Improving the effectiveness of watershed environmental management—dynamic coordination through government pollution control and resident participation

Abstract

To improve the effectiveness of envir onmental management of watersheds and improve the environmental management mechanism of cross-administrative watersheds, we develop a neoliberal framework for action using incentives, examine the cooperative strategies of local governments in watershed treatment and people-oriented environmental protection under central government subsidies, and analyze the cost effectiveness of multiple strategies in a dynamic perspective, and we have the following important findings: (1) Compared to vertical ecological compensation, the introduction of horizontal cost-sharing contracts is more effective in enhancing inter-local cooperative environmental governance. (2) When the marginal benefit of the downstream local government is greater than half of the upstream marginal benefit, the upstream local government’s pollution control investment and the effect of pollution control are improved, and the Pareto improvement of the environmental governance benefit of the watershed is realized, i.e., the cost-sharing contract driven by the downstream can achieve a win–win situation for both environmental and government governance benefits. (3) When the marginal benefit of downstream environmental advocacy is between 0.5 and 1.5 times the marginal benefit of upstream government, the cost-sharing contract is more effective in improving downstream benefits. Conversely, when the marginal benefit of downstream is greater than 1.5 times, the marginal benefit of upstream, the more effective the cost-sharing contract is in improving the marginal benefit of downstream. The results of the study provide useful insights for the government to develop reasonable pollution management cooperation mechanisms to improve environmental management performance and thus enhance the sustainable development of the watershed.

Graphical Abstract

Appendices

Appendix 1

Let ${R}_{T}^{{a}^{*}}\left({y}^{a}\right)={e}^{-\rho t}{F}_{T}\left({y}^{a}\right)$, according to the optimal control theory, ${F}_{T}\left({y}^{a}\right)$ satisfies the Hamilton–Jacobi-Bellman equation (HJB) equation for ${\forall y}^{a}\ge 0$,

$${F}_{T}\left({y}^{a}\right)=\underset{{I}_{u}^{b}\ge 0,{I}_{d}^{b}\ge 0}{\mathrm{max}}{e}^{-\rho t}\left\{\left({\prod }_{u}+{\prod }_{d}\right){y}^{a}-\left(1-{\varphi }_{u}\right)\frac{{w}_{u}}{2}{\left({I}_{u}^{a}\right)}^{2}-\left(1-{\varphi }_{d}\right)\frac{{w}_{d}}{2}{\left({I}_{d}^{a}\right)}^{2}+{F}^{^{\prime}}\left({y}^{a}\right)\left[\alpha \cdot {I}_{u}^{{a}^{*}}-\left(\gamma -r\right)y\right]\right\}$$

(35)

The Hessian matrix for ${I}_{u}^{{a}^{*}}$ and ${I}_{d}^{{a}^{*}}$ is

$$H=\left[\begin{array}{cc}-\left(1-{\varphi }_{u}\right){w}_{u}& 0\\ 0& -\left(1-{\varphi }_{D}\right){w}_{d}\end{array}\right]$$

(36)

Therefore, it can be known that the Hessian Matrix is semi-negative definite, that is, $\rho {F}_{T}\left({y}^{a}\right)$ is a concave function, so the maximum value can be obtained for ${I}_{u}^{a}$ and ${I}_{d}^{a}$. For ${F}_{T}\left({y}^{a}\right)$, find the first-order partial derivatives of ${I}_{u}^{a}$ and ${I}_{d}^{a}$ respectively and set them to 0 to obtain the maximization condition:

$$\left\{\begin{array}{c}{I}_{u}^{{a}^{*}}=\frac{\alpha {F}^{^{\prime}}\left({y}^{a}\right)}{\left(1-{\varphi }_{u}^{{a}^{*}}\right){w}_{u}}\\ {I}_{d}^{{a}^{*}}=\frac{({\prod }_{u}+{\prod }_{d})\varepsilon }{{w}_{d}\left(1-{\varphi }_{d}^{{a}^{*}}\right)}\end{array}\right.$$

(37)

