycling sector faces an upper limit regarding the size of the recycled stock $R$. This limit suggests that the speculative strategy for recycling is bounded. One can view this limit as the implicit storage cost of the recycled resource. To analyze this concept mor individually, we do not include the previously analyzed GHG ceiling. For this extension, we introduce a cap $\bar{R}$ such as $R(t) \le \bar{R}$. This new constraint leads to a new first order condition on stock $R$, introducing the associated co-state variable $\chi$:

$$ \dot{\lambda}_R = \rho\lambda_R + \chi \quad \text{with} \quad (\bar{R} - R(t))\chi(t) = 0 \quad \text{and} \quad \chi(t) \ge 0. \qquad (1.49) $$

The other optimal conditions refer to the basic case developed in Section 1.4. The optimal sequence of use has now the following characteristics:

**Proposition 11.** Regardless of the recycling rate $\beta(t)$, the optimal path with a cap on the recycled stock is such that:

1. While the recycled stock is not full, it follows the optimal program described in Proposition 1, with a sequential order of use.

2. When recycled stock is full, two scenarios can occur:

(a) If there is no recycling, we only use the virgin resource;

(b) If there is some recycling, there can be a stationary regime for $R$ where the two resources are used simultaneously such as $\beta v = (1 - \beta)r$.
*Proof.* While the cap $\bar{R}$ is not reached, equations from the general problem hold, especially $\dot{\lambda}_R = \rho\lambda_R$, thus proving 1. When $\bar{R}$ is reached, meaning a maximum for $R$, we have $\beta v = (1 - \beta)r$ from equation (1.2) (it can be only instantaneous and $R(t)$ starts decreasing again). Then this equality for stationarity on $R$ proves 2. □

The cap on the recycling stock offers the possibility of a stationary regime, thus with a simultaneous use of the two resource. At each instant, the recycling rate allows to perfectly compensate the depletion of $R$. However, this should not be interpreted as a sustainable scenario, as $\beta < 1$ gives an asymmetry in the material flows: if recycling is low ($\beta < 0, 5$) virgin production prevails, the opposite otherwise.

## Stationary regime for stock $R(t)$

As seen before, the stationary regime for stock $R(t)$ is defined by a simultaneous production from both inputs, and a ratio $v/r$ governed by the evolution of the recovery rate $\beta$. From Proposition 11 and the relationship between flows, we know that this ratio grows when rate $\beta$ declines: less recovery in recycling means that stock $R$ is less exploited ($r$ diminishes) and eventually consumption is sustained by higher extraction ($v$ increases).

To analyze further, let $T^*$ be the starting point of this stationary regime. From the new FOC on stock $R$ given by equation (1.49) and its associated transversality condition, we get expression for the shadow value of the recycled stock: $\lambda_R(t) = -\int_t^T \chi(s)e^{-\rho(s-t)}ds$. This cost is negative, as scarcity is not at stake here ($R(T) = \bar{R} > 0$) and the recycling stock is subject to an upper limit (similar to what was done for the GHG emissions stock). The model during the stationary regime is now governed by the following equations, after instant $T^*$:

$$ \forall t \ge T^*, \quad c_v + \lambda_{V0}e^{\rho t} = c_r - \int_t^T \chi(s)e^{-\rho(s-t)}ds \qquad (1.50) $$

$$ f'(b) = - \int_t^T \chi(s)e^{-\rho(s-t)}ds + \frac{c_W}{\rho + \alpha}(1 - e^{-(\rho+\alpha)(T-t)}). \quad (1.51) $$
Equation (1.50) comes from the simultaneous use of the resource, meaning that both marginal costs are equal. One necessary condition as seen here is that the cost for the recycled input $c_r$ should be higher than the cost for the virgin input $c_v$ when added to its scarcity rent $\lambda_V$. When expressed at the end of the program, it states that both should be equal. In that sense, either the virgin resource is scarce ($\lambda_{V0} > 0$) or both marginal delivery costs are equal.

