Optimal Abatement Technology Licensing in a Dynamic Transboundary Pollution Game: Fixed Fee Versus Royalty

Abstract

Transboundary pollution poses a major threat to environment and human health. An effective approach to addressing this problem is the adoption of long-term abatement technology; however, many developing regions are lacking in related technologies that can be acquired by licensing from developed regions. This study focuses on a differential game model of transboundary pollution between two asymmetric regions, one of which possesses advanced abatement technology that can reduce the abatement cost and licenses this technology to the other region by royalty or fixed-fee licensing. We characterize the equilibrium decisions in the regions and find that fixed-fee licensing is superior to royalty licensing from the viewpoint of both regions. The reason is that under fixed-fee licensing, the regions can gain improved incremental revenues and incur reduced environmental damage. Subsequently, we analyze the steady-state equilibrium behaviors and the effects of parameters on the licensing performance. The analysis indicates that the myopic view of the regions leads to short-term revenue maximization, resulting in an increase in total pollution stock. Moreover, a high level of abatement technology or emission tax prompts the licensee region to choose fixed-fee approach, which is beneficial both economically and environmentally for two regions.

Appendices

Appendix 1 for Proposition 1

The first-order conditions of $E_{iN}^{*}$, $E_{jN}^{*}$, $\alpha_{iN}^{*}$ and $\alpha_{jN}^{*}$ are as follows:

$$ E_{iN}^{*} (t) = A - \tau + V_{iN}^{\prime } (p),\;\alpha_{iN}^{*} (t) = \frac{{\tau - V_{iN}^{\prime } (p)}}{{(c - \varepsilon )[A - \tau + V_{iN}^{\prime } (p)]}} $$

(A1)

$$ E_{jN}^{*} (t) = \theta A - \tau + V_{jN}^{\prime } (p),\;\alpha_{jN}^{*} (t) = \frac{{\tau - V_{jN}^{\prime } (p)}}{{c[\theta A - \tau + V_{jN}^{\prime } (p)]}} $$

(A2)

Substituting (A1) and (A2) into (5a) and (5b), we then obtain the simplified HJB equations:

$$ \begin{aligned} \lambda V_{iN} (p) & = \frac{1}{2}[A - \tau + V_{iN}^{\prime } (p)]^{2} + \frac{{[\tau - V_{iN}^{\prime } (p)]^{2} }}{2(c - \varepsilon )} \\ &\quad+\, V_{iN}^{\prime } (p)\left[\theta A - \tau + V_{jN}^{\prime } (p) - \,\frac{{\tau - V_{jN}^{\prime } (p)}}{c}\right] - Dp(t) - \delta V_{iN}^{\prime } (p)p(t) \\ \end{aligned} $$

(A3)

$$ \begin{aligned} \lambda V_{jN} (p) & = \frac{1}{2}[\theta A - \tau + V_{jN}^{\prime } (p)]^{2} + \frac{{[\tau - V_{jN}^{\prime } (p)]^{2} }}{2c} \\ &\quad+\, V_{jN}^{\prime } (p)\left[A - \tau + V_{iN}^{\prime } (p) - \,\frac{{\tau - V_{iN}^{\prime } (p)}}{c - \varepsilon }\right] - \beta Dp(t) - \delta V_{jN}^{\prime } (p)p(t) \\ \end{aligned} $$

(A4)

For $m = i,j$, we guess the form the value functions $V_{mN} (p)$ to be linear in p, that is,

$$ V_{iN} (p) = k_{iN} + f_{iN} p,\quad V_{jN} (p) = k_{jN} + f_{jN} p $$

(A5)

where $f_{iN}$, $k_{iN}$, $f_{jN}$ and $k_{jN}$ are constant coefficients. Then, we obtain these constant coefficients to verify that our guess is correct, that means the exact expressions of the value functions are linear. Differentiating Eq. (A5) with respect to p, we obtain $V_{iN}^{\prime } (p) = f_{iN}$, $V_{jN}^{\prime } (p) = f_{jN}$ which when substituted into (A3) and (A4) gives

$$\begin{aligned} \lambda (k_{iN} + f_{iN} p) &= \frac{1}{2}[A - \tau + f_{iN} ]^{2} + \frac{{[\tau - f_{iN} ]^{2} }}{2(c - \varepsilon )}\\ & \quad + f_{iN} \left[ {\theta A - \tau + f_{jN} - \frac{{\tau - f_{jN} }}{c}} \right] - Dp(t) - \delta f_{iN} p(t)\end{aligned} $$