Substituting (37) into (35) and sorting out, you can get

$$\rho {F}_{T}\left({y}^{a}\right)=\frac{{\alpha }^{2}{\left({F}^{^{\prime}}\left({y}^{a}\right)\right)}^{2}}{2\left(1-{\varphi }_{u}\right){w}_{u}}+\frac{{\varepsilon }^{2}{\left({\prod }_{u}+{\prod }_{d}\right)}^{2}}{2\left(1-{\varphi }_{d}\right){w}_{d}}+\left({\prod }_{u}+{\prod }_{d}\right){E}_{0}+\left[\left({\prod }_{u}+{\prod }_{d}\right)s-\left(\gamma -r\right){F}^{^{\prime}}\left({y}^{a}\right)\right]{y}^{a}$$

(38)

Assume that ${F}_{T}\left({y}^{a}\right)$ has the following linear form:

$${F}_{T}\left({y}^{a}\right)={k}_{1}{y}^{a}+{b}_{1}$$

(39)

Among them, ${k}_{1}$ and ${b}_{1}$ are constants. Substituting ${F}_{T}\left({y}^{{a}^{*}}\right)$ and ${F}^{^{\prime}}\left({y}^{{a}^{*}}\right)$ into the formula (38), the solution is:

$${F}^{^{\prime}}\left({y}^{a}\right)={k}_{1}=\frac{\left({J}_{u}+{J}_{D}\right)s}{m}$$

(40)

At this time, substituting (40) into (37) can obtain ${I}_{u}^{{a}^{*}}$, substituting ${I}_{u}^{{a}^{**}}$ into (2) formula, can obtain ${y}^{{a}^{*}}$, and then we can find ${E}_{a}^{*}$. Substitute ${k}_{1}$ and ${b}_{1}$ into (17) to obtain ${R}_{T}^{{a}^{*}}$. The process of solving the central government's optimal subsidy coefficients ${\varphi }_{u}^{{a}^{*}}$ and ${\varphi }_{d}^{{a}^{*}}$ is similar to that of the following two decision-making equilibrium results. Due to space reasons, I will not repeat them here. Substitute ${\varphi }_{u}^{{a}^{*}}$, ${\varphi }_{d}^{{a}^{*}}$ into ${I}_{u}^{{a}^{*}}$, ${I}_{d}^{{a}^{*}}$ to get ${I}_{u}^{{a}^{**}}$, ${I}_{d}^{{a}^{**}}$, substitute in ${y}^{{a}^{*}}$, ${R}_{T}^{{a}^{*}}$ to get ${y}^{{a}^{**}}$, ${R}_{T}^{{a}^{**}}$.

Appendix 2

The optimal objective function of the upstream local government at time $t$ is

$${R}_{u}^{b}=\underset{{I}_{u}^{b}\ge 0}{\mathrm{max}}\;{\int\limits }_{0}^{\infty }{e}^{-\rho t}\left\{{\prod }_{u}\left(sy+\varepsilon {I}_{d}+{E}_{0}\right)+\left({\varphi }_{u}-1\right)\frac{{w}_{u}}{2}{\left({I}_{u}^{b}\right)}^{2}\right\}dt$$

(41)

Let ${R}_{u}^{{b}^{*}}\left({y}^{b}\right)={e}^{-\rho t}{F}_{u}\left({y}^{b}\right)$ according to the optimal control theory, ${F}_{u}\left({y}^{b}\right)$ satisfies the HJB equation for ${\forall y}^{{b}^{*}}\ge 0$,

$$\rho {F}_{u}\left({y}^{b}\right)\left({y}^{b}\right)=\underset{{I}_{u}^{b}\ge 0}{\mathrm{max}}\left[{\prod }_{u}\left(y+\varepsilon {I}_{d}+{E}_{0}\right)+\left({\varphi }_{u}-1\right)\frac{{w}_{u}}{2}{\left({I}_{u}^{b}\right)}^{2}+{F}_{u}^{^{\prime}}\left({y}^{b}\right)\left(\alpha {I}_{u}^{b}-\left(\gamma -r\right)y\right)\right]$$

(42)

In the same way, find the first-order partial derivative of ${I}_{u}^{b}$ for $\rho {F}_{u}\left({y}^{b}\right)$ and set it to 0 to obtain

$${I}_{u}^{{b}^{*}}=\frac{{F}_{u}^{^{\prime}}\left({y}^{b}\right)\alpha }{{w}_{u}\left(1-{\varphi }_{u}^{{b}^{*}}\right)m}$$