Another necessary condition for simultaneous use of the resources is that the recycling activity occurs during the whole stationary phase ($\beta > 0$). Equation (1.51) expresses the equality between cost and marginal benefit of recycling as described earlier. The optimal recycling program can be precised as follow:

**Proposition 12.** If the optimal path of the program enters a stationary regime for stock R, the optimal recycling strategy is to always recycle during this phase, and if $\beta$ reaches a maximum during this phase, $\hat{T}_{\beta}$ is defined by:

$$T^* < \hat{T}_{\beta} < T \quad \text{such that} \quad (c_r - c_v) = \frac{c_W}{\rho} e^{-\alpha(T - \hat{T}_{\beta})}. \qquad (1.52)$$

*Proof.* From the equality of FMCs (1.50) expressed at instant T of the program, we get $\lambda_{V0} = (c_r - c_v)e^{\rho T}$. This equality replaces $\lambda_R$ in the FOC for $\beta$, and is used in equality $f'(\beta) = \Phi(\beta)$. The maximum is reached at $\hat{T}_{\beta}$ when $\Phi(\beta) = 0$, thus giving equation (1.52). $\square$

During this specific phase, we have an exogenous expression for the maximum of recycling. This expression (1.52) shows that this instant occurs at a later time when the cost difference $c_r - c_v$ is relatively significant. On the other hand, when the damage of waste accumulation is relatively significant, $\hat{T}_{\beta}$ happens earlier. This prompts early recycling as we do not have a scarcity of R and the shadow cost of waste diminishes.
CHAPTER 2

SECTORAL, RESOURCE AND CARBON
IMPACTS OF INCREASED PAPER AND
CARDBOARD RECYCLING

***

The question of whether paper and cardboard recycling has a positive or negative carbon impact is crucial as the global forest sector plays a central role in climate change mitigation. In this chapter, with the help of an integrated assessment bio-economic model, we assess the effects of increased paper and cardboard recycling on the resource and the carbon balance of the forest sector. We show that these results are strongly linked to the hypothesis of substitution or complementarity between recycled and wood-pulp. We find increased emissions at the pulp sector level, while effects on other wood products are small. The impact on total net sequestration is positive when pulp products are substitutes, negative when complements.

***
## 2.1 Introduction

### 2.1.1 Context and motivations

Studying the underlying dynamics of the development of recycling in in chapter one brings to consider the different externalities linked to resource extraction and disposal. Historically, resource depletion and waste accumulation have been the principal motivation for the development of a circular economic model, mainly through recycling programs. However, recent developments led to a more holistic approach of environmental issues, as local and global pollution have become a focal point of environmental studies. The study of the circular economy is now strongly connected to climate change challenges, as a redefinition of our linear economic model could have impacts on greenhouse-gas emissions (GHG) (ADEME and FEDEREC, 2017; UNEP, 2019; ADEME, 2019).

This chapter focuses on the specific case of paper and cardboard recycling in France. We assess its effect on the forest sector with economic and carbon perspectives. We show that results are strongly linked to the hypothesis of substitution or complementarity between recycled and wood-pulp. In the case of substitutes, we account for a reduction of the carbon balance on the global forest sector, mainly driven by the sequestration mechanism of the forest and increased demand in fuel wood. However when products are complementary, increased recycling leads to more emissions.

More generally, the paper and cardboard recycling sector is well developed in France, with 66% of the production coming from recovered waste (ADEME and Bio by Deloitte, 2017). Besides, the paper pulp industry is directly connected to the global forest sector, which plays a crucial role in climate change mitigation. Unlike other recycling sectors like metals or plastics, there is little to no substantial difference in GHG emissions between recycled and wood-pulp industrial process (ADEME and FEDEREC, 2017; ADEME, 2019). Besides, complex cascading impacts on the resource and other wood products suggest unclear economic and environmental impacts of paper and cardboard recycling. Our objective is to
assess the effect of recycling by expending the usual scope of the study to the forest sector.