(A6)

$$\begin{aligned} \lambda (k_{jN} + f_{jN} p) &= \frac{1}{2}[\theta A - \tau + f_{jN} ]^{2} + \frac{{[\tau - f_{jN} ]^{2} }}{2c}\\ & \quad + f_{jN} \left[ {A - \tau + f_{iN} - \frac{{\tau - f_{iN} }}{c - \varepsilon }} \right] - \beta Dp(t) - \delta f_{jN} p(t)\end{aligned} $$

(A7)

Then, we can derive $f_{iN}$, $k_{iN}$, $f_{jN}$, $k_{jN}$ from (A6) and (A7), as follows:

$$ f_{iN} = \frac{ - D}{\lambda + \delta },\quad f_{jN} = \frac{ - \beta D}{\lambda + \delta } $$

(A8)

$$\begin{aligned} k_{iN} &= \frac{1}{2\lambda }\left( {A - \tau - \frac{D}{\lambda + \delta }} \right)^{2} + \frac{{[\tau (\lambda + \delta ) + D]^{2} }}{{2\lambda (\lambda + \delta )^{2} (c - \varepsilon )}}\\ & \quad - \frac{D}{\lambda (\lambda + \delta )}\left[ {\theta A - \tau - \frac{\beta D}{\lambda + \delta } - \frac{\tau (\lambda + \delta ) + \beta D}{c(\lambda + \delta )}} \right]\end{aligned} $$

(A9)

$$\begin{aligned} k_{jN} &= \frac{1}{2\lambda }\left( {\theta A - \tau - \frac{\beta D}{\lambda + \delta }} \right)^{2} + \frac{{[\tau (\lambda + \delta ) + \beta D]^{2} }}{{2c\lambda (\lambda + \delta )^{2} }}\\ & \quad - \frac{\beta D}{\lambda (\lambda + \delta )}\left[ {A - \tau - \frac{D}{\lambda + \delta } - \frac{\tau (\lambda + \delta ) + D}{(c - \varepsilon )(\lambda + \delta )}} \right]\end{aligned} $$

(A10)

Substituting (A8), (A9), and (A10) into $V_{mN}^{\prime } (p)$ and from (A1) and (A2), we can determine the Markov-perfect Nash equilibrium outputs, optimal proportions of pollutants that the regions choose to abate, and net revenues.

Appendix 2 for Corollary 1

The steady-state equilibrium decisions of $E_{iN}^{**}$, $E_{jN}^{**}$ and $\alpha_{iN}^{**}$, $\alpha_{jN}^{**}$ are the same as those in (6a) and (6b) in Scenario N. The steady state $V_{iN}^{**}$, $V_{jN}^{**}$ can be obtained by inserting $p_{N}^{**}$ into (6c) and (6d). Thus,

$$ \begin{aligned} V_{iN}^{**} (p) & = - \frac{D}{\lambda + \delta }p_{N}^{**} + \frac{1}{2\lambda }\left( {A - \tau - \frac{D}{\lambda + \delta }} \right)^{2} + \frac{{[\tau (\lambda + \delta ) + D]^{2} }}{{2\lambda (\lambda + \delta )^{2} (c - \varepsilon )}}\\ &\qquad - \frac{D}{\lambda (\lambda + \delta )}\left[ {\theta A - \tau - \frac{\beta D}{\lambda + \delta } - \frac{\tau (\lambda + \delta ) + \beta D}{c(\lambda + \delta )}} \right] \\ V_{jN}^{**} (p) & = - \frac{\beta D}{\lambda + \delta }p_{N}^{**} + \frac{1}{2\lambda }\left( {\theta A - \tau - \frac{\beta D}{\lambda + \delta }} \right)^{2} + \frac{{[\tau (\lambda + \delta ) + \beta D]^{2} }}{{2c\lambda (\lambda + \delta )^{2} }}\\ & \qquad - \frac{\beta D}{\lambda (\lambda + \delta )}\left[ {A - \tau - \frac{D}{\lambda + \delta } - \frac{\tau (\lambda + \delta ) + D}{(c - \varepsilon )(\lambda + \delta )}} \right] \\ \end{aligned} $$

