Non-cooperative Mode, Cost-Sharing Mode, or Cooperative Mode: Which is the Optimal Mode for Desertification Control?

Abstract

The problem of desertification is becoming increasingly severe, affecting the production and livelihood of people; this has led to keen interest in the issue, especially in developing countries. Based on the dynamic changes in desertification control scale and enterprise goodwill, we studied the game problem of desertification control between the government and enterprise under the non-cooperative, cost-sharing, and cooperative modes by constructing a differential game model. We put forward a revenue distribution mechanism with time consistency under the cooperation mode. The results show that government and enterprise control of desertification under the cooperative mode has the best effect, followed by the cost-sharing mode. The non-cooperative mode should be avoided. Lowering tax rates by the government is not always a good method of encouraging enterprises to increase desertification control investment. The tax rate should be adjusted according to the choice of governance mode as it can more effectively increase the investment level of enterprises in combating desertification. The optimal pricing of desert characteristic products is not affected by the governance mode.

Appendices

Appendix 1 for Proposition 1

The first-order conditions of $u_{gN}^{*}$, $u_{sN}^{*}$, and $p_{N}^{*}$ are as follows:

$$ u_{gN}^{*} (t) = \frac{{V^{\prime}_{gN} (S)e}}{{c_{g} }} $$

(26)

$$ u_{sN}^{*} (t) = \frac{{V^{\prime}_{sN} (S)f}}{{c_{s} }} $$

(27)

$$ p_{N}^{*} (t) = \frac{a}{{2b_{1} }} $$

(28)

Substituting Eqs. (26), (27), and (28) into Eqs. (7a) and (7b) of HJB to obtain the following.

$$ \begin{aligned} rV_{gN} (S,G) & = [B - \sigma V^{\prime}_{gN} (S) + {\kern 1pt} bV^{\prime}_{gN} (G)]S(t) + \left[ {T_{1} \pi \omega + \frac{{T_{2} \alpha a^{2} }}{{4b_{1} }} - {\kern 1pt} \delta V^{\prime}_{gN} (G){\kern 1pt} } \right]G(t) \\ & \quad + \,T_{1} \pi Q_{0} - BS_{\kappa } + \frac{{(V^{\prime}_{gN} (S))^{2} e^{2} }}{{2c_{g} }} + \frac{{f^{2} V^{\prime}_{gN} (S)V^{\prime}_{sN} (S){\kern 1pt} }}{{c_{s} }} \\ \end{aligned} $$

(29)

$$ \begin{aligned} rV_{sN} (S,G) & = [b{\kern 1pt} V^{\prime}_{sN} (G) - \sigma V^{\prime}_{sN} (S){\kern 1pt} ]S(t) + (1 - T_{1} )\pi Q_{0} + \frac{{(V^{\prime}_{sN} (S){\kern 1pt} )^{2} f^{2} }}{{2c_{s} }} \\ & \quad + \,\left[ {(1 - T_{1} )\pi \omega + \frac{{(1 - T_{2} )\alpha a^{2} }}{{4b_{1} }} - {\kern 1pt} \delta {\kern 1pt} V^{\prime}_{sN} (G)} \right]G{\kern 1pt} (t){\kern 1pt} {\kern 1pt} + \frac{{e^{2} V^{\prime}_{gN} (S){\kern 1pt} V^{\prime}_{sN} (S)}}{{c_{g} }} \\ \end{aligned} $$

(30)

From the structure of Eqs. (29) and (30), we can analyze that the expressions of $V_{gN} (S,G)$ and $V_{sN} (S,G)$ are linear functions, such that

$$ V_{gN} (S,G) = m_{1} S + m_{2} G + m_{3} ,\quad V_{sN} (S,G) = n_{1} S + n_{2} G + n_{3} $$

(31)

$$ {\text{then we can get}}:\quad V^{\prime}_{gN} (S) = m_{1} ,\quad V^{\prime}_{gN} (G) = m_{2} ,\quad V^{\prime}_{sN} (S) = n_{1} ,\quad V^{\prime}_{sN} (G) = n_{2} $$

(32)

where $m_{1}$, $m_{2}$, $m_{3}$, $n_{1}$, $n_{2}$, and $n_{3}$ are all constants. Substituting Eqs. (31) and (32) into Eqs. (29) and (30), according to the undetermined coefficient method, we obtain the following.

$$ \left\{ {\begin{array}{*{20}l} {m_{1} { = }\frac{{4b_{1} (r + \delta )B + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )}},m_{2} { = }\frac{{4b_{1} T_{1} \pi \omega + T_{2} \alpha a^{2} }}{{4b_{1} (r + \delta )}}} \hfill \\ {m_{3} = \frac{{T_{1} \pi Q_{0} - BS_{\kappa } }}{r} + \frac{{e^{2} [4b_{1} B(r + \delta ) + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b]^{2} }}{{32b_{1}^{2} c_{g} r(r + \delta )^{2} (r + \sigma )^{2} }}} \hfill \\ { + \frac{{f^{2} [4b_{1} B(r + \delta ) + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b][4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b]}}{{16b_{1}^{2} c_{s} r(r + \delta )^{2} (r + \sigma )^{2} }}} \hfill \\ \end{array} } \right. $$

