# Inverse Malthusianism and Recycling Economics: The Case of the Textile Industry

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## Abstract

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## 1. Introduction

- We introduce the concept of inverse Malthusianism, describe a paradigmatic case and explore the possible reasons behind this phenomenon.
- We show the conditions under what policies excluding recycled supplies are never optimal.
- We propose a method to demonetize user costs by means of MCDM.

## 2. Literature Review

## 3. A Formal Definition of Inverse Malthusianism

#### 3.1. Malthusianism and Inverse Malthusianism

#### 3.2. The Case of the Textile Industry

#### 3.3. Reasons for Inverse Malthusianism

## 4. The Equilibrium of Jevons and Recycling Economics

**Equal prices.**If marginal prices ${p}_{1}^{\prime}\left(t\right)$ and ${p}_{2}^{\prime}\left(t\right)$ are equal for conventional and recycled materials, Equation (22) becomes a comparison of marginal costs ${c}_{1}^{\prime}\left(t\right)$ and ${c}_{2}^{\prime}\left(t\right)$. Even though this price-equality situation may seem infrequent at first glance, it may arise in the textile industry for some polyester products whose differences in prices are sometimes negligible. In addition, it allows us to focus on costs as a first exploratory analysis. Producers should then allocate resources to conventional or recycled products by only focusing on differences in cost. The following alternative situations may turn out.- (a)
- ${c}_{1u}=0$, and ${c}_{1e}$ and ${c}_{2r}$ positive constants. When user costs are neglected or when there is no user cost due to the non-exhaustible nature of resources (e.g., solar energy), producers have an incentive to recycle only when ${c}_{1e}>{c}_{2r}$. Since this relation in constant, the equilibrium condition provides the optimal recycled quantity:$${q}_{2}^{*}=\frac{{c}_{1e}}{{c}_{2r}},$$
- (b)
- ${c}_{1e}$ constant, and ${c}_{1u}<{c}_{2r}$. Then, the marginal cost of recycling increases more rapidly than the marginal cost of producing conventional material. Then, there is an intersection point of marginal costs given by:$${q}_{0}=\frac{{c}_{1e}}{{c}_{2r}-{c}_{1u}}.$$However, this point does not provide the optimal policy $({q}_{1}^{*},{q}_{2}^{*})$. The optimal policy is given by the Jevons equilibrium condition:$${c}_{1e}+{c}_{1u}{q}_{1}^{*}={c}_{2r}{q}_{2}^{*}$$
- (c)
- ${c}_{1e}$ constant, and ${c}_{1u}>{c}_{2r}$. Then, the marginal cost of producing conventional materials increases more rapidly than the marginal cost of recycling. As a result, marginal costs never intersect. However, the equilibrium condition described in Equation (27) and the market restriction (20) provide the optimal policy $({q}_{1}^{*},{q}_{2}^{*})$. In this case, recycled quantity ${q}_{2}^{*}$ is always larger than conventional quantity ${q}_{1}^{*}$.

