Introduction

Environmental degradation and global warming over the past decades have accentuated the need to study how everyday activities of an economy affect the environmental quality and specifically the pollution levels. This is why the environmental Kuznets curve (EKC) is a very strong modeling tool that attempts to explain the relationship between output growth and pollution levels. The overall evidence of EKC shows that the relationship between environmental degradation and economic growth is characterized by an inverse-U curve, whereby as economic growth increases, the pollution level of the economy increases up to a specific level of average income and after that pollution starts to fall improving environmental quality. Grossman and Krueger [1] explored empirically the relationship between these two variables as they investigated the effects that trade liberalization and specifically the North American Free Trade Agreement (NAFTA) agreement would have on the income and pollution levels of the participant economies. They concluded that the NAFTA agreement can lead to higher production and income levels which will also in turn lead to higher pollution levels as well.

There are many papers in the literature that support the notion that the relationship between pollution and economic growth is characterized by an inverse-U curve, the EKC. However, there are some other papers which argue that an N curve would be a more suitable curve to characterize the relationship between environmental degradation and economic growth. The purpose of our study is to contribute to this debate by means of a theoretical model that is an extension of George et al. [2]. Our model assumes that there are only two factors of production in the economy producing the unique final good: the polluting resource and the recycling resource which is made by recycling part of the waste stock of the final product. The problem that the social planner faces is to maximize the social welfare of the economy under two constraints: the pollution accumulation path as well as the waste accumulation path. George et al. [2] assumed that the two factors of production are independent, something that is hard to justify. In our paper, we relax some of their assumptions and we also allow for standard utility and production functional form choices in order to make the derivation of our results apparent.

In addition, we explore the shape that characterizes the relationship between output and recycling as it emerges naturally from our theoretical model. This relationship is important because by studying the recycling level that an economy produces we can measure the current and future possible environmental quality improvements that may arise.

The structure of this study is as follows: in “2,” we discuss the existing literature of the EKC and the literature of recycling from an economic perspective. In the section “6,” we introduce our basic model, while in the section “7,” we present the solution of our model as well as our results and plots for the relationship between pollution and output, and recycling and output. In the section “8,” we present an extension of our basic model in which we add technological progress in the production of the polluting resource, while the section “9” presents a brief discussion on an additional extension model in which we add a negative dynamic effect on the polluting input. Finally, in the section “10,” we conclude followed up by the Appendix where we present more plots of our basic model and a detailed analysis of the second extension model and its results, as well as a section with details about the parameter value choices for each model.

Literature Review

The EKC hypothesis suggests that environmental degradation first increases with income per capita and then falls. Hence, in that sense, environmental pollutants cannot set a limit to economic growth, which can increase without causing any extra environmental damage. This topic is of central importance in environmental as well as growth economics since economic growth and the environment are inter-connected in a nonlinear way.

Classical EKC

EKC was first introduced to the literature in the early 1990s by Grossman and Krueger [1], who explored the empirical relationship between pollution and economic growth. They focused on the effects that trade liberalization and particularly the NAFTA agreement can have on income and pollution on the participating countries and they concluded that NAFTA can lead to larger scale of production and higher income which is accompanied by an increase in pollution as well. Andreoni and Levinson [3], with a theoretical and an empirical study, found that the inverse U pattern between pollution and income arises from the abatement technology, assuming that it is more costly to abate pollution when pollution is low before abatement as they focused on the case of increasing returns to scale in the abatement technology. Their main finding is that the technological connection which exists between the consumption of a desired good and the abatement of its undesirable by-product (pollution) is the key component in explaining the inverse-U relationship between pollution and income. Jeffords and Thompson [4], with an extension of the Andreoni and Levinson [3] model as well as a numerical exercise, find that the EKC is still described by an inverse-U curve. Their model incorporates consumers’ preferences and it also assumes that they have a minimum consumption requirement. Even under those new assumptions, the EKC holds with an inverse-U curve; however, higher minimum consumption requirement leads to higher pollution which affects the income level in which the turning point of the EKC happens. Finally, they show that meeting the minimum consumption requirement leads to higher pollution and higher cost to improve the environmental quality of the economy. Another paper that included both empirical and theoretical analyses is that of Abrate and Ferraris [5]. Their contribution to the existing literature was to offer additional support to the result that the relationship between pollution and income is an inverse-U curve using data from 547 Italian municipalities for the years 2004 to 2006. They differentiated their analysis from other papers by using waste as an indicator of environmental pollution and by arguing that recycling can improve welfare, since in their analysis consumption and non-recycled waste are the key components of the utility function of the representative agent and their focus was to find ways to stimulate households to recycle.