(33)

$$ \left\{ {\begin{array}{*{20}l} {n_{1} { = }\frac{{4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )}},n_{2} { = }\frac{{4b_{1} (1 - T_{1} )\pi \omega + (1 - T_{2} )\alpha a^{2} }}{{4b_{1} (r + \delta )}}} \hfill \\ {n_{3} = \frac{{(1 - T_{1} )\pi Q_{0} }}{r} + \frac{{f^{2} [4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b]^{2} }}{{32b_{1}^{2} c_{s} r(r + \delta )^{2} (r + \sigma )^{2} }}} \hfill \\ { + \frac{{e^{2} [4b_{1} B(r + \delta ) + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b][4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b]}}{{16b_{1}^{2} c_{g} r(r + \delta )^{2} (r + \sigma )^{2} }}} \hfill \\ \end{array} } \right. $$

(34)

By substituting Eqs. (33) and (34) into Eqs. (26), (27), and (28), we can obtain the expressions of $u_{gN}^{*} (t)$, $u_{sN}^{*} (t)$, and $p_{N}^{*} (t)$, and substituting Eqs. (33) and (34) into Eqs. (31), we can obtain $V_{gN}^{*} (S,G)$ and $V_{sN}^{*} (S,G)$. Then, Proposition 1 can be obtained.

Appendix 2 for Corollary 1

Taking the first derivatives of $u_{gN}^{*} (t)$, $u_{sN}^{*} (t)$, and $p_{N}^{*} (t)$ with respect to $T_{1}$ and $T_{2}$, we can obtain: $\frac{{\partial u_{gN}^{*} (t)}}{{\partial T_{1} }} = \frac{e\pi \omega b}{{(r + \delta )(r + \sigma )c_{g} }} > 0$, $\frac{{\partial u_{gN}^{*} (t)}}{{\partial T_{2} }} = \frac{{e\alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )c_{g} }} > 0$

$$ \frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{1} }} = - \frac{f\pi \omega b}{{(r + \delta )(r + \sigma )c_{s} }} < 0,\quad \frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{2} }} = - \frac{{f\alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )c_{s} }} < 0 $$

$$ \frac{{\partial p_{N}^{*} (t)}}{{\partial T_{1} }} = 0,\quad \frac{{\partial p_{N}^{*} (t)}}{{\partial T_{2} }} = 0 $$

$$ \left| {\frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{1} }}} \right| - \left| {\frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{2} }}} \right| = \frac{f\pi \omega b}{{(r + \delta )(r + \sigma )c_{s} }} - \frac{{f\alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )c_{s} }} $$

Then we can know that when $\pi > \frac{\alpha }{2\omega }p_{N}^{*} (t)$, $\left| {\frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{1} }}} \right| - \left| {\frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{2} }}} \right| > 0$, and when $\pi < \frac{\alpha }{2\omega }p_{N}^{*} (t)$, $\left| {\frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{1} }}} \right| - \left| {\frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{2} }}} \right| < 0$.

Taking the first derivatives of $u_{gN}^{*} (t)$, $u_{sN}^{*} (t)$, and $p_{N}^{*} (t)$ concerning $r$, we have:

$$ \frac{{\partial u_{gN}^{*} (t)}}{\partial r} = - \frac{{e\{ 4[\pi \omega T_{1} (\delta + \sigma + 2r)b + B(r + \delta )^{2} ]b_{1} + \alpha a^{2} bT_{2} (\delta + \sigma + 2r)\} }}{{4b_{1} (r + \delta )^{2} (r + \sigma )^{2} c_{g} }} < 0 $$

$$ \frac{{\partial u_{sN}^{*} (t)}}{\partial r} = - \frac{{[(1 - T_{2} )\alpha a^{2} + 4b_{1} \pi \omega (1 - T_{1} )](\delta + \sigma + 2r)bf}}{{4b_{1} (r + \delta )^{2} (r + \sigma )^{2} c_{s} }} < 0,\quad \frac{{\partial p_{N}^{*} (t)}}{\partial r} = 0 $$

Then, we can obtain the conclusions of Corollary 1.

Appendix 3 for Proposition 2

Taking the first derivatives of Eq. (14) with respect to $u_{gS} (t)$ and $\theta (t)$, we can obtain the optimal first-order condition of the government as follows:

$$ u_{gS}^{*} (t) = \frac{{V^{\prime}_{gS} (S)e}}{{c_{g} }} $$

(35)

$$ \theta^{ * } (t) = \frac{{2V^{\prime}_{gS} (S) - V^{\prime}_{sS} (S)}}{{2V^{\prime}_{gS} (S) + V^{\prime}_{sS} (S)}} $$

(36)

Substituting Eqs. (13a), (13b), (35), and (36) into Eqs. (12) and (14) to obtain the following.

$$ \begin{aligned} rV_{gS} (S,G) & = [B - \sigma V^{\prime}_{gS} (S) + {\kern 1pt} bV^{\prime}_{gS} (G)]S(t) + \left[ {T_{1} \pi \omega + \frac{{T_{2} \alpha a^{2} }}{{4b_{1} }} - {\kern 1pt} \delta V^{\prime}_{gS} (G)} \right]G(t) \\ & \quad + T_{1} \pi Q_{0} - BS_{\kappa } + \frac{{(V^{\prime}_{gS} (S))^{2} e^{2} }}{{2c_{g} }} + \frac{{f^{2} (2V^{\prime}_{gS} (S) + V^{\prime}_{sS} (S){\kern 1pt} )^{2} }}{{8c_{s} }} \\ \end{aligned} $$

