A Circular Economy Model of Economic Growth with Circular and Cumulative Causation and Trade

Abstract

The idea of a circular economy, first discussed by Kenneth Boulding in the 1960s and ‘70 s and reintroduced by environmental economists David Pearce and R. Kerry Turner in 1990, is a characterization of how goods and services can be produced and consumed in an ecologically sound and environmentally sustainable manner that meets concerns of overuse of resources, waste management, and climate change, inter alia, through the conscious interlinking of disparate economic activities. The notion of circular and cumulative causation (CCC), through which certain positive and negative effects are promoted and reinforced by positive feedbacks, also is not new. George et al. (2015) have presented a theoretical circular economy model of economic growth that leads them to infer that the maintenance or improvement of environmental quality is incompatible with economic growth. However, Donaghy (2021b) has demonstrated how, when sources of CCC are introduced to the model of George et al., a different conclusion may be reached; CCC may be harnessed to bring about desirable systems properties, such as those of a circular market economy. The present paper reviews the arguments and findings of the latter two studies and extends the analysis by introducing trade between three national or regional economies in an environmentally polluting resource, other materials, and recycling technologies. Results of numerical simulation exercises suggest that there could be gains from trade in terms of progress by multiple economies in aggregate towards an international or interregional circular growth economy. The paper also suggests how aspects of the Pearce and Turner model of a circular economy not presently included in theoretical circular economy models of economic growth (closed or open) can be accommodated.

Appendices

Appendix A

The current-value Hamiltonian for the social planner’s optimal control problem posed in Donaghy (2021b) can be written as follows²³:

$$H={e}^{-\rho t}\frac{1}{\gamma }{\left(\omega c\cdot {P}^{-\eta }\right)}^{\gamma }+ \lambda {e}^{-\rho t}[Q-c-\alpha z{-I}_{k}(1+\frac{{a}_{1{I}_{k}}}{2K}){-I}_{h}(1+\frac{{a}_{2{I}_{h}}}{2HC}) -\beta S-\dot{S}]$$

$$+\mu_1e^{-\rho t}\left[\vartheta z-\delta P+\left(1-\beta\right)S-\dot P\right]+\mu_2e^{-\rho t}\lbrack\beta S-\dot R\rbrack+q_1e^{-\rho t}\left(I_k-\dot K\right)+q_2e^{-\rho t}\left(I_h-\dot{HC}\right).$$

(31)

The zero-order optimality conditions for the control variables, costate equations, and transversality conditions can be obtained in a manner similar to that presented for the model of George et al.²⁴

$${H}_{c}={(\omega c\cdot {P}^{-\eta })}^{\gamma -1}{P}^{-\eta }-\lambda =0.$$

(32)

$${H}_{z}=\lambda \left(\kappa {\theta }_{2}{Q}^{\frac{\sigma +\kappa }{\kappa }}{z}^{-\left(1+\sigma \right)}-\alpha \right)+{\mu }_{1}\vartheta =0$$

(33)

$${H}_{{I}_{K}}=-\lambda \left(1+{a}_{1}\frac{{I}_{K}}{K}\right)+{q}_{1}=0.$$

(34)

$${H}_{{I}_{HC}}=-\lambda \left(1+{a}_{2}\frac{{I}_{HC}}{HC}\right)+{q}_{2}=0.$$

(35)

$${H}_{S}=-\dot{\lambda }+\rho \lambda =\lambda \left({Q}_{S}-\beta \right)+{\mu }_{1}\left(1-\beta \right)+{\mu }_{2}\beta ,$$

(36)

where ${Q}_{S}=\kappa {\theta }_{1}{Q}^{\frac{\sigma +\kappa }{\kappa }}{(\beta S)}^{-(1+\sigma )}.$ 

$${H}_{R}=-\dot{{\mu }_{2}}+ \rho {\mu }_{2}=\lambda {Q}_{R}-{\mu }_{1}{\beta }_{R}S+{\mu }_{2}{\beta }_{R}S,$$

