Recycling of multi-source waste in an aggregate circular economy

Abstract

We investigate how the relationship between capital accumulation and pollution is affected by the source of pollution: production or consumption. We are interested in polluting waste that cannot be naturally absorbed, but for which recycling efforts aim to avoid massive pollution accumulation with harmful consequences in the long run. Based on both environmental and social welfare perspectives, we determine how the interaction between growth and polluting waste accumulation is affected by the source of pollution, i.e., either consumption or production, and by the fact that recycling may or may not act as an income generator, i.e., either capital-improving or capital-neutral recycling efforts. Several new results are extracted regarding optimal recycling policy and the shape of the relationship between production and pollution. Beside the latter concern, we show both analytically and numerically that the optimal control of waste through recycling allows to reaching larger (resp., lower) consumption and capital stock levels under consumption-based waste compared to production-based waste while the latter permits to reach lower stocks of waste through lower recycling efforts.

Appendix

1.1 A1

The Legendre-Clebsch condition of joint concavity of the Hamiltonian with respect to the control variables is satisfied, because the Hessian $\left[ {\begin{array}{*{20}c} {H_{cc} } & {H_{cv} } \\ {H_{vc} } & {H_{vv} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 1} & 0 \\ 0 & { - 1} \\ \end{array} } \right]$ is negative definite. This guarantees a maximum of the Hamiltonian. Plugging the expressions of $c$ and $v$ respectively from (7) and (8) in (4) gives the maximized Hamiltonian:

$$H^{0} = \frac{{\left( {\theta - \lambda + \mu \beta } \right)^{2} }}{2} + \frac{{\left( {\lambda \varphi - \mu } \right)^{2} }}{2} + \lambda aK - \frac{{eW^{2} }}{2}$$

from which the Hessian $\left[ {\begin{array}{*{20}c} {H_{KK} } & {H_{KW} } \\ {H_{WK} } & {H_{WW} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & { - e} \\ \end{array} } \right]$ is negative semi-definite. This ensures that the necessary conditions are sufficient for optimality and thus the existence of a globally optimal solution. □

1.2 A2

The proof is by contradiction. Assume a time point $t$ such that the costate $\mu \ge 0$ and the transversality condition holds. Then, Eq. (6) implies that $\mu$ can grow only at a rate greater than $r$. Therefore, the transversality condition will never be met. Such a point cannot exist, i.e., $\mu < 0$. The result for $\lambda$ is derived in a similar way. Because we assume that $r < a$, starting from any $\lambda \left( 0 \right)$, $\lambda$ always tends to zero. □

1.3 A3

Differentiating (9), we find that $d\beta c - dv = 0$ if:

$$- d\lambda \left( {\beta + \varphi } \right) + d\mu \left( {1 + \beta^{2} } \right) = - \left( {r - a} \right)\lambda \left( {\beta + \varphi } \right)dt + \left( {r\mu dt + eWdt - d\eta } \right)\left( {1 + \beta^{2} } \right) = 0$$

and, with respect to $W = 0$, we have:

$$\dot{\eta } = r\mu - \frac{{\left( {\beta + \varphi } \right)\left( {r - a} \right)\lambda }}{{1 + \beta^{2} }}$$

(A3.1)

Since $r < a$, $\dot{\eta } \ge 0$ when $\lambda \ge 0$ and $\mu \ge 0$, which with respect to (9) requires:

$$\frac{\beta \theta }{{\beta + \varphi }} \le \lambda$$

(A3.2)

which holds only for positive $\lambda$. Such a solution is clearly feasible, if $c > 0$ and $v > 0$ and implies $\lambda$ remains positive and tends to zero while $\mu$ grows over time and satisfies the transversality condition by a growing negative jump. Next, we verify when the controls are feasible by substituting $\mu$ with respect to (9). Specifically, $c \ge 0$ if:

$$c = \theta - \lambda + \beta \mu = \theta - \lambda + \frac{{\beta \left[ {\left( {\beta + \varphi } \right)\lambda - \beta \theta } \right]}}{{1 + \beta^{2} }} = \theta \left( {1 - \frac{{\beta^{2} }}{{1 + \beta^{2} }}} \right) + \left( {\frac{\beta + \varphi }{{1 + \beta^{2} }} - 1} \right)\lambda \ge 0$$

or if:

$$\left( {1 + \beta^{2} - \beta - \varphi } \right)\lambda \le 0$$

(A3.3)

and $v = \varphi \lambda - \mu = \varphi \lambda - \frac{{\left( {\beta + \varphi } \right)\lambda - \beta \theta }}{{1 + \beta^{2} }} = \frac{{\varphi \left( {1 + \beta^{2} } \right)\lambda - \left( {\beta + \varphi } \right)\lambda + \beta \theta }}{{1 + \beta^{2} }}$, that is:

$$\left[ {\varphi \left( {1 + \beta^{2} } \right) - \left( {\beta + \varphi } \right)} \right]\lambda \ge - \beta \theta$$

