<figure><img src="image_12.png" /><figcaption>Figure 2.4: Relative changes in demand for other wood products between **baseline** and **med** in 2030</figcaption></figure>

### 2.3.2 Impacts on other forest products

The structure of the model, depicted in Figure 2.2, shows the link between the pulp industry and the rest of the forest products sector. Wood-pulp is indeed a transformed product of industrial wood, also used for pannels manufacturing and energy purposes. This induces a competition for the resource (the primary product). For this reason we can expect an effect of increased recycling on those products. At a wider scale, industrial wood competes with round-wood primary products for the forest resource. We can also expect an impact on these products. However, results from the work of Buongiorno et al. (2003) lead us to suspect very small effects in terms of relative changes in supply and demand of wood products.

Table 2.4 shows indeed very small relative effects on wood products. Fuel-wood, which is in direct competition with the pulp industry for primary industrial wood, shows higher impacts (from -0,14% to 0,4%). On the other hand, sawnwood shows negligible impacts as it is a transformed product from roundwood and thus far from the pulp industry (see Figure 2.2).

We also need to assess the climate impact that derives from these economic impacts. It regroups material and energy substitution related to the use of wood products, as well
as emissions linked to forest exploitation (forest operations and transport). As emissions of forest operations are negligible compared to energy and material substitution, they are not displayed in Table 2.3⁶. While they remain low in terms of relative changes, these sequestration mechanisms can potentially offset additional emissions of pulp transformation. This is especially true for energy substitution, derived from the differences in fuel-wood usage. It does not compensate higher emissions from transformation detailed above, but when pulps are substitutes, they amount 10% of the emissions variation displayed for the pulp in industry (in Table 2.2). However, when pulps are complementary products, sequestration is lower, thus deepening the negative climate impact of pulp recycling.

<table><caption>Table 2.3: Changes in carbon substitution for other wood products (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></thead><tbody><tr><td colspan="4">Energy substitution (MtCO<sub>2</sub>eq)</td></tr><tr><td>η = 0, 2</td><td>-0,012 (-0,08%)</td><td>-0,019 (-0,14%)</td><td>-0,030 (-0,18%)</td></tr><tr><td>η = -0, 2</td><td>0,010 (+0,07%)</td><td>0,017 (+0,12%)</td><td>0,023 (+0,16%)</td></tr><tr><td>η = -1</td><td>0,033 (+0,23%)</td><td>0,055 (+0,39%)</td><td>0,072 (+0,51%)</td></tr><tr><td colspan="4">Material substitution (MtCO<sub>2</sub>eq)</td></tr><tr><td>η = 0, 2</td><td>-0,004 (-0,01%)</td><td>-0,006 (-0,02%)</td><td>-0,008 (-0,03%)</td></tr><tr><td>η = -0, 2</td><td>0,003 (+0,01%)</td><td>0,005 (+0,02%)</td><td>0,007 (+0,03%)</td></tr><tr><td>η = -1</td><td>0,009 (+0,04%)</td><td>0,016 (+0,06%)</td><td>0,022 (+0,09%)</td></tr></tbody></table>

### 2.3.3 Impacts on the forest resource

The specification of resource impacts relies first on inventoried and non-inventoried biomass (branches and roots), as well as wood products (however this last one is negligible in terms of absolute values). We see from Table 2.4 that in the case of complementary pulp products, effects on stocks exacerbate the negative climate effect of increased recycling. Substitution in contrast leads to higher sequestrated carbon with up to a 1,5% increase in the most optimistic case (high substitution $η = -1$ and scenario high).

<footnote n="6">⁶Unlike Table 2.2 where we show GHG emissions amounts, we produce in Table 2.3, 2.4 and 2.6 GHG substitution and sequestration, where a negative sign means more emissions, and a positive sign more mitigation.</footnote>
<table><caption>Table 2.4: Changes in carbon stocks (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></thead><tbody><tr><td colspan="4">Carbon sequestration pool (MtCO<sub>2</sub>eq)</td></tr><tr><td>η = 0,2</td><td>-0,143 (-0,2%)</td><td>-0,232 (-0,3%)</td><td>-0,310 (-0,4%)</td></tr><tr><td>η = -0,2</td><td>0,141 (+0,2%)</td><td>0,245 (+0,4%)</td><td>0,336 (+0,5%)</td></tr><tr><td>η = -1</td><td>0,453 (+0,7%)</td><td>0,781 (+1,1%)</td><td>1,027 (+1,5%)</td></tr></tbody></table>

Changes for the forest resource are also reflected in landscapes evolutions, as increased recycling has impacts in terms of forest management after some decades. Results for the volume of resource in Table 2.5⁷ are coherent with results on carbon stocks for the sector, with an increase when substitutes and a decrease when complements. We can notice that this increase (resp. decrease when $η = 0,2$) mainly concerns the broadleaved cover. Changes in forest areas are also noticed. While the overall forest cover remains stable in France with no evolution in total, the ratio of species evolves with more broadleaves (resp. less when $η > 0$) and less conifers (resp. more). These results are in relative values limited, reflecting the overall stability of the forest sector. However, these landscape impacts can lead to modifications in terms of ecosystem services from the forest sector, with an overall reduction in some cases (Gamfeldt et al., 2013).

<table><caption>Table 2.5: Changes in forest volumes and areas between baseline and med in 2070</caption><thead><tr><th></th><th>Broadleaves</th><th>Conifers</th><th>Total</th></tr></thead><tbody><tr><td colspan="4">Forest areas (Mha)</td></tr><tr><td>η = 0,2</td><td>-1,980 (-0,02%)</td><td>1,730 (+0,04%)</td><td>0 (0%)</td></tr><tr><td>η = -0,2</td><td>1,749 (+0,02%)</td><td>-1,595 (-0,04%)</td><td>0 (0%)</td></tr><tr><td>η = -1</td><td>5,853 (+0,07%)</td><td>-5,420 (-0,14%)</td><td>0 (0%)</td></tr><tr><td colspan="4">Forest volumes (Mm<sup>3</sup>)</td></tr><tr><td>η = 0,2</td><td>-6,318 (-0,20%)</td><td>-1,695 (-0,18%)</td><td>-9,106 (-0,19%)</td></tr><tr><td>η = -0,2</td><td>5,264 (+0,17%)</td><td>1,297 (+0,14%)</td><td>7,441 (+0,16%)</td></tr><tr><td>η = -1</td><td>17,637 (+0,57%)</td><td>5,121 (+0,55%)</td><td>26,012 (+0,56%)</td></tr></tbody></table>

<footnote n="7">⁷Note that we do not display results for lands covered with a mix of conifers and broadleaves.</footnote>
### 2.3.4 Global carbon impact on the forest sector

Results on sequestration described above combined with changes in GHG emissions give the total CO₂ balance of the forest sector. Reduced sequestration with complementary pulp products lead to important diminutions in the global balance, as we find a reduction between 0,299 and 0,660 MtCO₂eq/year (Table 2.6). On the other hand, with low substitution ($\eta$ = 0,2) we find an equilibrium between additional emissions and sequestration, with slightly positive values for the carbon balance, although negligible compared with the use of other values for elasticity $\eta$. This effect is indeed higher when we examine the optimistic case of a high substitution between pulps, up to 0,8% of the total carbon balance of the French forest sector for scenario high.

While these global results are relatively small for the scale of the forest sector, it shows the central role of substitutability and complementarity to understand the environmental impact of paper recycling. With complementarity, reduced recycling costs and thus increased production of pulp overall (as shown in Table 2.3) leads to a higher usage of the virgin resource, coupled with increased forest utilization. On the other hand, substitution between pulps leads to lower forest exploitation, then higher sequestration compensating emissions of the pulp industry. It is interesting to notice that the forest sector only alleviates an increased carbon impact of the recycling sector when added recycled products are substitutes to wood-pulp. Besides, this is true for both substitution mechanisms (Table 2.3) and sequestration mechanisms (Table 2.4).

<table><caption>Table 2.6: Changes in global GHG balance (compared with scenario baseline) - yearly average for 2013-2080</caption><thead><tr><th rowspan="2">Scenario</th><th colspan="3">Global carbon balance (MtCO<sub>2</sub>eq)</th></tr><tr><th>low</th><th>med</th><th>high</th></tr></thead><tbody><tr><td>η = 0,2</td><td>-0,299 (-0,3%)</td><td>-0,489 (-0,5%)</td><td>-0,660 (-0,6%)</td></tr><tr><td>η = -0,2</td><td>0,027 (+0,03%)</td><td>0,041 (+0,04%)</td><td>0,050 (+0,05%)</td></tr><tr><td>η = -1</td><td>0,390 (+0,4%)</td><td>0,660 (+0,6%)</td><td>0,857 (+0,8%)</td></tr></tbody></table>
## 2.4 Conclusion

With this work, we contribute to the already existing literature focusing on recycling of wood products. The question of whether recycling has a positive or negative carbon impact is crucial when issues relative to climate change and circular economy are more and more prominent. Based on the present situation of the French forest sector, we examine what would be the impact of increased recycling, simulated through a cost reduction in the sector. The use of FFSM enables us to have a broader perspective for the climate impact. The market module and the recursive structure expands a standard LCA analysis with both economic impacts on other wood products sectors and variations in carbon substitution and sequestration.

First, we show the expected impact of recycling on wood-pulp industry. Expected qualitative results regarding the complementarity or substitution of both products are found, with respectively a positive or negative impact on demand. Beyond this economic outcome, GHG emissions of the pulp sector are always increasing, thus raising the question of their possible mitigation through the rest of the forest sector.

Second, we find small effects on other wood products. This is expected as the economic model combines small price-elasticities of products competing for timber resources and small cross-elasticities between virgin and recycled. The model produces small changes for transformed products from industrial wood (such as fuel-wood and pannels), with a higher demand when pulp recycling is a substitute (and vice versa when complements). However this effect is negligible for other wood products such as sawn wood and plywood. This involves small effects on carbon substitution mechanisms (material and energy).

Finally, our study of global carbon sequestration highlights small relative effects overall (between 0, 4% and 0, 8% in the additional carbon balance of the sector). However in terms of absolute evolution of the net sequestration potential, we observe a positive impact when considering substitution in the pulp industry, and a negative impact when considering complementarity. The already existing literature on the econometrics of paper and cardboard recycling shows that strongly substitute products would be a very optimistic scenario, while
low complementarity is a more realistic one. Our results show that the development of re-
cycling (and more generally of a circular economy) relies on the substitutability of natural
capitals to be environmentally efficient (here virgin and recycled capitals)(Ayres, 2007).

Our work has several implications in terms of public policies. First, in a global situa-
tion than leans toward weak complementarity between virgin and recycled paper, 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. It then appears than sobriety in the use of paper
and cardboard is an inevitable objective to reduce GHG emissions from the sector. Second,
there is a strong heterogeneity within the paper and cardboard industry: some products im-
ply complementarity between recycled and virgin material, while other ones allows stronger
substitutability. It follows that the sobriety objectives should be oriented toward products
where complementarity is stronger. In contrast, for products where substitution is possible,
the objective of sobriety may be less stringent and the use of recycling produce some climate
benefits.

This sectoral analysis on the economic and carbon impacts of paper and cardboard
recycling should be pursued with further investigation, including sensitivity analysis of other
parameters in the model, such as recovery rates and the introduction of policies promoting
recycling or a lower carbon footprint.
# CHAPTER 3

## AN ASSESSMENT OF THE EUROPEAN REGULATION ON BATTERY RECYCLING FOR ELECTRIC VEHICLES

* * *

Electric mobility is deemed necessary to cut CO2 emissions in the transport sector but the industrial and environmental impacts of lithium-ion battery manufacturing are controversial. 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. This chapter discusses the conditions required for that obligation to be fulfilled. Battery lifetimes are the most crucial parameters for recycling potential. This result could jeopardize avenues for extending battery lifetimes such as second-life battery usage. We recommend lower thresholds to improve the regulation credibility and the integration of other objectives in the regulation in order to address the lifetime of batteries.

* * *
## 3.1 Introduction

In a recent proposal for a regulation, the European Commission (EC) has introduced re-cycling as a key element for the development of an industry for electric vehicle (EV) and especially the battery industry (European Union: European Commission, 2020). It is supposed to be a central part of the European Union (EU) industrial strategy covering economic, social and environmental goals. It establishes a legislative framework for a more sustainable life-cycle of lithium-ion (Li-ion) batteries. One of its main initiative is the introduction of minimum thresholds for recycled material sourcing in new batteries.