(37)

where

${Q}_{K}=\kappa {\theta }_{3}{Q}^{\frac{\sigma +\kappa }{\kappa }}{K}^{-(1+\sigma )}$, and ${\vartheta }_{K}=\epsilon \chi {HC}^{\epsilon }{K}^{-\epsilon -1}$.

$$H_K=-\dot{{q}}+\rho{q}_{1}= \lambda \left({Q}_{K}+\frac{{a}_{1}{I}_{HC}^{2}}{2{HC}^{2}}\right)+{\mu }_{1}{\vartheta }_{K},$$

(38)

$${H}_{HC}=-\dot{{q}_{2}}+\rho {q}_{2}=\lambda \left({Q}_{HC}+\frac{{a}_{2}{I}_{HC}^{2}}{2{HC}^{2}}\right)+{\mu }_{1}{\vartheta }_{HC}.$$

(39)

where

${Q}_{HC}= \kappa {\theta }_{4}{Q}^{\frac{\sigma +\kappa }{\kappa }}{HC}^{-(1+\sigma )}$ and ${\vartheta }_{HC}=-\epsilon \chi {HC}^{\epsilon -1}{K}^{-\epsilon }$

$$\underset{t\to \infty }{\mathrm{lim}}\lambda S{e}^{-\rho t}=0, \underset{t\to \infty }{\mathrm{lim}}{\mu }_{1}P{e}^{-\rho t}=0, \underset{t\to \infty }{\mathrm{lim}}{\mu }_{2}R{e}^{-\rho t}=0,$$

$$\underset{t\to \infty }{\mathrm{lim}}{q}_{1}K{e}^{-\rho t}=0, \underset{t\to \infty }{\mathrm{lim}}{q}_{2}HC{e}^{-\rho t}=0.$$

(40)

Manipulation of these equations yield expressions for the rates of change in c and z that are homologous to Eqs. (11) and (12) in the model of George et al. (2015) considered in Sect. 3.

To conduct a deterministic dynamic simulation with the model of Donaghy (2021b), the optimality conditions and co-state equations were manipulated to obtain a mathematically equivalent set of conditions that are strictly in terms of ‘observable quantities’ and easier to solve numerically. (Essentially, information in the first-order necessary conditions for an optimal control solution was used to substitute for co-state variables with expressions in terms of state and control variables while ensuring that all necessary feedbacks were retained.) The simulated model comprises.

• 1) Eqs. (15) and (16) as previously defined,

• 2) the following modifications of Eqs. (14), (17) and (18):

  $$\dot{S}=Q-c-\alpha z-k\bullet K\left(1+\frac{{a}_{1}k\cdot K}{2K}\right)- h\cdot HC\left(1+\frac{{a}_{2}h\cdot HC}{2HC}\right) -\beta S,$$

  (41)
  $$\dot{K}=k\cdot K,$$

  (42)

where $k\equiv \frac{\dot{K}}{K},$

$$\dot{HC}=h\cdot HC,$$

(43)

where $h\equiv \dot{\frac{HC}{HC},}$

• 3) two new second-order equations, defining rates of change in the rates of physical and human capital formation:where.

  $$\dot k=\left(\frac1{a_1}+k\right)\left(\rho+\frac{\acute{\lambda }}{\lambda }\right)-\frac{Q_K}{a_1}-0.5k^2-\frac{\mu_1}\lambda\vartheta_Kz,$$

  (44)
  $$\dot{h}=\left(\frac{1}{{a}_{2}}+h\right)\left(\rho +\frac{\acute{\lambda }}{\lambda }\right)-\frac{{Q}_{HC}}{{a}_{2}}-0.5{h}^{2}-\frac{{\mu }_{1}}{\lambda }{\vartheta }_{HC}z,$$

  (45)

$\lambda \equiv {U}_{c}={\left(\omega c\cdot {P}^{-\eta }\right)}^{\gamma -1}{P}^{-\eta }, \frac{\dot{\lambda }}{\lambda }=\frac{{U}_{cc}\cdot \dot{c}}{{U}_{c}},{U}_{cc}=(\gamma -1){(\omega c\cdot {P}^{-\eta })}^{\gamma -2}{P}^{-2\eta }$, and ${\mu }_{1}=\frac{\lambda \left(\alpha -{Q}_{z}\right)}{\vartheta },$ and ${Q}_{z}=\kappa {\theta }_{2}{Q}^{(\sigma +\kappa )/\kappa }{z}^{-(1+\sigma )}$.