(37)

$$ \begin{aligned} rV_{sS} (S,G) & = [b{\kern 1pt} V^{\prime}_{sS} (G) - \sigma V^{\prime}_{sS} (S){\kern 1pt} ]S(t) + \frac{{V^{\prime}_{sS} (S)(2V^{\prime}_{gS} (S) + V^{\prime}_{sS} (S){\kern 1pt} )f^{2} }}{{4c_{s} }} \\ & \quad + (1 - T_{1} )\pi Q_{0} + \left[ {(1 - T_{1} )\pi \omega + \frac{{(1 - T_{2} )\alpha a^{2} }}{{4b_{1} }} - {\kern 1pt} \delta {\kern 1pt} V^{\prime}_{sS} (G)} \right]G{\kern 1pt} (t){\kern 1pt} {\kern 1pt} + \frac{{e^{2} V^{\prime}_{gS} (S){\kern 1pt} V^{\prime}_{sS} (S)}}{{c_{g} }} \\ \end{aligned} $$

(38)

From the structure of Eqs. (37) and (38), we can analyze that the expressions of $V_{gS} (S,G)$ and $V_{sS} (S,G)$ are linear functions, such that

$$ V_{gS} (S,G) = g_{1} S + g_{2} G + g_{3} ,\quad V_{sS} (S,G) = h_{1} S + h_{2} G + h_{3} $$

(39)

$$ {\text{then we can get}}:V^{\prime}_{gS} (S) = g_{1} ,\quad V^{\prime}_{gS} (G) = g_{2} ,\quad V^{\prime}_{sS} (S) = h_{1} ,\quad V^{\prime}_{sS} (G) = h_{2} $$

(40)

where $g_{1}$, $g_{2}$, $g_{3}$, $h_{1}$, $h_{2}$, and $h_{3}$ are all constants. Then, substituting Eqs. (39) and (40) into Eqs. (37) and (38), following the undetermined coefficient method, we obtain the following.

$$ \left\{ {\begin{array}{*{20}l} {g_{1} { = }\frac{{4b_{1} (r + \delta )B + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )}},g_{2} { = }\frac{{4b_{1} T_{1} \pi \omega + T_{2} \alpha a^{2} }}{{4b_{1} (r + \delta )}}} \hfill \\ {g_{3} = \frac{{T_{1} \pi Q_{0} - BS_{\kappa } }}{r} + \frac{{e^{2} [4b_{1} B(r + \delta ) + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b]^{2} }}{{32b_{1}^{2} c_{g} r(r + \delta )^{2} (r + \sigma )^{2} }}} \hfill \\ { + \frac{{f^{2} [8b_{1} B(r + \delta ) + 4b_{1} (1 + T_{1} )\pi \omega b + (1 + T_{2} )\alpha a^{2} b]}}{{128b_{1}^{2} c_{s} r(r + \delta )^{2} (r + \sigma )^{2} }}} \hfill \\ \end{array} } \right. $$

(41)

$$ \left\{ \begin{gathered} h_{1} = \frac{{4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )}},h_{2} = \frac{{4b_{1} (1 - T_{1} )\pi \omega + (1 - T_{2} )\alpha a^{2} }}{{4b_{1} (r + \delta )}} \hfill \\ h_{3} = \frac{{f^{2} [4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b][8b_{1} B(r + \delta ) + 4b_{1} (1 + T_{1} )\pi \omega b + (1 + T_{2} )\alpha a^{2} b]}}{{64b_{1}^{2} c_{s} r(r + \delta )^{2} (r + \sigma )^{2} }} \hfill \\ + \frac{{e^{2} [4b_{1} B(r + \delta ) + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b][4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b]}}{{16b_{1}^{2} c_{g} r(r + \delta )^{2} (r + \sigma )^{2} }} + \frac{{(1 - T_{1} )\pi Q_{0} }}{r} \hfill \\ \end{gathered} \right. $$

(42)

By substituting Eqs. (41) and (42) into Eqs. (13a), (13b), (35), and (36), we can obtain the expressions for $u_{sS}^{*} (t)$, $p_{S}^{*} (t)$, $u_{gS}^{*} (t)$, and $\theta^{*} (t)$, and substituting Eqs. (41) and (42) into Eq. (39), we can obtain $V_{gS}^{*} (S,G)$ and $V_{sS}^{*} (S,G)$. Then, Proposition 2 can be obtained.

Appendix 4 for Proposition 3

Taking the first derivatives of Eq. (15b) with respect to $T_{1}$ and $T_{2}$, we obtain:

$$ \frac{{\partial \theta^{ * } (t)}}{{\partial T_{1} }} = \frac{{2b_{1} \{ 4b_{1} [\pi \omega b + (r + \delta )B] + \alpha a^{2} b\} \pi \omega b}}{{\{ 8[\pi \omega (1 + T_{1} )b + 2(r + \delta )B]b_{1} + \alpha a^{2} b(1 + T_{2} )\}^{2} }} > 0 $$

$$ \frac{{\partial \theta^{ * } (t)}}{{\partial T_{2} }} = \frac{{\{ 4b_{1} [\pi \omega b + (r + \delta )B] + \alpha a^{2} b\} \alpha a^{2} b}}{{2\{ 8[\pi \omega (1 + T_{1} )b + 2(r + \delta )B]b_{1} + \alpha a^{2} b(1 + T_{2} )\}^{2} }} > 0 $$