This chapter aims to assess the conditions under which these thresholds may physically be reached. Our analysis of these conditions follows two steps. First, we set up 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 share of demand that can be covered by recycled materials in new batteries will be reduced by higher demand growth of electromobility, lower recycling efficiencies, longer lifetimes of batteries and technological changes that increase material use. Second, we estimate the sensitivity of recycling potentials to these mechanisms through a quantitative model. We set up a dynamic material flow analysis, calibrated using data from different reports from the International Energy Agency (IEA) and from the academic literature. For all materials, we find that battery lifespan is the most sensitive parameter driving the thresholds proposed by the EC. On the contrary, the growth rate of demand in batteries appears to bring little sensitivity. Regarding battery technological change, we find that only the recycling potential of cobalt is likely to be significantly impacted. Furthermore, the improvement of recycling processes is crucial to guarantee the sourcing in recycled lithium. Overall, our results indicate that the thresholds of recycled materials proposed by the EC may be difficult to comply with. At the light of this result, we discuss the eventual conflict in environmental goals that these minimum thresholds could cause. For instance, we highlight a trade-off between this
instrument and the development of a second life for batteries.

More generally, the proposal of regulation from the EC is motivated by the ongoing transformation of the automobile industry. Indeed, the penetration of EVs has grown steadily for a few years, with EU market shares increased from 0.4% in 2014 up to 4.6% in 2020. EVs have been advocated as one of the main technological option to decarbonize the transport sector and to decrease urban air pollution. This led government to provide a large policy support for electromobility. In parallel, Li-ion batteries - the central components of EVs - have been subject to remarkable technological progress. As a result, their costs have fallen from USD 1100 per kilowatt-hour (kWh) in 2010 to USD 156/kWh in 2020 (Nykvist et al., 2019; IEA, 2020). In this context, the growing importance of the Li-ion batteries also brings sustainability and strategic concerns. These include the carbon footprint of the life-cycle and waste management (Dai et al., 2019). These also include concerns regarding the availability of raw material, social impacts of the metal extraction as well as supply chain risk management (Hache et al., 2019; IEA, 2021b). Since 2008, the EU has taken into account this component of its industrial sectors. It mainly focuses on sustaining the access to those materials in the EU and worldwide, and developing recycling as well as resource efficiency. To address these issues, the EU has developed several initiatives, such as the European Battery Alliance<fnref n="1" /> and the Raw Materials Initiative.<fnref n="2" /> The proposal of the EC intends to bring a legislative framework to these initiatives.

Beyond the discussion of the regulation proposal of the European Commission, we contribute to two main strands of literature. First, we relate to the literature exploring the material usage for the development of electromobility. Ballinger et al. (2019) explore the risks for the EV sector related to material supply, in particular for graphite, lithium and cobalt used in batteries. Xu et al. (2020) and Sato and Nakata (2019) propose a long term analysis of the material demand for batteries. While the former show that the role of recy-

<footnote n="1">¹The European Battery Alliance is a consortium of industrial actors that aims at developing a European battery industry (https://www.eba250.com/).</footnote>

<footnote n="2">²https://ec.europa.eu/growth/sectors/raw-materials/policy-and-strategy-raw-materials_fr</footnote>
cling remains minor while the market is in expansion, the latter find between 30 and 50% of
mineral being supplied by recovery in 2035 in Japan. More generally, some articles address
the broad topic of mineral criticity in the energy transition. A thorough literature study
from Watari et al. (2020) covers the criticality of a wide range of material up to 2050. They
highlight the lack of both socio-environmental analysis and inclusion of recycling strategies.
While doing so, they link metal availability challenges to the key technologies of the energy
transition. In the same spirit, Seck et al. (2020) examine the importance of copper use in the
transition. They test different scenarios in a world model representing transport and energy
sectors and find that the increase in copper demand could lead to an important stress on
known mining resources. They use the same model for lithium and cobalt demand (Hache
et al., 2019; Seck et al., 2022). While they do not find the same stress on mineral resources
as for copper, they do highlight the risks caused by geographically concentrated resources,
geopolitical strategies, lack of investments and environmental consequences of extraction.
Helbig et al. (2018) also address risks in the supply chain of material used in Li-ion bat-
teries. They compute a level of risk for a range of battery technologies, depending on their
mineral composition. From the grey literature, reports from the IEA (IEA, 2020, 2021a)
present the global trends of EV markets. They also tackle issues of metal scarcity in the
energy transition, highlighting the important role that could be played by recycling in the
following years in order to ensure the sustainability of material supply chains IEA (2021b).
We contribute to this literature by disentangling the key-mechanisms of recycling potentials
and by estimating the relative magnitude of those effects. We also focus on the European
context.

Second, we relate to the literature on the aspects of the end of life of EV batteries. A
review from Lv et al. (2018) lists the different recycling technologies for lithium, and discusses
technological, environmental and economic perspectives for pyro-metallurgical and hydro-
metallurgical processes. Other authors have investigated on the possibility of extending
battery lifetimes through the re-utilization of used batteries. Such second life of batteries
could indeed provide stationary storage for the power grid and thus reducing their costs (Few
et al., 2018; Gur et al., 2018; Martinez-Laserna et al., 2018; Wu et al., 2020). We contribute
to this literature by showing potential incompatibilities when considering at the same time
recycled sourcing of materials and other sustainability objectives such as the reuse of EV
batteries.

The rest of this chapter is organized as follows. Section 3.2 presents the proposal of the EC and its context. Section 3.3 gives qualitative insights on the drivers of the level of recycled materials content from used batteries. Section 3.4 and 3.5 respectively present the methods and the results of the quantitative analysis. Sections 3.6 concludes.

## 3.2 Proposal of the European Commission

The current EU regulatory framework on batteries is established by a directive from the
European Commission in 2006 (Council of European Union, 2006).<fnref n="3" /> This directive aims at
providing common rules across the EU market that would limit environmental impacts of
battery wastes.<fnref n="4" /> It distinguishes three kinds of battery types to be regulated: portable (elec-
tronic equipment), automotive (for starting, lighting and ignition) and industrial (traction
for vehicles and other industrial applications). The directive applies from manufacturing
to end of life. It enforces the extended producer responsibility principle. More precisely,
it defines targets for the collection of portable battery wastes,<fnref n="5" /> and maximum contents of
cadmium and mercury in new batteries.

In the late 2020, the EC has proposed a new regulation for the regulation of batteries that
would replace the 2006 directive (European Union: European Commission, 2020).<fnref n="6" /> Indeed,
the disruption of Li-ion batteries fueled by the development of electromobility has outdated

<footnote n="3">³The 2006 directive is not limited to Li-ion batteries, but the whole battery sector.</footnote>

<footnote n="4">⁴Targeted sources of pollution involved cadmium, mercury and lead</footnote>

<footnote n="5">⁵25% by 2012, 45% by 2016</footnote>

<footnote n="6">⁶Unlike directives, regulations are automatically applied in EU members, without any transposition into national law. It results in a uniform effect for all the Union, in line with the objective of creating a coherent industrial sector at EU scale. See https://europa.eu/european-union/law/legal-acts_en for more details.</footnote>
made the former directive. Within the context of EU's Green New Deal, this proposal joins other measures pursuing the development of the sector in Europe, such as the initiative of the European Battery Alliance. It aims at providing a legislative framework that would gather industrial, strategical and environmental objectives. The proposal of the EC defines several environmental objectives that spread on the period 2025-2035. These objectives comprise the development of recycling, the limitation of the carbon footprint and the transparency of the supply chain.

<table><caption>Table 3.1: Targets on battery recycling of the 2020 EC proposal</caption><thead><tr><th></th><th></th><th>Cobalt</th><th>Copper</th><th>Lead</th><th>Lithium</th><th>Nickel</th></tr></thead><tbody><tr><td rowspan="2">Minimum recycled content in new</td><td>2030</td><td>12%</td><td>NA</td><td>85%</td><td>4%</td><td>4%</td></tr><tr><td>2035</td><td>20%</td><td>NA</td><td>85%</td><td>10%</td><td>12%</td></tr><tr><td>batteries</td><td>2026</td><td>90%</td><td>90%</td><td>90%</td><td>35%</td><td>90%</td></tr><tr><td>Recovery rate from scrapped batteries</td><td>2030</td><td>95%</td><td>95%</td><td>95%</td><td>70%</td><td>95%</td></tr></tbody></table>

This chapter focuses on the objectives regarding recycling. The main elements from the proposal establish minimum recycled contents for several materials in new batteries. These constraints only apply to cobalt, lithium and nickel for the Li-ion technology, and lead for other battery technologies. Minimum thresholds are defined in 2030 and 2035. Table 3.1 displays them. The proposal also defines mandatory target levels of recovered materials from scrapped batteries for cobalt, copper, lead, lithium and nickel, as displayed in Table 3.1. It is expressed in shares of these minerals as well as on total weight of the batteries.<fnref n="7" />

To the best of our knowledge, most of the proposal and in particular the minimum thresholds on recycling come from a feasibility study ordered by the EC (European Union: European Commission, Oeko Institut., Ramboll. and Umweltbundesamt., 2021; European Union: European Commission, Trinomics. and Oeko Institut e.V., 2021). They present a number of measures to improve the 2006 directive. They are based on a modelling assessment complemented with interviews with stakeholders. To compute the thresholds, authors set up a material flow model to assess the availability of minerals to be recycled in the upcoming

<footnote n="7">⁷After 2025, 75% of lead batteries, 65% of Li-ion batteries and 50% of other battery waste.</footnote>
years. By doing so, the proposed thresholds later used by the EC are set to match available
materials for recycling. It appears that the proposed minimum thresholds of recycled con-
tents match high estimates considering the availability of recycled material.<fnref n="8" /> Furthermore,
the impact of different parameters stemming from the technological and economical context
given by IEA reports are not quantitatively discussed in the report (IEA, 2020, 2021a,b). In
this chapter, we propose to fill this gap by investigating in particular the impact of demand
growth, lifespan of batteries, recycling efficiency and technological changes.

## 3.3 Qualitative Insights

Having clarified the European context, we now provide the intuitions that will serve the rea-
soning presented in the quantitative sections which follow. Assessing the minimum thresh-
olds of incorporated recycled materials given by the EC proposal can be done by computing
the maximum ratio of recycled waste material over demand (RMD ratio hereafter).<fnref n="9" /> As in
the impact studies that inspired the EC proposal, such ratios can be estimated with complex
dynamic material flow analysis, which requires a large number of hypothesis and data. We
argue that a simplified version of such models is useful to draw the essential drivers of this
dynamics.

We compute the RMD ratio in a two-period model ($t = 0, 1$). Each period, a quantity $Q_t$ of batteries for electric vehicles are produced (in kWh), given a technology mix that requires an average intensity $w_{k,t}$ for each material $k$ (in kg/kWh). At period 1, a quantity $P_{0,1}Q_0$ of battery is scrapped. These wastes are recovered and recycled with an overall efficiency $\epsilon_{k,1}$. For each material $k$, we define $RMD_{k,1}$ as the ratio of recycled material from battery wastes over demand at period 1. Such ratio can be decomposed as follows:

<footnote n="8">⁸Note a lower recovery rate for lithium. Indeed, the current state of lithium recycling technologies make it harder to recover it (a baseline of 10% in 2018 according to European Union: European Commission, Trinomics. and Oeko Institut e.V. (2021), matching other studies such as (Hache et al., 2019) mentioning low effective rates), thus involving lower recovery targets than nickel and cobalt.</footnote>

<footnote n="9">⁹Such ratio corresponds to the maximum amount of material that can be recycled from yearly scrapped batteries and incorporated in new manufactured batteries.</footnote>
$$RMD_{k,1} = \underbrace{P_{0,1}}_{\text{Scrapping probability}} \underbrace{\frac{Q_0}{Q_1}}_{\text{Demand change}} \underbrace{\frac{w_{k,0}}{w_{k,1}}}_{\text{Technological change}} \underbrace{\epsilon_{k,1}}_{\text{Recycling efficiency}} \quad (3.1)$$

Such decomposition, shown in equation (3.1) involves four terms, two of which being material-specific, while the other two are material-neutral.

* $P_{0,1}$ is the fraction of battery turned into waste and scrapped between the two periods. It is mainly linked to the average lifetime of batteries through the stochastic process of battery degradation. In particular, this fraction is reduced with higher average lifetimes, which can be caused by technological improvements or by the re-utilisation of used battery for other purpose.

* ratio $Q_0/Q_1$ of battery capacity demand at period 0 over period 1 relates to the evolution of demand for battery capacity. The larger is the development of electromobility,<fnref n="10" /> the lower this ratio is. This ratio is supposed to be large during the uptake of electromobility, and converge to one as fossil fuel cars are phased out.

* ratio $w_{k,0}/w_{k,1}$ in material intensities $k$ in period 0 over period 1 relates to the technological change of batteries. Such ratio decreases as battery technology requires lesser quantities of material $k$. Material footprint may decrease as a whole and there may be material substitutions, as displayed later when discussing our data (see Tables 3.3 and 3.4).