Among the many theoretical papers that attempt to examine the shape that best fits the EKC as an inverse-U curve, Stokey [6] developed three different theoretical models that generated the inverted-U curve of the EKC. Lopez and Mitra [7] introduced two game theoretic models, one of which is a cooperative Nash bargaining game, whereas the second one is a non-cooperative Stackelberg model. Both models find that pollution levels are higher than the socially optimal level and that the turning point of the inverted-U shape of the EKC takes place at income and pollution levels higher than the social optimum ones. They also found that these results will be worse for developing countries as the turning point of the EKC will be in a higher income and pollution level. In a framework different from the previous papers, Lieb [8] offers a very different yet important explanation of the inverted-U shape of the EKC. His analysis is based on an overlapping generations model with a myopic government who cares only about the present state of the country. There are two types of pollutants which are created by the production activities of the economy: one that causes damage immediately (flow pollutant) and one that causes damage in the future (stock pollutant). For that reason, the government sets a per capita tax and uses it in order to construct two different abatement technologies, one for each type of pollutant. Since the government is myopic, it does not care about the stock pollutant as its negative effects will be only apparent in the future, so it never abates this type of pollutant. In that framework, the inverted-U shape of the EKC is replicated but the explanation of it is totally different from that in most other papers. Even though the downward sloping part of the EKC may still arise, it happens only because the government neglects the stock pollutant. As a result, pollution levels may be higher in the future but that may not be visible in the plot of the EKC because the latter is about the flow pollutant only.

Contrarily, there are some papers which find that the EKC is not an inverse-U curve. An N-shaped curve is suggested by De Bruyn and Opschoor [9] to be a more appropriate shape that describes the relationship between pollution and economic growth. They based their analysis on a macroeconomic theoretical model and empirical research using data for 19 Western European countries for the period 1966 to 1990. Another paper that is not supporting the inverse-U shape of the EKC curve is that of Steger and Egli [10]. Their main aim was to find out which variables affect the turning point of the inverted-U shape of the EKC as well as to find which public policy can help reduce environmental pollution. They assumed that polluting units are generated by the consumption of the unique final good which can only be reduced by spending money on abatement policies. They also assume that there are two types of externalities in the economy with the first one being a negative externality due to pollution, while the second one is a positive one that comes from the environmental effort. They found that between taxing consumption and subsidizing abatement, the optimal policy in order to reduce pollution levels is by subsidizing environmental effort, and they also derived an N-shaped curve for the shape of EKC for some specific pollutants. Alvarez-Herranz and Balsalobre-Lorente [11] also found that an N-shaped curve best describes the EKC for OECD countries for the years between 1993 and 2010. Balsalobre-Lorente et al. [12], with an empirical and theoretical work, found that the EKC is described by an N-shaped curve in the long run. In particular, they showed that during developing stages, environmental pollution increases up to a point after which it stars decreasing until a second turning point where environmental degradation starts to increase again. The main characteristics of the third stage are low economic growth with high income levels and inefficient environmental measures. They also showed that energy innovation policies are an important instrument that helps economies solve air pollution problems, while economic growth by itself cannot lead to similar environmental improvements. In addition, Kasioumi [13], with a closed circular economy which takes into account the habit formation theory and uses a polluting as well as a recyclable input in the production process, showed that the EKC is best characterized by a downward sloping curve. Their model assumes that consumers form habits on recycling which leads them to receive higher utility levels if their current recycling is greater than the one of the previous period (habit stock). Finally, Bibi and Jamil [14] find that the EKC holds with an inverse U curve for the years between 2000 and 2018 for most regions of the world, and specifically for Latin America and the Caribbean, Europe and Central Asia, South Asia, East Asia and the Pacific, and finally for the Middle East and North Africa. However, the empirical analysis shows that there is no meaningful relationship between CO2 emissions and economic growth in the Sub-Saharan African region.

Finally, there are studies showing that the EKC might hold with an inverse U curve under specific cases and assumptions, while in general it is described by a different pattern. Özokcu and Özdemir [33] shows that the EKC does not hold with an inverse U pattern for the years between 1980 and 2010. Specifically, an inverted N-shaped curve is more appropriate for the EKC for 26 OECD countries, while an N-shaped curve is the best fit for 52 emerging countries. Another empirical analysis for seven European countries for the period 1980 to 2014 was contacted by Dogan and Inglesi-Lotz [15] who found that when economic growth is used to study changes on pollution, the inverse U shape of the EKC holds. On the other hand, when the industrial share pf the countries is used as a proxy to economic growth, the EKC turns out to be a U-shaped curve. In addition, Tenaw and Beyene [16], using human development as a proxy of economic growth, find an EKC described by an inverse U curve for resource-rich sub-Saharan African countries, and an increasing curve for non-resource-intensive sub-Saharan African countries. Finally, Isik et al. [17] show that the inverse U curve for the EKC does not hold but it can arise under specific cases only. Specifically, when the GDP per capita series includes only its increases to fit the theory more accurately, the EKC does not hold. On the other hand, when the series includes both its increases and decreases, the EKC holds with the usual inverted U pattern. Still, this result holds for only four out of eight OECD countries studied.