(A3.4)

Let $\varphi \left( {1 + \beta^{2} } \right) - \left( {\beta + \varphi } \right) \le 0$, then (A3.4) always holds and (A3.4) along with (A3.2) results in:

$$\frac{\beta \theta }{{\beta + \varphi }} \le \lambda \le \frac{\beta \theta }{{\beta + \varphi - \varphi \left( {1 + \beta^{2} } \right)}}$$

(A3.5)

Since $r < a$, $\lambda$ always decreases to zero and the left-hand side will not hold, i.e., $\beta c - v > 0$. Therefore, $W$ increases. Letting $1 + \beta^{2} - \beta - \varphi > 0$, then if $\varphi \left( {1 + \beta^{2} } \right) - \left( {\beta + \varphi } \right) \le 0$ still holds, we get the same result, $W$ increases. If $\varphi \left( {1 + \beta^{2} } \right) - \left( {\beta + \varphi } \right) > 0$, then (A3.4) holds. Consequently, (A3.3) along with (A3.2) results in:

$$\frac{\beta \theta }{{\beta + \varphi }} \le \lambda \le \frac{\theta }{{1 + \beta^{2} - \beta - \varphi }}$$

(A3.6)

Again, $W$ will eventually increase. Thus, $W = 0$ can occur only during a finite interval. Finally, while $W = 0$, $K$ may decrease to zero if $\varphi < \frac{1}{\beta }$.

Regarding the fact that $K\left( 0 \right) = 0$ can be maintained when $W\left( 0 \right) > 0$ only over a finite time interval, this requires a jump as given by (9). The jump is due to requiring $\dot{K} = 0$, that is:

$$\varphi v - c = - \left( {\theta - \lambda + \mu \beta } \right) + \varphi \left( {\lambda \varphi - \mu } \right) = - \theta + \left( {1 + \varphi^{2} } \right)\lambda - \left( {\varphi + \beta } \right)\mu = 0$$

(A3.7)

which, when differentiating to ensure $dc - d\varphi v = 0$, results in:

$$d\lambda \left( {1 + \varphi^{2} } \right) - d\mu \left( {\varphi + \beta } \right) = \left( {rdt - adt - d\eta } \right)\lambda \left( {1 + \varphi^{2} } \right) + \left( {r\mu + eW} \right)\left( {\varphi + \beta } \right)dt = 0$$

(A3.8)

Finally, to have $K = 0$ and $W = 0$ concurrently along with $v > 0$ and $c > 0$ over an interval of time, we need $\dot{K} = 0$ and $\dot{W} = 0$, i.e., $\varphi = 1/\beta$. Assuming that $\varphi \ne 1/\beta$, this would apparently not be possible. Let $\beta \varphi > 1$. If both $W = 0$ and $K = 0$ at a point in time, then from that point on, the nil solution, i.e., no recycling and no consumption, is optimal over the entire time horizon. Note that $\dot{K} = aK - c + \varphi v < 0$ and $\dot{W} = \beta c - v < 0$, if $\beta c < v < \frac{c - aK}{\varphi }$. This implies that if $K$ is small and $\beta < 1$, then it is possible that $\dot{W} < 0$ and $\dot{K} < 0$ occurs starting from some point in time so that both $K$ and $W$ will become equal to zero. Note that, in practice, this solution corresponds to a negative capital state. □

1.4 A4

Equating the RHS of (9)-(10)-(5)-(6) to 0 and solving by identification and substitution, we get $\lambda^{S} = 0$ and $\mu^{S} = - \frac{\beta \theta }{{1 + \beta^{2} }}$ and $K^{S}$ and $W^{S}$ as given in (11). Plugging these expressions in (7) and (8), respectively, and simplifying yields $c^{S}$ and $v^{S}$ in (11). From (11), it can be shown that the limiting transversality conditions are satisfied for the saddle-paths because:

$$\mathop {\lim }\limits_{t \to \infty } e^{ - rt} \lambda \left( t \right)K\left( t \right) = 0$$

$$\mathop {\lim }\limits_{t \to \infty } e^{ - rt} \mu \left( t \right)W\left( t \right) = \mathop {\lim }\limits_{t \to \infty } \frac{{ - r\beta^{2} \theta^{2} e^{ - rt} }}{{e\left( {1 + \beta^{2} } \right)^{2} }} = 0$$

This ensures the uniqueness of the globally optimal solution. □

1.5 A5

To analyze the stability of the canonical system (10)-(11)-(5)-(6), we write the Jacobian:

$$J = \left[ {\begin{array}{*{20}l} a \hfill & 0 \hfill & {1 + \varphi^{2} } \hfill & { - \beta - \varphi } \hfill \\ 0 \hfill & 0 \hfill & { - \beta - \varphi } \hfill & {1 + \beta^{2} } \hfill \\ 0 \hfill & 0 \hfill & {r - a} \hfill & 0 \hfill \\ 0 \hfill & e \hfill & 0 \hfill & r \hfill \\ \end{array} } \right]$$

