Differential game theoretic analysis of the dynamic emission abatement in low-carbon supply chains

Abstract

This paper studies the impact of bilateral participation strategy on the dynamic emission-reducing behaviors and associated performance of low-carbon supply chains using differential game models. We first consider a dyadic supply chain comprising one supplier and one manufacturer in the base model and derive the equilibrium solutions of dynamic emission abatement in decentralized and centralized systems, respectively. In the decentralized supply chain, we consider both unilateral sharing and bilateral participation contracts and examine the feedback equilibrium strategies and trajectories of emission abatement of all parties. By comparison, we find that the bilateral contract leads to lower emission abatement and higher profits of all players and the system than that under unilateral sharing contract. Meanwhile, we show that the properly-designed bilateral participation contract can coordinate the decentralized dyadic supply chain. By extending the base model, we further figure out the equilibria and optimal trajectories for the dynamic cooperative emission abatement and the associated mechanism design in a supply chain with two competing suppliers and one manufacturer. Interestingly, we find that the bilateral participation contract can also coordinate the competing channels with competitive or complementary components at the upstream level. We uncover that the bilateral participation contract is effective in coordinating supply chains with dynamic cooperative emission abatement since it can overcome the disadvantage of uneven distribution of emission-reducing capability endowments among chain members. Finally, we also conduct computational and sensitivity analyses to illustrate the previous results. These findings provide potential implications for firms’ dynamic cooperation in emission abatement.

Appendix

Proof of Propostion 1

Maximizing the right side of Eq. (9) produces

$$ Z_{S}^{C*} = \alpha V_{Tx} (x,y){/}\mu_{S} ; $$

(89)

$$ Z_{M}^{C*} = \delta V_{Ty} (x,y){/}\mu_{M} . $$

(90)

Substituting Eqs. (89) and (90) into Eq. (9), respectively gives

$$ \rho V_{T} (x,y) = (\gamma V_{Ty}^{{}} - \beta V_{Tx}^{{}} )x + [(U_{M} + U_{S} )\eta - V_{Ty}^{{}} \beta ]y + (U_{M} + U_{S} )D_{0} + \frac{{\alpha^{2} }}{{2\mu_{S} }}V_{Tx}^{2} + \frac{{\delta^{2} }}{{2\mu_{M} }}V_{Ty}^{2} . $$

(91)

Inferred from Eq. (91), the linear function of $x$ and $y$ is the solution of the HJB equation. Set

$$ V_{T} (x,y) = ax + by + c, $$

(92)

where $a_{1}$, $b_{1}$ and $c_{1}$ are constants. From Eq. (92), we know that

$$ \left\{ \begin{gathered} V_{Tx}^{{}} (x,y) = a_{1} \hfill \\ V_{Ty}^{{}} (x,y) = b_{1} \hfill \\ \end{gathered} \right.. $$

(93)

Substitute Eqs. (93) and (94) into Eq. (92) respectively, then collate and compare the coefficients of the similar terms on the left and right sides, and we have

$$ \left\{ \begin{aligned} a_{1} & = (U_{M} + U_{S} )\eta \gamma {\text{/}}(\rho + \beta )^{2} \\ b_{1} & = (U_{M} + U_{S} )\eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{1} & = \frac{{(U_{M} + U_{S} )D_{0} }}{\rho } + \frac{{\alpha ^{2} }}{{2\rho \mu _{S} }}[\frac{{(U_{M} + U_{S} )\eta \gamma }}{{(\rho + \beta )^{2} }}]^{2} + \frac{{\delta ^{2} }}{{2\rho \mu _{M} }}[\frac{{(U_{M} + U_{S} )\eta }}{{\rho + \beta }}]^{2} \\ \end{aligned} \right.. $$

(94)

Substituting Eq. (94) into Eqs. (89) and (90) respectively, the static feedback equilibrium strategy of emission reduction efforts of suppliers and manufacturer under centralized decision making is

$$ \left\{ \begin{gathered} Z_{S}^{C*} = (U_{M} + U_{S} )\alpha \eta \gamma {/[}\mu_{S} (\rho + \beta )^{2} {]} \hfill \\ Z_{M}^{C*} = (U_{M} + U_{S} )\delta \eta {/[}\mu_{M} (\rho + \beta ){]} \hfill \\ \end{gathered} \right.. $$

(95)

Thus, Proposition 1 can be proved.

Proof of Proposition 2

Substituting the optimal strategies $Z_{S}^{C*} { = }\frac{{(U_{M} + U_{S} )\alpha \eta \gamma }}{{\mu_{S} (\rho + \beta )^{2} }}$ and $Z_{M}^{C*} { = }\frac{{(U_{M} + U_{S} )\delta \eta }}{{\mu_{M} (\rho + \beta )}}$ into Eq. (2) in the case of centralized decision making of Proposition 1, we can solve the following differential equations

$$ \left\{ \begin{gathered} \dot{x}(t) = \alpha Z_{S} (t) - \beta x(t) \hfill \\ \dot{y}(t) = \gamma x(t) + \delta Z_{M} (t) - \beta y(t) \hfill \\ \end{gathered} \right.. $$

(96)

Eq. (103) is expressed in terms of vectors as follows.

$$ {\dot{\mathbf{x}}} = {\mathbf{AX}} + {\mathbf{b(t)}}. $$

(97)

Introduce vectors ${\mathbf{X}} = (x,y)^{T} {,}$ ${\mathbf{b(t)}} = (\alpha Z_{S} ,\delta Z_{M} )^{T}$ and matrix ${\mathbf{A = }}\left( {\begin{array}{*{20}c} { - \beta } & 0 \\ \gamma & { - \beta } \\ \end{array} } \right)$, where there are homogeneous system of differential equations.

$$ {\dot{\mathbf{x}}} = {\mathbf{AX}}. $$

(98)

Then the general solution of differential equations Eq. (97) consists of a special solution and the general solution of Eq. (98).

To find the general solution of Eq. (98), the eigenvalue of A is obtained firstly as double root $\lambda_{1} = \lambda_{2} = - \beta$. From the properties of the eigenvalue, one of the eigenvectors is ${\mathbf{v}} = (0,1)^{T}$. The corresponding linear independent solution can be solved as ${\mathbf{x}}_{{\mathbf{1}}} {\mathbf{(t)}} = \left( {_{1}^{0} } \right)e^{ - \beta t}$. Let another corresponding linear independent solution is ${\mathbf{x}}_{{\mathbf{2}}} {\mathbf{(t)}} = ({\mathbf{m}} + {\mathbf{n}}t)e^{ - \beta t}$. The conditions that ${\mathbf{x}}_{{\mathbf{2}}} {\mathbf{(t)}}$ is the solution of Eq. (97) are ${\mathbf{n + m}}l{\mathbf{ = Am}}$ and ${\mathbf{n}}l{\mathbf{ = An}}$, set ${\mathbf{n = v}}$, we can get ${\mathbf{m}} = (1{/}\gamma ,1)^{T}$. Here, the general solution of homogeneous system of differential equations Eq. (98) is

$$ {\mathbf{x(t)}} = C_{1} {\mathbf{x}}_{{\mathbf{1}}} {\mathbf{(t)}}{ + }C_{2} {\mathbf{x}}_{{\mathbf{2}}} {\mathbf{(t)}} = C_{1} \left( {_{1}^{0} } \right)e^{ - \beta t} + C_{2} \left[ {\left( \begin{gathered} 1{/}\gamma \hfill \\ 1 \hfill \\ \end{gathered} \right) + \left( {_{1}^{0} } \right)t} \right]e^{ - \beta t} . $$