* $\epsilon_{k,1}$ is the fraction of recovered material from battery wastes. It encompasses both the collection of battery wastes and the efficiency of recycling processes. Contrary to the other components, this parameter may be the most crucial in economic terms as

<footnote n="10">¹⁰Such development includes both the number of vehicles and battery size. Battery size will increase with the autonomy of electric vehicles, and should converge once autonomy of electric and thermal vehicles become close.</footnote>
collecting a high share of disposed batteries or recycling at higher rates can involve
very large marginal costs on recycling. Also, it has been indicated that it can be
difficult to reach high recycling efficiencies for some material, such as lithium, with the
same recycling process. One recycling process can foster the recovery of a material at
the expense of another (Xu et al., 2020).

This decomposition can be illustrated with a numerical example. Let's consider a 10
years time lapse. During this period, consider that demand quintuples ($Q_0/Q_1 \approx 0.2$),
material intensities remain steady $w_{k,0}/w_{k,1} \approx 1$, recycling is almost perfect ($\epsilon_k \approx 1$), and
40% of batteries produced in year 0 are scrapped in year 10 ($P_{0,1} \approx 0.4$). In this case, the
ratio $RMD_{k,1} \approx 0.08$. Such ratio is in the range of the minimum thresholds given by the
EC proposal.

## 3.4 Quantitative Analysis: Methods

### 3.4.1 Model

The previous section exhibited the main drivers for the ratios of recycled material over de-
mand. This following section aims at quantitatively estimating these levels and confronting
them to the EC targets. We model the material flow of Li-ion batteries in the EU at both
extremities of the lifecycle with yearly periods corresponding to 2020-2035: the demand of
batteries (by extension related to the market for EVs) and the waste aggregation following
the end-of-life. Note that this model can be seen as a sophisticated extension of the de-
composition previously discussed. The underlying variable for the flow of batteries is the
aggregate flow $F_{k,t}$ (in kg) for a specific material $k$ at year $t$:

$$F_{k,t} = \sum_{i} \sum_{j} Q_{i,t} m_{j,t} w_{j,k} \qquad (3.2)$$
This flow is summed for all types of EVs *i* and all technologies of Li-ion batteries *j*. $Q_{i,t}$ is the aggregated capacity (in kWh) from EVs of type *i* sold each year, and is computed as the product of the number of vehicle of type *i* with their related average battery size. $m_{j,t}$ is the market share of a battery technology and $w_{j,k}$ is the material *k* intensity of a specific technology *j* (kg/kWh).<fnref n="11" /> The definition of indexes {$i, j, k, t$} is summed up in Table 3.2. Differences between Li-ion battery technologies mainly stem from the conception of the cathode.

<table><thead><tr><th>Index</th><th>Name</th><th>Description</th></tr></thead><tbody><tr><td><i>i</i></td><td>EV types</td><td>cars BEV, cars PHEV, vans BEV, vans PHEV<sup>a</sup></td></tr><tr><td><i>j</i></td><td>Li-ion battery technology</td><td>ASSB, LMO, LFP, NMC333, NMC532, NMC622, NMC811, NCA+<sup>b</sup></td></tr><tr><td><i>k</i></td><td>material</td><td>copper, lithium, nickel, manganese, cobalt, graphite</td></tr><tr><td><i>t</i></td><td>time</td><td>2020 to 2035</td></tr></tbody></table>

<caption>Table 3.2: Indexes of the model</caption>

Each battery (*i*, *j*) in use has a probability $P_{i,j,\tau,t-\tau}$ to be scrapped, where $\tau$ is the year of production of the battery, hence $t-\tau$ is the age of the battery. This allows us to compute for each year *t* the flow of waste from production year $\tau$ for material *k* when no decay or recycling takes place:

$$B_{k,\tau,t} = \sum_{i} \sum_{j} V_{i,\tau} b_{i,\tau} m_{j,t} w_{j,k} P_{i,j,\tau,t-\tau} \quad (3.3)$$

The probability $P_{i,j,\tau,t-\tau}$ follows a Weibull distribution calibrated with minimum, maximum and most likely lifespans of Li-ion batteries. We use a Weibull distribution as the

<footnote>a BEV: battery electric vehicle; PHEV: plug-in hybrid electric vehicle</footnote>

<footnote>b ASSB: solid-state; LMO: Lithium-Manganese-Oxyde; LFP: Lithium-Iron-Phosphate; NMC: Nickel-Manganese-Cobalt; NCA: Nickel-Cobalt-Aluminium</footnote>

<footnote n="11">¹¹Contrary to the qualitative analysis, material intensity $w_{j,k}$ is not time related, as it relates to a specific technology whose market share will evolve through time.</footnote>
literature shows that it fits well the lifecycle of durable goods (Xu et al., 2020; Elshkaki, 2005; Melo, 1999; Spatari et al., 2005).<fnref n="12" /> Finally, we can compute for each year *t* the aggregated waste flow of material *k*:

$$W_{k,t} = \sum_{\tau=0}^{t-1} (B_{k,\tau,t} - B_{k,\tau,t-1}) \quad (3.4)$$

Finally, the ratio of recycled material over demand (RMD ratio) for each material *k* can be computed as follows:

$$RMD_{k,t} = \frac{\epsilon_{k,t} W_{k,t}}{F_{k,t}} \quad (3.5)$$

with $\epsilon_{k,t}$ the efficiency of the material recovery process. It includes physical collection of used batteries from used cars as well as the efficiency of recycling processes.<fnref n="13" />

Following the decomposition from the qualitative analysis, we elaborate several scenarios that aim at investigating the influence of demand evolution, battery lifespans, technological change and recycling efficiencies.

### 3.4.2 Data

The main source of data for the calibration of the model comes from recent IEA reports (IEA, 2020, 2021a). Projections of the EV market in 2025 and 2030 comes from (IEA, 2020). We use a quadratic interpolation to get the evolution of the total EV battery demand (in kWh) for the range 2020-2035, with the two different scenarios studied in IEA reports. Battery sizes are also calibrated using these reports. The Stated Policies Scenario (STEPS) relies on already existing and announced policies and their expected consequences on the development of electromobility. On the other hand, the Sustainable Development Scenario (SDS) is more

<footnote n="12">¹²We also computed our model with a gamma function, with very little difference in the results that do not alter the policy implications.</footnote>

<footnote n="13">¹³Due to the lack of data, we neglect material losses from the use of batteries.</footnote>
ambitious and built in order to meet the Paris Agreement objectives. The extension of the
demand curve up to 2035 is made with the hypothesis that the EV fleet does not yet reach
its final deployment. Figure 3.1 shows the projections for battery demand. Note that a
growing divergence between scenarios occurs after 2025, resulting for the SDS scenario in a
twice larger demand in 2030 than for STEPS, and even more for 2035. Comparing these
two scenarios allows to assess the effect of demand dynamics, which was the first component
of the decomposition of the qualitative analysis.

<figure><img src="image_13.png" /><figcaption>Figure 3.1: Projected EV batteries growth for two demand scenarios (extrapolated from IEA)</figcaption></figure>

Material intensities of each battery cell technology $w_{j,k}$ are taken from IEA reports and (Xu et al., 2020) and shown on Table 3.3. We linearly interpolate projected market shares $m_{j,t}$ of battery technologies from the IEA (2021b) as shown in Table 3.4, given for up to 2040, consisting in the main technological scenario.

The projections of the IEA in Table 3.4 show a technological change that shifts from cobalt/manganese-based toward nickel-based batteries. This is caused by the phase-out of
NMC333 batteries in favor of NMC811 with higher nickel incorporation rates. Another key feature of this technological change is the rise of solid-state batteries (ASSB), with an important reduction in material use among which cobalt and nickel.

<table><caption>Table 3.3: Battery Composition for each battery type. Values are given in kg/kWh.</caption><thead><tr><th>Battery Type</th><th>Lithium</th><th>Nickel</th><th>Cobalt</th><th>Manganese</th><th>Graphite</th><th>Copper</th></tr></thead><tbody><tr><td>ASSB</td><td>0.174</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.293</td></tr><tr><td>LMO</td><td>0.107</td><td>0.000</td><td>0.000</td><td>1.400</td><td>0.867</td><td>1.040</td></tr><tr><td>LFP</td><td>0.093</td><td>0.000</td><td>0.000</td><td>0.000</td><td>1.027</td><td>1.067</td></tr><tr><td>NMC811</td><td>0.093</td><td>0.587</td><td>0.080</td><td>0.067</td><td>0.907</td><td>0.853</td></tr><tr><td>NMC622</td><td>0.107</td><td>0.533</td><td>0.187</td><td>0.160</td><td>0.907</td><td>0.880</td></tr><tr><td>NMC532</td><td>0.160</td><td>0.507</td><td>0.200</td><td>0.280</td><td>0.907</td><td>0.907</td></tr><tr><td>NMC333</td><td>0.160</td><td>0.347</td><td>0.360</td><td>0.307</td><td>0.907</td><td>0.933</td></tr><tr><td>NCA+</td><td>0.107</td><td>0.773</td><td>0.027</td><td>0.000</td><td>0.907</td><td>0.853</td></tr></tbody></table>

<table><thead><tr><th>Scenario</th><th>Historic</th><th colspan="2">main</th><th colspan="2">status quo</th><th colspan="2">ASSB+</th><th colspan="2">LFP+</th></tr><tr><th>Year</th><th>2020</th><th>2030</th><th>2040</th><th>2030</th><th>2040</th><th>2030</th><th>2040</th><th>2030</th><th>2040</th></tr></thead><tbody><tr><td>ASSB</td><td>0</td><td>0.7</td><td>31</td><td>0</td><td>0</td><td>1.4</td><td>62</td><td>0.4</td><td>21.8</td></tr><tr><td>LMO</td><td>2</td><td>0</td><td>0</td><td>2</td><td>2</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>LFP</td><td>6</td><td>10</td><td>9</td><td>6</td><td>6</td><td>9.9</td><td>5</td><td>40</td><td>36</td></tr><tr><td>NMC811</td><td>15</td><td>60</td><td>53</td><td>15</td><td>15</td><td>59.6</td><td>29.2</td><td>40</td><td>37.3</td></tr><tr><td>NMC622</td><td>29</td><td>11</td><td>4</td><td>29</td><td>29</td><td>10.9</td><td>2.2</td><td>7.3</td><td>2.8</td></tr><tr><td>NMC532</td><td>22</td><td>7.6</td><td>0</td><td>22</td><td>22</td><td>7.5</td><td>0</td><td>5.1</td><td>0</td></tr><tr><td>NMC333</td><td>6</td><td>1</td><td>0</td><td>6</td><td>6</td><td>1</td><td>0</td><td>0.7</td><td>0</td></tr><tr><td>NCA+</td><td>20</td><td>9.7</td><td>3</td><td>20</td><td>20</td><td>9.7</td><td>1.6</td><td>6.5</td><td>2.1</td></tr></tbody></table>

Description of technological scenarios:

- main: Main scenario given used in IEA reports

- status-quo: market shares stay at the 2020 level

- ASSB+: Strong breakthrough of solid-state batteries (doubled market share)

- LFP+: Strong breakthrough of LFP batteries in 2030-2035 (four fold increase)

<caption>Table 3.4: Description technological scenarios with their respective market shares (in %).</caption>

In order to assess the impact of the evolution of material intensities, due to technological change, alternative scenarios are developed and summarized in Table 3.4, all considering a demand from the STEPS scenario. First, in scenario status-quo, market shares are maintained at their 2020 levels. Within it, battery manufacturing remains intensive in cobalt. Second, the scenario ASSB+ simulates a large penetration of ASSB batteries. It leads to
an important decrease in material use, especially nickel, manganese and cobalt, due to the technological breakthrough. Third, the scenario LFP+ is built to show an important market share of LFP batteries. It shows lower needs for nickel, manganese and cobalt. For the last two scenarios, proportions for other market shares are kept the same.

The Weibull distribution function is calibrated with the most likely lifetime for the battery, and minimum/maximum set at 99.7% of the distribution. There is a lack of knowledge regarding the lifespan of EV batteries, as the market is still at its early stage of development. For the specific case of electric vehicle, we find shorter lifespans in the literature, with 8 to 10 years (Casals et al., 2017, 2019). Lifespans of batteries are often high estimates in studies, with the example of Xu et al. (2020) who take an average of 15 years based on generic studies on automobiles. Our main scenario follows a most likely lifetime of 10 years. For sensitivity purposes, we also test scenarios with shorter lifespans, as described in Table 3.5, as well as longer as some emerging uses of batteries such as second life could postpone the year of recycled material availability (Martinez-Laserna et al., 2018). We use demand from the STEPS scenario for these ones.