Given that $\lambda$ and $\mu$ are evaluated at their steady state value, we compute the determinant:

$$\left| J \right| = ae\left( {a - r} \right)\left( {1 + \beta^{2} } \right)$$

which has a positive value for $r < a$. Using Dockner’s formula (Dockner 1985), we determine the sum of the principal minors of $J$ of order 2 minus the squared discounting rate, that is:

$${\Psi } = - a\left( {a - r} \right) - e\left( {1 + \beta^{2} } \right)$$

The necessary and sufficient conditions that ensure that two eigenvalues have negative real parts and two have positive real parts, which corresponds to the case of a two-dimensional stable manifold, are $\left| J \right| > 0$ and ${\Psi } < 0$. The sign of ${\Psi }$ is negative whenever $r < a$, which implies that a two-dimensional stable manifold (saddle-point) exists in the case of a sufficiently patient social planner. To determine whether the optimal path is monotonic or oscillatory, we compute the expression (Dockner 1985):

$${\Omega } = {\Psi }^{2} - 4\left| J \right| = \left[ {a\left( {a - r} \right) - e\left( {1 + \beta^{2} } \right)} \right]^{2}$$

A positive (negative) sign of ${\Omega }$ indicates that convergence to the saddle-point is monotonic (spiraling) near the steady state. Because ${\Omega } > 0$, the convergence to the saddle-point is monotonic near the steady state. □

1.6 A6

Differentiating (7) with respect to time, we get:

$$\dot{c} = - \dot{\lambda } + \beta \dot{\mu } = - \left( {r - a} \right)\lambda + \beta \left( {r\mu + eW} \right)$$

(A6.1)

which, when differentiating again, leads to:

$$\ddot{c} = - \left( {r - a} \right)^{2} \lambda + \beta \left[ {r\left( {r\mu + eW} \right) + e\dot{W}} \right]$$

(A6.2)

Substituting $\beta \left( {r\mu + eW} \right)$ from (A6.1) and $\dot{W}$ from the state Eq. (3), we have:

$$\ddot{c} = r\dot{c} + a\left( {r - a} \right)\lambda + \beta e\left( {\beta c - v} \right)$$

Similarly, from (8), we obtain:

$$\dot{v} = \varphi \left( {r - a} \right)\lambda - \left( {r\mu + eW} \right)$$

which, differentiated again, gives:

$$\ddot{v} = \varphi \left( {r - a} \right)^{2} \lambda - \left[ {r\left( {r\mu + eW} \right) + e\dot{W}} \right] = \varphi \left( {r - a} \right)^{2} \lambda + \left[ {r\varphi \left( {r - a} \right)\lambda - e\left( {\beta c - v} \right)} \right]$$

Thus, accounting for $\lambda \left( t \right) = Ge^{{\left( {r - a} \right)t}}$, we get (13)–(14). The transient path converging to a steady state can then be found with the boundary conditions:

$$\begin{aligned} & \dot{v}\left( 0 \right) = \varphi \left( {r - a} \right)G - \left( {r\mu \left( 0 \right) + eW\left( 0 \right)} \right),\quad \dot{c} = - \left( {r - a} \right)G + \beta \left( {r\mu \left( 0 \right) + eW\left( 0 \right)} \right), \\ & \ddot{c}\left( 0 \right) = r\dot{c}\left( 0 \right) + a\left( {r - a} \right)G + \beta e\left( {\beta c\left( 0 \right) - v\left( 0 \right)} \right),\quad c\left( 0 \right) = \theta - G + \beta \mu \left( 0 \right), \\ & \ddot{v}\left( 0 \right) = r\dot{v}\left( 0 \right) - a\varphi \left( {r - a} \right)G - e\left( {\beta c\left( 0 \right) - v\left( 0 \right)} \right),\quad v\left( 0 \right) = \varphi G - \mu \left( 0 \right), \\ & \ddot{c} = - \left( {r - a} \right)^{2} \lambda \left( 0 \right) + \beta \left[ {r\left( {r\mu \left( 0 \right) + eW\left( 0 \right)} \right) + e\dot{W}\left( 0 \right)} \right],\quad \dot{W}\left( 0 \right) = \beta c\left( 0 \right) - v\left( 0 \right), \\ & \ddot{v}\left( 0 \right) = \varphi \left( {r - a} \right)^{2} \lambda \left( 0 \right) - \left[ {r\left( {r\mu \left( 0 \right) + eW\left( 0 \right)} \right) + e\dot{W}\left( 0 \right)} \right].\quad \square \\ \end{aligned}$$