When the government does not undertake the enterprise’s governance cost, the impact of the tax rate $T_{1}$ and $T_{2}$ on the enterprise’s investment level is

$$ \frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{1} }} = - \frac{f\pi \omega b}{{(r + \delta )(r + \sigma )c_{s} }} < 0,\quad \frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{2} }} = - \frac{{f\alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )c_{s} }} < 0 $$

When the government undertake part of the enterprise’s governance cost, the impact of the government’s undertake proportion on the enterprise’s investment level is

$$ \frac{{\partial u_{sS}^{*} }}{\partial \theta } = \frac{{f\pi \omega bT_{1} }}{{(r + \delta )(r + \sigma )c_{s} (1 - \theta )^{2} }} + \frac{{f\alpha a^{2} bT_{2} }}{{4b_{1} (r + \delta )(r + \sigma )c_{s} (1 - \theta )^{2} }} + \frac{fB}{{(r + \sigma )c_{s} (1 - \theta )^{2} }} > 0 $$

Because we know $0 \le T_{1} \le 1$ and $0 \le \theta \le 1$, obviously $\left| {\frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{1} }}} \right| < \left| {\frac{{\partial u_{sS}^{*} }}{\partial \theta }} \right|$ and $\left| {\frac{{\partial u_{sN}^{*} (t)}}{{\partial T_{2} }}} \right| < \left| {\frac{{\partial u_{sS}^{*} }}{\partial \theta }} \right|$. Then, Proposition 3 can be obtained.

Appendix 5 for Corollary 3

Taking the first derivatives of Eqs. (15a), (15c), and (15d) with respect to $T_{1}$ and $T_{2}$, we can obtain:$\frac{{\partial u_{gS}^{*} (t)}}{{\partial T_{1} }} = \frac{e\pi \omega b}{{(r + \delta )(r + \sigma )c_{g} }} > 0$, $\frac{{\partial u_{gS}^{*} (t)}}{{\partial T_{2} }} = \frac{{e\alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )c_{g} }} > 0$

$$ \frac{{\partial u_{sS}^{*} (t)}}{{\partial T_{1} }} = \frac{f\pi \omega b}{{2(r + \delta )(r + \sigma )c_{s} }} > 0,\quad \frac{{\partial u_{sS}^{*} (t)}}{{\partial T_{2} }} = \frac{{f\alpha a^{2} b}}{{8b_{1} (r + \delta )(r + \sigma )c_{s} }} > 0 $$

$$ \frac{{\partial p_{S}^{*} (t)}}{{\partial T_{1} }} = 0,\quad \frac{{\partial p_{S}^{*} (t)}}{{\partial T_{2} }} = 0 $$

Based on the above discussion, Corollary 3 can be obtained.

Appendix 6 for Proposition 4

The optimal first-order conditions of $u_{gC}^{*}$, $u_{sC}^{*}$, and $p_{C}^{*}$ are as follows:

$$ u_{gC}^{*} (t) = \frac{{V^{\prime}_{C} (S)e}}{{c_{g} }} $$

(43)

$$ u_{sC}^{*} (t) = \frac{{V^{\prime}_{C} (S)f}}{{c_{s} }} $$

(44)

$$ p_{C}^{*} (t) = \frac{a}{{2b_{1} }} $$

(45)

Substituting Eqs. (43), (44), and (45) into Eq. (19) of HJB, we obtain:

$$ \begin{aligned} rV_{C} (S,G) & = [B - \sigma V^{\prime}_{C} (S) + {\kern 1pt} bV^{\prime}_{C} (G)]S(t) + \left[ {\pi \omega + \frac{{\alpha a^{2} }}{{4b_{1} }} - {\kern 1pt} \delta V^{\prime}_{C} (G)} \right]G(t) \\ & \quad + \pi Q_{0} - BS_{\kappa } + \frac{{(V^{\prime}_{C} (S))^{2} e^{2} }}{{2c_{g} }} + \frac{{f^{2} (V^{\prime}_{C} (S){\kern 1pt} )^{2} }}{{2c_{s} }} \\ \end{aligned} $$

(46)

From the structure of Eq. (46), we can analyze that the expression of $V_{C} (S,G)$ is a linear function, such that

$$ V_{C} (S,G) = l_{1} S + l_{2} G + l_{3} $$

(47)

$$ {\text{then we can get:}}V^{\prime}_{C} (S) = l_{1} ,\quad V^{\prime}_{C} (G) = l_{2} $$

(48)

where $l_{1}$, $l_{2}$, and $l_{3}$ are constants. Substituting Eqs. (47) and (48) into Eq. (46), according to the undetermined coefficient method, we obtain:

$$ \left\{ \begin{gathered} l_{1} = \frac{{4b_{1} B(r + \delta ) + 4b_{1} \pi \omega b + \alpha a^{2} b}}{{4b_{1} (r + \delta )(r + \sigma )}},l_{2} = \frac{{4b_{1} \pi \omega + \alpha a^{2} }}{{4b_{1} (r + \delta )}} \hfill \\ l_{3} = \frac{{\pi Q_{0} - BS_{\kappa } }}{r} + \frac{{[4b_{1} B(r + \delta ) + b_{1} \pi \omega b + \alpha a^{2} b]^{2} }}{{32b_{1}^{2} r(r + \delta )^{2} (r + \sigma )^{2} }}\left( {\frac{{e^{2} }}{{c_{g} }} + \frac{{f^{2} }}{{c_{s} }}} \right) \hfill \\ \end{gathered} \right. $$

(49)

Substituting Eq. (49) into Eqs.(43), (44), (45), and (47), we can obtain the expressions for $u_{gC}^{ * } (t)$, $u_{sC}^{ * } (t)$, $p_{C}^{ * } (t)$, and $V_{C}^{ * } (S,G)$. Then, Proposition 4 can be obtained.

