The effects of cap-and-trade regulation and sales effort on supply chain coordination

Abstract

Under cap-and-trade regulation, this paper firstly investigates the optimal production decisions and sales effort level with the cost-sharing and quantity-discount contracts. And we further explore the effects of the regulation and sales effort on supply chain coordination. We list some main conclusions here. First, with the two contracts, the manufacturer’s optimal profit firstly increases and then decreases as the cap increases. Both the manufacturer’s and retailer’s optimal profits are increasing in the marginal sales effort cost in some cases. Second, the two contracts partly coordinate the supply chain, and the quantity-discount contract is more flexible in coordinating the supply chain, which means that the condition of supply chain coordination through the quantity-discount contract is easier to meet. Third, for the two contracts, higher carbon trading price and stricter cap are beneficial to supply chain coordination, while the retailer’s sales effort damages supply chain coordination. Finally, we extend our model to consider production and retail competition to check the robustness of the coordination results.

Appendix

Proof of Theorem 1

From Eqs. (1) and (2), we know that $\Pi_{R}^{\psi } = (a - q + \theta s - w)q - {{(1 - \psi )ms^{2} } \mathord{\left/ {\vphantom {{(1 - \psi )ms^{2} } 2}} \right. \kern-0pt} 2}.$ Then we let the first partial derivatives of $\Pi_{R}^{\psi }$ with respect to $q$ and $s$ be equal to zero. That is, ${{\partial \Pi_{R}^{\psi } } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{\psi } } {\partial q}}} \right. \kern-0pt} {\partial q}} = a + \theta s - w - 2q = 0$ and ${{\partial \Pi_{R}^{\psi } } \mathord{\left/ {\vphantom {{\partial \Pi_{R}^{\psi } } {\partial s}}} \right. \kern-0pt} {\partial s}} = \theta q - (1 - \psi )ms = 0.$ And the Hessian matrix of $\Pi_{R}^{\psi }$ is as follows:

$$ H_{1} = \left( {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{R}^{\psi } }}{{\partial q^{2} }}} & {\frac{{\partial^{2} \Pi_{R}^{\psi } }}{\partial q\partial s}} \\ {\frac{{\partial^{2} \Pi_{R}^{\psi } }}{\partial s\partial q}} & {\frac{{\partial^{2} \Pi_{R}^{\psi } }}{{\partial s^{2} }}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { - 2} & \theta \\ \theta & { - (1 - \psi )m} \\ \end{array} } \right). $$

According to the assumption $2(1 - \psi )m > \theta^{2} ,$ the matrix $H_{1}$ is negative definite, then the optimal decisions from the first order conditions are as follows:

$$ q^{ * } = {{(1 - \psi )m(a - w)} \mathord{\left/ {\vphantom {{(1 - \psi )m(a - w)} {[2(1 - \psi )m - \theta^{2} }}} \right. \kern-0pt} {[2(1 - \psi )m - \theta^{2} }}],s^{ * } = {{(a - w)\theta } \mathord{\left/ {\vphantom {{(a - w)\theta } {[2(1 - \psi )m - \theta^{2} ]}}} \right. \kern-0pt} {[2(1 - \psi )m - \theta^{2} ]}}. $$

Then, we can derive that $p^{ * } = a - q^{ * } + \theta s^{ * } = {{[m(a + w) - \theta^{2} w]} \mathord{\left/ {\vphantom {{[m(a + w) - \theta^{2} w]} {[2(1 - \psi )m - \theta^{2} ]}}} \right. \kern-0pt} {[2(1 - \psi )m - \theta^{2} ]}}.$ According to Eq. (3), the manufacturer’s profit can be rewritten as follows:

$\Pi_{M}^{\psi } = \frac{(1 - \psi )m(a - w)(w - c - be)}{{[2(1 - \psi )m - \theta^{2} ]}} - \frac{{\psi m(a - w)^{2} \theta^{2} }}{{2[2(1 - \psi )m - \theta^{2} ]^{2} }} + bC.$ We let the first partial derivatives of $\Pi_{M}^{\psi }$ with respect to $w$ be equal to zero. That is $\frac{{\partial \Pi_{M}^{\psi } }}{\partial w} = \frac{(1 - \psi )(a + c + be - 2w)m}{{[2(1 - \psi )m - \theta^{2} ]}} + \frac{{\psi m(a - w)\theta^{2} }}{{[2(1 - \psi )m - \theta^{2} ]^{2} }} = 0.$ Since $\frac{{\partial^{2} \Pi_{M}^{\psi } }}{{\partial w^{2} }} = - \frac{2(1 - \psi )m}{{[2(1 - \psi )m - \theta^{2} ]}} - \frac{{\psi m\theta^{2} }}{{[2(1 - \psi )m - \theta^{2} ]^{2} }} < 0,$ $\Pi_{M}^{\psi }$ is a concave function of $w.$ Thus, we can derive that $w^{\psi * } = {{\{ (1 - \psi )[2(1 - \psi )m - \theta^{2} ](a + c + be)m + \psi ma\theta^{2} \} } \mathord{\left/ {\vphantom {{\{ (1 - \psi )[2(1 - \psi )m - \theta^{2} ](a + c + be)m + \psi ma\theta^{2} \} } {\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} }}} \right. \kern-0pt} {\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} }}.$ Therefore, we have the following solution group:

$q^{\psi * } = {{m(1 - \psi )^{2} (a - c - be)} \mathord{\left/ {\vphantom {{m(1 - \psi )^{2} (a - c - be)} {\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} }}} \right. \kern-0pt} {\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} }},$ $p^{\psi * } = \frac{{(1 - \psi )^{2} (3a + c + be)m + [\psi (2a + c + be) - (a + c + be)]\theta^{2} }}{{2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} }},$ $\Pi_{R}^{\psi * } = {{m(a - c - be)^{2} (1 - \psi )^{3} [2(1 - \psi )m - \theta^{2} ]} \mathord{\left/ {\vphantom {{m(a - c - be)^{2} (1 - \psi )^{3} [2(1 - \psi )m - \theta^{2} ]} {\{ 2\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \}^{2} }}} \right. \kern-0pt} {\{ 2\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \}^{2} }}\} ,$ $\Pi_{M}^{\psi * } = {{m(a - c - be)^{2} (1 - \psi )^{2} } \mathord{\left/ {\vphantom {{m(a - c - be)^{2} (1 - \psi )^{2} } {\{ 4(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + 2\psi \theta^{2} \} }}} \right. \kern-0pt} {\{ 4(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + 2\psi \theta^{2} \} }} + bC.$

Then, the manufacturer’s total carbon emission is $eq^{\psi * } = {{em(1 - \psi )^{2} (a - c - be)} \mathord{\left/ {\vphantom {{em(1 - \psi )^{2} (a - c - be)} {\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} }}} \right. \kern-0pt} {\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} }}.$ However, whether the value of $eq^{\psi * }$ is higher than the cap is not deterministic. If $eq^{\psi * } > C,$ the above results are true. If $eq^{\psi * } \le C,$ according to our assumption that $eq^{\psi * } \ge C,$ then, $eq^{\psi * } = C.$ It means that the manufacturer gets too many credits. Hence, he needs to use all the credits to make production. Under this circumstance, $q^{\psi * } = {C \mathord{\left/ {\vphantom {C e}} \right. \kern-0pt} e},$ we plug this value into the retailer’s response function and derive that $s^{\psi * } = {{\theta C} \mathord{\left/ {\vphantom {{\theta C} {[(1 - \psi )me]}}} \right. \kern-0pt} {[(1 - \psi )me]}},$ $w^{w * } = {{\{ (1 - \psi )ame - [2(1 - \psi )m - \theta^{2} ]C\} } \mathord{\left/ {\vphantom {{\{ (1 - \psi )ame - [2(1 - \psi )m - \theta^{2} ]C\} } {[(1 - \psi )me]}}} \right. \kern-0pt} {[(1 - \psi )me]}}.$ The firms’ optimal profits are $\Pi_{R}^{\psi * } = {{[2(1 - \psi )m - \theta^{2} ]C^{2} } \mathord{\left/ {\vphantom {{[2(1 - \psi )m - \theta^{2} ]C^{2} } {[2(1 - \psi )me^{2} ]}}} \right. \kern-0pt} {[2(1 - \psi )me^{2} ]}}$ and $\Pi_{M}^{\psi * } = {{\{ (1 - \psi )(a - c)meC - [2(1 - \psi )m - \theta^{2} ]C^{2} \} } \mathord{\left/ {\vphantom {{\{ (1 - \psi )(a - c)meC - [2(1 - \psi )m - \theta^{2} ]C^{2} \} } {[(1 - \psi )me^{2} ]}}} \right. \kern-0pt} {[(1 - \psi )me^{2} ]}} - {{\psi \theta^{2} C^{2} } \mathord{\left/ {\vphantom {{\psi \theta^{2} C^{2} } {[2(1 - \psi )^{2} me^{2} ]}}} \right. \kern-0pt} {[2(1 - \psi )^{2} me^{2} ]}},$ respectively.

