Production-based pollution versus deforestation: optimal policy with state-independent and-dependent environmental absorption efficiency restoration process

Abstract

An important yet largely unexamined issue is how the interaction between deforestation and pollution affects economic and environmental sustainability. This article seeks to bridge the gap by introducing a dynamic model of pollution accumulation where polluting emissions can be mitigated and the absorption efficiency of pollution sinks can be restored. We assume that emissions are due to a production activity, and we include deforestation both as an additional source of emissions and as a cause of the exhaustion of environmental absorption efficiency. To account for the fact that the switching of natural sinks to a pollution source can be either possible, and in such a case even reversible, or impossible, we consider that restoration efforts can be either independent from or dependent on environmental absorption efficiency, i.e., state-independent versus state-dependent restoration efforts. We determine (i) whether production or deforestation is the most detrimental from environmental and social welfare perspectives, and (ii) how state-dependent restoration process affects pollution accumulation and deforestation policies and the related environmental and social welfare consequences.

Appendix

A1. The Legendre-Clebsch condition holds for (4) since the Hessian is negative definite, that is:

$$ \left[ {\begin{array}{*{20}c} {H_{uu} } \\ {H_{vu} } \\ {H_{wu} } \\ \end{array} \begin{array}{*{20}c} {H_{uv} } \\ {H_{vv} } \\ {H_{wv} } \\ \end{array} \begin{array}{*{20}c} {H_{uw} } \\ {H_{vw} } \\ {H_{ww} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 1} \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ { - 1} \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ { - 1} \\ \end{array} } \right] $$

The Hamiltonian is therefore concave with respect to $\left( {u,v,w} \right)$, which guarantees a (local) maximum.

A2. The candidate arc segments of the optimal path are described in the following way.

1.1 Production-based emissions and deforestation arc

In this case, Eq. (7) yields $\left( {u^{*} , v^{*} , w^{*} } \right) = \left( {u^\circ \left( \lambda \right), v^\circ \left( {\lambda , \varphi } \right), w^\circ \left( {A^{*} , \varphi } \right)} \right)$ and $\rho_{1} = 0$, $\rho_{2} = 0$, $\rho_{3} = 0$, with:

$$ \left( {u^\circ v^\circ w^\circ } \right)^{t} \text{ := }\left( {a + \lambda b + \lambda \alpha - \varphi \varphi A^{\beta } } \right)^{t} $$

(A2.1)

being the solution of $H_{u} \left( {P, A, u^\circ , v^\circ , w^\circ , \lambda , \varphi } \right) = 0$, $H_{v} \left( {P, A, u^\circ , v^\circ , w^\circ , \lambda , \varphi } \right) = 0$ and $H_{w} \left( {P, A, u^\circ , v^\circ , w^\circ , \lambda , \varphi } \right) = 0$, respectively.

Plugging the expressions of $u^{*} , v^{*}$ and $w^{*}$ in (1)–(2) and (9)–(10), respectively, gives:

$$ \dot{P} = a + \left( {1 + \alpha^{2} } \right)\lambda + \alpha \left( {b - \varphi } \right) - AP $$

(A2.2)

$$ \dot{A} = \left( {1 + A^{2\beta } } \right)\varphi - b - \lambda \alpha - \gamma P $$

(A2.3)

$$ \dot{\lambda } = \left( {r + A} \right)\lambda + \gamma \varphi + cP $$

(A2.4)

$$ \dot{\varphi } = \left( {r - \beta \varphi A^{2\beta - 1} } \right)\varphi + \lambda P $$

(A2.5)

To analyze the behavior of the canonical system (A2.2)–(A2.5) in the neighborhood of the steady state, if it exists, we compute the Jacobian matrix:

$$ J = \left[ {\begin{array}{*{20}c} { - A} \\ { - \gamma } \\ c \\ \lambda \\ \end{array} \begin{array}{*{20}c} { - P} \\ {2\beta \varphi A^{2\beta - 1} } \\ \lambda \\ { - \beta \left( {2\beta - 1} \right)\varphi^{2} A^{2\beta - 2} } \\ \end{array} \begin{array}{*{20}c} {1 + \alpha^{2} } \\ { - \alpha } \\ {r + A} \\ P \\ \end{array} \begin{array}{*{20}c} { - \alpha } \\ {1 + A^{2\beta } } \\ \gamma \\ {r - \beta \varphi A^{2\beta - 1} } \\ \end{array} } \right] $$