Appendix 7 for Proposition 5

According to (8) in Proposition 1, (15) in Proposition 2, and (20) in Proposition 4, we can obtain: $p_{S}^{ * } (t) - p_{N}^{ * } (t) = 0$,$p_{C}^{ * } (t) - p_{S}^{ * } (t) = 0$

$$ u_{gN}^{ * } (t) - u_{gS}^{ * } (t) = 0,\quad u_{gC}^{ * } (t) - u_{gS}^{ * } (t) = \frac{{e[4b_{1} \pi \omega b(1 - T_{1} ) + \alpha a^{2} b(1 - T_{2} )]}}{{4b_{1} (r + \delta )(r + \sigma )c_{g} }} > 0 $$

$$ u_{sS}^{ * } (t) - u_{sN}^{ * } (t) = \frac{{f[4b_{1} \pi \omega b(3T_{1} - 1) + \alpha a^{2} b(3T_{2} - 1) + 8b_{1} (r + \delta )B]}}{{8b_{1} (r + \delta )(r + \sigma )c_{s} }} $$

Based on the condition $\theta^{*} (t) > 0$, we know that as long as the cost-sharing mode is established, $u_{sS}^{ * } (t) - u_{sN}^{ * } (t) > 0$.

$$ u_{sC}^{ * } (t) - u_{sS}^{ * } (t) = \frac{{f[b_{1} \pi \omega b(1 - T_{1} ) + \alpha a^{2} b(1 - T_{2} )]}}{{8b_{1} c_{s} (r + \delta )(r + \sigma )}} > 0 $$

We can get the conclusion:$p_{N}^{ * } (t) = p_{S}^{ * } (t) = p_{C}^{ * } (t)$,$u_{gN}^{ * } = u_{gS}^{ * } < u_{gC}^{ * }$, $u_{sN}^{ * } < u_{sS}^{ * } < u_{sC}^{ * }$.

Similarly, according to Eq. (9) in Proposition 1, (16) in Proposition 2, and (21) in Proposition 4, we have the following results.

$$ V_{gS}^{*} - V_{gN}^{*} = \frac{{f^{2} \{ 4b_{1} [2B(r + \delta ) + \pi \omega b(3T_{1} - 1)] + \alpha a^{2} b(3T_{2} - 1)\}^{2} }}{{64b_{1}^{2} c_{s} r(r + \delta )^{2} (r + \sigma )^{2} }} > 0 $$

$$ V_{sS}^{*} - V_{sN}^{*} = \frac{{4b_{1} \{ [2B(r + \delta ) + b\pi \omega (3T_{1} - 1)] + \alpha a^{2} b\} f^{2} b[4b_{1} \pi \omega (1 - T_{1} ) + \alpha a^{2} (1 - T_{2} )]}}{{64b_{1}^{2} c_{s} r(r + \delta )^{2} (r + \sigma )^{2} }} > 0 $$

$$ \begin{aligned} & & V_{C}^{*} - (V_{gN}^{*} + V_{sN}^{*} ) \\ & = \frac{{e^{2} b^{2} [4b_{1} \pi \omega (T_{1} - 1) + \alpha a^{2} (T_{2} - 1)]^{2} }}{{32b_{1}^{2} c_{g} r(r + \delta )^{2} (r + \sigma )^{2} }} + \frac{{f^{2} [4b_{1} B(r + \delta ) + \alpha a^{2} bT_{2} + 4b_{1} \pi \omega T_{1} b]^{2} }}{{32b_{1}^{2} c_{s} r(r + \delta )^{2} (r + \sigma )^{2} }} > 0 \\ \end{aligned} $$

$$ \begin{aligned} & V_{C}^{*} - (V_{gS}^{*} + V_{sS}^{*} ) \\ & = \frac{{e^{2} b^{2} [4b_{1} \pi \omega (T_{1} - 1) + \alpha a^{2} (T_{2} - 1)]^{2} }}{{32b_{1}^{2} c_{g} r(r + \delta )^{2} (r + \sigma )^{2} }} + \frac{{f^{2} b^{2} [\alpha a^{2} (T_{2} - 1) + 4b_{1} \pi \omega (T_{1} - 1)]^{2} }}{{128b_{1}^{2} c_{s} r(r + \delta )^{2} (r + \sigma )^{2} }} > 0 \\ \end{aligned} $$

We can then conclude $V_{gS}^{*} > V_{gN}^{*}$, $V_{sS}^{*} > V_{sN}^{*}$, $V_{C}^{*} > V_{gN}^{*} + V_{sN}^{*}$, and $V_{C}^{*} > V_{gS}^{*} + V_{sS}^{*}$.