(43)

Similarly, let the optimal objective function of the downstream local government at time $t={R}_{d}^{b}\left({y}^{b}\right){e}^{-\rho t}{F}_{d}\left({y}^{b}\right)$, which can be determined by the optimal control theory:

$$\rho {F}_{d}\left({y}^{b}\right)=\underset{{I}_{d}^{b}\ge 0}{\mathrm{max}}\left[{\prod }_{d}\left(sy+\varepsilon {I}_{d}+{E}_{0}\right)+\left({\varphi }_{u}-1\right)\frac{{w}_{d}}{2}{\left({I}_{d}^{b}\right)}^{2}+{F}_{d}^{^{\prime}}\left({y}^{b}\right)\left(\alpha {I}_{u}^{b}-\left(\gamma -r\right)y\right)\right]$$

(44)

We can find the first-order partial derivative of ${I}_{d}^{b}$ for $\rho {F}_{d}\left({y}^{{b}^{*}}\right)$ and set it to 0 to obtain

$${I}_{d}^{{b}^{*}}=\frac{\varepsilon {\prod }_{d}}{{w}_{d}\left(1-{\varphi }_{d}^{{b}^{*}}\right)}$$

(45)

Substituting Eqs. (43) and (46) into Eqs. (42) and (44), we can get

$$\rho {F}_{u}\left({y}^{b}\right)={\prod }_{u}{E}_{0}+\left({\prod }_{u}s\left(-r\right){F}_{d}^{^{\prime}}\left({y}^{b}\right){y}^{b}+\frac{{\prod }_{u}{\prod }_{d}{\varepsilon }^{2}}{{w}_{u}\left(1-{\varphi }_{u}^{{b}^{*}}\right)}+\frac{{\left({F}_{u}^{^{\prime}}\left({y}^{b}\right)\alpha \right)}^{2}}{{2w}_{u}\left(1-{\varphi }_{u}^{{b}^{*}}\right)}\right)$$

(46)

$$\rho {F}_{d}\left({y}^{b}\right)={\prod }_{d}{E}_{0}+\left({\prod }_{d}s-\left(\gamma -r\right){F}_{d}^{^{\prime}}\left({y}^{b}\right){y}^{b}+\frac{{\left({\prod }_{d}\varepsilon \right)}^{2}}{{2w}_{d}\left(1-{\varphi }_{d}^{{b}^{*}}\right)}+\frac{{\alpha }^{2}{F}_{u}^{^{\prime}}\left({y}^{b}\right){F}_{d}^{^{\prime}}\left({y}^{b}\right)}{{w}_{u}\left(1-{\varphi }_{u}^{{b}^{*}}\right)}\right)$$

(47)

Assume that ${F}_{u}\left({y}^{b}\right)$ and ${F}_{d}\left({y}^{b}\right)$ have the following linear form:

$${F}_{u}\left({y}^{b}\right)={k}_{2}{y}^{b}+{b}_{2}$$

(48)

$${F}_{d}\left({y}^{b}\right)={k}_{3}{y}^{b}+{b}_{3}$$

(49)

where,${k}_{2}{b}_{2}$,${k}_{3}$,${b}_{3}$ are constants. It is easy to know ${F}_{u}^{^{\prime}}\left({y}^{b}\right)$=${k}_{2}$ and${F}_{d}^{^{\prime}}\left({y}^{b}\right)={k}_{3}$. Substitute Eqs. (48) and (49) into Eqs. (46) and (47). We can obtain ${k}_{2}$ and${k}_{3}$. Substitute ${k}_{2}$ into (44), we can get${I}_{u}^{{b}^{*}}$.Then substitute ${I}_{u}^{{b}^{*}}$ into formula (2),we can get${y}^{{b}^{*}}$. Finally, substitute${k}_{2}$,${b}_{2}$,${k}_{3}$,${b}_{3}$ into ${F}_{u}\left({y}^{{b}^{*}}\right)$ and${F}_{d}\left({y}^{{b}^{*}}\right)$, we can get formula (19) and (20).