In the aforementioned equations, $p_{N}^{**}$ is given in Eq. (6e). We can subsequently derive

$$ \begin{aligned} & \frac{{\partial E_{iN}^{**} (t)}}{\partial \tau } = - 1 < 0,\quad \frac{{\partial E_{jN}^{**} (t)}}{\partial \tau } = - 1 < 0 \\ & \frac{{\partial \alpha_{iN}^{**} (t)}}{\partial \tau } = \frac{{A(\lambda + \delta )^{2} }}{{(c - \varepsilon )[(A - \tau )(\lambda + \delta ) - D]^{2} }} > 0,\\ & \frac{{\partial \alpha_{jN}^{**} (t)}}{\partial \tau } = \frac{{\theta A(\lambda + \delta )^{2} }}{{c[(\theta A - \tau )(\lambda + \delta ) - \beta D]^{2} }} > 0 \\ & \frac{{\partial \alpha_{iN}^{**} (t)}}{\partial \varepsilon } = \frac{\tau (\lambda + \delta ) + D}{{(c - \varepsilon )^{2} (\lambda + \delta )(A - \tau - \frac{D}{\lambda + \delta })}} > 0 \\ & \frac{{\partial s_{iN}^{**} }}{\partial \lambda } = - \frac{D}{{(c - \varepsilon )(\lambda + \delta )^{2} }} < 0,\quad \frac{{\partial s_{jN}^{**} }}{\partial \lambda } = - \frac{\beta D}{{c(\lambda + \delta )^{2} }} < 0 \\ & \frac{{\partial V_{iN}^{**} }}{\partial p} = - \frac{D}{\lambda + \delta } < 0,\quad \frac{{\partial V_{jN}^{**} }}{\partial p} = - \frac{\beta D}{\lambda + \delta } < 0 \\ \end{aligned} $$

Then, we can obtain the conclusions of Corollary 1.

Appendix 3 for Proposition 2

The first-order conditions of $E_{iR}^{*}$, $E_{jR}^{*}$, $\alpha_{iR}^{*}$, and $\alpha_{jR}^{*}$ are calculated as follows:

$$ E_{iR}^{*} (t) = A - \tau + V_{iR}^{\prime } (p),\quad \alpha_{iR}^{*} (t) = \frac{{\tau - V_{iR}^{\prime } (p)}}{{(c - \varepsilon )[A - \tau + V_{iR}^{\prime } (p)]}} $$

(A11)

$$ E_{jR}^{*} (t) = \theta A - \tau + V_{jR}^{\prime } (p),\quad \alpha_{jR}^{*} (t) = \frac{{\tau - \omega - V_{jR}^{\prime } (p)}}{{(c - \varepsilon )[\theta A - \tau + V_{jR}^{\prime } (p)]}} $$

(A12)

By substituting (A11) and (A12) into (8a) and (8b), we have

$$\begin{aligned} \lambda V_{iR} (p) &= \frac{1}{2}[A - \tau + V_{iR}^{\prime } (p)]^{2} + \frac{{[\tau - V_{iR}^{\prime } (p)]^{2} }}{2(c - \varepsilon )}\\ & \quad + V_{iR}^{\prime } (p)\left[ {\theta A - \tau + V_{jR}^{\prime } (p) - \frac{{\tau - V_{jR}^{\prime } (p) - \omega }}{c - \varepsilon }} \right] \\ & \quad- Dp(t) - \delta V_{iR}^{\prime } (p)p(t)\end{aligned} $$