Based on the above analysis Proposition 5 can be obtained.

Appendix 8 for Proposition 6

According to Eqs. (10), (17), and (22), we have:

$$ \begin{aligned} S_{S} (t) - S_{N} (t) & = (S_{S}^{SS} - S_{N}^{SS} )(1 - e^{ - \sigma t} ) \\ & = \frac{{f^{2} [8B(r + \delta )b_{1} + 4b_{1} \pi \omega b(3T_{1} - 1) + \alpha a^{2} b(3T_{2} - 1)](1 - e^{ - \sigma t} )}}{{8b_{1} c_{s} (r + \delta )(r + \sigma )\sigma }} > 0 \\ \end{aligned} $$

$$ \begin{aligned} S_{C} (t) - S_{S} (t) & = (D_{C}^{{SS}} - D_{S}^{{SS}} )(1 - e^{{ - \sigma t}} ) \\ & = \frac{{[4b_{1} \pi \omega b(1 - T_{1} ) + \alpha a^{2} b(1 - T_{2} )](2e^{2} c_{s} + f^{2} c_{g} )(1 - e^{{ - \sigma t}} )}}{{8b_{1} c_{s} c_{g} (r + \delta )(r + \sigma )\sigma }} > 0 \\ \end{aligned} $$

$$ \begin{aligned} G_{S} (t) - G_{N} (t) & = (G_{S}^{SS} - G_{N}^{SS} )(1 - e^{ - \delta t} ) + \frac{b}{\delta - \sigma }(S_{S}^{SS} - S_{N}^{S} )(e^{ - \delta t} - e^{ - \sigma t} ) \\ {\kern 1pt} & = b(S_{S}^{SS} - S_{N}^{S} )\left( {\frac{{e^{ - \delta t} }}{\delta - \sigma } - \frac{{e^{ - \delta t} }}{\delta } - \frac{{e^{ - \sigma t} }}{\delta - \sigma } + \frac{1}{\delta }} \right) \\ \end{aligned} $$

It is impossible to determine the sign of $G_{S} (t) - G_{N} (t)$ directly; we suppose $F(t) = G_{S} (t) - G_{N} (t)$, then we can get $F^{\prime}(t) = \frac{{b\sigma (S_{S}^{SS} - S_{N}^{SS} )}}{\delta - \sigma }(e^{ - \sigma t} - e^{ - \delta t} )$, $F^{\prime}(t) > 0$ as a constant. It can be seen that if $F(0) = 0$, then $F(t) = G_{S} (t) - G_{N} (t) > 0$. In the same way, we can conclude $G_{C} (t) - G_{S} (t) > 0$. From the above analysis, we can obtain Proposition 6.

Appendix 9 for Proposition 7

Substituting $u_{gN}^{ * }$ and $u_{sN}^{ * }$ in Proposition 1 and $u_{gC}^{ * }$ and $u_{sC}^{ * }$ in Proposition 4 into $V_{gN}$, $V_{sN}$, $W_{gC}$, and $W_{sC}$, we obtain the following expressions.

$V_{gN} = \int_{0}^{\infty } {e^{ - rt} \left\{ {T_{1} \pi Q_{0} - BS_{0} - \frac{{e^{2} (4b_{1} (r + \delta )B + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b)^{2} }}{{32b_{1}^{2} (r + \delta )^{2} (r + \sigma )^{2} c_{g} }}} \right.} + \left. {\frac{{4b_{1} T_{1} \pi \omega + T_{2} a^{2} \alpha }}{{4b_{1} }}G_{N} (t) + BS_{N} (t)} \right\}dt$$V_{sN} = \int_{0}^{\infty } {e^{ - rt} \left\{ {(1 - T_{1} )\pi Q_{0} - \frac{{f^{2} (4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b)^{2} }}{{32b_{1}^{2} (r + \delta )^{2} (r + \sigma )^{2} c_{s} }}} \right.} + \left. {\frac{{4b_{1} (1 - T_{1} )\pi \omega + (1 - T_{2} )a^{2} \alpha }}{{4b_{1} }}G_{N} (t)} \right\}dt$$W_{gC} = \int_{0}^{\infty } {e^{ - rt} \left\{ {T_{1} \pi Q_{0} - BS_{0} - \frac{{e^{2} (4b_{1} (r + \delta )B + 4b_{1} \pi \omega b + \alpha a^{2} b)^{2} }}{{32b_{1}^{2} (r + \delta )^{2} (r + \sigma )^{2} c_{g} }}} \right.} \, + \left. {\frac{{4b_{1} T_{1} \pi \omega + T_{2} a^{2} \alpha }}{{4b_{1} }}G_{C} (t) + BS_{C} (t)} \right\}dt$$W_{sC} = \int_{0}^{\infty } {e^{ - rt} \left\{ {(1 - T_{1} )\pi Q_{0} - \frac{{f^{2} (4b_{1} (r + \delta )B + 4b_{1} \pi \omega b + \alpha a^{2} b)^{2} }}{{32b_{1}^{2} (r + \delta )^{2} (r + \sigma )^{2} c_{s} }}} \right.} \, + \left. {\frac{{4b_{1} (1 - T_{1} )\pi \omega + (1 - T_{2} )a^{2} \alpha }}{{4b_{1} }}G_{C} (t)} \right\}dt$ According to (23), we can get $W_{g}^{NBS} (S_{C} (t),G_{C} (t))$, $W_{s}^{NBS} (S_{C} (t),G_{C} (t))$ as follows.