(99)

Let the special solution of Eq. (97) be $\tilde{x} = (f_{1} {; }f_{2} )$,where $f_{1}$ and $f_{2}$ are constants, and substitute them into Eq. (2), then we can get

$$ {\tilde{\mathbf{x}}} = \left( \begin{gathered} \, \alpha Z_{S} {/}\beta \hfill \\ \left( {\gamma \alpha Z_{S} {/}\beta + \delta Z_{M} } \right){/}\beta \hfill \\ \end{gathered} \right). $$

(100)

From Eqs. (99) and (100), the general solution of differential equations Eq. (97) is as follows:

$$ {\mathbf{x(t)}} = C_{1} \left( {_{1}^{0} } \right)e^{ - \beta t} + C_{2} \left[ {\left( \begin{gathered} 1{/}\gamma \hfill \\ 1 \hfill \\ \end{gathered} \right) + \left( {_{1}^{0} } \right)t} \right]e^{ - \beta t} + \left( {_{{{(}\gamma f_{1} + \delta Z_{M} {)/}\beta }}^{{ \, \alpha Z_{S} {/}\beta }} } \right). $$

(101)

When $t = 0$, substituting the initial values $x(0) = x_{0}$ and $y(0) = y_{0}$ into Eq. (101) yields

$$ C_{1} = y_{0} - y_{{_{SS} }}^{C*} - (x_{0} - x_{{_{SS} }}^{C*} )\gamma ; $$

(102)

$$ C_{2} = (x_{0} - x_{ss}^{{C^{*} }} )\gamma ; $$

(103)

where $x_{{_{SS} }}^{C*} = \alpha Z_{S} {/}\beta$ and $y_{{_{SS} }}^{{\text{C*}}} = (\gamma x_{ss}^{C*} + \delta Z_{M} ){/}\beta$.

Substitute Eqs. (102) and (103) into Eq. (101), and we can get

$$ x^{C} (t) = (x{}_{0} - x_{{_{SS} }}^{C*} )e^{ - \beta t} + x_{{_{SS} }}^{C*} ; $$

(104)

$$ y^{C} (t) = (y{}_{0} - y_{{_{SS} }}^{C*} )e^{ - \beta t} + \gamma (x{}_{0} - x_{SS} )te^{ - \beta t} + y_{{_{SS} }}^{{\text{C*}}} . $$

(105)

Thus, Proposition 2 has been proved.

Proof of Proposition 3

Maximizing the right side of Eqs. (18) and Eq. (19) gives

$$ Z_{S}^{D*} = \alpha V_{Sx}^{{}} (x,y){/}\mu_{S} ; $$

(106)

$$ Z_{M}^{D*} = \delta V_{My}^{{}} (x,y){/}\mu_{M} . $$

(107)

Substitute Eqs. (106) and (107) into Eqs. (18) and (19), respectively, and collate it. Then we can get

$$ \begin{aligned} \rho V_{S}^{D*} (x,y) & = U_{S} (D_{0} + \eta y) - \mu_{S} [\alpha V_{Sx}^{{}} (x,y){/}\mu_{S} ]^{2} {/2} + V_{Sx}^{D*} (x,y)[\alpha^{2} V_{Sx}^{{}} (x,y){/}\mu_{S} - \beta x] \\ & \quad + V_{Sy}^{D*} (x,y)[\delta^{2} V_{My}^{{}} (x,y){/}\mu_{M} + \gamma x - \beta y] \\ \end{aligned} $$

(108)

$$ \begin{aligned} \rho V_{M}^{D*} (x,y) & = U_{M} (D_{0} + \eta y) - \mu_{M} [\delta V_{My}^{{}} (x,y){/}\mu_{M} ]^{2} {/2} + V_{Mx}^{D*} (x,y)[\alpha^{2} V_{Sx}^{{}} (x,y){/}\mu_{S} - \beta x] \\ & \quad + V_{My}^{D*} (x,y)[\gamma x + \delta^{2} V_{My}^{{}} (x,y){/}\mu_{M} - \beta y] \\ \end{aligned} $$

(109)

It is inferred from the Eqs. (108) and (109) that the linear function of x and y is the solution of the HJB equation. Set

$$ V_{S} = a_{2} x + b_{2} y + c_{2} $$

(110)

$$ V_{M} = a_{3} x + b_{3} y + c_{3} $$

(111)

Then $V_{Sx} = a_{2}$, $V_{Sy} = b_{2}$, $V_{Mx} = a_{3}$, $V_{My} = b_{3}$. Substituting Eqs. (110) and (111) into Eqs. (108) and (109) respectively produces

$$ \rho (a_{2} x + b_{2} y + c_{2} ) = U_{S} (D_{0} + \eta y) - \mu_{S} (\alpha a_{2} {/}\mu_{S} )^{2} {/2} + a_{2} (\alpha^{2} a_{2} {/}\mu_{S} - \beta x) + b_{2} (\gamma x + \delta^{2} b_{3} {/}\mu_{M} - \beta y); $$

(112)

$$ \rho (a_{3} x + b_{3} y + c_{3} ) = U_{M} (D_{0} + \eta y) - \mu_{M} (\delta b_{3} {/}\mu_{M} )^{2} {/2} + a_{3} (\alpha^{2} a_{2} {/}\mu_{S} - \beta x) + b_{3} (\gamma x + \delta^{2} b_{3} {/}\mu_{M} - \beta y). $$

(113)

Collating Eq. (112) and comparing its coefficients of the counterpart terms in both sides yields

$$ \left\{ \begin{aligned} a_{2} & = U_{S} \eta \gamma {\text{/}}(\rho + \beta )^{2} \\ b_{2} & = U_{S} \eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{2} & = U_{S} D_{0} {\text{/}}\rho + \alpha ^{2} [U_{S} \eta \gamma {\text{/}}(\rho + \beta )^{2} ]^{2} {\text{/(}}2\rho \mu _{S} {\text{)}} + \delta ^{2} [U_{S} \eta {\text{/(}}\rho + \beta {\text{)}}]^{2} {\text{/(}}2\rho \mu _{M} {\text{)}} \\ \end{aligned} \right.. $$

(114)

Doing the same thing in Eq. (113) gives

$$ \left\{ \begin{aligned} a_{3} & = U_{M} \eta \gamma {\text{/}}(\rho + \beta )^{2} \\ b_{3} & = U_{M} \eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{3} & = U_{M} D_{0} {\text{/}}\rho + \alpha ^{2} [U_{M} \eta \gamma {\text{/}}(\rho + \beta )^{2} ]^{2} {\text{/(}}2\rho \mu _{S} {\text{)}} + \delta ^{2} [U_{M} \eta {\text{/(}}\rho + \beta {\text{)}}]^{2} {\text{/(}}2\rho \mu _{M} {\text{)}} \\ \end{aligned} \right.. $$

(115)

Then substitute them into Eqs. (106) and (107), we can get

$$ \left\{ \begin{gathered} Z_{S}^{D*} { = }\alpha a_{2} {/}\mu_{S} { = }U_{S} \alpha \eta \gamma {/[}\mu_{S} (\rho + \beta )^{2} {]} \hfill \\ Z_{M}^{D*} { = }\delta b_{3} {/}\mu_{M} { = }U_{M} \delta \eta {/[}\mu_{M} (\rho + \beta ){]} \hfill \\ \end{gathered} \right.. $$

(116)

So, the Proposition 3 has been proved.