<table><caption>Table 3.5: Description of scenarios on lifespans and other relevant parameter for the Weibull function of batteries (in years)</caption><thead><tr><th>Scenario</th><th>Minimum</th><th>Most likely</th><th>Maximum (99,7%)</th></tr></thead><tbody><tr><td>shorter lifetime</td><td>1</td><td>8</td><td>18</td></tr><tr><td>main</td><td>1</td><td>10</td><td>20</td></tr><tr><td>longer lifetime</td><td>1</td><td>12</td><td>22</td></tr></tbody></table>

Recycling efficiencies $\epsilon_{k,t}$ are set with the targets of the proposal of the EC (Table 3.1), and linearly interpolated between 2020 and 2026, and between 2026 and 2030, and are constant after 2030. Note that the EC aims at a recovery rate for lithium going from 35% in 2026 to 70% in 2030, while it is around 10% in 2020 (European Union: European Commission, Trinomics. and Oeko Institut e.V., 2021), hence an optimistic seven-fold increase in a decade. We also test a scenario low efficiency, with demand from STEPS, where we maintain recycling efficiencies at their current 2020 levels, with 80% efficiency for both cobalt
and nickel, and 10% for lithium.<fnref n="14" />

## 3.5 Quantitative Analysis: Results

This section presents the results of the quantitative analysis. It focuses on the computation of the ratios of recycled materials over demand in new batteries (RMD ratios) up to 2035. First, it presents the evolution of these shares for each material for the two IEA scenarii *STEPS* and *SDS*. Then, it successively shows the impacts of battery lifespans, battery technological change and recycling efficiencies on RMD ratios.

### 3.5.1 Evolution of RMD Ratios and Effect of Demand Growth

<figure><img src="image_14.png" /><figcaption>Figure 3.2: Ratios for Li, Ni and Co (policy objectives in dashed lines)</figcaption></figure>

Evolutions of RMD ratios are displayed for lithium, nickel and cobalt on Figure 3.2. For both demand scenarios, all RMD ratios are increasing. It indicates that the amount of available waste that can get recycled grows faster than the demand in new batteries. Indeed, demand grows at a fairly constant rate, while available wastes grow faster due to the specificity of the degradation process, where the probability of becoming wastes grows up to the most likely lifetime, around ten years here, then decreases.

<footnote n="14">¹⁴As indicated as the 2020 baseline for material recovery rate in (European Union: European Commission, Trinomics. and Oeko Institut e.V., 2021).</footnote>
Figure 3.2 also displays EC minimum thresholds for 2030 and 2035. These targets are only partially met. Regarding lithium, targets are only reached in 2030 under the STEPS scenario. It results from a combination of low recycling efficiency compared to other materials (70% after 2030) and the use of lithium intensive technologies. In the case of nickel, targets are met under STEPS scenario for both 2030 and 2035 targets, while under SDS they are barely met in 2030 and below the objective in 2035. Cobalt targets, which are much higher than for nickel and lithium, are in both cases not reached in 2030 and 2035, despite the early shift from cobalt/manganese to nickel. This transition makes relatively high amounts of cobalt available later in 2035 thanks to a decreasing demand, which explains that targets are almost reached for the STEPS scenario.

The comparison of STEPS and SDS scenarios highlights a conflict between a faster growth of electromobility and material sourcing from recycling. Faster demand growth in minerals makes it harder to achieve relatively high amounts of recycled inputs in new batteries. This indicates that while demand grows much faster in the SDS scenario than in STEPS, the difference between waste production growth is less significant.

### 3.5.2 Effect of the Battery Lifespan

<table><caption>Table 3.6: 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.</caption><thead><tr><th rowspan="2">Lifetime</th><th colspan="2">Lithium</th><th colspan="2">Nickel</th><th colspan="2">Cobalt</th></tr><tr><th>2030</th><th>2035</th><th>2030</th><th>2035</th><th>2030</th><th>2035</th></tr></thead><tbody><tr><td>8 years</td><td><strong>0,10</strong></td><td><strong>0,14</strong></td><td><strong>0,12</strong></td><td><strong>0.24</strong></td><td><strong>0.16</strong></td><td><strong>0.29</strong></td></tr><tr><td>10 years</td><td><strong>0.05</strong></td><td>0.09</td><td><strong>0.06</strong></td><td><strong>0.16</strong></td><td>0.08</td><td><strong>0,20</strong></td></tr><tr><td>12 years</td><td>0.02</td><td>0.06</td><td>0.03</td><td>0.09</td><td>0.04</td><td>0.13</td></tr></tbody></table>

We perform a sensitivity analysis on the most likely lifespan of batteries, according to Table 3.5. Results are shown in Tables 3.6 for the STEPS demand scenario. RMD ratios are very sensitive to average lifetime changes. For a lifespan up to 12 years, available metals
for recycling barely reach 4% in 2030 for all metal inputs. Indeed, longer lifetimes imply lower waste rates on the short run, as secondary materials are later available for recycling.

Oppositely, with shorter lifespans (8 years), meaning that batteries are more often replaced, EC targets can more easily be met.<fnref n="15" /> Hence, the impact of battery lifespans seems much larger than the effect of demand growth, while differences between *STEPS* and *SDS* scenarios were up to a factor two impact on the demand. In the case of lifespans, a 20% increase or decrease (from 12 to 8 years) leads to much larger differences in RMD ratios.

This result has implications regarding the development of second life of batteries. In this scenario, EV batteries are being refurbished when they do not meet EV power standards and are used for stationary storage, which could increase the economic value of batteries (IEA, 2020; Martinez-Laserna et al., 2018). A well developed second-life sector significantly extends battery lifetimes, and therefore limits the availability of recycled materials for reaching their sustainability objectives. Furthermore, lifespans of batteries could be increased a lot more than 12 years according to Martinez-Laserna et al. (2018), with technical requirements and economic profitability being the main drivers of life durations. However, this demand for second-life should be nuanced as stationary energy storage is expected to be a minor share of the overall EU demand in capacity in the following years (Martinez-Laserna et al., 2018).

### 3.5.3 Effect of Technological Change

<table><caption>Table 3.7: 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.</caption><thead><tr><th rowspan="2">Scenario</th><th colspan="2">Lithium</th><th colspan="2">Nickel</th><th colspan="2">Cobalt</th></tr><tr><th>2030</th><th>2035</th><th>2030</th><th>2035</th><th>2030</th><th>2035</th></tr></thead><tbody><tr><td><em>status-quo</em></td><td><strong>0.04</strong></td><td><strong>0.10</strong></td><td><strong>0.06</strong></td><td><strong>0.13</strong></td><td>0.06</td><td>0.13</td></tr><tr><td><em>main</em></td><td><strong>0.05</strong></td><td>0.09</td><td><strong>0.06</strong></td><td><strong>0.16</strong></td><td>0.08</td><td><strong>0.20</strong></td></tr><tr><td><em>ASSB+</em></td><td><strong>0.05</strong></td><td>0.09</td><td><strong>0.06</strong></td><td><strong>0.19</strong></td><td>0.08</td><td><strong>0.24</strong></td></tr><tr><td><em>LFP+</em></td><td><strong>0.05</strong></td><td><strong>0.10</strong></td><td><strong>0.08</strong></td><td><strong>0.18</strong></td><td>0.11</td><td><strong>0.25</strong></td></tr></tbody></table>

<footnote n="15">¹⁵Here we examine this effect *ceteris paribus*, meaning we do not consider an impact of lifespans on demand growth of batteries.</footnote>
After having discussed the influence of demand evolution and battery lifespans on the different RMD ratios, this section discusses alternative technological scenarios. Detailed in Table 3.4, each alternative scenario represents a characteristic trend, with demand following the *STEPS* scenario. Results show little differences in RMD ratios across scenarios as soon as batteries become less cobalt-intensive. This contrasts with the large sensitivity shown by battery lifespans.

As stated in the qualitative section, the effect of technological change is heterogeneous between materials. Indeed, all RMD ratios for lithium (resp. nickel) are in the range 0.04-0.05 (resp. 0.06-0.08) in 2030 and 0.09-0.10 (resp. 0.13-0.19) in 2035. In particular, despite the lithium intensity of ASSB batteries, similar RMD ratios are reached in 2035 for *ASSB+* and *main*, explainable by the early use of NMC333/532 with high lithium contents. The only deviation happens for the RMD ratio of cobalt under the *status-quo* scenario. In 2035, the RMD ratio is 0.13 compared to 0.20-0.25 for the other scenarios. Indeed, the *main* scenario as well as the *ASSB+* and *LFP+* scenarios show a significant decrease in cobalt intensity, especially after 2030. As this transition reduces its demand, RMD ratios for cobalt increase.

### 3.5.4 Effect of Recycling Efficiencies

<table><caption>Table 3.8: Ratios of recycled materials over demand (RMD ratios) for an alternative recycling efficiency scenario with the *STEPS* demand scenario. Values of RMD ratios above the respective EC minimum threshold are written in bold.</caption><thead><tr><th rowspan="2">Scenario</th><th colspan="2">Lithium</th><th colspan="2">Nickel</th><th colspan="2">Cobalt</th></tr><tr><th>2030</th><th>2035</th><th>2030</th><th>2035</th><th>2030</th><th>2035</th></tr></thead><tbody><tr><td><em>low efficiency</em></td><td>0.01</td><td>0.01</td><td><strong>0.05</strong></td><td><strong>0.13</strong></td><td>0.07</td><td>0.17</td></tr><tr><td><em>main</em></td><td><strong>0.05</strong></td><td>0.09</td><td><strong>0.06</strong></td><td><strong>0.16</strong></td><td>0.08</td><td><strong>0.20</strong></td></tr></tbody></table>

Finally, we also estimate the influence of recycling efficiencies in our results. As explained above, our *main* scenario is based on the EC targets for recovery rates in 2026 and 2030, with *STEPS* demand. They can include a very high increase, in particular for lithium (from 10% efficiency in 2020 to 70% in 2030). It implies key developments in terms of technology
and cost reductions. With scenario *low efficiency* (with STEPS demand too), we compute RMD ratios when recycling efficiencies do not change after 2020 (10% for lithium and 80% for both nickel and cobalt), thus remaining especially low for lithium. As expected, given the linearity of the recycling efficiency in the model (in equation (3.5)) important change occurs for lithium, with RMD ratios dropping to 0.01 (Table 3.8). To a lesser extent, RMD ratios for nickel and cobalt also drop, though recycling efficiencies were not expected to increase with the same magnitude.

## 3.6 Conclusion and Policy Implications

With its regulation proposal, the EC tries to cope with the fast-paced evolution of innovation and market development of Li-ion batteries. The ambition of the EC is to create conditions for the development of a local industry for electric vehicles while securing the different stages of its supply chain and limiting the life-cycle environmental impacts. This implies securing the sourcing of the necessary critical minerals to produce batteries. Developing a recycling industry would be a way to limit risks of geological scarcity and geopolitical tensions. This chapter discusses one of the main features of the proposal, which aims at enforcing the incorporation of minimum shares of recycled material in new batteries from 2030. Our approach is to disentangle the different key-assumptions for the calculations of such targets. Using a material flow model, we find that the targets given by the proposal are physically attainable under very specific assumptions on characteristics of battery technologies, recycling processes, and the evolution of demand. In particular, most of the targets given by the European Commission are not met for average battery lifetimes beyond 10 years. Keeping average battery lifetimes at 10 years might be unrealistic given the technological progress in the battery sector. To a lesser extent, we notice that meeting this target is even more difficult when the demand growth rate is high and recycling efficiencies are low. Heterogeneity between materials is also observed due to technological change and specific recycling
efficiencies.

This study sheds light on several consequences of EC regulation. First, the targets set by the EC are very likely to be too high to be realistic, given market growth, battery usage, and the expected technological framework. On the one hand, setting optimistic targets gives a powerful signal to industrial stakeholders, from battery and car manufacturers to the recycling industry. For instance, it guarantees producers of recycled material the existence of a demand for recycled input, thus encouraging a needed investment in the sector. On the other hand, setting unrealistic targets undermines the readability and stability of the regulation. We believe that too optimistic targets will severely increase the risk of needing downwards revisions, which would alter the credibility of the regulation. This study indicates that battery lifetime is the most sensitive parameter driving recycling. Hence, setting high targets could give incentives to decrease battery lifetimes. Indeed, shorter lifetimes would allow a faster circulation of material in the economy, thus increasing the potential use of recycled material. However, increasing lifetimes could be an objective in itself, as it would decrease material use.

Therefore, the policy recommendations of this study are twofold. First, we believe that the regulations should incorporate robust targets. Although it is stated that targets may be changed in the future, industrial actors of the sectors (battery and cars manufacturers, recycling industry) would need readable and credible regulations to make the right investments and technology choices. While this study indicates that the proposed targets may be difficult to reach, we believe it would be appropriate to propose lower targets for each material. Second, we believe the regulation could include other complementary indicators. Indeed, developing a sustainable battery sector should not rely on recycling and recycled content targets only. As our results indicate that battery lifetime is the main driver of recycled material incorporation in new batteries, we believe it would be adequate to set quantitative objectives on this matter. It could include precise targets on the lifetime of the battery, either regarding the lifetime in the electric vehicle only, or the total lifetime, which
could include eventual repurposing of used batteries.