• 4) two new first-order equations, defining the rates of change in consumption and use of the polluting resource:

  $$\dot{c}=\frac{\eta \gamma \left(\vartheta z-\delta P+\left(1-\beta \right)S\right)c}{P\left(\gamma -1-\frac{{U}_{cc}}{{U}_{c}}\right)},$$

  (46)
  $$\dot{z}=(\frac{\dot{{\mu }_{1}}}{{\mu }_{1}}-\frac{\dot{\lambda }}{\lambda })\left(\frac{{Q}_{z}-\alpha }{{Q}_{zz}}\right),$$

  (47)

where $\frac{\dot{{\mu }_{1}}}{{\mu }_{1}}=\rho +\delta -\eta {U}_{P}\vartheta /({U}_{c}\left({Q}_{z}-\alpha \right))$, ${U}_{P}=-\eta {\left(\omega c\cdot {P}^{-\eta }\right)}^{\gamma -1}c\cdot {P}^{-\eta -1},$ and ${Q}_{zz}=\left(\sigma +\kappa \right){\theta }_{2}{Q}^{\frac{\sigma }{\kappa }}{z}^{-\left(1+\sigma \right)}-\frac{\left(1+\sigma \right){Q}_{z}}{z},$ and.

• 5) the transversality conditions (40)

The modifications of Eqs. (14), (17) and (18) resulting in (41–43) just entail redefinition of variables.

To derive the second-order equation (44), divide (34) and (37) each by $\uplambda$ to obtain

$$-\left(1+{a}_{1}\frac{{I}_{K}}{K}\right)+{q}_{1}/\lambda =0,$$

(48)

and

$$-\dot{\frac{{q}_{1}}{\lambda }}+\frac{\rho {q}_{1}}{\lambda }=\left({Q}_{K}+\frac{{a}_{1}{I}_{K}^{2}}{2{K}^{2}}\right)+\left(\frac{{\mu }_{1}}{\lambda }\right){\vartheta }_{K}.$$

(49)

Then, defining k $\equiv$ I_K/K and q₁’$\equiv$ q₁/$\lambda$, note that $\dot{{q}_{1}{^{\prime}}}=\dot{({q}_{1}/\lambda )}=\frac{{\dot{q}}_{1}\lambda -{q}_{1}\dot{\lambda }}{{\lambda }^{2}}=\frac{\dot{{q}_{1}}}{\lambda }-\frac{{q}_{1}}{\lambda }\frac{\dot{\lambda }}{\lambda }=\dot{{q}_{1}}/\lambda -{q}_{1}^{^{\prime}}\frac{\dot{\lambda }}{\lambda }.$.

So $-\dot{{q}_{1}}/\lambda =-\dot{{q}_{1}^{^{\prime}}}+{q}_{1}^{^{\prime}}\frac{\dot{\lambda }}{\lambda }.$ Upon making substitutions for equivalent expressions in (37), we obtain

$$-\dot{{q}_{1}^{^{\prime}}}+{q}_{1}^{^{\prime}}\frac{\dot{\lambda }}{\lambda }+\rho {q}_{1}^{^{\prime}}={Q}_{K}+{a}_{1}{k}^{2}+\left(\frac{{\mu }_{1}}{\lambda }\right){\vartheta }_{K}.$$

(50)

Now differentiating (48) completely with respect to time and substituting into (50) $\left(1+{a}_{1}k\right)$ for ${q}_{1}^{^{\prime}}$ and $-{a}_{1}\dot{k}$ for $-\dot{{q}_{1}^{^{\prime}}}$ we obtain

$$-{a}_{1}\dot{k}+\left(\rho +\frac{\dot{\lambda }}{\lambda }\right)\left(1+{a}_{1}k\right)= {Q}_{K}+{a}_{1}{k}^{2}+\left(\frac{{\mu }_{1}}{\lambda }\right){\vartheta }_{K},$$

(51)

Which, upon rearranging, gives (44). 45) can be obtained by proceeding similarly.

