Collaborative Environmental Management

Abstract

After decades of rapid technological advancement and economic growth, alarming levels of pollution and environmental degradation are emerging globally. Due to the geographical diffusion of pollutants, the unilateral response of one nation or region is often ineffective. Reports portray the situation as an industrial civilization on the verge of suicide, destroying its environmental conditions of existence, with people being held as prisoners on a runaway catastrophe-bound train. Though global cooperation in environmental control holds out the best promise of effective action, limited success has been observed. This is the result of many hurdles, ranging from commitment, monitoring, and sharing of costs to disparities in future development under the cooperative plans. One finds it hard to be convinced that multinational joint initiatives, like the Kyoto Protocol, can offer a long-term solution because there is no guarantee that participants will always be better off within the entire extent of the agreement. More than anything else, it is due to the lack of these kinds of incentives that current cooperative schemes fail to provide an effective means to avert disaster. This is a “classic” game-theoretic problem.

Appendices

Appendix 1: Proof of Proposition 6.2

Using (6.34), (6.36), and (6.37), the system in (6.32) and (6.33) can be expressed as

(6.54)

(6.55)

For (6.54) and (6.55) to hold, it is required that

(6.56)

(6.57)

(6.58)

(6.59)

Equations (6.56)–(6.59) form a block recursive system of differential equations, with (6.56) and (6.57) being independent of (6.58) and (6.59).

Solving {A₁(t),A₂(t),…,A_n(t)} in (6.56) and (6.57) and substituting them into (6.58) and (6.59) yields a system of linear first-order differential equations

(6.60)

(6.61)

Since C_i(t) is independent of C_j(t) for i≠j,C_i(t) can be solved as

(6.62)

where

(6.63)

Hence Proposition 6.2 follows.

Appendix 2: Proof of Proposition 6.3

Substituting (6.44) and (6.46) into (6.42) and using (6.47) one obtains

(6.64)

(6.65)

For (6.64) and (6.65) to hold, it is required that

(6.66)

(6.67)

(6.68)

(6.69)

Equations (6.66)–(6.69) form a block recursive system of differential equations, with (6.66) and (6.67) being independent of (6.68) and (6.69). Moreover, (6.68) and (6.69) are a Riccati equation with constant coefficients, the solution to which can be obtained by standard methods as

(6.70)

where

is a particular solution of (6.66).

Substituting A^∗(t) above into (6.68), the system in (6.68) and (6.69) becomes a system of linear first-order differential equations

(6.71)

(6.72)

Solving (6.71) and (6.72) yields

(6.73)

where

$$C_{*}^{0}=\sum_{j = 1}^{n} g^{j}\bar{x}^{j}\mathrm{e}^{ - r(T - t_{0})}- \int_{t_{0}}^{T}F^{*}(y)\mathrm{e}^{ - r(y - t_{0})}\,\mathrm{d}y.$$

Hence Proposition 6.3 follows.