Proof of Proposition 4

Similar to the proof of Proposition 2, substituting optimal strategies $Z_{S}^{D*} { = }U_{S} \alpha \eta \gamma {/[}\mu_{S} (\rho + \beta )^{2} {]}$ and $Z_{M}^{D*} { = }U_{M} \delta \eta {/[}\mu_{M} (\rho + \beta ){]}$ into Eq. (2) respectively, in the case of decentralized decision making of Proposition 3, we can solve the following differential equations:

$$ \left\{ \begin{gathered} \dot{x}(t) = \alpha Z_{S} (t) - \beta x(t) \hfill \\ \dot{y}(t) = \gamma x(t) + \delta Z_{M} (t) - \beta y(t) \hfill \\ \end{gathered} \right.. $$

(117)

Then the optimal trajectory of Proposition 4 can be obtained.

Proof of Proposition 5

Maximizing the right side of Eqs. (34) and (35) shows us

$$ Z_{S}^{B*} = \alpha V_{Sx}^{{}} (x,y){/[}(1{ - }\omega )\mu_{S} {];} $$

(118)

$$ Z_{M}^{B*} = \delta V_{{M{\text{y}}}}^{{}} (x,y){/[}(1{ - }\psi )\mu_{M} {]}{\text{.}} $$

(119)

Substituting Eqs. (118) and (119) into Eqs. (34) and (35) respectively shows

$$ \begin{aligned} \rho V_{S}^{B*} (x,y) & = U_{S} (D_{0} + \eta y) - \frac{{\mu_{S} }}{2}(1 - \omega )\left[ {\frac{{\alpha V_{Sx}^{{}} (x,y)}}{{(1{ - }\omega )\mu_{S} }}} \right]^{2} - \frac{{\mu_{M} }}{2}\psi \left[ {\frac{{\delta V_{{M{\text{y}}}}^{{}} (x,y)}}{{(1{ - }\psi )\mu_{M} }}} \right]^{2} \\ & \quad + V_{Sx}^{B*} (x,y)\left[ {\alpha \frac{{\alpha V_{Sx}^{{}} (x,y)}}{{(1{ - }\omega )\mu_{S} }} - \beta x} \right] + V_{Sy}^{B*} (x,y)\left[ {\gamma x + \delta \frac{{\delta V_{{M{\text{y}}}}^{{}} (x,y)}}{{(1{ - }\psi )\mu_{M} }} - \beta y} \right]; \\ \end{aligned} $$

(120)

$$ \begin{aligned} \rho V_{M}^{B*} (x,y) & = U_{M} (D_{0} + \eta y) - \frac{{\mu_{M} }}{2}(1 - \psi )\left[ {\frac{{\delta V_{{M{\text{y}}}}^{{}} (x,y)}}{{(1{ - }\psi )\mu_{M} }}} \right]^{2} - \frac{{\mu_{S} }}{2}\omega \left[ {\frac{{\alpha V_{Sx}^{{}} (x,y)}}{{(1{ - }\omega )\mu_{S} }}} \right]^{2} \\ & \quad + V_{Mx}^{B*} (x,y)\left[ {\alpha \frac{{\alpha V_{Sx}^{{}} (x,y)}}{{(1{ - }\omega )\mu_{S} }} - \beta x} \right] + V_{My}^{B*} (x,y)\left[ {\gamma x + \delta \frac{{\delta V_{{M{\text{y}}}}^{{}} (x,y)}}{{(1{ - }\psi )\mu_{M} }} - \beta y} \right]. \\ \end{aligned} $$

(121)

It can be inferred from the Eqs. (120) and (121) that the linear function of x and y is the solution of the HJB equation. Set

$$ V_{S}^{B} = a_{4} x + b_{4} y + c_{4} ; $$

(122)

$$ V_{M}^{B} = a_{5} x + b_{5} y + c_{5} . $$

(123)

Then $V_{Sx}^{B} = a_{4}$, $V_{Sy}^{B} = b_{4}$, $V_{Mx}^{B} = a_{5}$, and $V_{My}^{B} = b_{5}$. Substituting Eqs. (122) and (123) into Eqs. (120) and (121), respectively, yields

$$ \begin{aligned} \rho (a_{4} x + b_{4} y + c_{4} ) & = U_{S} (D_{0} + \eta y) - \frac{{\mu_{S} }}{2}(1 - \omega )\left[ {\frac{{\alpha a_{4} }}{{(1{ - }\omega )\mu_{S} }}} \right]^{2} - \frac{{\mu_{M} }}{2}\psi \left[ {\frac{{\delta b_{5} }}{{(1{ - }\psi )\mu_{M} }}} \right]^{2} \\ & \quad + a_{4} \left( {\alpha \left[ {\frac{{\alpha a_{4} }}{{(1{ - }\omega )\mu_{S} }}} \right] - \beta x} \right) + b_{4} \left[ {\gamma x + \delta \frac{{\delta b_{5} }}{{(1{ - }\psi )\mu_{M} }} - \beta y} \right]; \\ \end{aligned} $$

(124)

$$ \begin{aligned} \rho (a_{5} x + b_{5} y + c_{5} ) & = U_{M} (D_{0} + \eta y) - \frac{{\mu_{M} }}{2}(1 - \psi )\left[ {\frac{{\delta b_{5} }}{{(1{ - }\psi )\mu_{M} }}} \right]^{2} - \frac{{\mu_{S} }}{2}\omega \left[ {\frac{{\alpha a_{4} }}{{(1{ - }\omega )\mu_{S} }}} \right]^{2} \\ & \quad + a_{5} \left( {\alpha \left[ {\frac{{\alpha a_{4} }}{{(1{ - }\omega )\mu_{S} }}} \right] - \beta x} \right) + b_{5} \left[ {\gamma x + \delta \frac{{\delta b_{5} }}{{(1{ - }\psi )\mu_{M} }} - \beta y} \right]. \\ \end{aligned} $$

(125)

Collate Eq. (124) and compare its coefficients of the similar terms on the left and right sides, and we can get

$$ \left\{ \begin{aligned} a_{4} & = U_{S} \eta \gamma {\text{/}}(\rho + \beta )^{2} \\ b_{4} & = U_{S} \eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{4} & = \frac{{U_{S} D_{0} }}{\rho } + \frac{{\alpha ^{2} }}{{2\rho \mu _{S} (1 - \omega )}}\left[ {\frac{{U_{S} \eta \gamma }}{{(\rho + \beta )^{2} }}} \right]^{2} - \frac{{\psi \delta ^{2} }}{{2\rho \mu _{M} (1 - \psi )^{2} }}\left[ {\frac{{U_{S} \eta }}{{\rho + \beta }}} \right]^{2} + \frac{{U_{S} U_{M} \delta ^{2} }}{{\rho \mu _{M} (1 - \psi )}}\left[ {\frac{\eta }{{\rho + \beta }}} \right]^{2} \\ \end{aligned} \right.. $$

(126)

Collating Eq. (125) and compare its coefficients of the similar terms in the left and right sides gives

$$ \left\{ \begin{aligned} a_{5} & = U_{M} \eta \gamma {\text{/}}(\rho + \beta )^{2} \\ b_{5} & = U_{M} \eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{5} & = \frac{{U_{M} D_{0} }}{\rho } + \frac{{\delta ^{2} }}{{2\rho \mu _{M} (1 - \psi )}}\left[ {\frac{{U_{M} \eta }}{{\rho + \beta }}} \right]^{2} - \frac{{\omega \alpha ^{2} }}{{2\rho \mu _{S} (1 - \omega )^{2} }}\left[ {\frac{{U_{S} \eta \gamma }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{U_{S} U_{M} \alpha ^{2} }}{{\rho \mu _{S} (1 - \omega )}}\left[ {\frac{{\eta \gamma }}{{(\rho + \beta )^{2} }}} \right]^{2} \\ \end{aligned} \right.. $$