Further research could complete our analysis in three directions. First, an analysis that incorporates strategies from the supply side to comply with the regulation can be made. For instance, battery wastes or recycled materials could be accumulated years before the start of the regulation and be incorporated in new batteries only once the regulation applies. There could also be strategies to meet recycling targets by extending the source of recovered material to other industries or geographical areas. However, this reasoning bears several limits: a significant part of total production of lithium and cobalt is already used for the battery industry, leaving relatively small amounts for recycling from other industries, especially with an exponentially growing sector; it is not in the spirit of the EC proposal which aims at regulating a specific sector, in this case the battery sector; it suggests coordination between different industries and the possibility of incompatible usage of materials; and, regarding environmental considerations, it would only result in a displacement of the material sourcing problem. Second, our sensitivity analysis could be enriched by accounting for the different links between the main parameters. For instance, larger battery lifetimes should also decrease battery renewal, hence new battery demand. Battery lifetimes should also differ between technologies. Third, the analysis could be improved by a resource economics approach, in order to assess how imposing recycling contents forces a demand for recycled material and how it affects prices on markets for primary and secondary materials.
CHAPTER 4

RECYCLING IN A SUSTAINABLE
FRAMEWORK

***

The development of recycling, and more generally the concept of a circular economy, has the ambition to help developing more sustainable economic paths. However, sustainability 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. In this chapter, we study a recycling model with the maximin criterion. We characterize an optimal path and its shadow values, and we study a new Hartwick investment rule which now depends on waste accumulation and the use of recycling in production.

***
## 4.1 Context and motivations

The ambition of developing a circular economy is directly tied to the idea of developing a more sustainable economic paradigm. It is supposed to overcome the "extract-produce-consume-throw away" linear model. In that sense, the concept can be viewed as a mean for economic development to remain sustainable, as defined by the Brundtland Report (WCED, 1987). In this chapter, we aim to explain how recycling, in particular, is a tool to redefine sustainability objectives.

With recycling at its center, the development of a circular economy first appeared as a standalone goal in itself. Its formalization by Braungart and McDonough (2002), and the consequential "grey literature" that followed, drew on the idea of a circular economy as the objective of creating a closed loop system. This conception can be linked to closed systems studied by Ayres (1999) who defends the idea of a "total recycling" claiming that all material in the economy can be recovered, provided that a sufficient amount of energy is available. Ayres' work is influenced by Georgescu-Roegen (Georgescu-Roegen, 1970), and he refutes the limitation imposed by his "fourth law of thermodynamics" applied to resource use. This controversy relies on the boundness of resource productivity and thermodynamical limits, with both arguments being subject to criticism. An important criticism is that material entropy is not a consistent concept (Young, 1991; Couix, 2019). Georgescu-Roegen indeed extends entropy from the field of thermodynamics to mechanics: matter is subject to conservation but also degradation, creating irreversible waste and thus making perfect circularity unattainable. These observations raise questions regarding a normative approach of circular economy, stating: is it still relevant to address circular economy as an ideal economy, or as one solution along the road towards a sustainable economic model? With this in mind, one should ask what are the links between recycling (and circular economy on a wider scale) and sustainability. The literature review by Geissdoerfer et al. (2017) describes the relationship between those two fields. He highlights that most of the existing work adopts a conditional or beneficial relationship. This implies that a circular economy is
either a condition to develop a sustainable economic model or a part of the solution to get
to such a state of the economy. This supports the idea of recycling not being a goal in itself
but rather a way towards sustainability.

Examining the recycling industry (a central element for closing material loops), chapter
one shows that while not being an objective per se, recycling can be a useful tool in re-
ducing pressure on exhaustible resources and reducing the production of polluting effluents
like waste and greenhouse gas (GHG). The underlying trade-off between consumption, ex-
haustion and externalities, all of those distributed through time, relates to the definition of
sustainability in economics, as developed by Baumgärtner and Quaas (2010): the consider-
ation of exploitation of resources while considering equality issues in terms of consumption
and environmental degradation due to these activities.

In this chapter, we advocate for applying the objectives and tools of sustainability eco-
nomics to the study of recycling, and, more broadly, the circular economy. We introduce a
simple model based on the Dasgupta-Heal-Solow model with a maximin rule. We character-
ize the optimal path when using recycling, especially with its shadow values. We discuss the
damage caused from waste in such a model, and we discuss the Hartwick investment rule.
This now more complex resource management rule depends on waste accumulation and the
use of recycling in production.

Current economic literature on sustainability proposes several criteria to define devel-
opment trajectories where current consumption decisions impact future trajectories of the
economy (Martinet, 2012). It addresses the issue of intergenerational equity by assigning
preferences to different time periods. Most of the literature in resource economics and recy-
cling in particular use discounted utility. However the use of discounted utility is criticized
for being a "dictatorship of the present" according to Chichilnisky (1995), resulting in de-
creasing consumption paths. Among other criteria for developing consumption trajectories
is the maximin which allows for maintaining a minimum sustainable consumption (Solow,
1974), provided that resource and capital are substitutable in production. It defines the
highest possible egalitarian path by maximizing the consumption of the poorest generation. An important underlying result that can be observed is the net investment needed in the economy for consumption to be sustained, with the notable Hartwick's rule at its center (Hartwick, 1977). This maximin criterion and the common analysis resulting from it will be the main focus in the modelling part of this chapter.

With regards to environmental aspects of sustainability, they are often taken into consideration through less equalitarian criteria like the previously mentioned discounted utility. In such an implementation, a price (direct or implicit) is placed on externalities. This is the method in our first chapter that also introduces a damage cost of waste accumulation as well as a cap on total GHG emissions. More ambitious criteria like maximin are less common, however. The notable examples of using this criteria have been in the context of environmental issues like amenities of natural capital (d'Autume and Schubert, 2008) and climate change (Stollery, 1998). Furthermore, in order to go beyond this sustainability paradigm called "weak sustainability", the concept of strong sustainability has been introduced (Neumayer, 2013). While weak sustainability relies on the objective of sustaining a certain level of utility, strong sustainability advocates that natural assets are essential to production, involving the need for their conservation. This central hypothesis insists that environmental quality should be maintained, thus not allowing pollution to go beyond certain thresholds. In this chapter, we discuss how recycling cannot be studied while ignoring this strong sustainability paradigm.

The rest of this chapter is organized as follows. Section 4.2 presents the literatures on recycling and on sustainability. Section 4.3 presents the maximin model of recycling and the results. Sections 4.4 concludes.
## 4.2 Recycling and sustainability

This chapter relates to several trends of literature: literature on sustainability and the maximin criterion; literature on the modeling of recycling and underlying sustainability issues; and, literature related to strong sustainability considerations in recycling.

Regarding weak sustainability considerations, we focus here on the maximin criterion. Primary work comes from Solow (1974), Hartwick (1977) and Dasgupta and Heal (1979). They describe optimal maximin paths in a modeling framework that considers capital and natural resources. Those standard-model paths have the property of being equalitarian, meaning that consumption remains the same for all generations. The discussion has been extended to the study of non-regular maximin paths, meaning that the consumption path is not constant, as opposed to the regular ones studied before. A taxonomy of possible regular and non-regular paths is discussed by Cairns and Martinet (2014) and examples of models describing non-regular paths can be found in the work of Asako (1980) and Cairns and Tian (2010). Furthermore, existing literature examines maximin paths in different specific modeling setups to investigate sustainability: the inclusion of a climate constraint (Stollery, 1998); the addition of an amenity value for the natural resource (d'Autune and Schubert, 2008); the development of an economy with two reproductible assets (Cairns et al., 2019); and, an application to growth and fishery models with the goal of reconciling growth and sustainability (Cairns and Martinet, 2021). To the best of our knowledge, there is no work studying recycling with a maximin criterion. While some of the previously mentioned work shows that a positive value of consumption can be sustained forever, with assumed substitution between capital and the natural resource, we can view the addition of recycling as a way to improve this sustained well-being.

In the existing literature on resource economics with a recycling component, discounted utility has been used in the utility function in order to characterize the optimal extraction and recycling paths of a resource. Early works from Plourde (1972), Smith (1972) and Lusky (1976) determine the optimal allocation of labor between the extracting activity and the re-
cycled production. They include resource scarcity or environmental pollution, sometimes both. With an optimality criterion that favors the consumption of present generations, the underlying optimal path often leads to an exhaustion of the resource at the end of the program, sometimes asymptotically. In the model of Hoel (1978), the rate of resource extraction decreases until recycling happens and then reaches a stationary level in the long run. Hoel presents intergenerational inequalities in terms of consumption as early generations take advantage of the virgin resource extraction. Furthermore, welfare analysis do not always take into account how negative externalities impact differently across generations. In yet another approach, Huhtala (1999) introduces both damage from waste accumulation and negative effects of pollution on the renewal of the resource. This kind of literature often focuses on steady state analysis through either the absence of a resource constraint (Plourde, 1972; Lusky, 1976; Kuhn et al., 2003; De Beir et al., 2007) or the use of a renewable resource (Smith, 1972; Huhtala, 1999; André and Cerdá, 2006). Another explored topic is the link between growth and recycling, with models assessing whether or not recycling can facilitate reaching balanced growth paths by alleviating resource scarcity and/or externalities (Di Vita, 2001, 2007; Pittel et al., 2010; Sorensen, 2017; Lafforgue and Rouge, 2019). In addition to accounting for the effects of technical change as well as the standard substitution between resource and capital, they also includes the substitution between virgin and recycled resource as a means to sustain a growth path. The work from André and Cerdá (2005, 2006) shows that the elasticity of substitution between resources is central in the ability to respond to resource scarcity. They show that in the long run, relying on renewable resources is essential to sustain a non decreasing consumption path. Material limitations are also a takeaway from the work of Fagnart and Germain (2011). They demonstrate that even with recycling, growth based on a material output can only be transitional. Long term paths rely on what they call "qualitative growth", as material inputs either remain constant or decrease. Finally, recycling has recently been introduced in resource models in the context of the energy transition (IEA, 2021b). They explore specific scarcity issues for minerals used
in energy production (Rosendahl and Rubiano, 2019; Fabre et al., 2020; Chazel et al., 2020; Pommeret et al., 2021) or extraction paths with a climate change related constraint (chapter one). These models describe an optimal depletion path for the resource, with a discounted utility rule and finite or infinite horizons. The method leads to intergenerational inequities through decreasing extraction paths. Overall, recycling appears to be a sustainable process that is mainly used in an unsustainable framework by design. Having shown that studies on other criteria like maximin benefit from the inclusion of recycling, economic literature on recycling could also benefit from the addition of sustainability concerns when determining optimal paths of resource extraction and consumption.

Finally, the above mentioned literature on sustainability relies on what is called "weak sustainability" (Neumayer, 2013). Conclusions based on weak sustainability remain valid while it is assumed that natural resources can be substituted with man-made capital (Das-gupta and Heal, 1979). Oppositely, "strong sustainability" does not support this hypothesis. It argues that a minimum physical input is needed to sustain production. To this point, the work of Ayres (2007) - who has defended an optimistic view of total recycling (Ayres, 1999) - argues that substitution is limited in the short term for many technologies; thus, a strong sustainability paradigm should be adopted. Beyond the discussion on capital and natural resources, substitution also relates to a discussion on natural resources and recycled ones. It remains true for metals as recycling processes allow for the production of physically identical or similar material inputs (Villalba et al., 2002, 2004), while bearing significant costs (Ayres, 1997). However, while substitution can be enhanced through technological change (Lafforgue and Rouge, 2019), studies show that materials like paper and cardboard do not show high substitutability between virgin and recycled resources, and can even be complementary inputs in production (Figge and Thorpe, 2019; Mansikkasalo et al., 2014). For these reasons, recycling should be included in the strong sustainability framework: first, to assess recycling's effect on sustainable consumption paths and determine under what conditions it can allow for a minimum level of consumption in the economy; and, second, to
examine what strong sustainability implies when considering the substitution of virgin and recycled resources.

## 4.3 A maximin model of recycling

### 4.3.1 Setup of the model

We develop a resource and waste model based on the study of Pittel et al. (2010). This model has the advantage of being based on a standard Dasgupta-Heal-Solow (DHS) model, which is central in the study of maximin problems (Dasgupta and Heal, 1974, 1979; Solow, 1974; d'Autume and Schubert, 2008; Cairns and Martinet, 2021). The economy has a stock of capital K, that follows the investment rule below:

$$ \dot{K} = F(K(t), z(t), r(t)) - c(t) \quad (4.1) $$

Production F(.) is either invested ($\dot{K}$) or consumed (c).<fnref n="1" /> It relies on capital K and two material resources: the extracted one z and the recycled one r. Production follows the standard neoclassical hypothesis:<fnref n="2" /> F is continuous, non decreasing and strictly concave on $\mathbb{R}_+^3$ ($F_x \ge 0$ and $F_x < 0$, $x \in \{K, z, r\}$); marginal productivity is bounded ($F_x(0) < +\infty$, $x \in \{K, z, r\}$); and, inputs are essential to production ($F(0, z, r) = F(K, 0, r) = F(K, z, 0) = 0$). We suppose that virgin and recycled resources are essential to production, in line with the previously mentioned literature on the substitutability of virgin resources by extracted resources (Ayres, 2007; Mansikkasalo et al., 2014; Figge and Thorpe, 2019).