Summarizing the above analysis, we can get Theorem 1.

Proof of Proposition 1

From Theorem 1, we get that, if $0 < C < C_{1} ,$ $p^{\psi * } = \frac{{(1 - \psi )^{2} (3a + c + be)m + [\psi (2a + c + be) - (a + c + be)]\theta^{2} }}{{2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} }},$ and if $C \ge C_{1} ,$ $p^{\psi * } = {{\{ (1 - \psi )ame - [(1 - \psi )m - \theta^{2} ]C\} } \mathord{\left/ {\vphantom {{\{ (1 - \psi )ame - [(1 - \psi )m - \theta^{2} ]C\} } {[(1 - \psi )me]}}} \right. \kern-0pt} {[(1 - \psi )me]}}.$ We can have that ${{\partial p^{\psi * } } \mathord{\left/ {\vphantom {{\partial p^{\psi * } } {\partial b}}} \right. \kern-0pt} {\partial b}} = {{[(1 - \psi )m - \theta^{2} ]} \mathord{\left/ {\vphantom {{[(1 - \psi )m - \theta^{2} ]} {\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} }}} \right. \kern-0pt} {\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} }}$ if $0 < C < C_{1} ,$ and if $C \ge C_{1} ,$ the optimal retail price is not relevant to carbon trading price. From Assumption 1, we know $m > {{\theta^{2} } \mathord{\left/ {\vphantom {{\theta^{2} } {[2(1 - \psi )]}}} \right. \kern-0pt} {[2(1 - \psi )]}}.$ Hence, we derive that ${{\partial p^{\psi * } } \mathord{\left/ {\vphantom {{\partial p^{\psi * } } \partial }} \right. \kern-0pt} \partial }b < 0$ when ${{\theta^{2} } \mathord{\left/ {\vphantom {{\theta^{2} } {[2(1 - \psi )] < }}} \right. \kern-0pt} {[2(1 - \psi )] < }}m \le {{\theta^{2} } \mathord{\left/ {\vphantom {{\theta^{2} } {(1 - \psi )}}} \right. \kern-0pt} {(1 - \psi )}};$ ${{\partial p^{\psi * } } \mathord{\left/ {\vphantom {{\partial p^{\psi * } } \partial }} \right. \kern-0pt} \partial }b > 0$ when $m > {{\theta^{2} } \mathord{\left/ {\vphantom {{\theta^{2} } {(1 - \psi )}}} \right. \kern-0pt} {(1 - \psi )}}.$

Proof of Theorem 2

Following a similar process in proving Theorem 1, we can get Theorem 2.

Proof of Proposition 2

Following a similar process in proving Proposition 1, we can get Proposition 2.