The main goal here is to investigate whether climate change mitigation mechanisms of the forest sector offset or not the small difference in climate impact between recycled and virgin pulp.

Our contribution here is the sectoral analysis of a recycling industry and its impacts in terms of GHG emissions. Instead of a standard accounting for GHG displacements using the LifeCycle Assessment methodology (LCA), we also use an Integrated Assessment Model (IAM)(the French Forest Sector Model - FFSM++)<fnref n="1" /> providing prospective scenarios of the French forest sector. Using an IAM allows us to account for the whole forest sector with a bioeconomic perspective, including dynamic mechanisms. Different economic hypotheses on the new recycling branch as well as a carbon accounting module allows us to give the climate impact of increased recycling at different scopes.

## 2.1.2 Related literature

The economic literature tackled the topic of recycling very early with a focus on social costs associated to waste accumulation and resource depletion, in order to find an optimal level of pollution (Hoel, 1978; Smith, 1972). Later on, some papers, like our first chapter, invested the topic of pollutant emissions and recycling: with a dynamic scope and a sectorial model like Huhtala (1999); with an endogenous growth model like Di Vita (2004); and, through a static micro-economic approach like Acuff and Kaffine (2013). These theoretical papers often isolate costs as the main barrier for development of a recycling industry, and accounting for environmental externalities can make this development more desirable. Considering recycling is also central for companies, with the influence of the reusability on the role of post consumer collection (Tsiliyannis, 2008), and solving supply chain problems (Vahdani et al., 2012). However, another central question when dealing with a recycled product, is the substitution with virgin material, especially for paper and cardboard.

<footnote n="1">¹https://ffsm-project.org/wiki/en/home</footnote>
The econometric literature highlights this issue of substitution, although studies do not appear to give a clear-cut result. Lee and Ma (2001) show that recycled and wood-pulp could be substitutes, however they find very low values for cross-elasticities that are not statistically significative. Some studies also show complementarity between inputs, with pulp industries hardly substituting materials in their transformation process (Lundmark and Olsson, 2015; Lundmark and Söderholm, 2003). A literature review from Mansikkasalo et al. (2014) shows that we can find variations between countries, that could be driven from technological differences. They also show that inputs can be substitutes or complements. Those heterogenous results can be explained by the wide diversity of products within the paper and cardboard industry. Beyond purely market reasons, these results can also be addressed by examining technical difficulties of swapping inputs in pulp production processes (ERPC, 2015). These considerations encouraged us to explore different scenarios where recycled and wood-pulp could be complementary or substitutable products. This question of substitution is actually a crucial point yet to be addressed in the circular economy literature. An undesirable scenario would be that recycling contributes to circular economy by creating a rebound effect in production when recycled products poorly substitute to others or even are complements, thus fulfilling an unsustainable goal (Zink and Geyer, 2017).

Beyond these investigations on substitution, econometric studies show low values for price elasticities of the demand for recycled pulp (Deadman and Turner, 1981; Edgren and Moreland, 1990; Lee and Ma, 2001). This low value is also found for the supply of recycled pulp (Mansikkasalo et al., 2014). However, differences can be found between the several use of transformed pulp, showing the difficulty to compute a composite value describing recycled pulp markets (Lundmark and Olsson, 2015; Lundmark and Söderholm, 2003). Another key parameter describing recycled pulp markets is the level of substitutability with, in this case, other wood products. Early studies tackle this question showing that wood-pulp is a complement to solid-wood (Newman, 1987).