$$ \begin{aligned} W_{g}^{NBS} (S_{C} (t),G_{C} (t)) & = \int_{0}^{\infty } {e^{ - rt} \left\{ {T_{1} \pi Q_{0} - \frac{{B^{2} (2c_{s} e^{2} + c_{g} f^{2} )}}{{4(r + \sigma )^{2} c_{g} c_{s} }}} \right.} + \frac{{B(S_{C} (t) - S_{N} (t) - 2S_{0} )}}{2} \\ & \quad + \,\frac{{\alpha a^{2} b}}{{64b_{1}^{2} (r + \delta )^{2} (r + \sigma )^{2} c_{g} c_{s} }}\left\{ {\alpha a^{2} b[c_{g} f^{2} (T_{2} - 2)T_{2} - c_{s} e^{2} (1 + T_{2}^{2} )]} \right. \\& \left. {\quad - \,8b_{1} (r + \delta )B(c_{s} e^{2} + c_{g} f^{2} ) - 8b_{1} \pi \omega b[c_{s} e^{2} (1 + T_{1} T_{2} ) - c_{g} f^{2} (T_{1} T_{2} - T_{1} - T_{2} )]} \right\} \\ & \quad + \,\frac{\pi \omega b}{{4(r + \delta )^{2} (r + \sigma )^{2} c_{g} c_{s} }}\left\{ {\pi \omega b[c_{g} f^{2} (T_{1} - 2)T_{1} - c_{s} e^{2} (1 + T_{1}^{2} )]} \right. \\ & \quad - \,\left. {2B(r + \delta )(c_{s} e^{2} (1 + T_{1} ) - c_{g} f^{2} )} \right\}\left. { + \frac{{(4b_{1} \pi \omega + \alpha a^{2} )[G_{C} (t) + G_{N} (t)(2T_{1} - 1)]}}{{8b_{1} }}} \right\}dt \\ \end{aligned} $$

(50)

$$ \begin{aligned} W_{s}^{NBS} (S_{C} (t),G_{C} (t)) & = \int_{0}^{\infty } {e^{ - rt} \left\{ {(1 - T_{1} )\pi Q_{0} + \frac{{B(S_{C} (t) - S_{N} (t))}}{2} - \frac{{f^{2} B^{2} c_{g} }}{{4(r + \sigma )^{2} c_{g} c_{s} }}} \right.} \\ & \quad + \frac{{\alpha a^{2} b}}{{64b_{1}^{2} (r + \delta )^{2} (r + \sigma )^{2} c_{g} c_{s} }}\left\{ {8b_{1} \pi \omega b[c_{s} e^{2} (T_{1} T_{2} - 1) - c_{g} f^{2} (2 - T_{1} - T_{2} + T_{1} T_{2} )]} \right. \\ & \quad + 8b_{1} (r + \delta )B(c_{s} e^{2} (T_{2} - 1) - c_{g} f^{2} ) + \left. {\alpha a^{2} b[c_{s} e^{2} (T_{2}^{2} - 1) - c_{g} f^{2} (T_{2}^{2} - 2T_{2} + 2)]} \right\} \\ & \quad - \frac{\pi \omega b}{{4(r + \delta )^{2} (r + \sigma )^{2} c_{g} c_{s} }}\left\{ {\pi \omega b[c_{s} e^{2} (T_{1}^{2} - 1) - c_{g} f^{2} (T_{1}^{2} - 2T_{1} + 2)]} \right. \\ & \quad + \left. {2B(r + \delta )(c_{s} e^{2} (T_{1} - 1) - c_{g} f^{2} )} \right\} + \frac{{[4b_{1} (1 - 2T_{1} )\pi \omega + (1 - 2T_{2} )\alpha a^{2} ]G_{N} (t)}}{{8b_{1} }} \\ & \quad + \left. {\frac{{(4b_{1} \pi \omega + \alpha a^{2} )G_{C} (t)}}{{8b_{1} }}} \right\}dt \\ \end{aligned} $$

(51)

The expressions of $S_{C} (t)$, $S_{N} (t)$, $G_{C} (t)$, and $G_{N} (t)$ in Eqs. (50) and (51) are as follows: $S_{C} (t){\text{ = S}}_{C}^{SS} + (S_{C} (\tau ) - S_{C}^{SS} )e^{ - \sigma (t - \tau )}$, $S_{N} (t){\text{ = S}}_{N}^{SS} + (S_{C} (\tau ) - S_{N}^{SS} )e^{ - \sigma (t - \tau )}$

$$ G_{C} (t) = G_{C}^{SS} + \frac{{b(S_{C} (\tau ) - S_{C}^{SS} )}}{\delta - \sigma }e^{ - \sigma (t - \tau )} + \left[ {G_{C} (\tau ) - G_{C}^{SS} - \frac{{b(S_{C} (\tau ) - S_{C}^{SS} )}}{\delta - \sigma }} \right]e^{ - \delta (t - \tau )} $$

$$ G_{N} (t) = G_{N}^{SS} + \frac{{b(S_{C} (\tau ) - S_{N}^{SS} )}}{\delta - \sigma }e^{ - \sigma (t - \tau )} + \left[ {G_{C} (\tau ) - G_{N}^{SS} - \frac{{b(S_{C} (\tau ) - S_{N}^{SS} )}}{\delta - \sigma }} \right]e^{ - \delta (t - \tau )} $$