(A13)

$$\begin{aligned} \lambda V_{jR} (p) &= \frac{1}{2}[\theta A - \tau + V_{jR}^{\prime } (p)]^{2} + \frac{{[V_{jR}^{\prime } (p) - \tau + \omega ]^{2} }}{2(c - \varepsilon )}\\ & \quad + V_{jR}^{\prime } (p)\left[A - \tau + V_{iR}^{\prime } (p) - \frac{{\tau - V_{iR}^{\prime } (p)}}{c - \varepsilon }\right] \\ & \quad- \beta Dp(t) - \delta V_{jR}^{\prime } (p)p(t)\end{aligned} $$

(A14)

From (A13) and (A14), we conjecture that the structures of (A13) and (A14) can be regarded as linear value functions:

$$ V_{iR} (p) = k_{iR} + f_{iR} p,\quad V_{jR} (p) = k_{jR} + f_{jR} p $$

(A15)

We then obtain $V_{iR}^{\prime } (p) = f_{iR}$, $V_{jR}^{\prime } (p) = f_{jR}$ and substitute $V_{iR}^{\prime } (p),V_{jR}^{\prime } (p)$ into (A13) and (A14):

$$\begin{aligned} \lambda (k_{iR} + f_{iR} p) &= \frac{1}{2}[A - \tau + f_{iR} ]^{2} + \frac{{[\tau - f_{iR} ]^{2} }}{2(c - \varepsilon )} + f_{iR} \left[\theta A - \tau + f_{jR} - \frac{{\tau - f_{jR} - \omega }}{c - \varepsilon }\right]\\ & \qquad + \frac{{\omega [\tau - f_{jR} - \omega ]}}{c - \varepsilon } - Dp(t) - \delta f_{iR} p(t)\end{aligned} $$

(A16)

$$\begin{aligned} \lambda (k_{jR} + f_{jR} p) &= \frac{1}{2}[\theta A - \tau + f_{jR} ]^{2} + \frac{{[f_{jR} - \tau + \omega ]^{2} }}{2(c - \varepsilon )}\\ & \quad + f_{jR} \left[A - \tau + f_{iR} - \frac{{\tau - f_{iR} }}{c - \varepsilon }\right] - \beta Dp(t) - \delta f_{jR} p(t)\end{aligned} $$

(A17)

We can determine $f_{iR}$, $k_{iR}$, $f_{jR}$, $k_{jR}$ from (A16) and (A17):

$$ f_{iR} = \frac{ - D}{\lambda + \delta },\quad f_{jR} = \frac{ - \beta D}{\lambda + \delta } $$

(A18)

$$\begin{aligned} k_{iR} &= \frac{1}{2\lambda }\left(A - \tau - \frac{D}{\lambda + \delta }\right)^{2} + \frac{{[\tau (\lambda + \delta ) + D]^{2} }}{{2\lambda (\lambda + \delta )^{2} (c - \varepsilon )}}\\ & \quad - \frac{D}{\lambda (\lambda + \delta )}\left[ {\theta A - \tau - \frac{\beta D}{\lambda + \delta } - \frac{\tau (\lambda + \delta ) + \beta D}{c(\lambda + \delta )}} \right] + \frac{{\tau \omega (\lambda + \delta ) + \beta \omega D - \omega^{2} (\lambda + \delta )}}{\lambda (c - \varepsilon )(\lambda + \delta )}\end{aligned} $$

(A19)

$$\begin{aligned} k_{jR} &= \frac{1}{2\lambda }\left(\theta A - \tau - \frac{\beta D}{\lambda + \delta }\right)^{2} + \frac{{[ - \tau (\lambda + \delta ) - \beta D + \omega (\lambda + \delta )]^{2} }}{{2\lambda (c - \varepsilon )(\lambda + \delta )^{2} }}\\ & \quad - \frac{\beta D}{\lambda (\lambda + \delta )}\left[ {A - \tau - \frac{D}{\lambda + \delta } - \frac{\tau (\lambda + \delta ) + D}{(c - \varepsilon )(\lambda + \delta )}} \right]\end{aligned} $$

(A20)

Substituting (A18), (A19), and (A20) into $V_{mR}^{\prime } (p)$ and referring to (A11) and (A12), we obtain the equilibrium outputs, optimal proportion of pollutants that the region chooses to reduce, and net revenues.