Proof of Proposition 3

After comparing the optimal production quantity and manufacturer’s optimal profit in Theorem 1 with those in the Theorem 2, we can derive that $q^{\psi * } - q^{\varphi * } = (a - c - be)m\left\{ {\frac{{(1 - \psi )^{2} }}{{2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} }} - \frac{1}{{2[(2 - \delta )m - \theta^{2} ]}}} \right\},$ and $\Pi_{M}^{\psi * } - \Pi_{M}^{\varphi * } = \frac{{(a - c - be)^{2} m}}{2}\left\{ {\frac{{(1 - \psi )^{2} }}{{2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} }} - \frac{1}{{2[(2 - \delta )m - \theta^{2} ]}}} \right\}.$ Thus, we only need to estimate the relationship between $\frac{{(1 - \psi )^{2} }}{{2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} }} - \frac{1}{{2[(2 - \delta )m - \theta^{2} ]}}$ and zero. $\frac{{(1 - \psi )^{2} }}{{2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} }} - \frac{1}{{2[(2 - \delta )m - \theta^{2} ]}} = \frac{{ - 2\delta (1 - \psi )^{2} m + \psi (1 - 2\psi )\theta^{2} }}{{\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \} \{ 2[(2 - \delta )m - \theta^{2} ]\} }}.$ Hence, we can derive that if ${1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2} \le \psi < 1,$ $q^{\psi * } - q^{\varphi * } < 0$ and $\Pi_{M}^{\psi * } - \Pi_{M}^{\varphi * } < 0.$ If $0 < \psi < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2},$ then $q^{\psi * } - q^{\varphi * } > 0$ and $\Pi_{M}^{\psi * } - \Pi_{M}^{\varphi * } > 0$ when $0 < \delta < {{\psi (1 - 2\psi )\theta^{2} } \mathord{\left/ {\vphantom {{\psi (1 - 2\psi )\theta^{2} } {[2m(1 - \psi )^{2} ]}}} \right. \kern-0pt} {[2m(1 - \psi )^{2} ]}}.$ Otherwise, $q^{\psi * } - q^{\varphi * } \le 0$ and $\Pi_{M}^{\psi * } - \Pi_{M}^{\varphi * } \le 0.$ In conclusion, if $\delta \in (0,{\text{max\{ }}0,{{\psi (1 - 2\psi )\theta^{2} } \mathord{\left/ {\vphantom {{\psi (1 - 2\psi )\theta^{2} } {[2m(1 - \psi )^{2} ]}}} \right. \kern-0pt} {[2m(1 - \psi )^{2} ]}}{\text{\} }}],$ $q^{\psi * } \ge q^{\varphi * }$ and $\Pi_{M}^{\psi * } \ge \Pi_{M}^{\varphi * } .$ Otherwise, $q^{\psi * } < q^{\varphi * }$ and $\Pi_{M}^{\psi * } < \Pi_{M}^{\varphi * } .$

Proof of Theorem 3

In the centralized supply chain, following the similar process in proving Theorems 1 and 2, we can get the results shown in Theorem 3.