(127)

Then substitute them into Eqs. (118) and (119), we can get

$$ Z_{S}^{B*} = U_{S} \alpha \eta \gamma {/[}\mu_{S} (1 - \omega )(\rho + \beta )^{2} {];} $$

(128)

$$ Z_{M}^{B*} = U_{M} \delta \eta {/[}\mu_{M} (1 - \psi )(\rho + \beta ){]}{\text{.}} $$

(129)

Coordinating the supply chain system can make the optimal emission reduction efforts of suppliers and manufacturer under bilateral decision making equal to those under centralized decision making, so $Z_{S}^{C*} = Z_{S}^{B*}$, $Z_{M}^{C*} = Z_{M}^{B*}$, i.e., $\frac{{(U_{M} + U_{S} )\alpha \eta \gamma }}{{\mu_{S} (\rho + \beta )^{2} }} = \frac{{U_{S} \alpha \eta \gamma }}{{(1 - \omega )\mu_{S} (\rho + \beta )^{2} }}$, $\frac{{(U_{M} + U_{S} )\delta \eta }}{{\mu_{M} (\rho + \beta )}} = \frac{{U_{M} \delta \eta }}{{(1 - \psi )\mu_{M} (\rho + \beta )}}$.

The comparison gives $U_{M} + U_{S} = U_{S} {/(}1 - \omega {)}$ and $U_{M} + U_{S} = U_{M} {/(}1 - \psi {)}$. Namely, $\omega^{*} = U_{M} {/(}U_{M} + U_{S} {)}$, and $\psi^{*} = U_{S} {/(}U_{M} + U_{S} {)}$. Thus, Proposition 5 can be proved.

Proof of Proposition 6

Maximizing the right side of Eq. (50) yields

$$ Z_{{S_{i} }}^{C*} = \alpha_{i} V_{{Tx_{i} }}^{{}} (x,y){/}\mu_{{S_{i} }} ; $$

(130)

$$ Z_{M}^{C*} = \delta V_{Ty}^{{}} (x,y){/}\mu_{M} . $$

(131)

Substituting Eqs. (130) and (131) into Eq. (50) derives

$$ \begin{aligned} \rho V_{T}^{C*} (x,y) & = \left( {U_{M} + \sum\nolimits_{i = 1}^{2} {v_{i} U_{{S_{i} }} } } \right)(D_{0} + \eta y) - \mu_{M} \left[ {\delta V_{Ty}^{{}} (x,y){/}\mu_{M} } \right]^{2} {/2} - \sum\nolimits_{i = 1}^{2} {\mu_{{S_{i} }} [\alpha_{i} V_{{Tx_{i} }}^{{}} (x,y){/}\mu_{{S_{i} }} ]^{2} {/2}} \\ & \quad + \sum\nolimits_{i = 1}^{2} {V_{Txi}^{C*} (x,y)\left[ {\alpha_{i}^{2} V_{{Tx_{i} }}^{{}} (x,y){/}\mu_{{S_{i} }} - \beta x_{i}^{{}} } \right]} + V_{Ty}^{C*} (x,y)\left[ {\sum\nolimits_{i = 1}^{2} {\gamma_{i} v_{i} x_{i} } + \delta^{2} V_{Ty}^{{}} (x,y){/}\mu_{M} - \beta {\text{y}}} \right]. \\ \end{aligned} $$

(132)

Inferred from the Eq. (132), the linear function of x and y is the solution of HJB equation. Let

$$ V_{T}^{C} = \sum\nolimits_{i = 1}^{2} {a_{6i} x} + b_{6} y + c_{6} . $$

(133)

Then $V_{{Tx_{i} }}^{C} = a_{6i}$ and $V_{Ty}^{C} = b_{6}$. Substituting Eq. (133) into Eq. (132) attains

$$ \begin{aligned} \rho (a_{6i} x + b_{6} y + c_{6} ) & = (b_{6} \gamma_{1} v_{1} - a_{61} \beta )x_{1} + (b_{6} \gamma_{2} v_{2} - a_{62} \beta )x_{2} + [(U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} )\eta - b_{6} \beta ]y \\ & \quad + (U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} )D_{0} - \mu_{M} (\delta b_{6} {/}\mu_{M} )^{2} {/2} - \mu_{{S_{1} }} (a_{61} \alpha_{1} {/}\mu_{{S_{1} }} )^{2} {/2} - \mu_{{S_{2} }} (a_{62} \alpha_{2} {/}\mu_{{S_{2} }} )^{2} {/2} \\ & \quad + a_{61} \alpha_{1} (a_{61} \alpha_{1} {/}\mu_{{S_{1} }} ) + a_{62} \alpha_{2} (a_{62} \alpha_{2} {/}\mu_{{S_{2} }} ) + b_{6} \delta^{2} b_{6} {/}\mu_{M} . \\ \end{aligned} $$

(134)

Comparing its coefficients of the similar terms of the left and right sides in Eq. (134) yields

$$ \left\{ \begin{aligned} a_{{61}} & = \left( {U_{M} + \sum\nolimits_{{j = 1}}^{2} {v_{i} U_{{S_{i} }} } } \right)\eta \gamma _{1} v_{1} {\text{/}}(\rho + \beta )^{2} \\ a_{{62}} & = \left( {U_{M} + \sum\nolimits_{{j = 1}}^{2} {v_{i} U_{{S_{i} }} } } \right)\eta \gamma _{2} v_{2} {\text{/}}(\rho + \beta )^{2} \\ b_{6} & = \left( {U_{M} + \sum\nolimits_{{j = 1}}^{2} {v_{i} U_{{S_{i} }} } } \right)\eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{6} & = \frac{{\left( {U_{M} + \sum\nolimits_{{j = 1}}^{2} {v_{i} U_{{S_{i} }} } } \right)D_{0} }}{\rho } + \left[ {\sum\nolimits_{{j = 1}}^{2} {\frac{{\alpha _{i} ^{2} (\gamma _{i} v_{i} )^{2} }}{{\mu _{{S_{i} }} (\rho + \beta )^{2} }} + \frac{{\delta ^{2} }}{{\mu _{M} }}} } \right]\frac{{\left( {U_{M} + \sum\nolimits_{{j = 1}}^{2} {v_{i} U_{{S_{i} }} } } \right)^{2} \eta ^{2} }}{{2\rho (\rho + \beta )^{2} }} \\ \end{aligned} \right.. $$

(135)

Then substituting them into Eqs. (130) and (131), we have

$$ Z_{{S_{i} }}^{C*} = (U_{M} + \sum\nolimits_{j = 1}^{2} {v_{j} U_{{S_{j} }} } )\alpha_{i} \eta \gamma_{i} v_{i} {/[}\mu_{{S_{i} }} (\rho + \beta )^{2} {];} $$

(136)

$$ Z_{{S_{2} }}^{C*} = \left( {U_{M} + \sum\nolimits_{j = 1}^{2} {v_{j} U_{{S_{j} }} } } \right)\alpha_{2} \eta \gamma_{2} v_{2} {/[}\mu_{{S_{2} }} (\rho + \beta )^{2} {];} $$

(137)

$$ Z_{M}^{C*} = \left( {U_{M} + \sum\nolimits_{j = 1}^{2} {v_{j} U_{{S_{j} }} } } \right)\eta \delta {/[}\mu_{M} (\rho + \beta ){]}{\text{.}} $$

(138)

Thus, Proposition 6 can be proved.