(A2.6)

where $\left( {P, A,\lambda ,\varphi } \right)$ are evaluated at their steady-state values, and whose determinant is given by:

$$ \left| J \right| = \beta \varphi^{2} A^{2\beta - 2} \left\{ {\left( {1 - 2\beta } \right)\left[ {\left( {\alpha \gamma + A} \right)\left( {r + \alpha \gamma + A} \right) + \gamma^{2} + c} \right] + \left( {1 + 2\beta } \right)A^{2\beta } \left[ {A\left( {r + A} \right) + c\left( {1 + \alpha^{2} } \right)} \right]} \right\} - 2\beta \varphi A^{2\beta - 1} \left\{ {r\left[ {A\left( {r + A} \right) + c\left( {1 + \alpha^{2} } \right)} \right] - \left( {r + 2A} \right)\left( {\alpha \lambda + \gamma P} \right) - 2\left[ {\gamma \lambda \left( {1 + \alpha^{2} } \right) - \alpha cP} \right]} \right\} - \alpha^{2} \lambda^{2} A^{2\beta } + \left( {1 + A^{2\beta } } \right)\left[ {cP^{2} - \lambda \left( {\lambda + rP + 2AP} \right)} \right] - r\lambda \left[ {\gamma \left( {1 + \alpha^{2} } \right) + \alpha A} \right] + P\left\{ {\alpha \left( {rc - 2\gamma \lambda } \right) + \gamma \left[ {\gamma P - r\left( {2r + A} \right)} \right]} \right\} $$

and the sum of the principal minors of $J$ of order 2 minus the squared discounting rate is:

$$ K = - A\left( {r + A} \right) - c\left( {1 + \alpha^{2} } \right) + 2\left( {\alpha \lambda - \gamma P} \right) + \beta \varphi A^{2\beta - 1} \left\{ {2r + \varphi A^{ - 1} \left[ {2\beta - 1 - \left( {2\beta + 1} \right)A^{2\beta } } \right]} \right\} $$

In the case of state-independent restoration efforts, $K_{{\left| {\beta = 0} \right.}} < 0$, which rules out the possibility of limit cycles (Dockner and Feichtinger 1991). For linearly state-dependent restoration efforts, the sign of $K_{{\left| {\beta = 1} \right.}} = - A\left( {r + A} \right) - c\left( {1 + \alpha^{2} } \right) + 2\left( {\alpha \lambda - \gamma P} \right) + \varphi \left[ {2rA + \varphi \left( {1 - A^{2} } \right)} \right]$ is not clear.

1.2 Production-based emissions-only arc

In this case, the control constraint $v\left( t \right) \ge 0$ is active. Using (5), the maximizing condition (6) yields $\left( {u^{*} , v^{*} , w^{*} } \right) = \left( {u^\circ \left( \lambda \right), 0, w^\circ \left( {A^{*} , \varphi } \right)} \right)$ and $\rho_{1} = 0$, $\rho_{2} \le 0$, $\rho_{3} = 0$, with:

$$ \left( {u^\circ w^\circ } \right)^{t} \text{ := }\left( {a + \lambda \varphi A^{\beta } } \right)^{t} \;{\text{and}}\;\rho_{2} = \varphi - b - \lambda \alpha $$

(A2.7)

being the solution of $L_{u} \left( {P, A, u^\circ , 0, w^\circ , \lambda , \varphi , 0, \rho_{2} , 0} \right) = 0$, $L_{v} \left( {P, A, u^\circ , 0, w^\circ , \lambda , \varphi , 0, \rho_{2} , 0} \right) = 0$ and $L_{w} \left( {P, A, u^\circ , 0, w^\circ , \lambda , \varphi , 0, \rho_{2} , 0} \right) = 0$, respectively. Plugging the expressions of $u^{*}$ and $w^{*}$ in (1)–(2) and (9)–(10) respectively gives the canonical system:

$$ \dot{P} = a + \lambda - AP $$

(A2.8)

$$ \dot{A} = \varphi A^{2\beta } - \gamma P $$

(A2.9)

$$ \dot{\lambda } = \left( {r + A} \right)\lambda + \gamma \varphi + cP $$

(A2.10)

$$ \dot{\varphi } = \left( {r - \beta \varphi A^{2\beta - 1} } \right)\varphi + \lambda P $$

(A2.11)

The steady state, if it exists, is obtained by solving (11) where:

$$ \left( {\varphi_{\infty } \lambda_{\infty } A_{\infty } } \right)^{t} = \left( {\frac{{\gamma a\left( {r + A_{\infty } } \right)}}{{{\varPhi }}} - \frac{{a\left( {\gamma^{2} + cA_{\infty }^{2\beta } } \right)}}{{{\varPhi }}} \frac{{a\left( {r + A_{\infty } } \right)A_{\infty }^{2\beta } }}{{{\varPhi }}}} \right)^{t} $$

with ${{\varPhi }} = A_{\infty }^{2\beta } \left[ {c + A_{\infty } \left( {r + A_{\infty } } \right)} \right] + \gamma^{2}$. It can be shown that the resolution of the system (A2.8)–(A2.11) for the case of state-independent restoration efforts ($\beta = 0$), leads to a steady state that is a saddle-point with either monotonic or spiraling convergence. On the other hand, if $\beta = 1$, we obtain $K_{{\left| {\beta = 1} \right.}} = - A\left[ {r + A\left( {1 + 3\varphi^{2} } \right)} \right] - 2\gamma P - c + 2r\varphi A$. Therefore, the possibility of limit cycles cannot be ruled out for linearly state-dependent restoration efforts.