These econometric considerations aside, GHG displacement factors of paper recycling
have often been studied with strong economic assumptions. Early assessments from Byström and Lönnstedt (1997) do not show that increased recycling could be environmentally friendly, while only taking into account the pulp sector. One paper from Merrild et al. (2009) gathers estimations on GHG emissions for different recycling technologies. By extending the analysis to the boundaries of the forest resource, they are able to take into account sequestration and substitution mechanisms. However, they assume substitution with wood-pulp and that unused wood is consumed for energy as a substitute for fossil fuel. They get then a high environmental performance of recycling, without taking into account the complexity of cascading effects in the forest sector. On the other hand, a French study from ADEME and FEDEREC (2017) uses LCA<fnref n="2" /> without feedbacks and competition in the rest of the forest sector. By comparing GHG emissions of virgin and recycled production, including avoided end-of-life, their static analysis concludes that paper and cardboard recycling emits the same or even more than producing wood-pulp. This study is extended in ADEME (2019) and finally shows GHG gains when adding assumptions on wastepaper anaerobic degradation. Finally, an Input/Output model applied to the Dutch case by Nakamura (1999) shows decreases in emissions while taking into account wide inter-industry effects, but not economic effects, being a quantity based model.

Including the forest sector in the analysis of the GHG impact of wood products recycling seems crucial as its relationship with climate change mitigation is consequential. It draws on two different mechanisms we take into account in this study. First, sequestration relies in the growth of forests in order to capture additional carbon and reduce GHG concentration. It depends strongly on climatic factors of the forest, thus varies around the world and is likely to be affected by ongoing and future climate variations (Cook-Patton et al., 2020). In addition to this environmental factor, management practices of the forest are also central as strategies on collected and replanted biomass can have significant impacts (Lobianco, Caurla, Delacote and Barkaoui, 2016). Second, the substitution mechanism consists in using wood

<footnote n="2">²ISO 14040 and ISO 14044</footnote>
instead of other products in industrial process. The GHG balance is impacted through different activities including replacing fossil fuel energy with fuel-wood, using less energy for manufacturing forest products instead of non-wood products as well as storing carbon in products (Sathre and O'Connor, 2010; Churkina et al., 2020). One theoretical analysis that includes recycling is carried out by Tatoutchoup and Gaudet (2011) and shows a reduction of the stock of trees, while not accounting for specific uses of wood and assuming perfect substitution between recycled and virgin products. Following this approach, Tatoutchoup (2016) computes an optimal recycling rate maximizing the forest area and the social net benefit. However in both papers they do not extend the analysis to formally compute the carbon balance of the model. Besides, econometric results stated before incite us to alleviate the hypothesis of perfect substitution and explore other possibilities.

These observations incited us to study the carbon impact of recycling while considering market interactions in the global forest sector. Such a study is carried out with an integrated assessment model representing the dynamic evolution of the french forest sector. A nice review of such models is carried out by Riviere et al. (2020), investigating their thematic focus, noticing a rise in climate change related topics. A similar model, the Global Forest Products Model, already shows that recycling has little impact on other wood products, however it does not compute GHG displacement factors (Buongiorno et al., 2003). A study on a smaller scale by Pieratti et al. (2019) using multi-criteria decision analysis with circular economy principles underlines under-optimal management practices for a single forest, while the global GHG balance is positive thanks to fossil fuel substitution. However they do not examine the specific effect of recycling. D'Amato et al. (2020) stress that LCA assessments of forest-based economy, while they very often include climate change impacts, lack waste and circularity considerations.

Our model here, FFSM++, has provided a wide stream of literature regarding long-term assessments of the sector, with market, resource or climate outlooks. Introduced by Caurla et al. (2010), it already produced results on the introduction of climate change mit-
igation policies and their possible crossed-effects (Caurla, 2012; Caurla, Delacote, Lecocq and Barkaoui, 2013; Caurla, Delacote, Lecocq, Barthès and Barkaoui, 2013). Climate impacts and risk aversion are also a useful scope for long-term assessments of the French forest dynamics, with Lobianco, Delacote, Caurla and Barkaoui (2016) showing overall stability of the volume stocks. Finally, FFSM++ also allows the observation of substitute products, with the introduction of crossed-elasticities (Caurla, Delacote, Lecocq, Barthès and Barkaoui, 2013; Petucco et al., 2019). We present here an extension of this model that links pulp production to waste accumulation and recycling.

The remaining of this chapter is structured as follows. Section 2.2 presents the extension we added to FFSM++ and the simulation strategy to analyse recycling. Section 2.3 describes the impact of increased recycling in the model, with a focus on carbon at different scales of the sector. Section 2.4 concludes.