Recycling

Even though the existing literature of the recycling output is not as rich as the literature of the standard EKC, there are some theoretical studies that deal with this topic. Selden and Song [18], using a simple growth model and assuming that the utility of the representative agent is affected positively by consumption and negatively by pollution levels, derived the shape of the EKC to be an inverted-U curve and that the corresponding curve for abatement to be a J-shaped curve. However, even though through the representation of these curves one might think that pollution follows an optimal path, they indicated that this is not the case as pollution may not be on that path. Another paper that worked on the same framework as Selden and Song [18] is that of Casino et al. [19]. Production creates units of pollution which affects negatively the utility of the consumers, but it also produces goods to be consumed which increases utility. In addition, countries can adopt some cleaning up habits which reduce the pollution level, but as these activities become more expensive their efficiency is being reduced. However, this study is extended to a two-country case which means that the pollution of one country acts as a negative externality on the other country. That also means that the benefit that each country has for adopting abatement activities is affected by the choices from both countries, while the cost is affected by the choice of each country independently. They concluded that even in the two-country case, the J curve between consumption and abatement can be replicated. This result has also been confirmed empirically by Kasioumi and Stengos [20]. One more study that dealt with recycling and growth is this of Di Vita [21], which showed that economic growth is higher when they introduce in the production process materials which come from waste recycling compared to the case that the economy uses only capital, labor, and technology to produce the final good. In addition, technological change is important in the model because it affects the quantity of recycled materials produced in the economy and furthermore affects the quantity of the final output. Finally, using values for the exogenous variables consistent with macroempirical evidence, they showed that developed countries increase their welfare by selling recyclable waste to developing countries which can produce secondary materials cheaper.

Pittel [22] introduced a new EKC model which varies in two main ways from the standard EKC model. Firstly, emissions are now substituted with recycling of the stock variables like household waste—particularly with iron and copper—and secondly, they showed that EKC can arise not only due to technological anomalies, increasing returns to abatement or sectoral switches as it was believed so far, but also due to recycling. His model assumed that there is a closed economy which functions under a material balance condition, that is materials cannot be destroyed and they can only be converted through recycling. What they found is that EKC might arise during the transition to the long-run balanced growth path when we work with non-renewable materials. In addition, the belief of the recycling firms about the kind of the materials (renewable or not) is also important, as it affects the speed with which an economy converges to its balanced growth path. This model is an extension of the initial model of Pittel et al. [34] which assumed that waste is an unprocessed input which can be transformed into a highly valuable resource if we take advantage of it and recycle it. Finally, George et al. [2] show that the EKC does not hold in a closed economy which uses two factors of production in the production process, namely a recyclable input as well as a polluting resource. Part of the final good is consumed in every period while the rest of it can either be recycled and reused in the production of the final good or remain in the economy as waste which leads to environmental pollution. They further assume that employing the recyclable input has no cost for the firm while the polluting input has one, which if increased a lot leads to the substitution of the more expensive polluting input by the cheapest recyclable input. What they show is that economic growth does not have a positive effect on environmental quality as the EKC suggests.

Circular Economy

Bernard [23] found that if firms collude in a duopolistic circular economy framework, when environmental regulation is not present, then the results for consumers and firms are positive. Specifically, both consumer’s surplus as well as firm’s profits increase. However, if firms collude while environmental regulation is present, then those positive results will not be present. Bonviu [24] indicated that a circular economy might be a better option for European markets, as linear economies lead to resource exhaustion followed up by an increase in their prices, lower demand for those, and hence lower profits. Ragossnig and Schneider [25] found that reusing unharmful waste is a necessary and powerful tool to ensure that an economy will reach a full circular economy. However, waste should have specific characteristics to be considered useful and hence undergo recycling. The four most important conditions that should hold include the existence of demand or a market for the waste recycled, environmental benefits from the use of the product or at least not deterioration of the environmental condition and human health, the object should be one that is commonly used, and finally, the object should meet the existing law, legislation, and standards applicable to products. Improved technology, a more expensive polluting input, and positive shocks to the refurbish productivity are some of the most important factors that can lead to a circular economy with high growth rate and high environmental benefits, according to Banionienea et al. [26].

