A Circular Model of Economic Growth and Waste Recycling

Abstract

In this paper, we are examining the environmental Kuznets curve (EKC) in the context of a circular economy recycling model. Our model assumes that there are only two factors of production in the production of the final good, a recyclable and a polluting input. We also present two extensions of this basic model by adding technological progress in the production of the polluting input and a dynamic reduction of the influence of the same input respectively. Our results suggest the presence of an inverse-U curve for the EKC confirming the findings of most of the current literature as well as an increasing curve for the recycling output nexus.

Appendices

Appendix 1 Plots

A Plots Based on Our Basic Model

In this section of our paper, we will present more plots showing that the relationship between pollution and output (EKC) as well as recycling and output does not change dramatically when we change the weight ($\gamma$) at which firm uses the two inputs in production.¹⁴ In addition, those plots hold under many other combinations of all the parameter values and initial values for our variables, but they are not included to conserve space (Figs. 5, 6, 7, and 8).

Fig. 5

Environmental Kuznets curve for g = 0.4

Fig. 6

Recycling–output relationship for g = 0.4

Fig. 7

Environmental Kuznets curve for g = 0.6

Fig. 8

Recycling–output relationship for g = 0.6

B Plots Based on the First Extension of the Basic Model

In that section, we will display some more plots based on the extension of our basic model (Section “8”), using different values for the wight ($\gamma$) that the firm uses. As before, the plots do not change in a high degree and they still hold under many different combinations of all the other parameter values as well as initial values for our main variables (Figs. 9, 10, 11, and 12).

Fig. 9

Environmental Kuznets curve for g = 0.2

Fig. 10

Recycling–output relationship for g = 0.2

Fig. 11

Environmental Kuznets curve for g = 0.5

Fig. 12

Recycling–output relationship for g = 0.5

Appendix 2 Extension: The Circular Economy Model with Technological Progress as the Effect of the Polluting Input Changes over Time

An interesting question that we should look into more detail is the importance of the effect that the polluting input has on this economy. Specifically, we want to investigate if a usage reduction of this input in time will change our results. In that case, we assume that there is consumer pressure to the firm for a greener economy, so the firm decides to use less amount of this input in the production process. In that way, the firm will be transformed into a greener firm leading to lower pollution levels coming not only from production but also from the waste stemming from production, satisfying consumer’s preferences in a higher degree.

Under those assumptions, the economy can end up following two totally different paths. In the first one, recycling might fall as less polluting input is used and less pollution units are created, confusing consumers’ who might create false beliefs about the environmental quality and start recycle less. On the other hand, the second scenario is the complete opposite of the previous one. In that one, people will still want to have an as clean as possible environment and continue recycling no matter what the firm decides to do. The difference between those two scenarios is that the first one assumes consumers who care about the environmental quality up to a specific degree and mainly for the current pollution, while the second one assumes consumers who care about the current and future environmental quality. In our model, we do not make any further assumptions about the consumers and we allow the model to reveal which scenario will be the dominant one under the assumptions made on Sections “6” and “8”

To examine whether recycling motives change or not under this economy, we assume that firms decide to constantly use less amount of the polluting input. For that reason, we introduce a new variable $\alpha$ which will represent the effect of the polluting input, and it will be reduced in time. So, the effect of this input is given by the following formula for which holds:

$${\alpha }_{t}={\alpha }_{0}{e}^{(-lt)} \mathrm{and} \frac{{\dot{\alpha }}_{t}}{{\alpha }_{t}}=-l$$

(25)

where ${\alpha }_{0}$ is the initial value of the effect and $l$ shows the rate at which the effect of the polluting input changes in time.

In that case, the new amount of the polluting input is equal to ${\alpha }_{t}z$ instead of $z$ that was previously leading to changes of some functions. Specifically, the production function, the waste accumulation path, and the pollution accumulation path have to change because of the addition of the dynamic effect of the polluting input, while the utility function and the instantaneous utility function will remain the same. Hence, the functions we are using to solve this problem are the following:

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

(26)

$$u(c,P)=wlnc-(1-w)lnP$$

(27)

$$\phi (x,z)=\phi (\tau \beta S,\alpha Az)=(\tau \beta S{)}^{(1-\gamma )}(\alpha Az{)}^{\gamma }$$

(28)

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

(29)

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

(30)

where A stands for ${A}_{t}={A}_{0}{e}^{(\eta t)}$ and $\alpha$ stands for ${\alpha }_{t}={\alpha }_{0}{e}^{(-lt)}$ and they are the technological progress and the dynamic effect of the polluting resource respectively.