Appendix 4 for Proposition 3

Similar to the aforementioned analysis, the first-order conditions of $E_{iF}^{*}$, $E_{jF}^{*}$, $\alpha_{iF}^{*}$, and $\alpha_{jF}^{*}$ are as follows:

$$ E_{iF}^{*} (t) = A - \tau + V_{iF}^{\prime } (p),\quad \alpha_{iF}^{*} (t) = \frac{{\tau - V_{iF}^{\prime } (p)}}{{(c - \varepsilon )[A - \tau + V_{iF}^{\prime } (p)]}} $$

(A21)

$$ E_{jF}^{*} (t) = \theta A - \tau + V_{jF}^{\prime } (p),\quad \alpha_{jF}^{*} (t) = \frac{{\tau - V_{jF}^{\prime } (p)}}{{(c - \varepsilon )[\theta A - \tau + V_{jF}^{\prime } (p)]}} $$

(A22)

Substituting (A21) and (A22) into (10a) and (10b), we obtain following HJB equations:

$$\begin{aligned} \lambda V_{iF} (p) &= \frac{1}{2}[A - \tau + V_{iF}^{\prime } (p)]^{2} + \frac{{[\tau - V_{iF}^{\prime } (p)]^{2} }}{2(c - \varepsilon )}\\ & \quad + V_{iF}^{\prime } (p)\left[ {\theta A - \tau + V_{jF}^{\prime } (p) - \frac{{\tau - V_{jF}^{\prime } (p)}}{c - \varepsilon }} \right] \\ & \quad- Dp(t) - \delta V_{iF}^{\prime } (p)p(t)\end{aligned} $$

(A23)

$$\begin{aligned} \lambda V_{jF} (p) &= \frac{1}{2}[\theta A - \tau + V_{jF}^{\prime } (p)]^{2} + \frac{{[\tau - V_{jF}^{\prime } (p)]^{2} }}{2(c - \varepsilon )}\\ & \quad + V_{jF}^{\prime } (p)\left[ {A - \tau + V_{iF}^{\prime } (p) - \frac{{\tau - V_{iF}^{\prime } (p)}}{c - \varepsilon }} \right] \\ & \quad- \beta Dp(t) - \delta V_{jF}^{\prime } (p)p(t)\end{aligned} $$

(A24)

Following the analysis for Scenario R, we then determine that the value function of the firm is linear in p; that is,

$$ V_{iF} (p) = k_{iF} + f_{iF} p,\quad V_{jF} (p) = k_{jF} + f_{jF} p $$

(A25)

Similar to the aforementioned analysis, we can obtain $V_{iF}^{\prime } (p) = f_{iF}$, $V_{jF}^{\prime } (p) = f_{jF}$, substitute it into (A23) and (A24), and determine $f_{iF}$, $k_{iF}$, $f_{jF}$, $k_{jF}$, as follows:

$$ f_{iF} = \frac{ - D}{\lambda + \delta },\quad f_{jF} = \frac{ - \beta D}{\lambda + \delta } $$

(A26)

$$\begin{aligned} k_{iF} &= \frac{1}{2\lambda }(A - \tau - \frac{D}{\lambda + \delta })^{2} + \frac{{[\tau (\lambda + \delta ) + D]^{2} }}{{2\lambda (\lambda + \delta )^{2} (c - \varepsilon )}}\\ & \quad - \frac{D}{\lambda (\lambda + \delta )}\left[ {\theta A - \tau - \frac{\beta D}{\lambda + \delta } - \frac{\tau (\lambda + \delta ) + \beta D}{(c - \varepsilon )(\lambda + \delta )}} \right]\end{aligned} $$