Proof of Proposition 7

Plugging optimal strategies $Z_{{S_{i} }}^{C*} = (U_{M} + \sum\nolimits_{i = 1}^{2} {v_{i} U_{{S_{i} }} } )\alpha_{i} \eta \gamma_{i} v_{i} {/[}\mu_{{S_{i} }} (\rho + \beta )^{2} {]}$ and $Z_{M}^{C*} = (U_{M} + \sum\nolimits_{j = 1}^{2} {v_{j} U_{{S_{j} }} } )\eta \delta {/[}\mu_{M} (\rho + \beta ){]}$ into Eq. (2) respectively, in the case of centralized decision making of Proposition 6, we can solve the following differential equations

$$ \left\{ \begin{gathered} \dot{x}(t) = \alpha Z_{S} (t) - \beta x(t) \hfill \\ \dot{y}(t) = \gamma x(t) + \delta Z_{M} (t) - \beta y(t) \hfill \\ \end{gathered} \right.. $$

(139)

Then the optimal trajectory of Proposition 7 can be obtained.

Proof of Proposition 8

Maximizing the right side of Eqs. (62) and (63) gives

$$ Z_{{S_{i} }}^{D*} { = }\alpha_{i} V_{{S_{i} x_{i} }}^{{}} (x,y){/}\mu_{{S_{i} }} ; $$

(140)

$$ Z_{M}^{D*} = \delta V_{My}^{{}} (x,y){/}\mu_{M} . $$

(141)

Substituting Eqs. (140) and (141) into Eqs. (62) and (63), respectively, we have

$$ \begin{aligned} \rho V_{{S_{i} }}^{D*} (x,y) & = v_{i} U_{{S_{i} }} (D_{0} + \eta y) - \mu_{{S_{i} }} [\alpha_{i} V_{{S_{i} x_{i} }}^{{}} (x,y){/}\mu_{{S_{i} }} ]^{2} {/2} + \sum\nolimits_{j = 1}^{2} {V_{{S_{i} x_{j} }}^{D} (\alpha_{j} \alpha_{i} V_{{S_{i} x_{i} }}^{{}} (x,y){/}\mu_{{S_{i} }} - \beta x_{j} )} \\ & \quad + V_{{S_{i} y}}^{D} (\sum\nolimits_{j = 1}^{2} {v_{j} x_{j} \gamma_{j} } + \delta^{2} V_{My}^{{}} (x,y){/}\mu_{M} - \beta y); \\ \end{aligned} $$

(142)

$$ \begin{aligned} \rho V_{M}^{D*} (x,y) & = U_{M} (D_{0} + \eta y) - \mu_{M} [\delta V_{My}^{{}} (x,y){/}\mu_{M} ]^{2} {/2} + \sum\nolimits_{i = 1}^{2} {V_{{Mx_{i} }}^{D} (\alpha_{i}^{2} V_{{S_{i} x_{i} }}^{{}} (x,y){/}\mu_{{S_{i} }} - \beta x_{i} )} \\ & \quad + V_{My}^{D} \left( {\sum\nolimits_{j = 1}^{2} {v_{j} x_{j} \gamma_{j} } + \delta^{2} V_{My}^{{}} (x,y){/}\mu_{M} - \beta y} \right). \\ \end{aligned} $$

(143)

Inferred from Eqs. (142) and (143), the linear function of x and y is the solution for HJB equation.

$$ V_{{S_{i} }}^{D} = \sum\nolimits_{j = 1}^{2} {a_{7ji} x_{j} } + b_{7i} y + c_{7i} ; $$

(144)

$$ V_{M}^{D} = \sum\nolimits_{i = 1}^{2} {a_{8i} x_{i} } + b_{8} y + c_{8} . $$

(145)

Then $V_{{S_{i} }}^{D} = a_{7ji}$ ($j \ne i$), $V_{{S_{i} y}}^{D} = b_{7i}$, $V_{{S_{i} x_{i} }}^{D} = a_{7ii}$, $V_{{Mx_{i} }}^{D} = a_{8i}$, and $V_{My}^{D} = b_{8}$.

Substituting Eqs. (144) and (145) into Eqs. (142) and (143), respectively, we have

$$ \begin{aligned} \rho (a_{711} x_{1} + a_{721} x_{2} + b_{71} y + c_{71} ) & = (b_{71} \gamma_{1} v_{1} - a_{711} \beta )x_{1} + (b_{71} \gamma_{2} v_{2} - a_{721} \beta )x_{2} + (v_{1} U_{{S_{1} }} \eta - b_{71} \beta )y \\ & \quad + v_{1} U_{{S_{1} }} D_{0} - \frac{{\mu_{{S_{1} }} }}{2}\left( {\frac{{a_{711} \alpha_{1} }}{{\mu_{{S_{1} }} }}} \right)^{2} + a_{711} \alpha_{1} \left( {\frac{{a_{711} \alpha_{1} }}{{\mu_{{S_{1} }} }}} \right) + a_{721} \alpha_{2} \left( {\frac{{a_{722} \alpha_{2} }}{{\mu_{{S_{2} }} }}} \right) + b_{71} \delta \frac{{\delta b_{8} }}{{\mu_{M} }}; \\ \end{aligned} $$

(146)

$$ \begin{aligned} \rho (a_{712} x_{1} + a_{722} x_{2} + b_{72} y + c_{72} ) & = (b_{72} \gamma_{1} v_{1} - a_{712} \beta )x_{1} + (b_{72} \gamma_{2} v_{2} - a_{722} \beta )x_{2} + (v_{2} U_{{S_{2} }} \eta - b_{72} \beta )y \\ & \quad + v_{2} U_{{S_{2} }} D_{0} - \frac{{\mu_{{S_{2} }} }}{2}\left( {\frac{{a_{722} \alpha_{2} }}{{\mu_{{S_{2} }} }}} \right)^{2} + a_{712} \alpha_{1} \left( {\frac{{a_{711} \alpha_{1} }}{{\mu_{{S_{1} }} }}} \right) + a_{722} \alpha_{2} \left( {\frac{{a_{722} \alpha_{2} }}{{\mu_{{S_{2} }} }}} \right) + b_{72} \delta \frac{{\delta b_{8} }}{{\mu_{M} }}; \\ \end{aligned} $$

(147)

$$ \begin{aligned} \rho (a_{81} x_{1} + a_{82} x_{2} + b_{8} y + c_{8} ) & = (b_{8} \gamma_{1} v_{1} - a_{81} \beta )x_{1} + (b_{8} \gamma_{2} v_{2} - a_{82} \beta )x_{2} + (U_{M} \eta - b_{8} \beta )y \\ & \quad + U_{M} D_{0} - \frac{{\mu_{M} }}{2}\left( {\frac{{b_{8} \delta }}{{\mu_{M} }}} \right)^{2} + a_{81} \alpha_{1} \left( {\frac{{a_{711} \alpha_{1} }}{{\mu_{{S_{1} }} }}} \right) + a_{82} \alpha_{2} \left( {\frac{{a_{722} \alpha_{2} }}{{\mu_{{S_{2} }} }}} \right) + b_{8} \delta \frac{{\delta b_{8} }}{{\mu_{M} }}. \\ \end{aligned} $$

(148)