## 2.2 Adding a paper recycling loop to a forest sector model

### 2.2.1 Modular structure of FFSM

FFSM is designed to explore the dynamics of markets and resources, as well as policies and their economic and environmental impacts<fnref n="3" />. Each year, the model computes prices and quantities for different primary and secondary wood products based on a Samuelson (1952) spatial equilibrium. This way, trade of products is represented on the national level, using prices in different regions, as well as on the international level, using Armington (1969) substitution model. Sauquet et al. (2011) investigate this last point with the FFSM framework, on the specific product sawnwood. This economic model is completed by a resource module representing the dynamics of the forest sector, and a carbon accounting module in

<footnote n="3">³See Caurla (2012) for a full description of the model, and https://ffsm-project.org/wiki/en/doc/home#published龍子女 for any further extension.</footnote>
order to assess the emissions balance of the sector. Carbon accounting consists in reporting sequestration mechanisms (through wood products, inventoried and extra forest biomass) and emissions (direct forest operations, energy and material substitution for transformed wood products with coefficients from the already existing literature) described in detail in Lobianco, Caurla, Delacote and Barkaoui (2016).

The dynamic recursive structure of FFSM is described in Figure 2.1, where previous year harvest and forest dynamics give the availability of the resource to the market module. The detailed description of the market module is given by Figure 2.2, with a supply in primary products upstream (roundwood and industrial wood) and a demand in products of first transformation downstream. The interface between production and consumption consists in the transformation industry, with an input-output logic (see Figure 2.2).

The addition of a recycling loop consists here in the creation of a pair of primary and transformed products. Recovered waste of the previous year (from both recycled and virgin pulp consumed in the model) is reprocessed into recycled pulp, linked to the demand in wood-pulp through complementarity or substitution.

<figure><img src="image_9.png" /><figcaption>Figure 2.1: Recursive structure of FFSM</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 2.2: Products structure of FFSM</figcaption></figure>

### 2.2.2 Dynamics of pulp recycling

The regional demand for recycled pulp is similar to demands for other transformed products introduced in FFSM (Caurla et al., 2010), expressed as a composite demand $D_{recy,i,t}$ (for region $i$ at year $t$) with a substitution between foreign and local shares of the total demand. Besides the recursive dependency with supply and prices, we introduce a cross elasticity between the recycled pulp and wood-pulp:<fnref n="4" />

$$D_{recy,i,t} = D_{recy,i,t-1} \left( \frac{P_{recy,i,t}}{P_{recy,i,t-1}} \right)^{\sigma_{recy}} \left( \frac{\frac{P_{recy,i,t}}{P_{pulp,i,t}}}{\frac{P_{recy,i,t-1}}{P_{pulp,i,t-1}}} \right)^{\eta_{recy,pulp}} \quad (2.1)$$

Where

<footnote n="4">⁴See Caurla, Delacote, Lecocq, Barthès and Barkaoui (2013).</footnote>
* $\sigma_{recy}$ is the price elasticity of demand;

* $P_{recy,i,t}$ is the price of composite recycled pulp, in region $i$ at year $t$;

* $P_{pulp,i,t}$ is the price of the substitute (or complementary) product, wood-pulp in region $i$ at year $t$;

* $\eta_{recy,pulp}$ is the cross elasticity of demand.

A similar and symmetric cross-elasticity $\eta_{pulp,recy}$ in the demand function for wood-pulp is introduced.

We also introduce a composite supply of recovered waste $S_{waste,i,t}$<fnref n="5" /> depending on its price and the stock of waste paper and cardboard that could be recovered for recycling purposes:

$$S_{waste,i,t} = S_{waste,i,t-1} \left( \frac{P_{waste,i,t}}{P_{waste,i,t-1}} \right)^{\epsilon_{waste}} \left( \frac{W_{i,t}}{W_{i,t-1}} \right)^{\beta_{waste}} \quad (2.2)$$

Where:

* $P_{waste,i,t}$ is the price of composite recovered waste paper and cardboard, in region $i$ at year $t$;

* $W_{i,t}$ is the volume of waste cardboard and paper that can be recovered;

* $\epsilon_{waste}$ is the price elasticity of supply;

* $\beta_{waste}$ is the stock elasticity of supply.