MacArthur et al. [27] point out that the basic principles of a circular economy include the design out waste (design products for which the biological and technical components can be recycled with minimal energy or composted), build resilience through diversity (diverse systems are more resilient to external shocks), rely on energy from renewable sources, think in systems (understand how parts influence one another and the whole, and vice versa), and waste is food (upcycling: reintroduce materials back into the biosphere). However, they support that even though a circular economy has great environmental benefits in the present, it can only delay the inevitable in the future, something that can be avoided if the entire operating system changes. According to Hartley et al. [28], reuse, recycling, and remanufacturing are the main pillars that should be improved in the European Union to achieve a circular economy. Specifically, the systems that should adapt to the needs of the circular economy are the product design and disposal systems, according to interviews taken from 47 public and private circular economy professionals. On the other hand, they found that cultural and psychological biases within both organizations and individuals are the key barriers to the adoption of a circular economy; and thus, new mechanisms should arise to enable the adoption of a circular economy. In addition, Patwa et al. [29], using a sample of 183 consumer responses, show that both government and consumer behavior are key factors on the adoption of a circular economy in emerging economies. Platforms for design thinking, educational programs, infrastructure development from government, and behavioral change by consumers are the most important aspects that affect the adoption of a circular economy. Nevertheless, renewable energy, waste to energy, and resource utilization are concepts that can lead to a circular economy and at the same time economic efficiency.

On the other hand, Zink and Geyer [30] showed that under an economic perspective, a circular economy cannot guarantee environmental improvement. Specifically, closing material loops can lead to higher production levels which offset the benefits of a circular economy either by imposing lower prices leading to higher consumption or by failing to compete with primary production. In other words, they showed that under a circular economy, both secondary and primary production increase something that can offset the benefits of the circular economy. That problem could be avoided if firms decided to adopt specific strategies; however, they do not because that would reduce their profits. Additionally, Allwood [31] supports that some characteristics of the circular economy, and specifically the continuous material recycling, is one of the most unattractive ways of improving the environmental quality of an economy. Instead, reusing materials and products and reducing the demand for them are factors that can be applied to all materials, and they can lead to environmental improvements. Finally, Zhou and Smulders [32] found that a circular economy can lead to economic losses in some cases and particularly when innovation is focused on resource saving. Their model includes a two-sector economy one of which refurbishes waste and one that produces the final output of the economy.

The Circular Economy Model

We focus our analysis on a quite simple but very useful model which gives us insight into the problem of characterizing an optimal growth path in a circular economy. We follow the main structure of the model of George et al. [2], but with some substantial differences as we relax some of their main assumptions. The importance of this analysis is to address the significance of recycling for the growth of an economy in a simple benchmark framework.

Suppose we have a centralized closed economy in which the social planner wants to maximize a social welfare function with concave preferences of the following form:

$$U={\int }_{0}^{\infty }{e}^{-\rho t}u(c,P)dt$$
(1)

where \(u(c,P)\) is the instantaneous utility function which depends positively on consumption c and negatively on the stock of pollution P.Footnote 1Hence, we assume that \({u}_{c}>0\) and \({u}_{P}<0\) and we further assume that \({u}_{cc}<0\) and \({u}_{PP}>0\). We also assume that \({u}_{cP}\) as well as \({u}_{Pc}\) are both equal to 0, that is, there are no cross effects and, in addition, \(\rho\) is the rate of time preference.

We suppose for simplicity that the consumption good does not pollute the environment. Hence, we can assume a product like reusable bottles/bags/straws/dryer balls, e-readers, solar chargers, LED light bulbs, biodegradable trash bags, and more. In addition, we focus on local air pollutants which remain in the geographical region where they were produced, like \({\mathrm{SO}}_{2}\) and \({\mathrm{NO}}_{\mathrm{x}}\) (see Millimet et al. [35] ).

The social planner wants to maximize the discounted present value of the future utility stream \(u(c,P)\), for which we assume a utility function that satisfies all the above assumptions, described by the following formula:Footnote 2 \(u(c,P)=wlnc-(1-w)\;lnP\), where \(w\) is the weight that the representative household assigns to consumption. Finally, we assume that the intertemporal elasticity of substitution is constant at each time point and due to the above choice of utility function we can easily see that it is equal to 1:\(\sigma =\frac{-{u}_{c}}{{u}_{cc}c}=1\).