(A27)

$$\begin{aligned} k_{jF} &= \frac{1}{2\lambda }\left( {\theta A - \tau - \frac{\beta D}{\lambda + \delta }} \right)^{2} + \frac{{[\tau (\lambda + \delta ) + \beta D]^{2} }}{{2(c - \varepsilon )\lambda (\lambda + \delta )^{2} }}\\ & \quad - \frac{\beta D}{\lambda (\lambda + \delta )}\left[ {A - \tau - \frac{D}{\lambda + \delta } - \frac{\tau (\lambda + \delta ) + D}{(c - \varepsilon )(\lambda + \delta )}} \right]\end{aligned} $$

(A28)

Substituting (A26), (A27), and (A28) into $V_{mF}^{\prime } (p)$, and referring to (A21) and (A22), we can derive the equilibrium outputs, optimal proportion of pollutants that the region chooses to reduce, and net revenues.

Appendix 5 for Proposition 4

$$ \begin{aligned} V_{jR}^{*} - V_{jN}^{*} & = \frac{{[\omega (\lambda + \delta ) - \tau (\lambda + \delta ) - \beta D]^{2} }}{{2(c - \varepsilon )(\lambda + \delta )^{2} \lambda }} - \frac{{[\tau (\lambda + \delta ) + \beta D]^{2} }}{{2c(\lambda + \delta )^{2} \lambda }} \\ & = \frac{{c[\omega (\lambda + \delta ) - \tau (\lambda + \delta ) - \beta D]^{2} - (c - \varepsilon )[\tau (\lambda + \delta ) + \beta D]^{2} }}{{2c(c - \varepsilon )(\lambda + \delta )^{2} \lambda }} \\ \end{aligned} $$

(A29)

We determine the optimal per-unit royalty licensing fee $\omega^{*}$ when $V_{jR}^{*} (p) - V_{jN}^{*} (p) = 0$. For region i, we have

$$ \begin{aligned} V_{iR}^{*} - V_{iN}^{*} & = \frac{D[(\tau - \omega )(\lambda + \delta ) + \beta D]}{{\lambda (c - \varepsilon )(\lambda + \delta )^{2} }} - \frac{D[\tau (\lambda + \delta ) + \beta D]}{{\lambda c(\lambda + \delta )^{2} }} + \frac{\omega [(\tau - \omega )(\lambda + \delta ) + \beta D]}{\lambda (c - \varepsilon )(\lambda + \delta )} \\ & = \frac{D[\varepsilon \tau (\lambda + \delta ) - c\omega (\lambda + \delta ) + \varepsilon \beta D]}{{\lambda c(c - \varepsilon )(\lambda + \delta )^{2} }} + \frac{\omega [(\tau - \omega )(\lambda + \delta ) + \beta D]}{(c - \varepsilon )(\lambda + \delta )\lambda } \\ \end{aligned} $$

(A30)

Substituting the steady-state $\omega^{*}$ into (A30), we obtain

$$\begin{aligned} V_{iR}^{*} - V_{iN}^{*} &= \frac{{[ - c + \varepsilon + \sqrt {c(c - \varepsilon )} ]D[\tau (\lambda + \delta ) + \beta D]}}{{c\lambda (c - \varepsilon )(\lambda + \delta )^{2} }} \\ & \quad + \frac{{[c - \sqrt {c(c - \varepsilon )} ][\sqrt {c(c - \varepsilon )} ][\tau (\lambda + \delta ) + \beta D]^{2} }}{{\lambda c^{2} (c - \varepsilon )(\lambda + \delta )^{2} }}\end{aligned} $$

(A31)

Evidently, the polynomial $[ - c + \varepsilon + \sqrt {c(c - \varepsilon )} ] > 0$, $c - \sqrt {c(c - \varepsilon )}$ when $c > \varepsilon > 0$; thus, $V_{iR}^{*} - V_{iN}^{*} > 0$. For $\alpha_{iR}^{*}$, $\alpha_{iN}^{*}$, $\alpha_{jR}^{*}$, $\alpha_{jN}^{*}$,

$$ \alpha_{iR}^{*} (t) - \alpha_{iN}^{*} (t) = \frac{\tau (\lambda + \delta ) + D}{{(c - \varepsilon )(\lambda + \delta )(A - \tau - \frac{D}{\lambda + \delta })}} - \frac{\tau (\lambda + \delta ) + D}{{(c - \varepsilon )(\lambda + \delta )(A - \tau - \frac{D}{\lambda + \delta })}} = 0 $$