To derive equation (46), first differentiate zero-order condition (32) totally with respect to time to get, upon cancellation of terms,

$$\frac{\left(\gamma -1\right)\dot{c}}{c}-\frac{\gamma \eta \dot{p}}{p}=\frac{\dot{\lambda }}{\lambda }.$$

(52)

Isolating $\dot{c}$ yields

$$\dot{c}=\frac{1}{(\gamma -1)}\left(\frac{\dot{\lambda }}{\lambda }+\gamma \eta \frac{\dot{P}}{P}\right)c.$$

(53)

Substituting $\frac{{U}_{cc\bullet }\dot{c}}{{U}_{c}}$ for $\frac{\dot{\lambda }}{\lambda }$ and isolating $\dot{c}$ once again yields

$$\dot{c}=\frac{\frac{\gamma \eta \dot{p}}{p}}{\left(1-\frac{{U}_{cc\bullet }c}{{U}_{c}\left(\gamma -1\right)}\right)},$$

(54)

where $\dot{p}$ is defined by (15).

To derive equation (47), rewrite (33) as

$${H}_{z}=\lambda \left({Q}_{z}-\alpha \right)+{\mu }_{1}\vartheta =0.$$

(55)

Isolate ${\mu }_{1}$ to obtain

$${\mu }_{1}=\frac{\lambda \left(\alpha -{Q}_{z}\right)}{\vartheta }.$$

(56)

Differentiate the resulting expression totally with respect to time to obtain

$$\dot{{\mu }_{1}}=\frac{\dot{\lambda }\left(\alpha -{Q}_{z}\right)}{\vartheta }+\frac{\lambda \left(-{Q}_{zz}\cdot \dot{z}\right)}{\vartheta }.$$

(57)

Dividing (57) by (56) yields

$$\frac{\dot{{\mu }_{1}}}{{\mu }_{1}}=\frac{\dot{\lambda }}{\lambda }+\frac{\left(-{Q}_{zz}\cdot\dot{z}\right)}{\left(\alpha -{Q}_{z}\right)}.$$

(58)

Isolating $\dot{z}$ in this expression yields (47).

Similar algebraic manipulations were performed to solve the model in the simulations discussed in Sect. 5 of this paper.

Appendix B

The current-value Hamiltonian for the problem faced by the social planner in each of the three countries, i = A, B, and C, in the absence of trade can be written as follows:

$$H_i=e^{-\rho t}\frac1\gamma\left(\omega c_i\cdot P_i^{-\eta}\right)^\gamma+\lambda_ie^{-\rho t}\left[Q_i-c_i-\alpha_{zi}z{-\alpha_{OMi}{MO}_i-I}_{ki}\left(1+\frac{a_{1I_{ki}}}{2K_i}\right)-\overline{h_i}{HC}_i\left(1+0.5a_2\overline{h_i}\right)-\beta S_i-\dot{S_i}\right]+\mu_{1i}e^{-\rho t}\left[\vartheta_iz_i-\delta P_i+\left(1-\beta_i\right)S_i-\dot{P_i}\right]+\mu_{2i}e^{-\rho t}\left[\beta_iS_i-\dot{R_i}\right]$$

$$+{q}_{i}{e}^{-\rho t}\left[{I}_{ki}-\dot{{K}_{i}}\right].$$

(59)

The zero-order optimality conditions for the control variables, the costate equations, and transversality conditions can be obtained by appropriate differentiation of (5.7) as follows.

$${H}_{ci}={(\omega {c}_{i}\cdot {{P}_{i}}^{-\eta })}^{\gamma -1}{{P}_{i}}^{-\eta }-{\lambda }_{i}=0.$$

