Commitment-Based Equilibrium Environmental Strategies Under Time-Dependent Absorption Efficiency

Abstract

This paper investigates how current and future generations are affected by commitment-based Nash equilibrium environmental strategies when the environmental absorption efficiency is susceptible to switch from a pollution sink to a source. We formulate a two-player differential game model of transboundary pollution that includes the environmental absorption efficiency as a state variable that can be enhanced thanks to restoration efforts. Based on a logarithmic specification for the instantaneous revenue function, we characterize the cooperative solution and the commitment-based Nash equilibrium strategy, and examine their differences in terms of steady state and transient behavior. We notably show that a commitment-based Nash equilibrium strategy makes it possible to prevent a definitive switching of the environmental absorption efficiency from a pollution sink to a source but imposes greater economic sacrifices on current generations than on future generations. In comparison, the cooperative solution imposes greater sacrifices on current generations in terms of revenues but it imposes lower environmental costs on both current and future generations than commitment-based Nash equilibrium strategy.

Appendix

1.1 Sketch of the proofs of the results presented in Sect. 3

For the optimal control problem related to the CS case, the current-value Hamiltonian is:

$$\begin{aligned} H=a\sum \nolimits _i {\ln e_i } -cP^{2}-\sum \nolimits _i {{w_i^2 }/2} +\eta _1 \left( {\sum \nolimits _i {e_i } -AP} \right) +\eta _2 \left( {\sum \nolimits _i {w_i } -\gamma P} \right) \nonumber \\ \end{aligned}$$

(A1)

in which $\eta _j \equiv \eta _j (t)$ are (current-value) costate variables, $j=1,2$.

Assuming an interior solution, the application of Pontryagin’s principle gives the following necessary conditions:

$$\begin{aligned} H_{e_i }= & {} a/{e_i }+\eta _1 =0\Rightarrow e_i =-a/{\eta _1 } \end{aligned}$$

(A2)

$$\begin{aligned} H_{w_i }= & {} -w_i +\eta _2 =0\Rightarrow w_i =\eta _2 \end{aligned}$$

(A3)

$i=1,2,$ while the costate variables evolution is described by

$$\begin{aligned} \dot{\eta }_1= & {} 2cP+\left( {r+A} \right) \eta _1 +\gamma \eta _2 \end{aligned}$$

(A4)

$$\begin{aligned} \dot{\eta }_2= & {} r\eta _2 +\eta _1 P \end{aligned}$$

(A5)

Equations (A2) and (A3) show that the players’ CS policies are indeed symmetric, as expected. Using (A2)–(A3), formulas (1)–(2) can be rewritten as:

$$\begin{aligned} \dot{P}= & {} -{2a}/{\eta _1 }-AP \end{aligned}$$

(A6)

$$\begin{aligned} \dot{A}= & {} 2\eta _2 -\gamma P \end{aligned}$$

(A7)

To get further insights into the optimal control problem at hand, let us assume a steady state to study how an optimal control is at such a state. Setting $\dot{P} =\dot{A}=0$ provides:$\sum _i {e_i } =AP$ and $\sum _i {w_i } =\gamma P$. Differentiating the two latter equations with respect to time and recalling $\dot{P} =\dot{A}=0$ provides steady efforts: $\dot{e}_i =\dot{w}_i =0,\quad i = 1, 2$. Moreover, differentiating (A2)–(A3) over any time interval and using $\dot{e}_i =\dot{w}_i =0$, $i=1,\;2$, the costate variables are also time-invariant. Hence, if the absorption efficiency and the pollution stock are both steady, then the associated costate variables are also steady, and each player’s efforts are time-invariant.

Given this framework, the theoretical results presented in Sect. 3 are obtained by the resolution of the equations (A1–A7). It is noteworthy that the limiting transversality conditions are satisfied by the CS steady state in (5).

1.2 Sketch of the proofs of the results presented in Sect. 4

For the differential game problem related to the OLNE case, the current-value Hamiltonian of player ican be expressed as:

$$\begin{aligned} H^{i}=a\ln e_i -{cP^{2}}/2-{w_i^2 }/2+\lambda _{i1} \left( {\sum _h {e_h } -AP} \right) +\lambda _{i2} \left( {\sum _h {w_h } -\gamma P} \right) \nonumber \\ \end{aligned}$$

(A8)

in which $\lambda _{ij} \equiv \lambda _{ij} (t)$ denote player i’s (current-value) costate variables, $i=1,2$, $j=1,2$.

Assuming an interior solution, player i’s necessary conditions for the OLNE are:

$$\begin{aligned} H_{e_i }^i= & {} a/{e_i }+\lambda _{i1} =0\Rightarrow e_i^{on} =-a/{\lambda _{i1} } \end{aligned}$$

(A9)

$$\begin{aligned} H_{w_i }^i= & {} -w_i+\lambda _{i2} =0\Rightarrow w_i^{on} =\lambda _{i2} \end{aligned}$$

(A10)

$i=1,2,$ in which the superscript “on” is used to denote the OLNE.

The costate variables evolve according to:

$$\begin{aligned} \dot{\lambda }_{i1} =cP+\left( {r+A} \right) \lambda _{i1} +\gamma \lambda _{i2} \dot{\lambda }_{i2} =r\lambda _{i2} +\lambda _{i1} P \end{aligned}$$

(A11)

In the present case, the costate variables of both players are identical; that is, $\lambda _{i1} \equiv \lambda _1$ and $\lambda _{i2} \equiv \lambda _2$, $i=1,2$. Hence, (A9) and (A10) show that the players’ equilibrium strategies are symmetric, as expected. As a consequence, only two costate equations have to be considered in the analysis.

Exploiting (A9)–(A10), (1)–(2) can be rewritten as

$$\begin{aligned} \dot{P}= & {} -{2a}/{\lambda _1 }-AP \end{aligned}$$

(A12)

$$\begin{aligned} \dot{A}= & {} 2\lambda _2 -\gamma P \end{aligned}$$

(A13)

Hence, if the absorption efficiency and the pollution stock are both steady, then each player’s control efforts are time-invariant.

Given this framework, the theoretical results presented in Sect. 4 are obtained by the resolution of the equations (A8)–(A13). It is also noteworthy that the limiting transversality conditions are satisfied by the OLNE steady state (11).