(A32)

$$ \alpha_{jR}^{*} (t) - \alpha_{jN}^{*} (t) = \frac{{[\sqrt {c(c - \varepsilon )} - c + \varepsilon ][\tau (\lambda + \delta ) + \beta D]}}{{c(c - \varepsilon )(\lambda + \delta )(\theta A - \tau - \frac{\beta D}{\lambda + \delta })}} $$

(A33)

In accordance with (A33), $\alpha_{iR}^{*} - \alpha_{iN}^{*} > 0$ in the steady state, and $E_{iN}^{*} = E_{iR}^{*}$, $E_{jN}^{*} = E_{jR}^{*}$.

On the basis of (A29)–(A33), we can determine the condition of Proposition 4.

Appendix 6 for Proposition 5

The proof is similar to that of Proposition 4, which is derived from (6c), (6d) and (12c), (12d), gives as follows:

$$ \begin{aligned} V_{jF}^{*} - V_{jN}^{*} & = \frac{{[\tau (\lambda + \delta ) + \beta D]^{2} }}{{2\lambda (c - \varepsilon )(\lambda + \delta )^{2} }} - \frac{{[\tau (\lambda + \delta ) + \beta D]^{2} }}{{2\lambda c(\lambda + \delta )^{2} }} - M \\ & = \frac{{\varepsilon [\tau (\lambda + \delta ) + \beta D]^{2} }}{{2\lambda c(c - \varepsilon )(\lambda + \delta )^{2} }} - M \\ \end{aligned} $$

(A34)

$$ V_{iF}^{*} - V_{iN}^{*} = \frac{\varepsilon D[\tau (\lambda + \delta ) + \beta D]}{{\lambda c(c - \varepsilon )(\lambda + \delta )^{2} }} + M > 0 $$

(A35)

For $V_{jF}^{*} - V_{jN}^{*} = 0$, we can calculate the equilibrium $M^{*} = \frac{{\varepsilon [\tau (\lambda + \delta ) + \beta D]^{2} }}{{2\lambda c(c - \varepsilon )(\lambda + \delta )^{2} }}$ in the steady state.

Similarly, referring to (6a), (6b) and (12a), (12b), we derive

$$ \begin{aligned} \alpha_{jF}^{*} - \alpha_{jN}^{*} & = \frac{\tau (\lambda + \delta ) + \beta D}{{(c - \varepsilon )(\lambda + \delta )(\theta A - \tau - \frac{\beta D}{\lambda + \delta })}} - \frac{\tau (\lambda + \delta ) + \beta D}{{c(\lambda + \delta )(\theta A - \tau - \frac{\beta D}{\lambda + \delta })}} \\ & = \frac{\varepsilon [\tau (\lambda + \delta ) + \beta D]}{{c(c - \varepsilon )(\lambda + \delta )(\theta A - \tau - \frac{\beta D}{\lambda + \delta })}} > 0 \\ \end{aligned} $$

(A36)

$$ \alpha_{iF}^{*} - \alpha_{iN}^{*} = \frac{\tau (\lambda + \delta ) + D}{{(c - \varepsilon )(\lambda + \delta )(A - \tau - \frac{D}{\lambda + \delta })}} - \frac{\tau (\lambda + \delta ) + D}{{(c - \varepsilon )(\lambda + \delta )(A - \tau - \frac{D}{\lambda + \delta })}} = 0 $$

(A37)

The total outputs of the two regions remain unchanged ($E_{iN}^{*} = E_{iF}^{*}$, $E_{jN}^{*} = E_{jF}^{*}$); thus, Proposition 5 can be directly obtained from the aforementioned proofs.