(60)

$${H}_{zi}={\lambda }_{i}\left(\kappa {\theta }_{2}{{Q}_{i}}^{\frac{\sigma +\kappa }{\kappa }}{{z}_{i}}^{-\left(1+\sigma \right)}-\alpha \right)+{\mu }_{1}{\vartheta }_{i}=0.$$

(61)

$${H}_{{I}_{Ki}}=-{\lambda }_{i}\left(1+{a}_{1}\frac{{I}_{Ki}}{{K}_{i}}\right)+{q}_{1i}=0.$$

(62)

$${H}_{Si}=-\dot{{\lambda }_{i}}+\rho {\lambda }_{i}={\lambda }_{i}\left({Q}_{Si}-{\beta }_{i}\right)+{\mu }_{1}\left(1-{\beta }_{i}\right)+{\mu }_{2}{\beta }_{i},$$

(63)

where

$${Q}_{Si}=\kappa {\theta }_{1}{{Q}_{i}}^{\frac{\sigma +\kappa }{\kappa }}{({\beta }_{i}{S}_{i})}^{-(1+\sigma )}.$$

$${H}_{Pi}=-\dot{{\mu }_{1i}}+\rho {\mu }_{1i}=-\eta {\left(\omega {c}_{i}\cdot {{P}_{i}}^{-\eta }\right)}^{\gamma -1}{c}_{i}\cdot {P}^{-\eta -1}-{\mu }_{1i}.$$

(64)

$${H}_{Ri}=-\dot{{\mu }_{2i}}+ \rho {\mu }_{2i}=\lambda {Q}_{Ri}-{\mu }_{1}{\beta }_{Ri}{S}_{i}+{\mu }_{2}{\beta }_{Ri}{S}_{i},$$

(65)

where

${Q}_{Ri}=\frac{\kappa {\theta }_{1}{Q}^{\frac{\sigma +\kappa }{\kappa }}{\left({\beta }_{i}{S}_{i}\right)}^{-\left(1+\sigma \right)}{{\beta }_{i}}^{2}\phi {S}_{i}}{{{R}_{i}}^{2}},$ and ${\beta }_{Ri}=\frac{{{\beta }_{i}}^{2}\phi }{{{R}_{i}}^{2}}$.

$${H}_{Ki}= -\dot{{q}_{1i}}+\rho {q}_{1i}=\lambda \left({Q}_{Ki}+\frac{{a}_{1}{I}_{Ki}^{2}}{2{{K}_{i}}^{2}}\right)+{\mu }_{1i}{\vartheta }_{Ki},$$

(66)

where

${Q}_{Ki}=\kappa {\theta }_{3}{{Q}_{i}}^{\frac{\sigma +\kappa }{\kappa }}{{K}_{i}}^{-(1+\sigma )},$ and ${\vartheta }_{Ki}=\epsilon \chi {{HC}_{i}}^{\epsilon }{{K}_{i}}^{-\epsilon -1}$.

$$\underset{t\to \infty }{\mathrm{lim}}{\lambda }_{i}{S}_{i}{e}^{-\rho t}=0, \underset{t\to \infty }{\mathrm{lim}}{\mu }_{1}{P}_{i}{e}^{-\rho t}=0, \underset{t\to \infty }{\mathrm{lim}}{\mu }_{2}{R}_{i}{e}^{-\rho t}=0, \underset{t\to \infty }{\mathrm{lim}}{q}_{1i}{K}_{i}{e}^{-\rho t}=0.$$

(67)

In the second simulation in which there is trade between the three economies, the zero-order optimality condition for the polluting resource, (B.3) is modified for economies A, and C as follows:

$${H}_{zi}={\lambda }_{i}\left(\kappa {\theta }_{2}{{Q}_{i}}^{\frac{\sigma +\kappa }{\kappa }}{{z}_{i}}^{-\left(1+\sigma \right)}-{\alpha }_{zi}\right)+{\mu }_{1}\left({\vartheta }_{i}+{\boldsymbol{\varphi }}_{{\varvec{z}}{\varvec{i}}}\right)=0, i=\mathrm{A\; and\; C}.$$

(68)