Summarizing, the previous cases analyze different situations in which user, extraction and recycling costs determine the optimal material selection policy. Even in the unrealistic case in which user costs are neglected, recycled materials seem to should be playing a more important role than they currently do in the context of inverse Malthusiansim. In the USA, the recycling rate for all textiles was 15% in 2017 [44]. In Europe, about 15 to 20% of disposed textiles are collected and about 50% is downcycled and 50% is reused [13]. If we assume, for example, an average recycling rate of 20%, the implication is that marginal costs of recycling ${c}_{2r}$ must be, at least, four times larger than user cost ${c}_{1u}$ to guarantee an optimal policy in the sense of Jevons according to Equation (27). This assumption seems to be unrealistic for the most consumed fibers (polyester and cotton) so that we argue that the textile industry is far from following an optimal conventional-recycled path. This conclusion can be extended to each sector situation in which recycling rates and total consumption are known to analyze the degree of suboptimality of current materials selection policies.**Different prices.**Quality differences, taxes and many other factors may provoke that marginal prices ${p}_{1}^{\prime}\left({q}_{1}\right)={a}_{1}-{b}_{1}{q}_{1}$, for conventional materials, and marginal prices ${p}_{2}^{\prime}\left({q}_{2}\right)={a}_{2}-{b}_{2}{q}_{2}$, for recycled materials, are not equal. As a result, profit functions obtained as the difference between marginal price and cost functions in Equation (22) play a key role to derive the optimal conventional-recycled materials policy. Next, we consider two alternative situations.- (a)
- Prices for recycled materials above prices for conventional materials. This situation arises when ${a}_{1}<{a}_{2}$, and ${b}_{1}\ge {b}_{2}$. Benefits for conventional and recycled materials are, respectively, decreasing functions of ${q}_{1}$ and ${q}_{2}$. This fact implies the existence of two limiting quantities equal to the intersection of the marginal profit functions with the horizontal axis:$${q}_{1,max}=\frac{{a}_{1}-{c}_{1e}}{{b}_{1}+{c}_{1u}}$$$${q}_{2,max}=\frac{{a}_{2}}{{b}_{2}+{c}_{2r}}$$Beyond these quantities, profits are negative. Then, an additional constraint that must be satisfied is ${q}_{1,max}+{q}_{2,max}\ge {C}_{0}{e}^{rt}$. Otherwise, the market restriction is impossible to fulfill. The Jevonsian equilibrium in the case of different prices for conventional and recycled materials with linear dependence on quantities for marginal price and cost functions is as follows:$${a}_{1}-{c}_{1e}-({b}_{1}+{c}_{1u}){q}_{1}^{*}={a}_{2}-({b}_{2}+{c}_{2r}){q}_{2}^{*}$$In this case, if ${a}_{1}-{c}_{1e}<{a}_{2}$ and ${b}_{1}+{c}_{1u}>{b}_{2}+{c}_{2r}$, the marginal benefit of conventional materials decreases more rapidly with quantities than the marginal benefit of recycling. Both marginal functions never intersect. Then, ${q}_{2}^{*}$ will always be larger than ${q}_{1}^{*}$. The optimal policy implies producing more recycled than conventional materials. On the contrary, if ${a}_{1}-{c}_{1e}<{a}_{2}$, but ${b}_{1}+{c}_{1u}<{b}_{2}+{c}_{2r}$, there is another intersection point:$${q}_{0}=\frac{{a}_{2}-{a}_{1}+{c}_{1e}}{{b}_{2}+{c}_{2r}-{b}_{1}-{c}_{1u}}.$$Together with the market restriction ${q}_{1}^{*}+{q}_{2}^{*}={C}_{0}{e}^{rt}$, if ${q}_{0}>0.5{C}_{0}{e}^{rt}$, then ${q}_{1}^{*}<{q}_{2}^{*}$ and the optimal policy recommends to produce more recycled materials than conventional. On the contrary, if ${q}_{0}<0.5{C}_{0}{e}^{rt}$, then ${q}_{1}^{*}>{q}_{2}^{*}$ and the optimal policy recommends to produce more conventional materials than recycled.
- (b)
- Prices for recycled materials below prices for conventional materials. This situation arises when ${a}_{1}>{a}_{2}$, and ${b}_{1}\le {b}_{2}$. Again, benefits are decreasing functions of quantities and there are two limiting points that must satisfy the restriction ${q}_{1,max}+{q}_{2,max}\ge {C}_{0}{e}^{rt}$. Otherwise, the market restriction is impossible to fulfill. The optimal policy can be computed by means of the Jevonsian equilibrium described in Equation (30). Similarly to point 2.(a) above, if ${a}_{1}-{c}_{1e}>{a}_{2}$ and ${b}_{1}+{c}_{1u}<{b}_{2}+{c}_{2r}$, the optimal policy implies producing more conventional than recycled materials. On the contrary, if ${a}_{1}-{c}_{1e}>{a}_{2}$, but ${b}_{1}+{c}_{1u}>{b}_{2}+{c}_{2r}$, there is another intersection point that can be again computed by means of Equation (31). Together with the market restriction ${q}_{1}^{*}+{q}_{2}^{*}={C}_{0}{e}^{rt}$, if ${q}_{0}>0.5{C}_{0}{e}^{rt}$, then ${q}_{1}^{*}>{q}_{2}^{*}$ and the optimal policy recommends to produce more conventional materials than recycled. On the contrary, if ${q}_{0}<0.5{C}_{0}{e}^{rt}$, then ${q}_{1}^{*}<{q}_{2}^{*}$ and the optimal policy recommends to produce more recycled materials than conventional.
- (c)
- Mixed marginal price functions when ${a}_{1}>{a}_{2}$, but ${b}_{1}>{b}_{2}$, or ${a}_{1}<{a}_{2}$, but ${b}_{1}<{b}_{2}$. As a result, marginal benefits may produce different optimal policies that, however, can be computed by means of the Jevonsian equilibrium described in Equation (30).