From the dispersive use of material and the technological limits associated to recycling and recovery, stock $W_{i,t}$ is a share of the total pulp (from wood and recycled) that has been produced at $t-1$. Following a common assumption in literature concerning recycling of resources, we do not account for the duration of use of the pulp produced that would induce

<footnote n="5">⁵see Caurla et al. (2010) for the detailed construction of the composite supply of primary products</footnote>
lagged effects in the model (Palmer et al., 1997; Huhtala, 1999). This means that each year, paper is produced, consumed and then recycled or disposed of the following year.

$$W_{i,t} = \gamma_{i,t} (D_{pulp,i,t-1} + D_{recy,i,t-1}) \quad (2.3)$$

Where:

* $\gamma_{i,t}$ is the share of pulp (recycled and wood) consumed in region $i$ at year $t-1$ that can be recovered in region $i$ at year $t$;

* $D_{pulp,i,t-1}$ is the demand in wood-pulp in region $i$ at year $t-1$;

* $D_{recy,i,t-1}$ is the demand in recycled pulp in region $i$ at year $t-1$;

Recovered waste paper and cardboard are transformed by a recycling industry at a constant cost of production $c_{recy}$ for each unit of recycled pulp produced. This cost enters in the surplus expression, as the model is solved through a static Samuelson spatial equilibrium maximizing the surplus function (detailed at length in Caurla (2012)).

### 2.2.3 Calibration and simulations strategy

The model is run through a numerical solver with a calibration relying on previous calibrations detailed in the FFSM literature (Caurla et al., 2010; Caurla, 2012). Values on supply and demand parameters come from sectorial analysis of the French forest sector (Lenglet et al., 2017; Montagné and Niedzwiedz, n.d.) and the recycling sector (Copacel, 2016; ADEME and Bio by Deloitte, 2017). Price elasticities of supply and demand come from the relative literature mentioned above (Lee and Ma, 2001; Buongiorno et al., 2003; Mansikkasalo et al., 2014).

As the literature shows conflicting results for the cross-elasticity between recycled and wood-pulp, three cases are designed with different values to examine the sensitivity to this
parameter. As there seems to be a common consensus toward low values (Lee and Ma, 2001; Lundmark and Olsson, 2015), we introduced low substitution ($\eta = -0,2$) and low complementarity ($\eta = 0,2$) between transformed products. A third case is also explored with an optimistic hypothesis on substitution, considering a high negative cross-elasticity ($\eta = -1$). As we do not investigate asymmetries between those products, we use the same crossed elasticity $\eta = \eta_{recy,pulp} = \eta_{pulp,recy}$. The paper and cardboard industry encompasses a wide variety of different products, which creates various types of substitutability/complementarity between virgin and recycled pulp. Thus the elasticity we use should be understood as a synthetic indicator of those many types of relationships for the various products within the industry.

GHG emissions linked to the transformation activity of the pulp industry are included in the carbon balance module. Based on the LCA of ADEME and FEDEREC (2017), we assign to the recycling sector emissions of collection, sorting and avoided end of life. Besides, there is a difference between cardboard and paper recycling (cardboard pollutes more): we take into account this difference by creating a weighted coefficient relying on the share of French production, around 40% paper and 60% cardboard (ADEME and Bio by Deloitte, 2017; Lenglet et al., 2017). The resulting GHG coefficients are 350 kg CO2eq/t for wood-pulp and 530 kg CO2eq/t for recycled pulp. Note that these coefficients relate to the paper/recycled paper industry and do not account for effects on the rest of the forest sector, which is in fact the purpose of this work.