Moving on to the firm’s side, the output q is produced using two factors of production: x which is the input produced by recycling and z which is the environmentally polluting input,Footnote 3 both of which evolve over time. Our first assumption which differentiates our model from the one of George et al. [2] is the fact that we no longer assume that the above two factors of production are independent. Since there are only two factors of production, we should assume that for a given output level they are substitutes,Footnote 4 something that will lead to some extra marginal cross effects that will affect our final results, which was ignored by George et al. [2]. In a growing economy, the level of recycled input cannot catch up with new production since the available material stock is always less than the market demand for new products. Therefore, although the input of raw materials can be reduced, it cannot be eliminated; thus, the polluting factor can only be moderated but not minimized. The equilibrium of recycled material used in new production can be achieved only in a steady-state economy or degrowth. However, our model assumes continued growth of the economy; hence, recycling can only limit the input of virgin raw materials but not totally substitute it.Footnote 5 The production of the final output q is characterized by a concave production function \(\phi\), given by the following formula:

$$q=\phi (x,z)$$
(2)

We assume for convenience and simplicity to have a Cobb–Douglas production function which takes the following form: \(\phi (x,z)=A{x}^{(1-\gamma )}{z}^{\gamma }\), where A is a constant technology factor that affects the production process.

Continuing with the constraints, we assume that the final good can be used to employ the polluting input or it can be consumed. The final output that is not used accumulates as waste which can be either recycled or not. We assume that the recyclable proportion of the waste stock is \(\beta\) while the proportion which cannot be recycled is \((1-\beta )\). The second assumption which differentiates our model is that the decision about the magnitude of recycling we are choosing to do is also affected by the importance we attach to recycling. If consumers greatly dislike pollution, they will choose to recycle with higher intensity to avoid pollution from waste, while if they do not care much about the environment, they will choose to recycle less. Hence, the total recycling level is \(\tau \beta S\) where \(\tau\) is the intensity of recycling.

Thus, the low of motion of the waste accumulation can be described by the following differential equation:

$$\dot{S}=\phi (x,z)-c-z-\tau \beta S$$
(3)

We assume that part of the waste stock is being recycled and is transformed into a useful factor that can be used again for the production of the final output. In other words, the recyclable resource x is equal to the total recycling of the waste stock that takes place each period (\(x=\tau \beta S\)). By substituting this to Eq. 3, we get the following equation which characterizes the dynamics of waste accumulation in the economy: •Footnote 6

$$\dot{S}=\phi (\tau \beta S,z)-c-z-\tau \beta S$$
(4)

Apart from the waste accumulation path, we also have the pollution accumulation path of this economy. Part of the environmental pollution comes from the use of the polluting resource. In particular, we assume that each unit of this input leads to \(\theta\) units of pollution. Moreover, the pollution level increases by the part of the waste stock that cannot be recycled which generates polluting emissions in a one for one form, that is by \((1-\beta )S\). Last but not least, we assume that the stock of pollution is automatically reduced at each period by a rate \(\delta\) since the environment is assumed to be self-renewed. Even though we assume that the polluting input is an extractable resource, pollution from extraction is not part of our definition of pollution for simplicity.Footnote 7

Thus, the low of motion of the pollution accumulation in this economy can be described by the following differential equation:

$$\dot{P}=\theta z+(1-\beta )S-\delta P$$
(5)

The Optimal Growth in the Economy

The problem that the social planner faces is to maximize the social welfare function (Eq. 1) subject to the two laws of motion for waste and pollution accumulation (Eqs. 4 and 5). The Hamiltonian function taking into account that P and S are the state variables, while c and z are the control variables that take the following form:

$$H={e}^{-\rho t}u(c,P)+\lambda [\phi (\tau \beta S,z)-c-z-\tau \beta S]+\mu [\theta z-\delta P+(1-\beta )S]$$
(6)

where \(\lambda\) and \(\mu\) are the shadow prices of the two constraints we have, that is for waste accumulation and pollution accumulation respectively. The first-order conditions of the above problem are given below as:

$${H}_{c}={e}^{-\rho t}{u}_{c}-\lambda =0$$
(7)
$${H}_{z}=\lambda ({\phi }_{z}-1)+\mu \theta =0$$
(8)

or

$$\mu =\lambda \left(\frac{1-{\phi }_{z}}{\theta }\right)$$
(9)
$${H}_{S}=\lambda \tau \beta ({\phi }_{x}-1)+\mu (1-\beta )=-\dot{\lambda }$$
(10)
$${H}_{P}={e}^{-\rho t}{u}_{P}-\mu \delta =-\dot{\mu }$$
(11)

Differentiating 7 with respect to time t and substituting 10 for \(\dot{\lambda }\), 7 for \(\lambda\), and 9 for \(\mu\), we get the following Euler equation for consumption:

$$\frac{\dot{c}}{c}=\left(-\frac{{u}_{c}}{{u}_{cc}c}\right)\left[\frac{(\beta -1)({\phi }_{z}-1)}{\theta }+\tau \beta ({\phi }_{x}-1)-\rho \right]$$
(12)