1.7 A7

The Legendre–Clebsch condition of concavity of the Hamiltonian with respect to the control variables is satisfied, as $\left[ {\begin{array}{*{20}c} {H_{cc} } & {H_{cv} } \\ {H_{vc} } & {H_{vv} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 1} & 0 \\ 0 & { - 1} \\ \end{array} } \right]$ is negative definite. This guarantees a maximum of the Hamiltonian. Plugging the respective expressions of $c$ and $v$ in (4) gives the maximized Hamiltonian:

$$H^{0} = \frac{{\left( {\theta - \lambda } \right)^{2} }}{2} + \frac{{\left( {\lambda \varphi - \mu } \right)^{2} }}{2} + \left( {\lambda + \kappa \mu } \right)aK - \frac{{eW^{2} }}{2}$$

from which the Hessian matrix: $\left[ {\begin{array}{*{20}c} {H_{KK} } & {H_{KW} } \\ {H_{WK} } & {H_{WW} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & { - e} \\ \end{array} } \right]$ is negative semi-definite. This ensures that the necessary conditions are also sufficient for optimality. □

1.8 A8

The proof is identical to that for Lemma 2. □

1.9 A9

Let $W = 0$ and $K = 0$, $v = 0$ and $c = 0$ due to $\varphi \lambda - \mu \le 0$ and $c = \theta - \lambda \le 0$, that is:

$$\theta \le \lambda \le \mu /\varphi$$

(A9.1)

Then, both $\lambda > 0$ and $\mu > 0$ to meet (A9.1) and if:

$$\mu /\varphi > \theta$$

(A9.2)

then (A9.1) can be feasible. Accordingly, to maintain $W = 0$ and $K = 0$, we need to keep $v = 0$ and $c = 0$ which, with respect to (18) and (19), leads to:

$$dv = \varphi d\lambda - d\mu = \varphi \left[ {\left( {r - a} \right)\lambda dt - \mu \kappa adt - d\eta_{1} } \right] - \left( {r\mu dt - d\eta_{2} } \right) \le 0$$

$dc = - d\lambda = - \left( {r - a} \right)\lambda dt + \mu \kappa adt + d\eta_{1} \le 0$

Solving this system of inequalities we find:

$$\left( {r - a} \right)\lambda - \mu \kappa a - \frac{{r\mu - \dot{\eta }_{2} }}{\varphi } \le \dot{\eta }_{1}$$

(A9.3)

$$0 \le \dot{\eta }_{2} \le r\mu$$

(A9.4)

where $\left( {r - a} \right)\lambda - \mu \kappa a - \frac{{r\mu - \dot{\eta }_{2} }}{\varphi } < 0$ due to (A9.4). Thus, $\dot{\eta }_{1} \ge 0$ and $0 \le \dot{\eta }_{2} \le r\mu$ hold. Further, from (6) we observe that, starting from a positive value, the growth of $\mu$ can be fully offset by the maximal jump $\dot{\eta }_{2} = r\mu$, so that, $\dot{\mu } = r\mu - r\mu = 0$ which ensures the transversality condition. Further, although according to (A9.3) the optimality conditions can be met with a zero jump, $\dot{\eta }_{1} = 0$, $\lambda$ always decreases. Starting from a positive $\lambda$ that satisfies (A9.3), although $\mu$ is constant, $\dot{\mu } = 0$, (satisfying (A9.2)), we observe that decreasing $\lambda$ will inevitably violate (A9.1). However, for an optimal solution to be feasible in terms of the state constraints, we need the capital to grow by reducing consumption, which is already zero. However, an increase in recycling will immediately lead to a negative waste. Thus, there is no feasible solution to complete the paths. □

1.10 A10

Equating the RHS of (20)-(21)-(16)-(17) to 0 and solving by identification and substitution, we get:

$$\lambda^{S} = \frac{{\kappa^{2} \theta a}}{{\left( {1 + \varphi \kappa } \right)\left[ {\left( {1 + \varphi \kappa } \right)a - r} \right] + \kappa^{2} a}}$$

$$\mu^{S} = - \frac{{\kappa \theta \left( {a - r} \right)}}{{\left( {1 + \varphi \kappa } \right)\left[ {\left( {1 + \varphi \kappa } \right)a - r} \right] + \kappa^{2} a}}$$

along with $K^{S}$ and $W^{S}$ as given in (11). Note that $r < a$ implies that $r < \left( {1 + \kappa a} \right)a$, which ensures a feasible steady state. Plugging the expressions in (18) and (19), respectively, and simplifying, yields $c^{S}$ and $v^{S}$ in (22). From (22), the limiting transversality conditions are satisfied for the saddle-paths because:

$$\mathop {\lim }\limits_{t \to \infty } e^{ - rt} \lambda \left( t \right)K\left( t \right) = \mathop {\lim }\limits_{t \to \infty } \frac{{\kappa^{2} \theta^{2} a\left[ {\left( {1 + \varphi \kappa } \right)a - r} \right]e^{ - rt} }}{{a\left\{ {\left( {1 + \varphi \kappa } \right)\left[ {\left( {1 + \varphi \kappa } \right)a - r} \right] + \kappa^{2} a} \right\}^{2} }} = 0$$

$$\mathop {\lim }\limits_{t \to \infty } e^{ - rt} \mu \left( t \right)W\left( t \right) = \mathop {\lim }\limits_{t \to \infty } \frac{{ - r\kappa^{2} \theta^{2} \left( {a - r} \right)^{2} e^{ - rt} }}{{e\left\{ {\left( {1 + \varphi \kappa } \right)\left[ {\left( {1 + \varphi \kappa } \right)a - r} \right] + \kappa^{2} a} \right\}^{2} }} = 0$$