Our objective in this paper is to assess the effect of paper recycling on the forest sector, and especially its GHG balance. To this end, we test four scenarios with different recycling intensities: a baseline, lower increase in recycling, medium increase and higher increase. For these increases we use the proxy of the transformation cost $c_{recy}$, as shown in Table 2.1. This change in costs of the recycling process can come from different sources: a technological breakthrough making the recycling technology broadly available (Lafforgue and Rouge, 2019) or policy choices focused on subsidizing recycling (Palmer et al., 1997) As explained above,
we set out this simulation in three different cases: low complementarity, low substitutability and high substitutability, and we compare scenarios with baseline.

<table><caption>Table 2.1: Tested recycling scenarios</caption><thead><tr><th>Scenario name</th><th>baseline</th><th>low</th><th>med</th><th>high</th></tr></thead><tbody><tr><td>c<sub>recy</sub> (€/m<sup>3</sup>)</td><td>70</td><td>50</td><td>30</td><td>10</td></tr></tbody></table>

## 2.3 Impact of paper recycling on the carbon balance

The model is solved numerically over a period of 67 years (from 2013 to 2080). It computes supply/demand/prices of products, as well as resource dynamics and a detailed carbon accounting. In the following part, yearly results, averaged over the simulation period, are given.

### 2.3.1 Impacts on the pulp industry

What we observe for the pulp sector is the impact on production, given a cost reduction. This impact is direct for the recycling industry, and indirect for wood-pulp. The wood-pulp industry is indeed affected through the cross-elasticity between the two products, with a higher demand in the case of complementarity, a lower with substitution. However, in any case, this absolute change in wood-pulp demand is smaller than the absolute increase of demand for recycled pulp. This is illustrated in Table 2.3 showing the evolution in demand for pulp between baseline and med in 2030. For this reason, Table 2.2 shows that emissions in the pulp industry are always higher with a cost reduction: increased recycling prevails. Besides, from the same reasoning, we show that increasing substitution (from $\eta = -0,2$ to $\eta = -1$) leads to a smaller emissions difference.

One interpretation of complementarity comes from paper and cardboard industries introducing fixed proportions of wood-pulp and recycled pulp in their transformed products, according to the use and the level of quality they look for. There is indeed a lower quality of
<figure><img src="image_11.png" /><figcaption>Figure 2.3: Changes in demand for pulp between baseline and med for substitutability ($\eta < 0$) and complementarity ($\eta > 0$) in 2030</figcaption></figure>

<table><caption>Table 2.2: Changes in emissions linked to transformation in the pulp industry (compared with scenario baseline) - yearly average for 2013-2080</caption><thead><tr><th>Scenario</th><th>low</th><th>med</th><th>high</th></tr><tr><td colspan="4">Emissions variations (MtCO<sub>2</sub>eq) of the transformation industry</td></tr></thead><tbody><tr><td>$\eta = 0,2$</td><td>0,140 (+4%)</td><td>0,231 (+6%)</td><td>0,315 (+8%)</td></tr><tr><td>$\eta = -0,2$</td><td>0,127 (+3%)</td><td>0,225 (+6%)</td><td>0,315 (+8%)</td></tr><tr><td>$\eta = -1$</td><td>0,105 (+3%)</td><td>0,192 (+5%)</td><td>0,265 (+6%)</td></tr></tbody></table>

the fiber for each recycling process, thus imposing a limit of cycles and use (ERPC, 2015). With this hypothesis, recycling fosters the demand for wood-pulp. In contrast, when products are substitutes, introducing recycled pulp implies a stronger competition for wood-pulp, thus decreasing the quantities. High substitution also reflects technical possibilities, where recycled pulp has an equivalent quality and consumption pattern as wood-pulp.

The purpose of our analysis here is to go beyond this straightforward economic result of higher direct GHG emissions. Structural links between the pulp industry and the rest of the forest sector are expected to induce economic and carbon impact when recycling is more prevalent. These results will figure out whether the wood sector can alleviate the increased climate impact of the pulp industry.