Proof of Theorem 4

We use a contract $(w^{c\psi * } ,s^{c\psi * } )$ to explore the supply chain coordination, where $s^{ * } = {{(a - c - be)\theta } \mathord{\left/ {\vphantom {{(a - c - be)\theta } {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}$ when $0 < C < C_{3}$ and $s^{ * } = {{\theta C} \mathord{\left/ {\vphantom {{\theta C} {(me)}}} \right. \kern-0pt} {(me)}}$ when $C \ge C_{3} .$ If the contract $(w^{c\psi * } ,s^{c\psi * } )$ can facilitate the retailer to order $q^{ * }$ presented in Theorem 3, then the contract coordinates the supply chain. That is, $q^{ * } = {{m(a - c - be)} \mathord{\left/ {\vphantom {{m(a - c - be)} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}$ when $0 < C < C_{3} ;$ $q^{ * } = {C \mathord{\left/ {\vphantom {C e}} \right. \kern-0pt} e}$ when $C \ge C_{3} ,$ where $C_{3} = {{me(a - c - be)} \mathord{\left/ {\vphantom {{me(a - c - be)} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}.$ In a decentralized supply chain, for the given wholesale price, $\Pi_{R}^{\psi } = (p - w)q - {{(1 - \psi )ms^{ * 2} } \mathord{\left/ {\vphantom {{(1 - \psi )ms^{ * 2} } 2}} \right. \kern-0pt} 2} = (a - q + \theta s^{ * } - w)q - (1 - \psi ){{ms^{ * 2} } \mathord{\left/ {\vphantom {{ms^{ * 2} } 2}} \right. \kern-0pt} 2},$ it is a concave function of $q,$ Then, we can find that $q^{\psi } = {{(a + \theta s^{ * } - w)} \mathord{\left/ {\vphantom {{(a + \theta s^{ * } - w)} 2}} \right. \kern-0pt} 2}.$ In order to let $q^{\psi } = q^{ * } ,$ we derive that $w^{c\psi * } = c + be$ when $0 < C < C_{3} ;$ $w^{c\psi * } = {{[ame - (2m - \theta^{2} )C]} \mathord{\left/ {\vphantom {{[ame - (2m - \theta^{2} )C]} {(me)}}} \right. \kern-0pt} {(me)}}$ when $C \ge C_{3} .$ Thus, the optimal profits of the two firms are $\Pi_{R}^{c\psi * } = {{m(a - c - be)^{2} [2m - (1 - \psi )\theta^{2} ]} \mathord{\left/ {\vphantom {{m(a - c - be)^{2} [2m - (1 - \psi )\theta^{2} ]} {[2(2m - \theta^{2} )^{2} ]}}} \right. \kern-0pt} {[2(2m - \theta^{2} )^{2} ]}}$ and $\Pi_{M}^{c\psi * } = bC - {{\psi m(a - c - be)^{2} \theta^{2} } \mathord{\left/ {\vphantom {{\psi m(a - c - be)^{2} \theta^{2} } {[2(2m - \theta^{2} )^{2} ]}}} \right. \kern-0pt} {[2(2m - \theta^{2} )^{2} ]}}$ when $0 < C < C_{3} ;$ $\Pi_{R}^{c\psi * } = {{[2m - (1 - \psi )\theta^{2} ]C^{2} } \mathord{\left/ {\vphantom {{[2m - (1 - \psi )\theta^{2} ]C^{2} } {(2me^{2} }}} \right. \kern-0pt} {(2me^{2} }})$ and $\Pi_{M}^{c\psi * } = {{\{ 2(a - c)meC - [4m - (2 - \psi )\theta^{2} ]C^{2} \} } \mathord{\left/ {\vphantom {{\{ 2(a - c)meC - [4m - (2 - \psi )\theta^{2} ]C^{2} \} } {(2me^{2} )}}} \right. \kern-0pt} {(2me^{2} )}}$ when $C \ge C_{3} .$ When $C \ge C_{3} ,$ whether $w^{c\psi * } = {{[ame - (2m - \theta^{2} )C]} \mathord{\left/ {\vphantom {{[ame - (2m - \theta^{2} )C]} {(me)}}} \right. \kern-0pt} {(me)}}$ is higher than production cost $c$ is not deterministic. Comparing these two parameters, we can derive that, if $C_{3} \le C \le C_{4} ,$ $w^{c\psi * } \ge c;$ if $C > C_{4} ,$ $w^{c\psi * } < c,$ where $C_{4} = {{me(a - c)} \mathord{\left/ {\vphantom {{me(a - c)} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}.$ Thus, when $C > C_{4} ,$ the supply chain is non-coordinated.

Summarizing the above analysis, we can get Theorem 4.

Proof of Theorem 5

Following a similar process in proving Theorem 3, we can get Theorem 5.