This ensures the uniqueness of the globally optimal solution. □

1.11 A11

The Jacobian matrix of the canonical system (21)-(22)-(16)-(17) is:

$$J = \left[ {\begin{array}{*{20}l} a \hfill & 0 \hfill & {1 + \varphi^{2} } \hfill & { - \varphi } \hfill \\ {\kappa a} \hfill & 0 \hfill & { - \varphi } \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & {r - a} \hfill & { - \kappa a} \hfill \\ 0 \hfill & e \hfill & 0 \hfill & r \hfill \\ \end{array} } \right]$$

Given that $\lambda$ and $\mu$ are evaluated at their steady state value, we compute the determinant:

$$\left| J \right| = ae\left\{ {\left( {1 + \varphi \kappa } \right)\left[ {\left( {1 + \varphi \kappa } \right)a - r} \right] + \kappa^{2} a} \right\}$$

which is positive for $r < a$, and the sum of the principal minors of $J$ of order 2 minus the squared discounting rate:

$${\Psi } = - a\left( {a - r} \right) - e$$

which is negative. Both signs imply that a two-dimensional stable manifold (saddle-point) exists. To check whether the optimal path is monotonic or cyclical, we compute:

$${\Omega } = \left[ {a\left( {a - r} \right) + e} \right]^{2} - 4ae\left\{ {\left( {1 + \varphi \kappa } \right)\left[ {\left( {1 + \varphi \kappa } \right)a - r} \right] + \kappa^{2} a} \right\}$$

Because the sign of ${\Omega }$ is ambiguous, a limit value analysis highlights the role of $a$, $e$, $\varphi$, and $\alpha$ in the sign of ${\Omega }$ for a given $r < a$ (see Table

Table 4 Limit value analysis

4). The results suggest that ${\Omega }$ can be either positive or negative depending on the parameters value. For any given $a$ and $e$, convergence is monotonic near the steady state. However, $\mathop {\lim }\limits_{{\kappa \to 0^{ + } }} {\Omega } > 0$ and $\mathop {\lim }\limits_{\kappa \to \infty } {\Omega } = - \infty$, as well as $\partial {\Omega }/\partial \alpha < 0$, which implies that there is a threshold $\tilde{\kappa } > 0$ such that for any $\kappa > \tilde{\kappa }$ ($\kappa < \tilde{\kappa }$), we have ${\Omega } < 0$ (${\Omega } > 0$).

A similar result is obtained for $\varphi$, i.e., for a high value of $\tilde{\varphi } > 0$ such that for any $\varphi > \tilde{\varphi }$, we get ${\Omega } < 0$. □

1.12 A12

Comparing $c_{{\left| {\kappa = 0} \right.}}^{S}$ from (12) and $c_{{\left| {\beta = 0} \right.}}^{S}$ from (22), for $\varphi = 0$ and $\kappa = \beta > 0$, $\kappa^{2} a > \beta^{2} \left( {a - r} \right)$, proves (22)–(26). □

1.13 A13

To compute the numerical solution of the consumption and production-based waste generation optimal control problems, we solve the related Hamilton–Jacobi–Bellman (HJB) equations, that is:

$$rV^{1} = c\left( {\theta - \frac{c}{2}} \right) - \frac{{eW^{2} }}{2} - \frac{{v^{2} }}{2} + V_{K}^{1} \left( {aK - c + \varphi v} \right) + V_{W}^{1} \left( {\beta c - v} \right)$$

for consumption-based waste generation, and:

$$rV^{2} = c\left( {\theta - \frac{c}{2}} \right) - \frac{{eW^{2} }}{2} - \frac{{v^{2} }}{2} + V_{K}^{2} \left( {aK - c + \varphi v} \right) + V_{W}^{2} \left( {\kappa aK - v} \right)$$

for production-based waste generation, where $V^{i} \left( {K,W} \right)$, $i = 1,2$, are the value functions respectively associated with the consumption and production-based waste generation optimal control problems. Assuming interior solutions, the necessary conditions are:

$$c = \theta - V_{K}^{1} + \beta V_{W}^{1} ,\quad v = \varphi V_{K}^{1} - V_{W}^{1}$$

for consumption-based waste generation, and:

$$c = \theta - V_{K}^{2} ,\quad v = \varphi V_{K}^{2} - V_{W}^{2}$$

for production-based waste generation. Using the necessary conditions, respectively, we get:

$$rV^{1} = \frac{{\left( {\theta + \beta V_{W}^{1} - V_{K}^{1} } \right)^{2} }}{2} - \frac{{\left( {\varphi V_{K}^{1} - V_{W}^{1} } \right)^{2} }}{2} - \frac{{eW^{2} }}{2} + aV_{K}^{1} K$$

for consumption-based waste generation, and:

$$rV^{2} = \frac{{\left( {\theta - V_{K}^{2} } \right)^{2} }}{2} - \frac{{\left( {\varphi V_{K}^{2} - V_{W}^{2} } \right)^{2} }}{2} - \frac{{eW^{2} }}{2} + a\left( {V_{K}^{2} + \kappa V_{W}^{2} } \right)K$$

for production-based waste generation. For the value functions, we consider the following conjectures:

$$V^{1} = \psi_{1} + \psi_{2} K + \psi_{3} W + \psi_{4} K^{2} + \psi_{5} W^{2} + \psi_{6} KW$$

for consumption-based waste generation, where $\psi_{1} , \ldots ,\psi_{6}$ are real parameters, and:

$$V^{2} = \xi_{1} + \xi_{2} K + \xi_{3} W + \xi_{4} K^{2} + \xi_{5} W^{2} + \xi_{6} KW$$

for production-based waste generation, where $\xi_{1} , \ldots ,\xi_{6}$ are real parameters.

Depending on whether the waste generation results from consumption or production, one solves a system of 6 algebraic equations in 6 unknowns in each case, that is:

$$r\xi_{1} - \frac{{\left( {\theta + \xi_{2} } \right)^{2} }}{2} - \frac{{\left( {\varphi \xi_{2} - \xi_{3} } \right)^{2} }}{2} - \beta \left( {\theta - \xi_{2} + \frac{{\beta \xi_{3} }}{2}} \right)\xi_{3} = 0$$

$$r\xi_{3} - 2\left[ {\beta \theta - \left( {\beta + \varphi } \right)\xi_{2} + \left( {1 + \beta^{2} } \right)\xi_{3} } \right]\xi_{5} + \left[ {\theta - \left( {1 + \varphi^{2} } \right)\xi_{2} + \left( {\beta + \varphi } \right)\xi_{3} } \right]\xi_{6} = 0$$

$$\left( {r - a} \right)\xi_{2} + 2\left[ {\theta - \left( {1 + \varphi^{2} } \right)\xi_{2} + \left( {\beta + \varphi } \right)\xi_{3} } \right]\xi_{4} - \left[ {\beta \theta + \left( {\beta + \varphi } \right)\xi_{2} + \left( {1 + \beta^{2} } \right)\xi_{3} } \right]\xi_{6} = 0$$

$$\left( {r - a} \right)\xi_{6} + 2\left[ {2\left( {\beta + \varphi } \right)\xi_{5} - \left( {1 + \varphi^{2} } \right)\xi_{6} } \right]\xi_{4} - \left[ {2\left( {1 + \beta^{2} } \right)\xi_{5} - \left( {\beta + \varphi } \right)\xi_{6} } \right]\xi_{6} = 0$$

$$r\xi_{4} - 2\left[ {a + \left( {1 + \varphi^{2} } \right)\xi_{4} - \left( {\beta + \varphi } \right)\xi_{6} } \right]\xi_{4} - \left( {1 + \beta^{2} } \right)\frac{{\xi_{6}^{2} }}{2} = 0$$

$$\left[ {r - 2\left( {1 + \beta^{2} } \right)\xi_{5} } \right]\xi_{5} + \left[ {2\left( {\beta + \varphi } \right)\xi_{5} - \left( {1 + \varphi^{2} } \right)\frac{{\xi_{6} }}{2}} \right]\xi_{6} + \frac{e}{2} = 0$$

for consumption-based waste generation, and:

$$r\psi_{1} - \frac{{\left( {\theta + \psi_{2} } \right)^{2} }}{2} - \frac{{\left( {\varphi \psi_{2} + \psi_{3} } \right)^{2} }}{2} = 0$$

$$r\psi_{3} - \left( {1 + \varphi^{2} } \right)\psi_{2} \psi_{6} + \left( {\theta + \varphi \psi_{3} } \right)\psi_{6} - 2\left( {\psi_{3} - \varphi \psi_{2} } \right)\psi_{5} = 0$$

$$\left[ {r - a - 2\left( {1 + \varphi^{2} } \right)\psi_{4} + \varphi \psi_{6} } \right]\psi_{2} - \left( {\kappa a + \psi_{6} } \right)\psi_{3} + 2\left( {\theta + \varphi \psi_{3} } \right)\psi_{4} = 0$$

$$- 2\left( {\kappa a - 2\varphi \psi_{4} } \right)\psi_{5} + \left[ {r - a - 2\left( {1 + \varphi^{2} } \right)\psi_{4} - 2\psi_{5} + \varphi \psi_{6} } \right]\psi_{6} = 0$$