Similarly, We do the same for the optimal objective function of the central government, let ${R}_{g}^{b}={e}^{-\rho t}{F}_{g}\left({y}^{{b}^{*}}\right)$,${F}_{g}\left({y}^{b}\right)$ satisfies the HJB equation for ${\forall y}^{b}\ge 0$,

$$\rho {F}_{g}\left({y}^{b}\right)=\underset{{I}_{u}^{b},{I}_{d}^{b}}{\mathrm{max}}\left[{\prod }_{u}+{\prod }_{d}\left(sy\left(t\right)+\varepsilon {I}_{d}\left(t\right)+{E}_{0}\right)-\frac{{w}_{u}}{2}{\left({I}_{u}^{b}\right)}^{2}-\frac{{w}_{d}}{2}{\left({I}_{d}^{b}\right)}^{2}+{F}_{g}^{^{\prime}}\left({y}^{b}\right)\left(\alpha {I}_{u}^{b}-\left(\gamma -r\right)y\right)\right]$$

(50)

Substitute ${I}_{u}^{b}$,${I}_{d}^{b}$ into formula (51), and find the first-order partial derivatives of ${\varphi }_{u}$ and ${\varphi }_{d}$, we can get

$${\varphi }_{u}^{{b}^{*}}=1-\frac{{\prod }_{d}s}{{F}_{g}^{^{\prime}}\left({y}^{b}\right)m}$$

(51)

$${\varphi }_{d}^{{b}^{*}}=\frac{{\prod }_{u}}{{\Pi }_{u}+{\Pi }_{d}}$$

(52)

We substitute (51), (52) into (50), we can obtain

$$\rho {F}_{g}\left({y}^{b}\right)=\left({\prod }_{u}+{\prod }_{d}\right){E}_{0}+\left[\left({\prod }_{u}+{\prod }_{d}\right)s-\left(\gamma -r\right){F}_{g}^{^{\prime}}\left({y}^{b}\right)\right]y+\frac{{\left({F}_{g}^{^{\prime}}\left({y}^{b}\right)\alpha \right)}^{2}}{2{w}_{u}}+\frac{{\left(\varepsilon \left({\prod }_{u}+{\prod }_{d}\right)\right)}^{2}}{{2w}_{d}}$$

(53)

In the same way, we assume that ${F}_{g}\left({y}^{{b}^{*}}\right)$ has the following linear form:

$${F}_{g}\left({y}^{{b}^{*}}\right)={k}_{4}{y}^{b}+{b}_{4}$$

(54)

It is easy to know that ${F}_{g}^{^{\prime}}\left({y}^{b}\right)={k}_{4}$. Substitute (54) into (53), we can get${k}_{4} , {b}_{4}$.Then we substitute ${k}_{4}$ into (51), we can obtain${\varphi }_{u}^{{b}^{*}}$. Substitute${\varphi }_{u}^{{b}^{*}}$,${\varphi }_{d }^{{b}^{*}}$ into${I}_{u}^{{b}^{*}}{,I}_{d}^{{b}^{*}},{y}^{{b}^{*}}$, we can get formulas (19) and (21).

Appendix C

The optimal objective function of the upstream local government at time $t$ is

$${R}_{u}^{c}=\underset{{I}_{u}^{c}\ge 0}{max}\;{\int\limits }_{0}^{\infty }{e}^{-\rho t}\left\{{\prod }_{u}\cdot E-\left(1-{\varphi }_{u}^{c}-\theta \right){C}_{u}\right\}dt$$

(55)

Let ${R}_{u}^{c}={e}^{-\rho t}{F}_{u}\left({y}^{{c}^{*}}\right)$, according to the optimal control theory, ${F}_{u}\left({y}^{{c}^{*}}\right)$ satisfies the HJB equation for ${\forall y}^{{c}^{*}}\ge 0$,

$$\rho {F}_{u}\left({y}^{c}\right)=\underset{{I}_{u}^{c}\ge 0}{\mathrm{max}}\left[{\prod }_{u}\left(sy+\varepsilon {I}_{d}+{E}_{0}\right)+\left({\varphi }_{u}+\mu -1\right)+\frac{{w}_{u}}{2}{\left({I}_{u}^{c}\right)}^{2}+{F}_{u}^{^{\prime}}\left({y}^{c}\right)\left(\alpha {I}_{u}^{c}-\left(\gamma -r\right)y\right)\right]$$