Proof of Theorem 6

Based on Theorem 1, we know that $\Pi_{R}^{\psi * } = {{m(a - c - be)^{2} (1 - \psi )^{3} [2(1 - \psi )m - \theta^{2} ]} \mathord{\left/ {\vphantom {{m(a - c - be)^{2} (1 - \psi )^{3} [2(1 - \psi )m - \theta^{2} ]} {\{ 2\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \}^{2} }}} \right. \kern-0pt} {\{ 2\{ 2(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + \psi \theta^{2} \}^{2} }}\}$ and $\Pi_{M}^{\psi * } = {{m(a - c - be)^{2} (1 - \psi )^{2} } \mathord{\left/ {\vphantom {{m(a - c - be)^{2} (1 - \psi )^{2} } {\{ 4(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + 2\psi \theta^{2} \} }}} \right. \kern-0pt} {\{ 4(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + 2\psi \theta^{2} \} }} + bC$ when $0 < C < C_{1} ;$ $\Pi_{R}^{\psi * } = {{[2(1 - \psi )m - \theta^{2} ]C^{2} } \mathord{\left/ {\vphantom {{[2(1 - \psi )m - \theta^{2} ]C^{2} } {[2(1 - \psi )me^{2} ]}}} \right. \kern-0pt} {[2(1 - \psi )me^{2} ]}}$ and $\Pi_{M}^{\psi * } = {{m(a - c - be)^{2} (1 - \psi )^{2} } \mathord{\left/ {\vphantom {{m(a - c - be)^{2} (1 - \psi )^{2} } {\{ 4(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + 2\psi \theta^{2} \} }}} \right. \kern-0pt} {\{ 4(1 - \psi )[2(1 - \psi )m - \theta^{2} ] + 2\psi \theta^{2} \} }} + bC$ when $C \ge C_{1}$ with the cost-sharing contract. Based on Theorem 3, we know that $\Pi_{R}^{c\psi * } = {{m(a - c - be)[2m - (1 - \psi )\theta^{2} ]} \mathord{\left/ {\vphantom {{m(a - c - be)[2m - (1 - \psi )\theta^{2} ]} {[2(2m - \theta^{2} )^{2} ]}}} \right. \kern-0pt} {[2(2m - \theta^{2} )^{2} ]}}$ and $\Pi_{M}^{c\psi * } = bC - {{\psi m(a - c - be)^{2} \theta^{2} } \mathord{\left/ {\vphantom {{\psi m(a - c - be)^{2} \theta^{2} } {[2(2m - \theta^{2} )^{2} ]}}} \right. \kern-0pt} {[2(2m - \theta^{2} )^{2} ]}}$ when $0 < C < C_{3} ;$ $\Pi_{R}^{c\psi * } = {{[2m - (1 - \psi )\theta^{2} ]C^{2} } \mathord{\left/ {\vphantom {{[2m - (1 - \psi )\theta^{2} ]C^{2} } {(2me^{2} }}} \right. \kern-0pt} {(2me^{2} }})$ and $\Pi_{M}^{c\psi * } = {{\{ 2(a - c)meC - [4m - (2 - \psi )\theta^{2} ]C^{2} \} } \mathord{\left/ {\vphantom {{\{ 2(a - c)meC - [4m - (2 - \psi )\theta^{2} ]C^{2} \} } {(2me^{2} )}}} \right. \kern-0pt} {(2me^{2} )}}$ when $C_{3} \le C \le C_{4} .$ To achieve a win–win result for the profits of the manufacturer and retailer, it needs $\Pi_{R}^{c\psi * } - T_{1} > \Pi_{R}^{\psi * }$ and $\Pi_{M}^{c\psi * } + T_{1} > \Pi_{M}^{\psi * } .$ Hence, we can derive the upper bound and lower bound of $T_{1}$ which can be seen in Theorem 6. Similarly, we can derive the boundary conditions of $T_{2}$ with the quantity-discount contract.

Thus, there exists an interval $(\underline {T}_{i} ,\overline{T}_{i} )$ of $T_{i}$ to achieve Pareto improvement for the profits of the manufacturer and retailer with the cost-sharing and quantity-discount contracts.