However, we know that in our case the intertemporal elasticity of substitution, given by the term in the first parentheses, is equal to 1, so by substituting it to the above equation we get:

$$\frac{\dot{c}}{c}=\frac{(\beta -1)({\phi }_{z}-1)}{\theta }+\tau \beta ({\phi }_{x}-1)-\rho$$
(13)

From the above equation, we can clearly see that the consumption time path is affected by the marginal product of both recyclable and polluting resources. In particular, the marginal product of the recycling input (\({\phi }_{x}\)) affects positively the growth rate of consumption while the marginal product of the polluting input (\({\phi }_{z}\)) affects it negatively. Hence, the final output increases when the firm uses more units of the recyclable input and decreases with more units of the polluting input. Similarly, \(\beta\) (the recycling ratio) and \(\theta\) (the additional pollution that the environmental polluting resource creates when used), as well as \(\tau\) (recycling’s intensity parameter), all affect the consumption growth rate positively. The addition of the recycling’s intensity parameter differentiate our results from the results of George et al. [2]. As we can see, by including that parameter, we make the growth rate of consumption to increase faster. On the other hand, \(\rho\), the rate of time preference, affects the consumption path in a negative way.

An important point of departure from George et al. [2] is the fact that now the substitution between the two production inputs becomes harder. In our case, polluting input (\(z\)) does not have a unit cost, but now the recyclable input (\(x\)) has an indirect cost. Since we included the intensity of recycling in our analysis, the recycling level that takes place at each period is affected by this parameter. If agents care a lot for the environmental quality and so for the recycling level, they recycle more and the recyclable input is easier to be found and used. On the other hand, if agents do not care for the environmental quality, they recycle less and the recyclable factor of production is more rare to find which corresponds to an indirect cost. George et al. [2] assumed that when the cost of the polluting input increases, which is the only input that has a cost in their model, the economy will substitute this expensive input with the free recyclable input, something that will lead to an increase in economic growth. This argument does not hold anymore because in our case we take into account that the recyclable resource might not be easy to find and use in production, leading to limitations of its usage, similar to the case were the recyclable input were to have a unit cost.

In order to derive the growth path of the environmental polluting resource \(z\), we differentiate Eq. 8 with respect to time and we substitute Eq. 9 for \(\theta\). That gives us the following equation:

$$\dot{z}=\frac{\left(\frac{\dot{\mu }}{\mu }-\frac{\dot{\lambda }}{\lambda }\right)({\phi }_{z}-1)}{{\phi }_{zz}}-\frac{{\phi }_{zx}}{{\phi }_{zz}}\tau \beta \dot{S}$$
(14)

Substituting in the above equation, Eqs. 10 for \(\dot{\lambda }\), 9 for \(\mu\), 11 for \(\dot{\mu }\), 7 for \(\lambda\), and 4 for \(\dot{S}\), we get the optimal path of the polluting input:

$$\dot{z}=\left(\frac{{\phi }_{z}-1}{{\phi }_{zz}}\right)\left[\delta +\tau \beta ({\phi }_{x}-1)+\frac{(\beta -1)({\phi }_{z}-1)}{\theta }\right]+\frac{{u}_{P}\theta }{{u}_{c}{\phi }_{zz}}$$
$$-\frac{{\phi }_{zx}}{{\phi }_{zz}}\tau \beta \left[\phi (\tau \beta S,z)-z-c-\tau \beta S\right]$$
(15)

According to Eq. 15, the quantity of the polluting resource used in the production of the final good depends on the intensity of recycling (\(\tau\)); \(\theta\), the rate at which pollution is created by using the polluting input in the production process; \(\delta\), the rate in which the pollution decays naturally; \(\beta\), the proportion of the waste stock that can be recycled, consumption \(c\), waste stock \(S\), and the polluting resource \(z\), as well as the marginal products of the polluting resource and the marginal utilities with respect to pollution and consumption.

The solution to the system consisting of Eqs. 4, 5, and 13 and 15 allows us to find numerical values for waste (\(S\)), pollution (\(P\)), consumption (\(c\)), and the polluting input (\(z\)) over time, which then allow us to calculate the recycling level (\(x=\tau \beta S\)) and the output level (\(q\)) given from Eq. 2. The above solution can be seen in the following plots where we present the EKC as well as the relationship between recycling and output. Footnote 8Footnote 9

Fig. 1
figure 1

Environmental Kuznets curve

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Fig. 2
figure 2

Recycling–output relationship

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The inverted U-shaped curve depicted in Fig. 1 shows the relationship between pollution and output level. We can see that the shape which is supported to a great extent by the empirical literature characterizing the environmental Kuznets curve (EKC) holds under our theoretical model. In other words, as output increases in time, pollution initially increases dramatically up to a point after which it starts to fall with every additional unit of output.