From Eq. (146), comparing its coefficients of the similar terms on the left and right sides shows us

$$ \left\{ \begin{aligned} a_{{711}} & = v_{1} U_{{S_{1} }} \eta \gamma _{1} v_{1} {\text{/}}(\rho + \beta )^{2} \\ a_{{721}} & = v_{1} U_{{S_{1} }} \eta \gamma _{2} v_{2} /(\rho + \beta )^{2} \\ b_{{71}} & = v_{1} U_{{S_{1} }} \eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{{71}} & = \frac{{v_{1} U_{{S_{1} }} D_{0} }}{\rho } + \frac{{\alpha _{1} ^{2} }}{{2\rho \mu _{{S_{1} }} }}\left[ {\frac{{v_{1} U_{{S_{1} }} \eta \gamma _{1} v_{1} }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{\alpha _{2} ^{2} v_{1} \gamma _{1} \gamma _{2} v_{2} }}{{\rho \mu _{{S_{2} }} }}\left[ {\frac{{v_{1} U_{{S_{1} }} \eta }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{U_{{M_{1} }} v_{1} U_{{S_{1} }} }}{{\rho \mu _{M} }}\left[ {\frac{{\delta \eta }}{{\rho + \beta }}} \right]^{2} \\ \end{aligned} \right.. $$

(149)

Collating Eq. (147) and comparing its coefficients of the similar terms on the left and right sides gives

$$ \left\{ \begin{aligned} a_{{712}} & = v_{2} U_{{S_{2} }} \eta \gamma _{1} v_{1} {\text{/}}(\rho + \beta )^{2} \\ a_{{722}} & = v_{2} U_{{S_{2} }} \eta \gamma _{2} v_{2} {\text{/}}(\rho + \beta )^{2} \\ b_{{72}} & = v_{2} U_{{S_{2} }} \eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{{72}} & = \frac{{v_{2} U_{{S_{2} }} D_{0} }}{\rho } + \frac{{v_{1} U_{{S_{1} }} v_{2} U_{{S_{2} }} \alpha _{1} ^{2} }}{{2\rho \mu _{{S_{1} }} }}\left[ {\frac{{\eta \gamma _{1} v_{1} }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{\alpha _{2} ^{2} }}{{2\rho \mu _{{S_{2} }} }}\left[ {\frac{{v_{2} U_{{S_{2} }} \eta \gamma _{2} v_{2} }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{U_{M} v_{2} U_{{S_{2} }} }}{{\rho \mu _{M} }}\left[ {\frac{{\delta \eta }}{{\rho + \beta }}} \right]^{2} \\ \end{aligned} \right.. $$

(150)

Comparing the coefficients of terms on the left and right sides in Eq. (148) shows us

$$ \left\{ \begin{aligned} a_{{81}} & = U_{M} \eta \gamma _{1} v_{1} {\text{/}}(\rho + \beta )^{2} \\ a_{{82}} & = U_{M} \eta \gamma _{2} v_{2} {\text{/}}(\rho + \beta )^{2} \\ b_{8} & = U_{M} \eta {\text{/(}}\rho + \beta {\text{)}} \\ c_{8} & = \frac{{U_{M} D_{0} }}{\rho } + \frac{{v_{1} U_{{S_{1} }} U_{M} \alpha _{1} ^{2} }}{{\rho \mu _{{S_{1} }} }}\left[ {\frac{{\eta \gamma _{1} v_{1} }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{v_{2} U_{{S_{2} }} U_{M} \alpha _{2} ^{2} }}{{\rho \mu _{{S_{2} }} }}\left[ {\frac{{\eta \gamma _{2} v_{2} }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{\delta ^{2} }}{{2\rho \mu _{M} }}\left[ {\frac{{U_{M} \eta }}{{\rho + \beta }}} \right]^{2} \\ \end{aligned} \right.. $$

(151)

Then substitute them into Eqs. (140) and (141), we can get

$$ Z_{{S_{1} }}^{D*} { = }U_{{S_{1} }} \alpha_{1} \eta \gamma_{1} v_{1}^{2} {/[}\mu_{{S_{1} }} (\rho + \beta )^{2} {];} $$

(152)

$$ Z_{{S_{2} }}^{D*} { = }U_{{S_{2} }} \alpha_{2} \eta \gamma_{2} v_{2}^{2} {/}[\mu_{{S_{2} }} (\rho + \beta )^{2} ]; $$

(153)

$$ Z_{M}^{D*} { = }U_{M} \delta \eta {/[}\mu_{M} (\rho + \beta ){]}{\text{.}} $$

(154)

Thus, Proposition 8 can be proved.

Proof of Proposition 9

Similar to the proof of Proposition 2, substituting the optimal strategies $Z_{{S_{i} }}^{D*} { = }U_{{S_{i} }} \alpha_{i} \eta \gamma_{i} v_{i}^{2} {/[}\mu_{{S_{i} }} (\rho + \beta )^{2} {]}$ and $Z_{M}^{D*} { = }U_{M} \delta \eta {/[}\mu_{M} (\rho + \beta ){]}$ into Eq. (2) in the case of decentralized decision making of Proposition 8, we can solve the following differential equations

$$ \left\{ \begin{gathered} \dot{x}(t) = \alpha Z_{S} (t) - \beta x(t) \hfill \\ \dot{y}(t) = \gamma x(t) + \delta Z_{M} (t) - \beta y(t) \hfill \\ \end{gathered} \right.. $$

(155)

Then the optimal trajectory of Proposition 9 can be obtained.

Proof of Proposition 10

Maximizing the right side of Eqs. (77) and (78) gives

$$ Z_{{S_{i} }}^{B*} { = }\alpha_{i} V_{{S_{i} x_{i} }}^{{}} (x,y){/[}\mu_{{S_{i} }} (1 - \omega_{i} ){];} $$

(156)

$$ Z_{M}^{B*} = {{\delta V_{My}^{{}} (x,y)} \mathord{\left/ {\vphantom {{\delta V_{My}^{{}} (x,y)} {\left[ {\mu_{M} \left( {1 - \sum\nolimits_{i = 1}^{2} {\psi_{i} } } \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\mu_{M} \left( {1 - \sum\nolimits_{i = 1}^{2} {\psi_{i} } } \right)} \right]}}{.} $$

(157)

Substituting Eqs. (156) and (157) into Eqs. (77) and (78) respectively attains

$$ \begin{aligned} \rho V_{{S_{i} }}^{B*} (x,y) & = U_{{S_{i} }} v_{i} (D_{0} + \eta y) - \frac{{\mu_{{S_{i} }} }}{2}(1 - \omega_{i} )\left[ {\frac{{\alpha_{i} V_{{S_{i} x_{i} }}^{{}} (x,y)}}{{\mu_{{S_{i} }} (1 - \omega_{i} )}}} \right]^{2} - \frac{{\mu_{M} }}{2}\psi_{i} \left[ {\frac{{\delta V_{{M_{1} y}}^{{}} (x,y)}}{{\mu_{M} (1 - \sum\nolimits_{j = 1}^{2} {\psi_{j} } )}}} \right]^{2} \\ & \quad + \sum\limits_{j = 1}^{2} {V_{{S_{i} x_{j} }}^{B*} \left[ {\alpha_{j} \frac{{\alpha_{i} V_{{S_{i} x_{i} }}^{{}} (x,y)}}{{\mu_{{S_{i} }} (1 - \omega_{i} )}} - \beta x_{j} } \right]} + V_{{S_{i} y}}^{B*} \left[ {\sum\limits_{j = 1}^{2} {\gamma_{j} x_{j} v_{j} } + \frac{{\delta^{2} V_{My}^{{}} (x,y)}}{{\mu_{M} (1 - \sum\nolimits_{j = 1}^{2} {\psi_{j} } )}} - \beta y} \right]; \\ \end{aligned} $$