Proof of Theorem 7

The demand functions and the profit function of the centralized supply chain are $\hat{q}_{1} = a - \hat{p}_{1} + r(\hat{p}_{2} - \hat{p}_{1} ) + \theta \hat{s},$ $\hat{q}_{2} = a - \hat{p}_{2} + r(\hat{p}_{1} - \hat{p}_{2} ) + \theta \hat{s},$ $\hat{\pi } = \hat{p}_{1} \hat{q}_{2} + \hat{p}_{2} \hat{q}_{2} - c(\hat{q}_{1} + \hat{q}_{2} ) - b[e(\hat{q}_{1} + \hat{q}_{2} ) - C] - m\hat{s}^{2} .$ Then we let the first partial derivatives of $\hat{\pi }$ with respect to $\hat{q}_{1} ,$ $\hat{q}_{2}$ and $\hat{s}$ be equal to zero. That is, ${{\partial \hat{\pi }} \mathord{\left/ {\vphantom {{\partial \hat{\pi }} {\partial \hat{q}_{1} }}} \right. \kern-0pt} {\partial \hat{q}_{1} }} = 0,$ ${{\partial \hat{\pi }} \mathord{\left/ {\vphantom {{\partial \hat{\pi }} {\partial \hat{q}_{2} }}} \right. \kern-0pt} {\partial \hat{q}_{2} }} = 0$ and ${{\partial \hat{\pi }} \mathord{\left/ {\vphantom {{\partial \hat{\pi }} {\partial \hat{s}}}} \right. \kern-0pt} {\partial \hat{s}}} = 0.$ And the Hessian matrix of $\hat{\pi }$ is $H_{2} = \left( {\begin{array}{*{20}c} {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{q}_{1}^{2} }}} & {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{q}_{1} \partial \hat{q}_{2} }}} & {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{q}_{1} \partial \hat{s}}}} \\ {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{q}_{2} \partial \hat{q}_{1} }}} & {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{q}_{2}^{2} }}} & {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{q}_{2} \partial \hat{s}}}} \\ {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{s}\partial \hat{q}_{1} }}} & {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{s}\partial \hat{q}_{2} }}} & {\frac{{\partial^{2} \hat{\pi }}}{{\partial \hat{s}^{2} }}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { - \frac{2(1 + r)}{{1 + 2r}}} & { - \frac{2r}{{1 + 2r}}} & \theta \\ { - \frac{2r}{{1 + 2r}}} & { - \frac{2(1 + r)}{{1 + 2r}}} & \theta \\ \begin{gathered} \hfill \\ \theta \hfill \\ \end{gathered} & \begin{gathered} \hfill \\ \theta \hfill \\ \end{gathered} & \begin{gathered} \hfill \\ - 2m \hfill \\ \end{gathered} \\ \end{array} } \right),$ which is verified to be negative definite. The optimal decisions from the first order conditions are $\hat{q}_{{_{1} }}^{*} = \hat{q}_{{_{2} }}^{*} = {{m(a - c - be)} \mathord{\left/ {\vphantom {{m(a - c - be)} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}},$ $\hat{s}^{*} = {{\theta (a - c - be)} \mathord{\left/ {\vphantom {{\theta (a - c - be)} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}.$ Then, submitting them into the demand functions and profit function of the centralized supply chain, we obtain $\hat{p}_{{_{1} }}^{*} = \hat{p}_{{_{2} }}^{*} = c + be + {{m(a - c - be)} \mathord{\left/ {\vphantom {{m(a - c - be)} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}},$$\hat{\pi }^{*} = {{[2mbC + m(a - c - be)^{2} - bC\theta^{2} ]} \mathord{\left/ {\vphantom {{[2mbC + m(a - c - be)^{2} - bC\theta^{2} ]} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}.$