On the other hand, Fig. 2 depicts the relationship between recycling and output. In that case, we can see that we are getting a J curve as the literature suggests, showing that recycling increases with output. Specifically, more output leads to more recycling of the waste pile and higher production of the recyclable input. Hence, we can see that our expectation about this curve is satisfied.

By relaxing some assumptions of the paper of George et al. [2] and including a new parameter which affects the recycling level (\(\tau\)), we are able to reverse their results and get the inverse-U pattern of the EKC that the main literature suggests, and also prove that the relationship between recycling and output is indeed increasing, and specifically a J curve, as some papers already found.Footnote 10

Extension: the Circular Economy Model with Technological Progress

There are two ways that we can add technological progress in this model. The first one is by adding it in the production of the recyclable input (\(x\))Footnote 11 and the second one is by adding it in the production of the polluting input (\(z\)). We will analyze the second one as it is a more interesting case and for that we will assume, by which the efficiency of the polluting input increases, transforming the production into a more polluting input-saving one. In that case, the polluting input \(z\) will increase faster than the available amount of it, allowing the firm to use less units of it, but producing the same amount of final good \(q\) as before. Specifically, we assume that technological progress (\({A}_{t}\)) increases exponentially with time by the following formula:

$${A}_{t}={A}_{0}{e}^{(\eta t)} \mathrm{and} \frac{{\dot{A}}_{t}}{{A}_{t}}=\eta$$
(16)

where \({A}_{0}\) is the initial price of technology and \(\eta\) shows the rate at which technological progress changes in time.

The addition of technological progress does not affect the instantaneous utility function that we use nor the pollution accumulation path of the economy. In other words, the following formulas are the same as in the basic model that we discussed earlier:

$$U={\int }_{0}^{\infty }{e}^{-\rho t}u(c,P)dt$$
(17)
$$u(c,P)=wlnc-(1-w)lnP$$
(18)
$$\dot{P}=\theta z-\delta P+(1-\beta )S$$
(19)

However, the production function and the waste accumulation path of this economy both change due to the introduction of the technological progress. Specifically, the production function includes the variable technology now instead of the fixed parameter that we had earlier, something that will also affect the waste accumulation path, leading to the following two new formulas:

$$\phi (x,z)=\phi (\tau \beta S,{A}_{t}z)=(\tau \beta S{)}^{(1-\gamma )}({A}_{t}z{)}^{\gamma }$$
(20)
$$\dot{S}=\phi (\tau \beta S,{A}_{t}z)-c-z-\tau \beta S$$
(21)

The solution to this new model follows similar steps as the solution of the benchmark model that we discussed in the section “7.” We now have to maximize the social welfare function (Eq. 17) subject to the new waste accumulation path (Eq. 21) and the old pollution accumulation path (Eq. 19). We are solving this problem with a Hamiltonian function as we did previously, where \(c\) and \(z\) are the control variables while \(P\) and \(S\) are the state variables, taking the following form:

$$H={e}^{-\rho t}u(c,P)+\lambda [\phi (\tau \beta S,{A}_{t}z)-c-z-\tau \beta S]+\mu [\theta z-\delta P+(1-\beta )S]$$
(22)

where \(\lambda\) is the shadow price of the waste accumulation path and \(\mu\) is the shadow price of the pollution accumulation path. Under these assumptions, the Euler equation for consumption and the optimal path of the polluting input are characterized by the following two equations respectively:

$$\frac{\dot{c}}{c}=\frac{(\beta -1)({A}_{t}{\phi }_{z}-1)}{\theta }+\tau \beta ({\phi }_{x}-1)-\rho$$
(23)
$$\dot{z}=\left(\frac{{A}_{t}{\phi }_{z}-1}{{\phi }_{zz}{A}_{t}^{2}}\right)\left[\delta -\frac{{u}_{P}\theta }{{u}_{c}(1-{A}_{t}{\phi }_{z})}+\tau \beta ({\phi }_{x}-1)+\frac{(\beta -1)({A}_{t}{\phi }_{z}-1)}{\theta }\right]$$
$$-\eta z-\frac{\eta {\phi }_{z}}{{A}_{t}{\phi }_{zz}}-\frac{{\phi }_{zx}}{{A}_{t}{\phi }_{zz}}\tau \beta \left[\phi (\tau \beta S,{A}_{t}z)-z-c-\tau \beta S\right]$$
(24)

As can be seen, the above two equations are similar to the corresponding two equations of our basic model, but with some important differences. First of all, since we have technological progress in this model, it affects both equations. In addition, the technological progress variable also includes the rate at which it increases in time (\(\eta\)) which affects only the optimal path of the polluting input in a negative way.