The flow of extracted input comes from a stock S of exhaustible resource with the

<footnote n="1">¹In order to simplify notations, we will hide the time subscript whenever it is convenient and clear from the context.</footnote>

<footnote n="2">²We denote by $G_x$ the partial derivative of a function G according to variable x, $\frac{\partial G}{\partial x}$.</footnote>
following dynamics:

$$ \dot{S} = -z(t) \qquad (4.2) $$

The flow of recycled input comes from a stock W of waste with the following dynamics:

$$ \dot{W} = -r + \pi c - \rho W \qquad (4.3) $$

Waste is accumulated through the share $\pi$ of consumption, with $\pi$ representing the physical share of consumption. We keep this share constant and exogenous ($\pi \in [0; 1]$). This waste stock is either recycled into an input r, or follows a natural resorption mechanism at a constant and exogenous rate $\rho$, $\rho \in [0; 1]$.

The economy is initially endowed with amounts $K_0$, $S_0$ and $W_0$ of stocks K, S and W ($\{K_0, S_0, W_0\} \in \mathbb{R}_+ \times \mathbb{R}_+^* \times \mathbb{R}_+$).

The economy includes identical consumers distributed along infinite generations, living instantaneously in continuous time. They derive utility $U(c, W)$ from consumption c and waste accumulation W. Utility $U(.)$ is twice differentiable, strictly increasing and strictly concave with respect to consumption ($U_c > 0$, $U_{cc} < 0$). It is twice differentiable and strictly decreasing with respect to waste accumulation ($U_W < 0$).<fnref n="3" />

**The maximin value: definition** – The maximin value m of a state $(K, S, W)$ of the economy is defined as the highest level of utility U that can be sustained forever, starting from that state:

$$ m(K, S, W) = \max_{u,c,z,r} u \qquad (4.4) $$

$$ \text{s.c. (4.1), (4.2), (4.3) and } U(c, W) \geq u. \qquad (4.5) $$

<footnote n="3">³This theoretical framework has been first studied for a maximin path by Stollery (1998), with the specific case of increasing temperature having an impact on production and utility. It brought further investigations on the topic to understand the role of negative externalities and find closed form solutions (d'Autume et al., 2010; Bazhanov, 2012).</footnote>
In other words, the maximin criterion maximizes the utility of the poorest generation, with an equivalent mathematical expression being:

$$ \max_{c,z,r} \left( \min_{t} U(c(t), W(t)) \right). $$

*Optimal control problem* – As demonstrated by Cairns and Long (2006) and Cairns et al. (2019), we can express a maximin problem through a standard optimal control problem. This approach consists in introducing the minimum level of utility *u* as an additional state variable *u*(*t*), with *u̇* = 0 and a free initial value *u* (0). The problem is expressed as follows:

$$
\begin{array}{ll}
\displaystyle \max_{c(t),z(t),r(t)} & u(0) \\
\text{s.c.} & \dot{K} = F(K(t), z(t), r(t)) - c(t) \\
& \dot{S} = -z(t) \\
& \dot{W} = -r + \pi c - \rho W \\
& \dot{u} = 0 \\
& U(c(t), W(t)) - u(t) \ge 0 \\
& K(0) = K_0; S(0) = S_0; W(0) = W_0 \\
& \{c(t), z(t), r(t)\} \in \mathbb{R}_+^3 \\
& \{K(t), S(t), W(t), u(t)\} \in \mathbb{R}_+^4
\end{array}
\qquad (4.6)
$$

From Proposition 2 in (Cairns et al., 2019), we know that an optimal solution complies with the following conditions.<fnref n="4" /> There exist a number $p_0$, $p_0 \in \{0; 1\}$, costate variables ($\lambda(t), \mu(t), \eta(t), \gamma(t)$) associated respectively to state variables ($K, S, W, u$), and multiplier $\omega(t)$ associated to constraint (4.5) such that the following Hamiltonian and Lagrangian are

<footnote n="4">⁴For a detailed methodology and proof of the approach, see the work of Cairns et al. (2019). In particular, they use a the restrictive case of a upper-bounded production function that leads to a bounded maximin value. However, previous work on DHS models proves the existence for convex production functions (Mitra, 2013), given some specifications on the production function.</footnote>
given by:

$$ \mathcal{H} = \lambda(F(K, z, r) - c) + \mu(-z) + \eta(-r + \pi c - \rho W) + \gamma \times 0 \quad (4.7) $$

$$ \mathcal{L} = \mathcal{H} + \omega(U(c, W) - u). \quad (4.8) $$

The optimal path follows the first order conditions:

$$ \frac{\partial \mathcal{L}}{\partial c} = 0 \qquad \Leftrightarrow \lambda = \pi\eta + \omega U_c \quad (4.9) $$

$$ \frac{\partial \mathcal{L}}{\partial z} = 0 \qquad \Leftrightarrow \lambda F_z = \mu \quad (4.10) $$

$$ \frac{\partial \mathcal{L}}{\partial r} = 0 \qquad \Leftrightarrow \lambda F_r = \eta \quad (4.11) $$

$$ \frac{\partial \mathcal{L}}{\partial K} = -\dot{\lambda} \qquad \Leftrightarrow \dot{\lambda} = -\lambda F_K \quad (4.12) $$

$$ \frac{\partial \mathcal{L}}{\partial S} = -\dot{\mu} \qquad \Leftrightarrow \dot{\mu} = 0 \quad (4.13) $$

$$ \frac{\partial \mathcal{L}}{\partial W} = -\dot{\eta} \qquad \Leftrightarrow \dot{\eta} = \rho\eta - \omega U_W \quad (4.14) $$

$$ \frac{\partial \mathcal{L}}{\partial u} = -\dot{\gamma} \qquad \Leftrightarrow \dot{\gamma} = \omega \quad (4.15) $$

the slackness condition:

$$ U(c, W) - u \geq 0; \omega \geq 0; \omega(U(c, W) - u) = 0 \quad (4.16) $$

transversality conditions:<fnref n="5" />

$$ \lim_{\eta \to \infty} \eta \geq 0 \quad \text{and} \quad \lim_{\lambda \to \infty} \lambda = \lim_{\mu \to \infty} \mu = \lim_{\eta \to \infty} \eta W = \lim_{\gamma \to \infty} \gamma = 0 \quad (4.17) $$

<footnote n="5">⁵Unlike Cairns et al. (2019), we need a pure state constraint on the shadow values of W, as the stock could be exhausted by recycling and resorption while being refurnished by consumption. This leads to the possibility of a non zero shadow value at infinity, following Eq. (215b) Seierstad and Sydsaeter (1987), Theorem 16, p. 244-245.</footnote>
and optimality condition on the free initial state $u(0)$:

$$ \gamma(0) = -p_0 \qquad (4.18) $$

As shown by Cairns and Long (2006) and Cairns et al. (2019), the Hamiltonian of the program is the net investment and is equal to zero along an optimal path. Furthermore, from Cairns and Martinet (2014), this net investment is the time derivative of the maximin value<fnref n="6" />

$$ \mathcal{H}(K, S, W, u, c, z, r, \lambda, \mu, \eta, \gamma) = 0 \qquad (4.19) $$

*Regularity of the path* – Let us examine the behavior of shadow value $\gamma$ associated with the constraint of utility parameter $u$. By integrating equation (4.15) and using transversality conditions (4.17) and initial state condition (4.18), we get:

$$ \int_{0}^{\infty} \omega(t)dt = p_{0} \qquad (4.20) $$

Variable $\omega$ relates to the constraint on a minimal level of utility to sustain, thus equal zero when the constraint holds. Equation (4.20) expresses whether equity holds during the program, i.e. whether utility varies from the constant path given by parameter $u$. Cairns and Long (2006) and Cairns et al. (2019) relate to this cost $\omega$ as the shadow value of equity, the opportunity cost maintaining a constant level of utility during the program. Constant utility paths are called regular paths and are conditioned by $\omega > 0$ thus $p_0 = 1$. Conditions for the emergence of non-regularity are given and interpreted by Cairns and Martinet (2014). Following their analysis, the path of our model is regular as utility is not locally bounded and the investment can be controlled in the model (see equation (4.1)).

<footnote n="6">⁶Cairns and Long (2006) show the correspondence between costate variables and shadow values of the stocks: $\lambda = \frac{\partial m}{\partial K}$, $\mu = \frac{\partial m}{\partial S}$, $\eta = \frac{\partial m}{\partial W}$ and $\gamma = \frac{\partial m}{\partial u}$.</footnote>
### 4.3.2 Characterization of the maximin path

We suppose that the set of maximin path is not empty.<fnref n="7" /> From first order condition (4.9), the shadow value of equity is linked to shadow values of the capital and the waste stock. Indeed, the shadow value of the resource $\lambda$ is equal to the marginal utility of consumption weighed by shadow value of equity $\omega U_c$, plus the shadow value of waste weighed by the material share of consumption $\pi\eta$. In other words, the value of the resource is the sum of a term relative to the opportunity cost of maintaining a minimum level of utility at a certain marginal utility of consumption, and a term relative to the value of waste at the level of the material share of production.

**Proposition 13.** *Shadow values of stocks have the same sign and a necessary and sufficient condition to be negative is that the marginal productivity of the recycled resource must be higher than the inverse of the material share in consumption:*

$$ \frac{1}{\pi} < F_r \qquad (4.21) $$

*Proof.* From (4.10) and (4.11), we know that shadow values $\lambda$, $\mu$ and $\eta$ have the same sign. Rearranging (4.9) with (4.11) gives $\eta(\frac{1}{F_r} - \pi) = \omega U_c$. As $\omega > 0$ according to the slackness conditions and marginal utility of consumption is positive, it gives the necessary and sufficient condition $1 < \pi F_r$ for negative shadow values. $\square$

If the shadow value of waste weighs negatively on the resource, with for instance an important disutility of waste, the resource and capital shadow values become negative. With a production function that does not rely a lot on the use of the recycled resource (meaning high marginal productivity due to concavity), or having a consumption that is highly material: the economy produces a lot of waste and that waste is not very useful for production. Waste weighs negatively on the economy through disutility. In this case, all stocks

<footnote n="7">⁷From Mitra (2013) and Cairns et al. (2019), we know that it is the case when waste has no effect on utility drawn from consumption.</footnote>
contribute negatively to sustainability, as waste is too much of a burden in the economy: a marginal increase in capital or in the resource stock leads to a reduction of the maximin value as the subsequent production will lead to more harmful waste ($\frac{\partial m}{\partial X} > 0, X \in \{K, S\}$).

The combination of (4.10) and (4.11) give us the relative value of the resource and waste, equal to the marginal rate of substitution in production between waste and extracted resource:

$$ \frac{\mu}{\eta} = \frac{F_z}{F_r} \qquad (4.22) $$

According to equation (4.12), when positive the shadow value of capital decreases at the level of its productivity. The resource stock shadow value is constant according to equation (4.13). Rearranging (4.14) with first order conditions (4.9) and (4.11) give us the growth rate of the shadow value of waste:

$$ \frac{\dot{\eta}}{\eta} = \rho - \left(\frac{1}{F_r} - \pi\right) \frac{U_W}{U_c} \qquad (4.23) $$

**Proposition 14.** There is an instant $T_W$ such as $W(t) > 0$ while $t < T_W$ and $W(t) = 0$ when $t > T_W$ (with possibly $T_W = \infty$).

* When $t < T_W$, shadow values of stocks are always negative or always positive.

* When $t > T_W$, shadow values of stocks are positive, $\eta$ increases and $\lambda$ decreases.

*Proof.* Transversality conditions on $W$ give us that the waste stock is either zero at infinity, or its shadow value is zero. Suppose that the waste stock is not depleted. Cairns and Long (2006) precise that continuity of the shadow values is guaranteed only while the problem does not have pure state constraints that are reached (following the work of Seierstad and Sydsaeter (1987)). $\mu$ would then be constant and equal to zero (according to its transversality condition), thus all three shadow values of stocks are (according to equations (4.10) and (4.11)). This contradicts equation (4.9) and (4.20) as the path is regular ($\omega > 0$). By
contraposition, the waste stock must reach a phase of depletion. Transversality condition on η give us that shadow values are positive.

While $t < T_W$, equation (4.23) and first order conditions (4.10) and (4.11) give us the optimal behavior. At $T_W$, a discontinuity can occur (especially if shadow values are negative)(Cairns and Long, 2006). When $t > T_W$, as shadow values are positive, equations ruling their dynamics give us that η increases and λ decreases. □

Proposition 14 tells us that the previously described phase where shadow costs are negative can only happen during a transitory phase. From other first order conditions (4.12) and (4.13), we know that after $T_W$, the shadow value of the extracted resource is constant, and the shadow value of capital decreases at a rate equal to its marginal productivity.