(158)

$$ \begin{aligned} \rho V_{M}^{B*} (x,y) & = U_{{M_{1} }} (D_{0} + \eta y) - \frac{{\mu_{M} }}{2}(1 - \sum\nolimits_{j = 1}^{2} {\psi_{j} } )\left[ {\frac{{\delta V_{My}^{{}} (x,y)}}{{\mu_{M} (1 - \sum\nolimits_{j = 1}^{2} {\psi_{j} } )}}} \right]^{2} - \sum\limits_{i = 1}^{2} {\frac{{\mu_{{S_{i} }} }}{2}\omega_{i} \left[ {\frac{{\alpha_{i} V_{{S_{i} x_{i} }}^{{}} (x,y)}}{{\mu_{{S_{i} }} (1 - \omega_{i} )}}} \right]}^{2} \\ & \quad + \sum\nolimits_{j = 1}^{2} {V_{{Mx_{i} }}^{B*} \left( {\frac{{\alpha_{i}^{2} V_{{S_{i} x_{i} }}^{{}} (x,y)}}{{\mu_{{S_{i} }} (1 - \omega_{i} )}} - \beta x_{i} } \right)} + V_{My}^{B*} \left[ {\sum\limits_{i = 1}^{2} {\gamma_{i} x_{i} v_{i} } + \frac{{\delta^{2} V_{My}^{{}} (x,y)}}{{\mu_{M} (1 - \sum\nolimits_{j = 1}^{2} {\psi_{j} } )}} - \beta y} \right]. \\ \end{aligned} $$

(159)

From Eqs. (158) and (159), The HJB equation has the linear function of $x$ and $y$ as the solution. Set

$$ V_{{S_{i} }}^{B} = \sum\nolimits_{j = 1}^{2} {a_{9ji} x_{j} } + b_{9i} y + c_{9i} ; $$

(160)

$$ V_{M}^{B} = \sum\nolimits_{i = 1}^{2} {a_{10i} x_{i} } + b_{10} y + c_{10} . $$

(161)

Then $V_{{S_{i} x_{i} }}^{B} = a_{9ii}$, $V_{{S_{i} x_{j} }}^{B} = a_{9ji}$ Eq. ($j \ne i$); $V_{{S_{i} y}}^{B} = b_{9i}$, $V_{{M_{1} x_{i} }}^{B} = a_{10i}$, $V_{{M_{1} y}}^{B} = b_{10}$.

Plugging Eqs. (160) and (161) into Eqs. (158) and (159) respectively gives

$$ \begin{aligned} \rho (a_{911} x_{1} + a_{921} x_{2} + b_{91} y + c_{91} ) & = (b_{91} \gamma_{1} v_{1} - a_{911} \beta )x_{1} + (b_{91} \gamma_{2} v_{2} - a_{921} \beta )x_{2} + (v_{1} U_{{S_{1} }} \eta - b_{91} \beta )y \\ & \quad + v_{1} U_{{S_{1} }} D_{0} - \frac{{\mu_{{S_{1} }} }}{2}(1 - \omega_{1} )\left[ {\frac{{a_{911} \alpha_{1} }}{{\mu_{{S_{1} }} (1 - \omega_{1} )}}} \right]^{2} + a_{911} \alpha_{1} \left[ {\frac{{a_{911} \alpha_{1} }}{{\mu_{{S_{1} }} (1 - \omega_{1} )}}} \right] \\ & \quad + a_{921} \alpha_{2} \left[ {\frac{{a_{922} \alpha_{2} }}{{\mu_{{S_{2} }} (1 - \omega_{2} )}}} \right] + b_{91} \delta \frac{{\delta b_{10} }}{{\mu_{M} (1 - \psi_{1} - \psi_{2} )}} - \frac{{\mu_{M} }}{2}\psi_{1} \left[ {\frac{{\delta b_{10} }}{{\mu_{M} (1 - \psi_{1} - \psi_{2} )}}} \right]^{2} ; \\ \end{aligned} $$

(162)

$$ \begin{aligned} \rho (a_{912} x_{1} + a_{922} x_{2} + b_{92} y + c_{92} ) & = (b_{92} \gamma_{1} v_{1} - a_{912} \beta )x_{1} + (b_{92} \gamma_{2} v_{2} - a_{922} \beta )x_{2} + (v_{2} U_{{S_{2} }} \eta - b_{92} \beta )y \\ & \quad + v_{2} U_{{S_{2} }} D_{0} - \frac{{\mu_{{S_{2} }} }}{2}(1 - \omega_{2} )\left[ {\frac{{a_{922} \alpha_{2} }}{{\mu_{{S_{2} }} (1 - \omega_{2} )}}} \right]^{2} + a_{912} \alpha_{1} \left[ {\frac{{a_{911} \alpha_{1} }}{{\mu_{{S_{1} }} (1 - \omega_{1} )}}} \right] \\ & \quad + a_{922} \alpha_{2} \left[ {\frac{{a_{922} \alpha_{2} }}{{\mu_{{S_{2} }} (1 - \omega_{2} )}}} \right] + b_{92} \delta \frac{{\delta b_{10} }}{{\mu_{M} (1 - \psi_{1} - \psi_{2} )}} - \frac{{\mu_{M} }}{2}\psi_{2} \left[ {\frac{{\delta b_{10} }}{{\mu_{M} (1 - \psi_{1} - \psi_{2} )}}} \right]^{2} ; \\ \end{aligned} $$

(163)

$$ \begin{aligned} \rho (a_{101} x_{1} + a_{102} x_{2} + b_{10} y + c_{10} ) & = (b_{10} \gamma_{1} v_{1} - a_{101} \beta )x_{1} + (b_{10} \gamma_{2} v_{2} - a_{102} \beta )x_{2} + (U_{M} \eta - b_{10} \beta )y \\ & \quad + U_{M} D_{0} - \frac{{\mu_{M} }}{2}(1 - \psi_{1} - \psi_{2} )\left[ {\frac{{b_{10} \delta }}{{\mu_{M} (1 - \psi_{1} - \psi_{2} )}}} \right]^{2} - \frac{{\mu_{{S_{1} }} }}{2}\omega_{1} \left[ {\frac{{a_{911} \alpha_{1} }}{{\mu_{{S_{1} }} (1 - \omega_{1} )}}} \right]^{2} - \frac{{\mu_{{S_{2} }} }}{2}\omega_{2} \left[ {\frac{{a_{922} \alpha_{2} }}{{\mu_{{S_{2} }} (1 - \omega_{2} )}}} \right]^{2} \\ & \quad + a_{101} \alpha_{1} \left[ {\frac{{a_{911} \alpha_{1} }}{{\mu_{{S_{1} }} (1 - \omega_{1} )}}} \right] + a_{102} \alpha_{2} \left[ {\frac{{a_{922} \alpha_{2} }}{{\mu_{{S_{2} }} (1 - \omega_{1} )}}} \right] + b_{10} \delta \frac{{\delta b_{10} }}{{\mu_{M} (1 - \psi_{1} - \psi_{2} )}}. \\ \end{aligned} $$

(164)