1.3 Deforestation-only arc

In this case, the control constraint $u\left( t \right) \ge 0$ is active. Using (5), the maximizing condition (6) yields $\left( {u^{*} , v^{*} , w^{*} } \right) = \left( {0, v^\circ \left( {\lambda ,\varphi } \right), w^\circ \left( {A^{*} , \varphi } \right)} \right)$ and $\rho_{1} \le 0$, $\rho_{2} = 0$, $\rho_{3} = 0$, with:

$$ \left( {v^\circ w^\circ } \right)^{t} \text{ := }\left( {b + \lambda \alpha - \varphi \varphi A^{\beta } } \right)^{t} \;{\text{and}}\;\rho_{1} = - \lambda - a $$

(A2.12)

being the solution of $L_{u} \left( {P, A, 0,v^\circ , w^\circ , \lambda , \varphi , \rho_{1} , 0, 0} \right) = 0$, $L_{v} \left( {P, A, 0,v^\circ , w^\circ , \lambda , \varphi , \rho_{1} , 0, 0} \right) = 0$ and $L_{w} \left( {P, A, 0,v^\circ , w^\circ , \lambda , \varphi , \rho_{1} , 0, 0} \right) = 0$, respectively. Plugging the expressions of $v^{*}$ and $w^{*}$ in (1)–(2) and (9)–(10) respectively gives the canonical system:

$$ \dot{P} = \alpha \left( {b + \lambda \alpha - \varphi } \right) - AP $$

(A2.13)

$$ \dot{A} = \varphi \left( {1 + A^{2\beta } } \right) - b - \lambda \alpha - \gamma P $$

(A2.14)

$$ \dot{\lambda } = \left( {r + A} \right)\lambda + \gamma \varphi + cP $$

(A2.15)

$$ \dot{\varphi } = \left( {r - \beta \varphi A^{2\beta - 1} } \right)\varphi + \lambda P $$

(A2.16)

The steady state, if it exists, is obtained by solving (11) where:

$$ \left( {\varphi_{\infty } \lambda_{\infty } A_{\infty } } \right)^{t} = \left( {\frac{{b\left( {r + A_{\infty } } \right)\left( {\alpha \gamma + A_{\infty } } \right)}}{{{\varPsi }}} - \frac{{b\left[ {\gamma \left( {\alpha \gamma + A_{\infty } } \right) + \alpha cA_{\infty }^{2\beta } } \right]}}{{{\varPsi }}} \frac{{\alpha b\left( {r + A_{\infty } } \right)A_{\infty }^{2\beta } }}{{{\varPsi }}}} \right)^{t} $$

with ${{\varPsi }} = \left( {A_{\infty } + \alpha \gamma } \right)\left( {r + A_{\infty } + \alpha \gamma } \right) + A_{\infty }^{2\beta } \left[ {c\alpha^{2} + A_{\infty } \left( {r + A_{\infty } } \right)} \right]$. If $\beta = 0$, there can be no limit cycle. If $\beta = 1$, $K_{{\left| {\beta = 1} \right.}} = - A\left[ {r + A\left( {1 + 3\varphi^{2} } \right)} \right] - \alpha^{2} c + 2\left( {\alpha \lambda - \gamma P} \right) + \varphi \left( {\varphi + 2rA} \right)$, which implies that limit cycles are possible for linearly state-dependent restoration efforts.

A3. Using (A2.1) gives:

$$ w^\circ = A^{\beta } \left( {\alpha u^\circ - v^\circ + b - \alpha a} \right) $$

A4. The steady state, if it exists, is obtained by solving (A2.8)–(A2.11) to zero. Equations (A2.8)–(A2.10) are linear in $\varphi$, $\lambda$ and $P$, and after substituting their solutions into (A2.11), we get (11).

A5. The time decomposition method used for the numerical resolution follows the main steps below:

• Step 1 Set a large enough time horizon $t \in \left[ {0,T} \right]$. Set the initial states $P\left( 0 \right)$ and $A\left( 0 \right)$ at the given values. Set the terminal values of the costate variables at the steady-state values, $\lambda \left( T \right) = \lambda_{\infty }$ and $\varphi \left( T \right) = \varphi_{\infty }$.

• Step 2 Guess feasible control functions, $u\left( t \right)$, $v\left( t \right)$ and $w\left( t \right)$.

• Step 3 Integrate the state system from left to right.

• Step 4 Integrate the costate system from right to left.

• Step 5 If the optimality conditions are satisfied (the Hamiltonian is maximized at each $t$) with a required tolerance, stop. Otherwise, go to step 6.

• Step 6 At each $t$, where the Hamiltonian is not maximized, change $u\left( t \right)$, $v\left( t \right)$ and $w\left( t \right)$ to $u\left( t \right) + \delta u\left( t \right)$, $v\left( t \right) + \delta v\left( t \right)$ and $w\left( t \right) + \delta w\left( t \right)$ where $\delta u\left( t \right)$, $\delta v\left( t \right)$ and $\delta w\left( t \right)$ are small enough positive/negative increments to make the Hamiltonian rise.

• Step 7 Go to Step 3.