The solution to this new system consisting of Eqs. 19, 21, 23, and 24 can be seen in the following plots (Figs. 3 and 4).Footnote 12 The above system of differential equations allows us to find the numerical values of consumption, pollution, waste, and polluting resource, all of which help us to calculate the recycling level and the output level over time. We are interested in the relationships between our main variables (recycling and pollution) with output; hence, we will focus only on those plots.

Fig. 3
figure 3

Environmental Kuznets curve

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Fig. 4
figure 4

Recycling–output relationship

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Figure 3 depicts the relationship between pollution and output (EKC) which is once again being characterized by an inverse-U curve, while Fig. 4 shows that the relationship between recycling and output is again positive. It is clear that our results do not change noticeably and offer robustness to the initial simpler model.Footnote 13

Extension: the Circular Economy Model with Technological Progress as the Effect of the Polluting Input Changes over Time

In this section, we would like to see whether attaching a declining coefficient to the polluting input (\(z\)) can change the previous results. In other words, we reduce the influence that this input has both in production and in the two constraints. Hence, we want to focus more on recycling and on an economy that decides to be greener. Now, there is consumer pressure for greener economy, so the firm decides to transition from a semi-polluting firm to a green firm and as a result to use less amount of the polluting resource and over time reduce it to the minimum. That will have an effect on production and on the allocation of the final good, and also on the pollution stemming from that input. We want to examine if consumers will continue to recycle as much as they were doing before, or they will recycle less as there will be less pollution. A detailed explanation of this model, of the functions, and of the solution can be found in Appendix 12 where we also present the results of this extension.

Even though we add one more dynamic variable in our model which affects three of our main formulas, our results do not change significantly. In particular, we get an inverse-U-shaped curve for the EKC while the recycling-output curve is characterized by an upward slope curve. Once again, our results confirm the existence of the EKC as the main literature suggests (inverse-U curve) and we also offer robustness to the papers studying the relationship between recycling and output.

Conclusion

In this paper, we examined the relationship between pollution and output as well as the relationship between recycling and output. We focused our analysis on the environmental Kuznets curve (EKC) for which the main literature shows that the relationship between the pollution level of an economy and the output is characterized by an inverse-U curve. In addition, we studied the relationship between the recycling level of an economy and it’s output level. Previous papers dealing with this topic suggest that the relationship between abatement and output is characterized by a J curve. Our findings reproduce those patterns for the two plots and offer robustness to the existing literature supporting the inverse-U curve and the J curve, under a theoretical model.

Our theoretical model offers an extension of the analytical framework of George et al.[2] Our circular economy model suggests that the environmental Kuznets curve is characterized by an inverse-U curve, coming into agreement with other empirical and theoretical models of the literature. In addition, the shape that best characterizes the relationship between recycling and output is always an increasing curve, approaching the results of the literature which suggest a J curve for this relationship. Our predictions hold under three different theoretical models, which makes our results even stronger, showing that no matter how many additional variables we are using in our model, the curves remain the same.

Our results suggest that when the output of an economy increases, so does the recycling level, while the pollution level falls. We consider recycling to be the opposite of pollution, since recycling can contribute to a cleaner environment and is a measure of pollution abatement. As economies become richer (higher output level), they are able to spend a higher proportion of their resources on environmental friendly procedures (like recycling), leading to a reduction of the pollution and to a cleaner environment.

The model represents a circular economy that reuses raw materials through recycling of waste created during the production process. That general framework can represent many markets which use extracted and recycled material in their production. However, a constraint of the markets that it can represent is the assumption that the final good of this economy cannot lead to higher pollution levels. Hence, some examples of the markets we can represent are the following: jewelry market, firms producing reusable bottles/bags/straws/dryer balls, e-reader, and more. In addition, the materials used in the production should have the following characteristics: they should be extracted from earth, and they should have a non-renewable character. That being said, materials like metals (iron, coper, gold, etc.) could be an example, in addition to coal, natural gas, oil, and nuclear energy. However, our model does not take into account materials like biomass (wood, plant-based materials), fossil fuels (coal), specific metals (iron, copper), and non-metallic minerals (sand, limestone) as those are renewable. Moreover, our model cannot represent economies for which trade is available. Finally, an additional drawback is that we do not account for greenhouse gas emissions (GHG) produced during the production process while we focus on local air pollutants.

Further extensions can be made to this model in order to examine whether the relationships that we found for both the EKC and the recycling-output curve hold. One can use more inputs in the production process like capital or labor and also include a government which by imposing taxes on the firm for creating pollution units or by investing in abatement technology can lead to different results and it may cause the shape of the curves to change.