(56)

In the same way, find the first-order partial derivative of ${I}_{u}^{c}$ for $\rho {F}_{u}\left({y}^{{c}^{*}}\right)$ and set it to 0 to obtain

$${I}_{u}^{{c}^{*}}=\frac{{F}_{u}^{^{\prime}}\left({y}^{c}\right)\alpha }{{w}_{u}\left(1-{\varphi }_{u}^{{c}^{*}}-\mu \right)}$$

(57)

Similarly, let the optimal objective function of the downstream local government at time $t={R}_{d}^{c}\left({y}^{c}\right){e}^{-\rho t}{F}_{d}\left({y}^{c}\right)$, which can be determined by the optimal control theory:

$$\rho {F}_{d}\left({y}^{c}\right)=\underset{{I}_{d}^{c}\ge 0}{\mathrm{max}}\left[{\prod }_{d}\left(sy+\varepsilon {I}_{d}+{E}_{0}\right)+\left({\varphi }_{d}-1\right)\frac{{w}_{d}}{2}{\left({I}_{d}^{c}\right)}^{2}-\mu \frac{{w}_{u}}{2}{\left({I}_{u}^{c}\right)}^{2}+{F}_{d}^{^{\prime}}\left({y}^{c}\right)\left(\alpha {I}_{u}^{c}-\left(\gamma -r\right)y\right)\right]$$

(58)

We can find the first-order partial derivative of ${I}_{d}^{c}$ for $\rho {F}_{d}\left({y}^{c}\right)$ and set it to 0 to obtain

$${I}_{d}^{{c}^{*}}=\frac{\varepsilon {\prod }_{d}}{{w}_{d}\left(1-{\varphi }_{d}^{{c}^{*}}\right)}$$

(59)

$${\mu }^{*}=\frac{(2{F}_{d}^{^{\prime}}\left({y}^{c}\right)-{F}_{u}^{^{\prime}}\left({y}^{c}\right))(1-{\varphi }_{u})}{2{F}_{d}^{^{\prime}}\left({y}^{c}\right)+{F}_{u}^{^{\prime}}\left({y}^{c}\right)}$$

(60)

Substitute (57), (59), (60) into (56) and (58), we can obtain

$$\rho {F}_{u}\left({y}^{c}\right)={\prod }_{u}{E}_{0}+\left[{\prod }_{u}s-\left(\gamma -r\right){F}_{u}^{^{\prime}}\left({y}^{c}\right)\right]y+\frac{{\prod }_{u}{\prod }_{d}{\varepsilon }^{2}}{{w}_{u}\left(1-{\varphi }_{u}^{{c}^{*}}\right)} +\frac{{\alpha }^{2}{F}_{u}^{^{\prime}}\left({y}^{c}\right)\left(2{F}_{d}^{^{\prime}}\left({y}^{c}\right){+F}_{u}^{^{\prime}}\left({y}^{c}\right)\right)}{{4w}_{u}\left(1-{\varphi }_{u}^{{c}^{*}}\right)}$$

(61)

$$\rho {F}_{d}\left({y}^{c}\right)={\prod }_{d}{E}_{0}+\left[{\prod }_{d}s-\left(\gamma -r\right){F}_{d}^{^{\prime}}\left({y}^{c}\right)\right]y+\frac{{\left({\prod }_{d}\varepsilon \right)}^{2}}{{2w}_{d}\left(1-{\varphi }_{d}^{{c}^{*}}\right)} +\frac{{\alpha }^{2}{\left({F}_{d}^{^{\prime}}\left({y}^{c}\right)+{F}_{u}^{^{\prime}}\left({y}^{c}\right)\right)}^{2}}{{8w}_{u}\left(1-{\varphi }_{u}^{{c}^{*}}\right)}$$

(62)