Collating Eq. (162) and compare its coefficients of the similar terms on both sides yields

$$ \left\{ \begin{aligned} a_{911} & = v_{1} U_{{S_{1} }} \eta \gamma_{1} v_{1} {/}(\rho + \beta )^{2} \\ a_{921} & = v_{1} U_{{S_{1} }} \eta \gamma_{2} v_{2} {/}(\rho + \beta )^{2} \\ b_{91} & = v_{1} U_{{S_{1} }} \eta {/(}\rho + \beta {)} \\ c_{91} & = \frac{{v_{1} U_{{S_{1} }} D_{0} }}{\rho } + \frac{{\alpha_{1}^{2} }}{{2\rho \mu_{{S_{1} }} (1 - \omega_{1} )}}\left[ {\frac{{v_{1} U_{{S_{1} }} \eta \gamma_{1} v_{1} }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{\alpha_{2}^{2} v_{1} U_{{S_{1} }} v_{2} U_{{S_{2} }} }}{{\rho \mu_{{S_{2} }} (1 - \omega_{2} )}}\left[ {\frac{{\eta \gamma_{2} v_{2} }}{{(\rho + \beta )^{2} }}} \right]^{2} \\ & \quad + \frac{{v_{1} U_{{S_{1} }} U_{M} \delta^{2} }}{{\rho \mu_{M} (1 - \psi_{1} - \psi_{2} )}}\left[ {\frac{\eta }{\rho + \beta }} \right]^{2} + \frac{{\psi_{1} \delta^{2} }}{{2\rho \mu_{M} }}\left[ {\frac{{U_{M} \eta }}{{(1 - \psi_{1} - \psi_{2} )(\rho + \beta )}}} \right]^{2} \\ \end{aligned} \right.. $$

(165)

Collating Eq. (163) and comparing its coefficients of the terms on the left and right sides gives

$$ \left\{ \begin{aligned} a_{912} & = v_{2} U_{{S_{2} }} \eta \gamma_{1} v_{1} {/}(\rho + \beta )^{2} \\ a_{922} & = v_{2} U_{{S_{2} }} \eta \gamma_{2} v_{2} {/}(\rho + \beta )^{2} \\ b_{92} & = v_{2} U_{{S_{2} }} \eta {/(}\rho + \beta {)} \\ c_{92} & = \frac{{v_{2} U_{{S_{2} }} D_{0} }}{\rho } + \frac{{\alpha_{2}^{2} }}{{2\rho \mu_{{S_{2} }} (1 - \omega_{2} )}}\left[ {\frac{{v_{2} U_{{S_{2} }} \eta \gamma_{2} v_{2} }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{\alpha_{1}^{2} v_{1} U_{{S_{1} }} v_{2} U_{{S_{2} }} }}{{\rho \mu_{{S_{1} }} (1 - \omega_{1} )}}\left[ {\frac{{\eta \gamma_{1} v_{1} }}{{(\rho + \beta )^{2} }}} \right]^{2} \\ & \quad + \frac{{v_{2} U_{{S_{2} }} U_{M} \delta^{2} }}{{\rho \mu_{M} (1 - \psi_{1} - \psi_{2} )}}\left[ {\frac{\eta }{\rho + \beta }} \right]^{2} + \frac{{\psi_{2} \delta^{2} }}{{2\rho \mu_{M} }}\left[ {\frac{{U_{M} \eta }}{{(1 - \psi_{1} - \psi_{2} )(\rho + \beta )}}} \right]^{2} \\ \end{aligned} \right.. $$

(166)

Comparing its coefficients of the similar terms on both sides in Eq. (164) produces

$$ \left\{ \begin{aligned} a_{101} & = U_{M} \eta \gamma_{1} v_{1} {/}(\rho + \beta )^{2} \\ a_{102} & = U_{M} \eta \gamma_{2} v_{2} {/}(\rho + \beta )^{2} \\ b_{10} & = U_{M} \eta {/(}\rho + \beta {)} \\ c_{10} & = \frac{{U_{M} D_{0} }}{\rho } - \frac{{\omega_{1} \alpha_{1}^{2} }}{{2\rho \mu_{{S_{1} }} (1 - \omega_{1} )}}\left[ {\frac{{v_{1} U_{{S_{1} }} \eta \gamma_{1} v_{1} }}{{(\rho + \beta )^{2} }}} \right]^{2} - \frac{{\omega_{2} \alpha_{2}^{2} }}{{2\rho \mu_{{S_{2} }} (1 - \omega_{2} )^{2} }}\left[ {\frac{{v_{2} U_{{S_{2} }} \eta \gamma_{2} v_{2} }}{{(\rho + \beta )^{2} }}} \right]^{2} \\ & \quad + \frac{{\delta^{2} }}{{2\rho \mu_{M} (1 - \psi_{1} - \psi_{2} )}}\left[ {\frac{{U_{{M_{1} }} \eta }}{\rho + \beta }} \right]^{2} + \frac{{v_{1} U_{{S_{1} }} U_{M} \alpha_{1}^{2} }}{{\rho \mu_{{S_{1} }} (1 - \omega_{1} )}}\left[ {\frac{{\eta \gamma_{1} v_{1} }}{{(\rho + \beta )^{2} }}} \right]^{2} + \frac{{v_{2} U_{{S_{2} }} U_{M} \alpha_{2}^{2} }}{{\rho \mu_{{S_{2} }} (1 - \omega_{2} )^{2} }}\left[ {\frac{{\eta \gamma_{2} v_{2} }}{{(\rho + \beta )^{2} }}} \right]^{2} \\ \end{aligned} \right.. $$

(167)

Coordinating the supply chain system can make the optimal emission reduction efforts of suppliers and manufacturer under bilateral contract equal to those under centralized case, so $Z_{{S_{1} }}^{C*} = Z_{{S_{1} }}^{B*}$,$Z_{{S_{2} }}^{C*} = Z_{{S_{2} }}^{B*}$,$Z_{M}^{C*} = Z_{M}^{B*}$. Namely, $\frac{{(U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} )\alpha_{1} \eta \gamma_{1} v_{1} }}{{\mu_{{S_{1} }} (\rho + \beta )^{2} }} =$$\frac{{v_{1} U_{{S_{1} }} \alpha_{1} \eta \gamma_{1} v_{1} }}{{(1 - \omega_{1} )\mu_{{S_{2} }} (\rho + \beta )^{2} }}$, $\frac{{(U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} )\alpha_{2} \eta \gamma_{2} v_{2} }}{{\mu_{{S_{2} }} (\rho + \beta )^{2} }} = \frac{{v_{2} U_{{S_{2} }} \alpha_{2} \eta \gamma_{2} v_{2} }}{{(1 - \omega_{2} )\mu_{{S_{2} }} (\rho + \beta )^{2} }}{,}$$\frac{{(U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} )\delta \eta }}{{\mu_{{M_{1} }} (\rho + \beta )}}$$= \frac{{U_{M} \delta \eta }}{{(1 - \psi_{1} - \psi_{2} )\mu_{M} (\rho + \beta )}}{.}$ By comparison, we have $U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} = v_{1} U_{{S_{1} }} {/(}1 - \omega_{1} {)}$, $U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} = v_{2} U_{{S_{2} }} {/(}1 - \omega_{2} {),}$ $U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} = U_{M} {/(}1 - \psi_{1} - \psi_{2} {)}$. So we get $\omega_{1}^{*} = \frac{{U_{M} + v_{2} U_{{S_{2} }} }}{{U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} }}$, $\omega_{2}^{*} = \frac{{U_{M} + v_{1} U_{{S_{1} }} }}{{U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} }}$, $\psi_{1}^{*} + \psi_{2}^{*} = \frac{{v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} }}{{U_{M} + v_{1} U_{{S_{1} }} + v_{2} U_{{S_{2} }} }}$. Thus, Proposition 10 can be proved.