**Corollary 1.** After $T_W$, the marginal productivity of the recycled resource is bounded: $F_r < \frac{1}{\pi}$.

The previous corollary involves that at given capital level and resource extraction, the use of the recycled resource has a lower bound.

By log-derivation of (4.10), we get that at any time on the optimal path, the marginal productivity of the resource increases at the level of the marginal productivity of capital:

$$F_K = \frac{\dot{F}_z}{F_z} \qquad (4.24)$$

As the resource is depleted, meaning the use of extractive inputs in production must decrease, the marginal productivity grows due to concavity. As capital accumulates, its marginal productivity decreases. Equality (4.24) reflects the compensation by capital to sustain production. We can note that this relation does not depend on the marginal productivity of the recycled resource. Recycling has an effect hidden in the marginal productivity of the resource according to the following proposition.
**Proposition 15.** Marginal productivities of material resources are related through the following equation:

$$ \frac{\dot{F}_r}{F_r} - \frac{\dot{F}_z}{F_z} = \rho + \left(\pi - \frac{1}{F_r}\right) \frac{U_W}{U_c} \qquad (4.25) $$

*Proof.* The relation comes from the combination of equation (4.23) and the log-derivation of (4.11). □

After $T_W$, the right-hand side of equation (4.25) is positive: marginal productivity of the recycled resource grows quicker than the marginal productivity one of the virgin one. This difference, explained by the right hand side of the equation relies on the self resorption of waste $\rho$ and a term relative to the reduction of the waste stock and its associated negative externality $\left(\pi - \frac{1}{F_r}\right) \frac{U_W}{U_c}$.

As explained by Cairns and Long (2006) and Cairns et al. (2019), the shadow value of equity $\omega$ can be interpreted as an endogenous discount factor, with the intertemporal maximization of $\omega(t)U(c(t), W(t))$ (Cairns and Long (2006), Proposition 3 p. 281). However $\omega$ is here the opportunity cost of maintaining the sustained level of utility, and not the preference between two time periods. Therefore we can isolate a discount rate that follows a pseudo Keynes-Ramsey rule by log-derivating equation (4.9) and replacing shadow values:

8

$$ -\frac{\dot{\omega}}{\omega} = F_K + \frac{\dot{U}_c}{U_c} + \frac{\pi \dot{F}_r}{1 - \pi F_r} \qquad (4.26) $$

Equation (4.26) shows that this virtual discount rate decreases at rate equal to two the sum of two standard terms already found by Cairns et al. (2019), as well as a third term relative to recycling. The first two terms are the marginal productivity of capital $F_K$ and the rate of change in marginal utility in consumption $\dot{U}_c/U_c$. A third term $\pi \dot{F}_r/(1 - \pi F_r)$

<footnote n="8">⁸Replacing $\eta$ by (4.11) in (4.9) gives $\lambda(1 - \pi F_r) = \omega U_c$. The we log-derivate the expression and we use (4.12) to replace $\lambda$.</footnote>
relates to the rate of change in marginal productivity of the recycled resource weighed by the material part of consumption $\pi$. In a sense, one could consider that

### 4.3.3 Hartwick rule and sustainability

#### Hartwick rule

If an optimal sustainable path exists, it follows the Hartwick investment rule:

**Proposition 16.** *The optimal maximin path has an optimal Hartwick investment rule such as:*

$$ \dot{K} = zF_z - \dot{W}F_r \qquad (4.27) $$

*Proof.* We use the nil net investment condition (4.19) and combine it with first order conditions (4.10) and (4.11) to obtain equation (4.27). □

With the addition of the waste stock $W$, we can see that the Hartwick rule makes you invest more or less. The first term $zF_z$ is common to standard models: we invest at the level of the productivity of the natural resource. However we must add here a second part, modulating the investment. The idea here is that recycling allows to invest less/more in order to sustain utility, at the level $F_r$, the marginal productivity of the recycled resource, times the apparent waste accumulation $\dot{W}$. When accumulated waste increases ($\dot{W} > 0$), more resource is available and we can invest less ceteris paribus. However, to this availability of the "waste resource" that allows less investment in capital must also be confronted the weigh of waste through disutility. This effect can be expected to intervene in the marginal productivity of recycling $F_r$, as shown in equation (4.25).

Starting from the Hartwick rule, we now want to assess the effect of recycling on strong sustainability criteria. Is a maximin path exists, it follows the Hartwick rule. However in the DHS model, this rule imposes an upper bound on the share of production to invest. If
this share given by the Hartwick rule is too high, a minimum level of consumption cannot be sustained. By giving Cobb Douglas forms to production functions ($F(K, z) = K^{\alpha}z^{\beta}$), work from Dasgupta and Heal (1979) give us that a maximin solution is attainable while $\alpha > \beta$.<fnref n="9" /> It means that production should rely more on capital than resource for a sustainable maximin path to exist. Such a production function (where $\alpha < \beta$) without a maximin solution is one way of expressing that resource is essential to production, thus being in line with the strong sustainability paradigm (Neumayer, 2013).<fnref n="10" /> With this kind of hypothesis in our model, we do not know if a maximin path exists, as it does not in the case without recycling.

We introduce a Cobb-Douglas function for production:

$$F(K, z, r) = K^{\alpha}z^{\beta}r^{\delta} \qquad (4.28)$$

with $\alpha, \beta, \delta > 0$ and $\alpha + \beta + \delta < 1$.

We use the following utility function:

$$U(c(t), W(t)) = \frac{v(t)^{1-1/\sigma}}{1-1/\sigma} \quad \text{and} \quad v(t) = c(t)W(t)^{-\epsilon} \qquad (4.29)$$

with $\sigma > 0$, $\sigma \neq 1$ and $\epsilon \ge 0$.

The investment rule is now:

$$\dot{K} = \left( \beta + \frac{-\delta}{r/\dot{W}} \right) F \equiv \theta F \qquad (4.30)$$

Similarly to the standard case, equation (4.30) shows that a share $\theta$ of consumption should be invested in capital. In the recycling case, this share is not constant and an additional term $\frac{-\delta}{r/\dot{W}}$ is added to the elasticity $\beta$ of production with respect to the resource.

<footnote n="9">⁹In DHS models, the Hartwick rule can be reduced to a fixed share of production, equal to the elasticity of production with respect to the resource: $\dot{K} = \beta F$. It follows that for the maximin to exist, we shall have to invest too much.</footnote>

<footnote n="10">¹⁰As highlighted by Neumayer (2013), other possibilities would be to allow for time dependent production elasticities or using a constant elasticity of substitution (CES) function with an elasticity of substitution below one, meaning that there is complementarity between the resource and capital.</footnote>
<table><caption>Table 4.1: Share of reinvested production $\theta(t)$</caption><thead><tr><th></th><th>dot W > 0</th><th>dot W < 0</th><th>dot W = 0</th></tr></thead><tbody><tr><td>r > |dot W|</td><td>β - δ < θ < β<br>Not optimal</td><td>β < θ < β + δ</td><td>θ = β</td></tr><tr><td>r < |dot W|</td><td>θ < β - δ<br>Not optimal</td><td>β + δ < θ</td><td>θ = β<br>Impossible</td></tr><tr><td>r = |dot W|</td><td>β - δ<br>Not optimal</td><td>θ = β + δ<br>Standard DHS model with two resources</td><td>θ = β<br>Standard DHS model with one resource (not optimal)</td></tr></tbody></table>

$r/\dot{W}$ is the share of recycled input that contributes to the net variation of the waste stock. When it is equal to -1, $\pi c = \rho W$: self resorption exactly compensates the flow of waste and production can only rely on the existing stock. The waste stock behaves like a standard resource stock and we find the usual investment rule with two natural stocks: $\theta = \beta + \delta$ (see Table 4.1).

As previously mentioned, in the standard maximin problem for DHS models with a Cobb Douglas production function, the share of production that is reallocated to the capital is the elasticity of production with respect to the resource. For a sustainable path to exist, it must be lower than the elasticity of capital in production. This gives us insights to discuss sustainability of a maximin path with recycling: does the Hartwick rule impose a lower share of production to have a sustainable trajectory, in which case making the problem more likely to have a solution. Table 4.1 gathers the possible values this share $\theta$ can take during an sustainable path. $\theta$ is not fixed and can vary through time. However as shown previously, the optimal path leads to an exhaustion of the waste stock, in finite or infinite time. For this reason, first column where the waste stock increases cannot be optimal in the long run, while the underlying consumption path is trivially sustainable. If the stock decreases, the question of the feasibility is raised. When the rate is constant and matches a standard DHS model with two resources, the standard condition $\alpha > \theta$ applies in the long run for a sustainable path to exist. In the two other cases, sustainability is not trivial.
In the case of the waste stock reaching a steady state ($\dot{W} = 0$), $\theta$ is always equal to $\beta$. In this phase, the use of recycled material cannot be lower than the variation of waste, for obvious positivity reasons. When both are equal ($r = \dot{W} = 0$), self-resorption exactly compensates the waste flow $\pi c$. The recycled resource is not used and the model becomes a standard DHS one with a single depletable resource. However in the long run, it is not optimal as the transversality condition involves that $W$ reaches zero, and consumption with it. Finally, the case where recycling is used in production involves that, in the long run with a depleted waste stock, we instantaneously use in production what comes from production ($r = \pi c$).

### Closed form solution without externalities

In order to expand this analysis, we know suppose that time $T_W$ where the waste stock is depleted is reached in finite time ($T_W < \infty$). It follows that during this phase, the utility function does not depend on waste: $U(c)$. The dynamics of the system are given by:

$$ \dot{K} = \beta F \qquad (4.31) $$

$$ \dot{S} = -F^{1/\beta} K^{-\alpha/\beta} r^{-\delta/\beta} \qquad (4.32) $$

**Proposition 17.** When the waste stock is depleted in a finite time $T_W$, the optimal level of consumption is given by:

$$ c^* = \left(\pi^{\delta}(1-\beta)^{1-\beta}\right)^{1/(1-(\beta+\delta))} \left( (\alpha-\beta)S_{T_W} K_{T_W}^{\frac{\alpha-\beta}{\beta}} \right)^{\beta/(1-(\beta+\delta))} \qquad (4.33) $$

where $S_{T_W}$ and $K_{T_W}$ are the levels of stocks at $T_W$. In this case, a necessary condition for the production function is $\alpha > \beta$.

*Proof.* In order to find a closed-form solution, we follow the methodology of d'Autume and Schubert (2008). We introduce $c(U)$ consumption as a function of the utility $U$ only (as $W = 0$). For a regular maximin path, it is constant. The investment rule and the dynamics
of capital give us that $F = c(U)/(1 - \beta)$. As the waste stock is depleted, we have $r = \pi c(U)$. Replacing in (4.31) and (4.32), and combining both it gives us:

$$-\pi^{\delta/\beta}(1 - \beta)^{\frac{1-\beta}{\beta}} c(U)^{-\frac{1-(\beta+\delta)}{\beta}} dS = \frac{1}{\beta} K^{-\alpha/\beta} dK$$

Integration between $S_{T_W}$ and $S$ on the left, and $K_{T_W}$ and $K$ on the right gives:

$$\pi^{\delta/\beta}(1 - \beta)^{\frac{1-\beta}{\beta}} c(U)^{-\frac{1-(\beta+\delta)}{\beta}} (S_{T_W} - S) = \frac{1}{\alpha - \beta} (K_{T_W}^{1-\alpha/\beta} - K^{1-\alpha/\beta})$$

At this stage, for a solution to exist, there must be $\alpha > \beta$, otherwise the resource difference diverges towards infinity. Isolating $c(U)$ we get the optimal level $c^*$ given by (4.33). □

An important take of proposition 17 is that despite the existence of recycling, the weak sustainability constraint cannot be relaxed: we must have $\alpha > \beta$. However, it changes the sustainable level of consumption given by (4.33). When $\delta = 0$, we find the known value of sustainable consumption from Dasgupta and Heal (1979). Two effects of recycling are in play here.<fnref n="11" /> First, consumption itself allows a higher level of utility as it can be reintegrated in production through recycling. Mathematically, it appears with exposants $\beta/(1 - (\beta + \delta)) > \beta/(1 - \beta)$. However, as consumption is partially material, the material share $\pi$ also intervenes through the factor $\pi^{\delta/(1-(\beta+\delta))} < 1$. It attenuates the increase in sustainable consumption (compared with the level in the standard case without recycling). It should also be kept in mind that this level of consumption intervenes after a transitory phase with the depletion of the waste stock. However, one could assume that the optimality problem could start with an already depleted waste stock. In that case, equation (4.33) only includes exogenous parameters as stock levels are initial ones $S_0$ and $K_0$. Note that the effect of $\delta$ is ambiguous: one can assume that it alleviates part of the scarcity as recycling can be considered as a partially renewable resource. However it also leads to a dependency on a

<footnote n="11">¹¹Note that the addition of recycling can be compared with elasticities $\alpha$ and $\beta$ remaining the same. A proper comparison could also consider that initial elasticity $\tilde{\beta}$ in a model without recycling is distributed between elasticities of recycling and virgin resource: $\tilde{\beta} = \beta + \delta$.</footnote>
resource depending on the level of consumption and its material share. Taking for instance a highly dematerialized consumption ($\pi$ is low), the sustained level of consumption could be lower with recycling, as the economy relies on a poorly available resource. Note that as the previous reasoning is done when it has become optimal to keep waste at zero, so the rate of self resorption does not intervene. Analyzing the effect of recycling for an optimal path with a non zero waste stock involves less tractable equations as seen in equation (4.30).