Assume that ${F}_{u}\left({y}^{{c}^{*}}\right)$ and ${F}_{d}\left({y}^{{c}^{*}}\right)$ have the following linear form:

$${F}_{u}\left({y}^{c}\right)={k}_{5}{y}^{{c}^{*}}+{b}_{5}$$

(63)

$${F}_{d}\left({y}^{c}\right)={k}_{6}{y}^{{c}^{*}}+{b}_{6}$$

(64)

where ${k}_{5}$,${b}_{5}$,${k}_{6}$,${b}_{6}$ are constants. It is easy to know ${F}_{u}^{^{\prime}}\left({y}^{c}\right)$=${k}_{5}$ and ${F}_{d}^{^{\prime}}\left({y}^{c}\right)={k}_{6}$. Substitute Eqs. (63) and (64) into Eqs. (61) and (62). We can obtain ${k}_{5}$ and ${k}_{6}$. Substitute ${k}_{5}$ into substitute, we can get ${I}_{u}^{{c}^{*}}$.Then substitute ${I}_{u}^{{c}^{*}}$ into formula (2),we can get ${y}^{{C}^{*}}$. Finally, substitute ${k}_{5}$,${b}_{5}$,${k}_{6}$,${b}_{6}$ into ${F}_{u}\left({y}^{{c}^{*}}\right)$ and ${F}_{d}\left({y}^{c}\right)$,we can get formula (28) and (30).

Similarly, let the optimal objective function of the central government at time ${t=R}_{g}^{c}\left({y}^{{c}^{*}}\right){e}^{-\rho t}{F}_{g}\left({y}^{{c}^{*}}\right)$, which can be determined by the optimal control theory:

$$\rho {F}_{g}\left({y}^{c}\right)=\underset{{\varphi }_{u}{\varphi }_{d}}{\mathrm{max}}\left[\left({\prod }_{u}+{\prod }_{d}\right)\left(sy\left(t\right)+\varepsilon {I}_{d}\left(t\right)+{E}_{0}\right)-\frac{{w}_{u}}{2}{\left({I}_{u}^{c}\right)}^{2}-\frac{{w}_{d}}{2}{\left({I}_{d}^{c}\right)}^{2}+{F}_{g}^{^{\prime}}\left({y}^{c}\right)\left(\alpha {I}_{u}^{c}-\left(\gamma -r\right)y\right)\right]$$

(65)

Substitute (57), (69) into (65), we can get

$${\varphi }_{u}^{{c}^{*}}=1-\frac{\left(2{\prod }_{u}+{\prod }_{d}\right)s}{2{F}_{g}^{^{\prime}}\left({y}^{c}\right)m}$$

(66)

$${\varphi }_{d}^{{c}^{*}}=\frac{{\prod }_{u}}{{\prod }_{u}+{\prod }_{d}}$$

(67)

Substitute (66), (67) into (65), we can get

$$\rho {F}_{g}\left({y}^{c}\right)=\left({\prod }_{u}+{\prod }_{d}\right)\left(sy\left(t\right)+\varepsilon {I}_{d}\left(t\right)+{E}_{0}\right)-\frac{{w}_{u}}{2}{\left({I}_{u}^{c}\right)}^{2}-\frac{{w}_{d}}{2}{\left({I}_{d}^{c}\right)}^{2}+\left({F}_{g}^{^{\prime}}\left({y}^{c}\right)\alpha {I}_{u}^{c}-\left(\gamma -r\right)y\right)$$

(68)

In the same way, we assume that ${F}_{g}\left({y}^{c}\right)$ has the following linear form:

$${F}_{g}\left({y}^{c}\right)={k}_{7}{y}^{c}+{b}_{7}$$

(69)

It is easy to know that ${F}_{g}^{^{\prime}}\left({y}^{c}\right)={k}_{7}$. Substitute (69) into (68), we can get ${k}_{7} , {b}_{7}$.Then, we substitute ${k}_{7}$ into (67), we can obtain${\varphi }_{u}^{{c}^{*}}$. Substitute${\varphi }_{u}^{{c}^{*}}$,${\varphi }_{d }^{{c}^{*}}$ into${I}_{u}^{{c}^{*}}{,I}_{d}^{{c}^{*}},{y}^{{c}^{*}}$, we can get formulas (28) and (30).