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 5. Demonetizing User Costs by Means of MCDM

- Global warming potential (kg CO${}_{2}$ eq) is a measure of how much heat a greenhouse gas traps in the atmosphere and indicates the potential change in climate patterns.
- Eutrophication (kg PO${}_{4-}$ eq) occurs when excessive nutrients enter a body of water and causes a dense growth of plant life and death of animal life from lack of oxygen.
- Abiotic resource depletion of fossil fuels (MJ) is the consumption of a resource such as carbon or oil faster than it can be replenished.
- Water scarcity (m${}^{3}$) measures the environmental impacts of freshwater consumption in three areas: human health, ecosystem quality and resources.
- Chemicals used (points) are qualitatively assessed by considering finishes applied to a given material and chemical standards such as Oeko-tex.

## 6. Conclusions

**On the impacts of inverse Malthusianism.**The increasing resource consumption due to population growth and economic development imposes a cost on future generations due to associated environmental impact. The novel concept of inverse Malthusianism serves to illustrate the situation in which production increases exponentially and population grows linearly. This situation implies a tremendous environmental challenge. One suitable option to face this challenge is the combination of conventional and recycled materials that we here discuss.

**On the conventional-recycled materials dilemma.**The set of conditions characterizing optimal materials selection policies considering both conventional and recycled materials is one the main concerns of producers along the whole supply chain. What mix of materials should they use to maximize utility within a context of inverse Malthusianism? Relying on the concept of Jevonsian equilibrium, we show that materials selection policies excluding recycled supplies are never optimal under some well-defined conditions. Current recycling rates in the textile industry seem to indicate that there is room for improvement in long-term sustainability through recycling.

**On the difficulties to monetize user costs.**The conditions for optimal conventional-recycled policies are based on user costs as an attempt to monetize the impact of current consumption in future generations. To overcome the difficulties of reducing environmental aspects to monetary units, we propose a multicriteria approach linking the concept of equimarginality and Pareto efficiency. We argue that further insights can be derived by approaching the challenges of environmental economics from a multiobjective perspective.

## Author Contributions

## Funding

## Conflicts of Interest

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Time (t) | Year | Population (Milions) | Consumption per Capita (kg/person) | Consumption (kg) | Variation (kg) |
---|---|---|---|---|---|

1 | 1950 | 2518 | 3.7 | 9317 | - |

2 | 1960 | 2982 | 4.9 | 14,612 | 464 |

3 | 1970 | 3692 | 5.9 | 21,783 | 710 |

4 | 1980 | 4434 | 6.6 | 29,264 | 742 |

5 | 1990 | 5263 | 7.7 | 40,525 | 829 |

6 | 2000 | 6070 | 8.7 | 52,809 | 807 |

7 | 2010 | 6863 | 10.5 | 72,062 | 793 |

8 | 2020 | 7700 * | 13.0* | 100,100 * | 837 * |

Fibre | Conventional Higg Index | Recycled Higg Index | Production 2017 (%) | Weighted ${\mathit{h}}_{1}$ | Weighted ${\mathit{h}}_{2}$ |
---|---|---|---|---|---|

Polyester | 44 | 35 | 54 | 24 | 19 |

Cotton | 98 | 39 | 26 | 25 | 10 |

Viscose | 62 | 43 | 7 | 4 | 3 |

Polyamide | 60 | 36 | 6 | 4 | 2 |

Wool | 82 | 49 | 1 | 1 | 1 |

Others | − | − | 6 | − | − |

Global | 58 | 35 |

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## Share and Cite

**MDPI and ACS Style**

Salas-Molina, F.; Pla-Santamaria, D.; Vercher-Ferrándiz, M.L.; Reig-Mullor, J.
Inverse Malthusianism and Recycling Economics: The Case of the Textile Industry. *Sustainability* **2020**, *12*, 5861.
https://doi.org/10.3390/su12145861

**AMA Style**

Salas-Molina F, Pla-Santamaria D, Vercher-Ferrándiz ML, Reig-Mullor J.
Inverse Malthusianism and Recycling Economics: The Case of the Textile Industry. *Sustainability*. 2020; 12(14):5861.
https://doi.org/10.3390/su12145861

**Chicago/Turabian Style**

Salas-Molina, Francisco, David Pla-Santamaria, Maria Luisa Vercher-Ferrándiz, and Javier Reig-Mullor.
2020. "Inverse Malthusianism and Recycling Economics: The Case of the Textile Industry" *Sustainability* 12, no. 14: 5861.
https://doi.org/10.3390/su12145861