Once again, we have to maximize the social welfare function (Eq. 26) subject to the two constraints that we have: the pollution accumulation path (Eq. 29) and the waste accumulation path (Eq. 30), both of which differ compared to the previous models we discussed. To solve this problem, we will follow similar steps as in the initial model and we will also use the following Hamiltonian function:

$$H={e}^{-\rho t}u(c,P)+\lambda [\phi (\tau \beta S,\alpha Az)-c-\alpha z-\tau \beta S]$$

$$+\mu [\alpha \theta z-\delta P+(1-\beta )S]$$

(31)

where $\lambda$ is the shadow price of the waste accumulation path and $\mu$ is the shadow price of the pollution accumulation path. For the above problem, we assume again that $c$ and $z$ are the control variables while $P$ and $S$ are the state variables. Under these assumptions, the Euler equation for consumption and the optimal path of the polluting input can be seen in the following two equations respectively:

$$\frac{\dot c}c=\frac{(\beta-1)(A_t\phi_z-1)}\theta+\tau\beta(\phi_x-1)-\rho$$

(32)

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

$$+(l-\eta )z+\frac{\eta z}{{A}_{t}{\alpha }_{t}(1-\gamma )}-\frac{\tau \beta {\phi }_{zx}}{{A}_{t}{\alpha }_{t}{\phi }_{zz}}\left[\phi (\tau \beta S,{\alpha }_{t}{A}_{t}z)-{\alpha }_{t}z-c-\tau \beta S\right]$$

(33)

As we can see, these two functions are similar to the two functions we obtained in Section “Extension: the Circular Economy Model with Technological Progress” in the extension of the initial model. First of all, we see that the Euler function for consumption does not change at all since it does not include the effect of the polluting resource. However, it is affected by some indirect effects coming from the substitution of the derivatives of the production function. On the other hand, the optimal path of the polluting input has a similar function as in the extension model one (Eq. 24), with some differences stemming from the introduction of the polluting input’s effect.

The solution of the system consisting of Eqs. 29, 30, 32, and 33 can be seen in the following plots.¹⁵ The way of construction of the following plots is similar to the previous two cases: the system of differential equations allows us to calculate directly the numerical values of consumption, pollution, waste, and polluting input which then allow us to calculate recycling and output level over time.

The relationship between pollution and output (EKC) (Fig. 13) is given by an inverse-U curve, while the relationship between recycling and output (Fig. 14) is still an increasing curve.¹⁶ In both plots, we obtain similar curves as in the plots we had earlier, offering robustness to our previous results. We see that no matter how many variables we add or make them change in time, the plots we are getting are similar to our initial findings, supporting the existing literature as well. As we can see, as economy grows pollution falls and recycling increases, showing that the consumers decide to recycle even under the case that the firm does not pollute the environment in a high degree, which means that the second scenario discussed earlier comes true.

Fig. 13

Environmental Kuznets curve

Fig. 14

Recycling–output relationship

Appendix 3 Parameter Values

In this section, we will provide some more information about the values of the parameters used on all models discussed on this paper. The following table (Table 1) presents the values of the parameters we used to generate the plots of our initial model described in Section “The Circular Economy Model,” while Tables 2 and 3 present the parameter values of the extension models 1 and 2 described in Sections “8” and “9” respectively.¹⁷ Even though we chose the following specific values, our results are robust to many other combinations of the parameters values and initial values for our variables.

Table 1 Parameter values of the initial model

Table 2 Parameter values of extension model 1

Table 3 Parameter values of extension model 2

Unfortunately, we are not able to double check if the values we chose for the parameters are consistent with the theory, since the theoretical literature on that topic is quite thin. However, we believe that this is not a problem and it does not reduce the significance of our results since we are able to get similar plots under many other combinations of the parameter values.