## Substitutable material resources

Another way of modeling resource input is to consider waste and virgin resource as complementary inputs (Di Vita, 2001, 2007). $\beta$ is now the elasticity of all material inputs in production, and we introduce $\zeta$ ($0 < \zeta \le 1$), the technological substitutability between both material inputs. Perfect substitutability occurs when $\zeta = 1$. Note that now, the strict combination of both material inputs are not essential in production. The production function becomes:

$$F(K, z, r) = K^{\alpha} (z + \zeta r)^{\beta} \qquad (4.34)$$

In this case, the Hartwick rule is now the following:

$$\dot{K} = \left( 1 - \frac{\zeta(\pi c - \rho W)}{z + \zeta r} \right) \beta F \qquad (4.35)$$

The share $\theta(t)$ of consumption invested in capital relies on the ratio of input that comes from net production at the level of its technological efficiency $\zeta(\pi c - \rho W)$ in the global use of material input $z + \zeta r$. This ratio is positive, as we have seen that it is not optimal to increase the waste stock. Comparing with the standard Hartwick rule, it is lowered as net residuals of consumption $\zeta(\pi c - \rho W)$ are included in production. We find a similar result as before: recycled input allows for less investment to maintain a sustainable level of utility while the waste stock is not depleted.
## 4.4 Conclusion

While recycling and sustainability economics seem to be topics with strong connections, a framework of analysis that includes both is slow to emerge. In particular, despite many economic models representing recycling as a way to ensure optimal consumption paths (mainly through waste reduction and a less stringent resource scarcity), modeling, in its vast majority, relies on intertemporal criteria that do not favour equity and are thus unsustainable. This goes indeed in contradiction with the idea of sustainability as defined in diverse reports and in the economic literature (WCED, 1987; Baumgärtner and Quaas, 2010). These observations advocate for a more thorough study of recycling with the use of the concepts and mathematical tools of sustainability economics: this goes from a diversity of sustainability criteria to weak and strong sustainability frameworks (Martinet, 2012; Neumayer, 2013).

Studying a maximin problem with a growth model that includes waste and recycling is one first step in this direction. This model, while being simple, 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. The rule now depends on waste accumulation and the use of recycling in production. In particular, the share of production that has to be dedicated to capital accumulation can be studied in light of the already existing results for models with maximin objectives (Dasgupta and Heal, 1979). 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, we show that the assumption of having an elasticity of capital higher than the elasticity of resource in production is crucial for a sustainable path to exist. This weakens the idea that recycling could be, in this theoretical framework, a way to get a sustainable path when assuming the strong sustainability hypothesis.

Further research could go in two different directions. First, the study of transitory dynamics should be added, especially when it is optimal to deplete the waste stock asymptotically.
cally. It seems mathematically challenging to get explicit solutions, however it would greatly
improve the comparative analysis between maximin with recycling and maximin without re-
cycling. Second, the topic of recycling and strong sustainability is also central. Results here
are not very optimistic regarding the impact of recycling on the long run. However, it could
be expanded to other functional forms such as CES functions with elasticities lower than one,
meaning that inputs are complements. This hypothesis is central for the resource/capital
combination in production and allows to determine the sustainability framework considered
in the model. Moreover, it is also the case for the extraction/recycling combination in pro-
duction, as for many materials, empirical studies do not find perfect substitution between
both inputs. This raises the question wether substitution between recycled and extracted
material should also be a central hypothesis of the strong sustainability paradigm.
GENERAL CONCLUSION

This dissertation explores the economics of recycling when taking into account the different externalities linked to resource extraction, transformation and consumption. It includes the depletion of natural stocks, the environmental impacts of waste accumulation and GHG emissions. We examine the trade-offs between recycling and extraction in different cases, with different methodologies. For this purpose, we combine theoretical approaches of resource economics as well as simulation and prospective approaches using calibrated data.

In the first chapter, we consider a model in resource economics allowing us to specify under which conditions it is optimal to recycle and use a recycled resource in production. This analysis takes into account the social costs of extraction, consumption and recycling. In order to set optimal consumption paths, private costs and GHG emissions from recycling are also taken into account.

In the second chapter, this take on recycling is enhanced using an integrated bio-economic assessment model showing how increasing recycling for the paper and cardboard industry can lead to negative climate impacts. This counterintuitive result prevails when both resources are complementary for paper and cardboard production.

Focusing on the specific sector of Li-ion batteries, chapter three examines the feasibility of a policy imposing recycled contents thresholds with a resource flow model. Once again, we show that the optimal level of recycled material used in production is the function of
a trade-off between several parameters across time. In particular, another sustainability parameter - the lifetime of batteries - can hinder the objective of a high recycled content.

As recycling is strongly linked to sustainability objectives, chapter four assesses the lack of sustainability considerations in the macroeconomic literature of recycling. We introduce the first model that examines recycling through the use of the maximin criterion.

The rest of this conclusive section is divided in two parts. The first part examines policy implications of the different chapters and tries to discuss them in a transversal analysis. The second part proposes leads for future research.

# Transversal and policy considerations

## An holistic approach of recycling

The introduction to this dissertation describes how material use in the economy is informed by many different parameters related to resource stocks, the environment, climate change and social considerations. For this reason, a thorough study of recycling cannot be done without a holistic approach. This is especially true when addressing the climate impact of recycling. As shown in chapter one, trade-offs between inputs in production are influenced by the externalities we take into account. The use of recycled material can be encouraged in order to reduce GHG emissions (chapter one and three), but can also be at the origin of negative impacts on climate, given the specificities of some production sectors (chapter two). By considering recycling as a means for achieving sustainability, policies aiming at encouraging recycling have to take into consideration the multidimensional aspects behind it. This means accounting for trade-offs between different private and social costs, defining sectorial and spatial boundaries, and including the temporal dimension of the system (chapter four).
Can recycling be bad?

Some results from this dissertation can lead the reader to consider that recycling is not such a good thing, or can even be a bad thing in terms of climate change or other sustainability objectives (chapter two and three). In fact, as with any economic decision, the net social outcome of choosing between recycling or not recycling depends on the assumptions made. In the case of perfect substitution, higher GHG emissions, and not so high private costs, recycling is the favored option (chapter one). However, when recycling can in many cases be the socially optimal choice, properly defining spatial and temporal boundaries of the system, industries and markets can lead to negative impacts (chapter two and three). In the specific case of paper and cardboard recycling with a complementarity between recycled paper pulp and wood pulp, the marginal effect of an increase in recycling that has a negative climate impact. One cannot conclude that all paper recycling is bad, but given the evolution of the industry through the years, we are in a situation where more recycling can be harmful. These results advocate for more broadly defined sustainability objectives to be used, thus not only recycling objectives, when putting in place new policies.

Substitution and complementarity

From this dissertation, a central parameter when evaluating the impact of recycling and whether or not it should be promoted appears to be the substitutability between recycled and extracted materials. Perfect substitution, a feasible assumption for metals, allows for the switch from one input to the other without productivity considerations (chapter one and three). However, complementarity forces the industrial apparatus to produce more output when recycling is encouraged (chapter two), a result some authors called the circular economy rebound. In a way, the substitutability issue here is comparable to the debate addressed by sustainability economics: can accumulated capital be a substitute for natural capital. The economic literature shows that this question is crucial for the feasibility of economic paths that preserve natural resources. In the case of complementarity, part of what
is called the strong sustainability paradigm, long term consumption must be guaranteed
by renewable resources. For this reason, policies regarding recycling should address the
different sustainability paradigms (chapter four) and thus take into account, for each material
considered, how it can be substituted.

Recycling and circular economy as a goal?

In parts of this dissertation, recycling is not taken as a goal, but as a production choice that comes from the trade-off between different private costs and externalities (chapter one and four). When recycling is a goal in itself (chapter three) or a consequence of a market equilibrium (chapter two), we show that it can lead to adverse results. Thus, we advocate for policy makers not to consider recycling as an objective per se, but one option, admittedly often favorable, for industries. Indeed, when considering the concept of a circular economy, recycling only appears as one of its principles (although a central one as it allows to close the material loop in the economy). It can be completed by other practices that are part of the circular economy such as reuse, eco-design or product-service systems. Recycling, and the circular economy on a larger scale, should be studied as means for a pathways towards sustainability. When not considering recycling as and end in itself, it involves reflecting on its meaning, relevance and impact. This also means coming back to the definition of sustainability as given by the Brundtland report, “Our Common Future” (WCED 1987): ensuring that humanity “meets the needs of the present without compromising the ability of future generations to meet their own needs”, and designing policies that support this goal.

Future work

Recycling and sustainability accounting

Chapter three is a first attempt at reconciling the macroeconomics of recycling and sustain-
ability accounting. The costs and benefits of recycling are indeed distributed across time.
An optimal recycling rate should take into account inequities across generations when allowing for more available resources as well as a reduction in waste accumulation. Furthermore, sustainability concerns cannot be separated from the topic of climate change: future generations can have welfare gains when current ones use less carbon intensive resources like, in most cases, recycled ones.

Sustainable accounting for an optimal depletion of resources is also a crucial topic for critical raw materials in the energy transition. It is necessary to assess the impact of recycling when those minerals will be crucial to develop low carbon infrastructures in the future decades.

In this dissertation we limit ourselves to the maximin criterion. Other frameworks of analysis could be used to study optimal recycling programs, such as a golden rule (maximizing long term utility) or the Chichilnisky criterion (a trade off between present and future generations).

## Oligopolistic market structures for critical raw materials

This dissertation deals with several aspects of recycling with the scope of an arbitration between costs, resource availability and externalities. However it does not examine the market structure of recycling and mining industries. In particular, critical raw materials for the energy transition are managed by oligopolistic markets that operate on a worldwide scale. The emergence of recycling was an important topic of research in the second half of the 20th century, with the example of the Alcoa antitrust case for aluminum. This branch of research is now subject to renewed interest as recycling industries are expected to gain in size. In particular, this poses interesting questions in the context of the increasing environmental and social pressures on these industries.
## Substituting virgin and recycled resources

Being able to substitute virgin resources with recycled ones appears to be a central question in assessing the environmental and economical impact of recycling. Most of the theoretical work relies on arbitrary hypothesis such as perfect substitution, which is really only valid for non ferrous metals. In reality, combining both resources in industrial processes relies on complex physical and economical constraints. Having complementarity between inputs could be dramatic for the environmental impact of recycling as it could lead to a rebound effect of material use. Moreover, substitutability is interesting for policy makers as it can have an impact on policy instruments. It questions practices of industrials and waste managers through the choice of waste treatments and the quality of the recycled output.

Econometric work already exists for the paper and cardboard industry, as the sector is already mature in developed countries (in France, 66% of the production comes from recycled waste). However, the substitutability of recycled plastic resins is under-researched, while plastic recycling remains to be a critical challenge. In both economic and technical literature, it appears that categories of plastics encompass a wide range of resins with different properties, uses and the degree of recyclability. The prices of resins follow distinct trajectories as they are not linked to the same industrial sectors. As such, resins must be approached separately as heterogeneous products, since they generally are not interchangeable. Thus, I propose econometric research on plastic products as the next step towards understanding the potential of recycling for addressing sustainability in the plastics industry.
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Si notre or est fumier, en revanche, notre fumier est or.

Que fait-on de cet or fumier? On le balaye à l'abîme.

[...]

Ces tas d'ordures du coin des bornes, ces tombereaux de boue cahotés la nuit dans les rues, ces affreux tonneaux de la voirie, ces fétides écoulements de fange souterraine que le pavé vous cache, savez-vous ce que c'est?

C'est de la prairie en fleur, c'est de l'herbe verte, c'est du serpolet et du thym et de la sauge, c'est du gibier, c'est du bétail, c'est le mugissement satisfait des grands bœufs le soir, c'est du foin parfumé, c'est du blé doré, c'est du pain sur votre table, c'est du sang chaud dans vos veines, c'est de la santé, c'est de la joie, c'est de la vie.

Victor Hugo - *Les Misérables* - Livre V

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