$$\left[ {r - 2\left( {1 + a + \varphi^{2} \psi_{4} } \right)} \right]\psi_{4} - \left( {\kappa a - 2\varphi \psi_{4} - \frac{{\psi_{6} }}{2}} \right)\psi_{6} = 0$$

$$\left[ {r - 2\left( {\psi_{5} - \varphi \psi_{6} } \right)} \right]\psi_{5} - \frac{{\left( {1 + \varphi^{2} } \right)\psi_{6}^{2} }}{2} + \frac{e}{2} = 0$$

for production-based waste generation. In general, each system of algebraic equations above may admit zero, one or more solutions. An admissible solution for each system, if any, should satisfy the global asymptotic stability criterion. The necessary conditions for consumption-based waste generation rewrite:

$$c = \theta - \psi_{2} + \beta \psi_{3} - \left( {2\psi_{4} - \beta \psi_{6} } \right)K - \left( {\psi_{6} - 2\beta \psi_{5} } \right)W$$

$$v = \varphi \psi_{2} - \psi_{3} + \left( {2\varphi \psi_{4} - \psi_{6} } \right)K + \left( {\varphi \psi_{6} - 2\psi_{5} } \right)W$$

and the necessary conditions related to production-based waste generation become:

$$c = \theta - \xi_{2} - 2\xi_{4} K - \xi_{6} W$$

$$v = \varphi \xi_{2} - \xi_{3} + \left( {2\varphi \xi_{4} - \xi_{6} } \right)K + \left( {\varphi \xi_{6} - 2\xi_{5} } \right)W$$

Plugging the optimal expressions into the corresponding system of state equations, we get:

$$\dot{K} = - \theta + \left( {1 + \varphi^{2} } \right)\psi_{2} - \left( {\beta + \varphi } \right)\psi_{3} + \left[ {a + 2\left( {1 + \varphi^{2} } \right)\psi_{4} - \left( {\beta + \varphi } \right)\psi_{6} } \right]K - \left[ {2\left( {\beta + \varphi } \right)\psi_{5} - \left( {1 + \varphi^{2} } \right)\psi_{6} } \right]W$$

$$\dot{W} = \beta \theta - \left( {\beta + \varphi } \right)\psi_{2} + \left( {1 + \beta^{2} } \right)\psi_{3} - \left[ {2\left( {\beta + \varphi } \right)\psi_{4} - \left( {1 + \beta^{2} } \right)\psi_{6} } \right]K + \left[ {2\left( {1 + \beta^{2} } \right)\psi_{5} - \left( {\beta + \varphi } \right)\psi_{6} } \right]W$$

for consumption-based waste generation, and:

$$\dot{K} = - \theta + \left( {1 + 2\varphi^{2} } \right)\xi_{2} - \varphi \xi_{3} + \left[ {a + 2\left( {1 + 2\varphi^{2} } \right)\xi_{4} - \varphi \xi_{6} } \right]K + \left[ {\left( {1 + 2\varphi^{2} } \right)\xi_{6} - 2\varphi \xi_{5} } \right]W$$

$$\dot{W} = - \varphi \xi_{2} + \xi_{3} + \left( {\kappa a - 2\varphi \xi_{4} + \xi_{6} } \right)K + \left( {2\xi_{5} - \varphi \xi_{6} } \right)W$$

for production-based waste generation. Imposing a steady state for each state equations system yields:

$$K^{S} = \frac{{\theta \left[ {2\left( {2\beta^{2} + \beta \varphi + 1} \right)\psi_{5} - \left( {\beta \varphi^{2} + 2\beta + \varphi } \right)\psi_{6} } \right] - \left( {1 - \beta \varphi } \right)^{2} \left( {2\psi_{2} \psi_{5} - \psi_{3} \psi_{6} } \right)}}{{\left( {1 - \beta \varphi } \right)^{2} \left( {4\psi_{4} \psi_{5} - \psi_{6}^{2} } \right) + a\left[ {2\left( {1 + \beta^{2} } \right)\psi_{5} - \left( {\beta + \varphi } \right)\psi_{6} } \right]}}$$

$$W^{S} = \frac{{a\left[ {\beta \theta + \left( {\beta + \varphi } \right)\psi_{2} } \right] - a\left( {1 + \beta^{2} } \right)\psi_{3} + \theta \left[ {2\left( {\beta \varphi^{2} + 2\beta + \varphi } \right)\psi_{4} - \left( {2\beta^{2} + \beta \varphi + 1} \right)\psi_{6} } \right] + \left( {1 - \beta \varphi } \right)^{2} \left( {\psi_{2} \psi_{6} - 2\psi_{3} \psi_{4} } \right)}}{{\left( {1 - \beta \varphi } \right)^{2} \left( {4\psi_{4} \psi_{5} - \psi_{6}^{2} } \right) + a\left[ {2\left( {1 + \beta^{2} } \right)\psi_{5} - \left( {\beta + \varphi } \right)\psi_{6} } \right]}}$$

for consumption-based waste generation, and:

$$K^{S} = \frac{{\left( {1 + \varphi^{2} } \right)\left( {2\xi_{2} \xi_{5} - \xi_{3} \xi_{6} } \right) + \theta \left( {\varphi \xi_{6} - 2\xi_{5} } \right)}}{{a\left[ {\left( {\kappa + \varphi + 2\kappa \varphi^{2} } \right)\xi_{6} - 2\left( {1 + \kappa \varphi } \right)\xi_{5} } \right] - \left( {1 + \varphi^{2} } \right)\left( {4\xi_{4} \xi_{5} - \xi_{6}^{2} } \right)}}$$

$$W^{S} = \frac{{a\left[ {\kappa \theta - \left( {2\kappa \varphi^{2} + \kappa + \varphi } \right)\xi_{2} + \left( {1 + \kappa \varphi } \right)\xi_{3} } \right] - \theta \left( {2\varphi \xi_{4} - \xi_{6} } \right) - \left( {1 + \varphi^{2} } \right)\left( {\xi_{2} \xi_{6} - 2\xi_{3} \xi_{4} } \right)}}{{a\left[ {\left( {\kappa + \varphi + 2\kappa \varphi^{2} } \right)\xi_{6} - 2\left( {1 + \kappa \varphi } \right)\xi_{5} } \right] - \left( {1 + \varphi^{2} } \right)\left( {4\xi_{4} \xi_{5} - \xi_{6}^{2} } \right)}}$$

for production-based waste generation. Clearly, the positivity of the steady-state values should hold. Also, the steady-state solution given for each system of state equations, if any, should be globally asymptotically stable. The Jacobian matrix for consumption-based waste generation is:

$$J^{1} = \left[ {\begin{array}{*{20}l} {a + 2\left( {1 + 2\varphi^{2} } \right)\psi_{4} - \left( {\beta + \varphi } \right)\psi_{6} } \hfill & { - 2\left( {\beta + \varphi } \right)\psi_{5} - \left( {1 - \varphi^{2} } \right)\psi_{6} } \hfill \\ { - 2\left( {\beta + \varphi } \right)\psi_{4} + \left( {1 + \beta^{2} } \right)\psi_{6} } \hfill & {2\left( {1 + \beta^{2} } \right)\psi_{5} - \left( {\beta + \varphi } \right)\psi_{6} } \hfill \\ \end{array} } \right]$$

whose trace and determinant are respectively:

$$Tr J^{1} = a + 2\left( {1 + 2\varphi^{2} } \right)\psi_{4} + 2\left( {1 + \beta^{2} } \right)\psi_{5} - 2\left( {\beta + \varphi } \right)\psi_{6}$$

$$\left| { J^{1} } \right| = \left[ {a + 2\left( {1 + 2\varphi^{2} } \right)\psi_{4} - \left( {\beta + \varphi } \right)\psi_{6} } \right]\left[ {2\left( {1 + \beta^{2} } \right)\psi_{5} - \left( {\beta + \varphi } \right)\psi_{6} } \right] - \left[ {2\left( {\beta + \varphi } \right)\psi_{5} + \left( {1 - \varphi^{2} } \right)\psi_{6} } \right]\left[ {2\left( {\beta + \varphi } \right)\psi_{4} - \left( {1 + \beta^{2} } \right)\psi_{6} } \right]$$

On the other hand, the Jacobian matrix for production-based waste generation is:

$$J^{2} = \left[ {\begin{array}{*{20}l} {a + 2\left( {1 + 2\varphi^{2} } \right)\xi_{4} - \varphi \xi_{6} } \hfill & {\left( {1 + 2\varphi^{2} } \right)\xi_{6} - 2\varphi \xi_{5} } \hfill \\ {\kappa a - 2\varphi \xi_{4} + \xi_{6} } \hfill & {2\xi_{5} - \varphi \xi_{6} } \hfill \\ \end{array} } \right]$$

whose trace and determinant are respectively:

$$Tr J^{2} = a + 2\left( {1 + 2\varphi^{2} } \right)\xi_{4} + 2\left( {\xi_{5} - \varphi \xi_{6} } \right)$$

$$\left| { J^{2} } \right| = \left[ {a + 2\left( {1 + 2\varphi^{2} } \right)\xi_{4} - \varphi \xi_{6} } \right]\left( {2\xi_{5} - \varphi \xi_{6} } \right) - \left[ {\left( {1 + 2\varphi^{2} } \right)\xi_{6} - 2\varphi \xi_{5} } \right]\left( {\kappa a - 2\varphi \xi_{4} + \xi_{6} } \right)$$

Based on the global asymptotic stability conditions, that is, $Tr J^{i} < 0$ and $\left| { J^{i} } \right| > 0$, $i = 1,2$, the solution of $\psi_{1} , \ldots ,\psi_{6}$ for consumption-based waste generation, and $\xi_{1} , \ldots ,\xi_{6}$ for production-based waste generation can be selected.