where $S_{N}^{SS} = \frac{{e^{2} [4b_{1} B(r + \delta ) + 4b_{1} T_{1} \pi \omega b + T_{2} \alpha a^{2} b]}}{{4b_{1} c_{g} (r + \delta )(r + \sigma )\sigma }} + \frac{{f^{2} [4b_{1} (1 - T_{1} )\pi \omega b + (1 - T_{2} )\alpha a^{2} b]}}{{4b_{1} c_{s} (r + \delta )(r + \sigma )\sigma }}$, $S_{C}^{SS} = \frac{{[4b_{1} (r + \delta )B + 4b_{1} \pi \omega b + \alpha a^{2} b]}}{{4b_{1} (r + \delta )(r + \sigma )\sigma }}\left( {\frac{{e^{2} }}{{c_{g} }} + \frac{{f^{2} }}{{c_{s} }}} \right)$, $G_{N}^{SS} = \frac{b}{\delta }S_{N}^{SS}$, $G_{C}^{SS} = \frac{b}{\delta }S_{C}^{SS}$. Because the specific expression is too long, we will not expand it here.

Substituting Eqs. (50) and (51) into Eq. (25), we obtain the revenue distribution mechanism $\xi_{g} (t)$ and $\xi_{s} (t)$ as follows.

$\begin{aligned} \xi_{g} (t) & = T_{1} \pi Q_{0} - \frac{{B^{2} (2c_{s} e^{2} + c_{g} f^{2} )}}{{4(r + \sigma )^{2} c_{g} c_{s} }} + \frac{{\alpha a^{2} b}}{{64b_{1}^{2} (r + \delta )^{2} (r + \sigma )^{2} c_{g} c_{s} }}\left\{ {\alpha a^{2} b[c_{g} f^{2} (T_{2} - 2)T_{2} } \right. \\ & \quad - c_{s} e^{2} (1 + T_{2}^{2} )] - 8b_{1} (r + \delta )B(c_{s} e^{2} + c_{g} f^{2} ) - 8b_{1} \pi \omega b[c_{s} e^{2} (1 + T_{1} T_{2} ) - \left. {c_{g} f^{2} (T_{1} T_{2} - T_{1} - T_{2} )]} \right\} \\ & \quad + \frac{\pi \omega b}{{4(r + \delta )^{2} (r + \sigma )^{2} c_{g} c_{s} }}\left\{ {\pi \omega b[c_{g} f^{2} (T_{1} - 2)T_{1} } \right. - c_{s} e^{2} (1 + T_{1}^{2} )] - \left. {2B(r + \delta )(c_{s} e^{2} (1 + T_{1} ) - c_{g} f^{2} )} \right\} \\ & \quad + r\int_{0}^{\infty } {e^{ - rt} \left\{ {\frac{{B(S_{C} (t) - S_{N} (t) - 2S_{0} )}}{2} + \frac{{(4b_{1} \pi \omega + \alpha a^{2} )[G_{C} (t) + G_{N} (t)(2T_{1} - 1)]}}{{8b_{1} }}} \right\}} dt \\ \end{aligned}$ $\begin{aligned} \xi_{s} (t) & = (1 - T_{1} )\pi Q_{0} - \frac{{f^{2} B^{2} c_{g} }}{{4(r + \sigma )^{2} c_{g} c_{s} }} + \frac{{\alpha a^{2} b}}{{64b_{1}^{2} (r + \delta )^{2} (r + \sigma )^{2} c_{g} c_{s} }}\left\{ {8b_{1} \pi \omega b[c_{s} e^{2} (T_{1} T_{2} - 1)} \right. \\ & \quad - c_{g} f^{2} (2 - T_{1} - T_{2} + T_{1} T_{2} )] + 8b_{1} (r + \delta )B(c_{s} e^{2} (T_{2} - 1) - c_{g} f^{2} ) + \alpha a^{2} b[c_{s} e^{2} (T_{2}^{2} - 1) \\ & \quad - \left. {c_{g} f^{2} (T_{2}^{2} - 2T_{2} + 2)]} \right\} - \frac{\pi \omega b}{{4(r + \delta )^{2} (r + \sigma )^{2} c_{g} c_{s} }}\left\{ {\pi \omega b[c_{s} e^{2} (T_{1}^{2} - 1) - c_{g} f^{2} (T_{1}^{2} - 2T_{1} + 2)]} \right. \\ & \quad + \left. {2B(r + \delta )(c_{s} e^{2} (T_{1} - 1) - c_{g} f^{2} )} \right\} + r\int_{0}^{\infty } {e^{ - rt} \left\{ {\frac{{B(S_{C} (t) - S_{N} (t))}}{2} + \frac{{(4b_{1} \pi \omega + \alpha a^{2} )G_{C} (t)}}{{8b_{1} }}} \right.} \\ & \quad + \left. {\frac{{[4b_{1} (1 - 2T_{1} )\pi \omega + (1 - 2T_{2} )\alpha a^{2} ]G_{N} (t)}}{{8b_{1} }}} \right\}dt \\ \end{aligned}$ Based on the above analysis, Proposition 7 can be obtained.