Similar to the analysis of Theorem 1, the above results are true if $e(\hat{q}_{{_{1} }}^{*} + \hat{q}_{{_{2} }}^{*} ) > C.$ If $e(\hat{q}_{{_{1} }}^{*} + \hat{q}_{{_{2} }}^{*} ) \le C,$ according to the assumption $e(\hat{q}_{{_{1} }}^{*} + \hat{q}_{{_{2} }}^{*} ) \ge C,$ the manufacturer uses all the allocated credits (i.e., the cap) to make production, i.e., $\hat{q}_{{_{1} }}^{*} + \hat{q}_{{_{2} }}^{*} = {C \mathord{\left/ {\vphantom {C e}} \right. \kern-0pt} e}.$ We simplify the profit function $\hat{\pi } = \hat{p}_{1} \hat{q}_{2} + \hat{p}_{2} \hat{q}_{2} - c(\hat{q}_{1} + \hat{q}_{2} ) - b[e(\hat{q}_{1} + \hat{q}_{2} ) - C] - m\hat{s}^{2}$ and get $\hat{\pi } = - m\hat{s}^{2} - {{[e(\hat{q}_{1}^{2} + \hat{q}_{2}^{2} ) + {{C^{2} r} \mathord{\left/ {\vphantom {{C^{2} r} e}} \right. \kern-0pt} e} + C(1 + 2r)(c - a - \hat{s}\theta )]} \mathord{\left/ {\vphantom {{[e(\hat{q}_{1}^{2} + \hat{q}_{2}^{2} ) + {{C^{2} r} \mathord{\left/ {\vphantom {{C^{2} r} e}} \right. \kern-0pt} e} + C(1 + 2r)(c - a - \hat{s}\theta )]} {[e(1 + 2r)]}}} \right. \kern-0pt} {[e(1 + 2r)]}}.$ It is acknowledged that $\hat{q}_{1}^{2} + \hat{q}_{2}^{2} \ge {{(\hat{q}_{1} + \hat{q}_{2} )^{2} } \mathord{\left/ {\vphantom {{(\hat{q}_{1} + \hat{q}_{2} )^{2} } 2}} \right. \kern-0pt} 2}$ and $\hat{q}_{1}^{2} + \hat{q}_{2}^{2}$ obtains the minimum value ${{(\hat{q}_{1} + \hat{q}_{2} )^{2} } \mathord{\left/ {\vphantom {{(\hat{q}_{1} + \hat{q}_{2} )^{2} } 2}} \right. \kern-0pt} 2}$ when $\hat{q}_{1} = \hat{q}_{2} .$ Then we can obtain $\hat{q}_{1} = \hat{q}_{2} = {C \mathord{\left/ {\vphantom {C {(2e)}}} \right. \kern-0pt} {(2e)}}.$ That means $\hat{\pi }$ obtains the maximum value when $\hat{q}_{{_{1} }}^{*} = \hat{q}_{{_{2} }}^{*} = {C \mathord{\left/ {\vphantom {C {(2e)}}} \right. \kern-0pt} {(2e)}}.$ Submitting the optimal production quantities into the demand functions and profit function of the centralized supply chain, we solve the equation ${{\partial \hat{\pi }} \mathord{\left/ {\vphantom {{\partial \hat{\pi }} {\partial \hat{s}}}} \right. \kern-0pt} {\partial \hat{s}}} = 0$ and obtain $\hat{s}^{*} = {{C\theta } \mathord{\left/ {\vphantom {{C\theta } {(2me)}}} \right. \kern-0pt} {(2me)}}.$ Submitting $\hat{q}_{{_{1} }}^{*} ,$ $\hat{q}_{{_{2} }}^{*}$ and $\hat{s}^{*}$ into related demand and profit functions, we obtain $\hat{p}_{{_{1} }}^{*} = \hat{p}_{{_{2} }}^{*} = [{{2ame - (m - \theta^{2} )C]} \mathord{\left/ {\vphantom {{2ame - (m - \theta^{2} )C]} {(2me)}}} \right. \kern-0pt} {(2me)}},$ $\hat{\pi }^{*} = {{[4(a - c)meC - (2m - \theta^{2} )C^{2} ]} \mathord{\left/ {\vphantom {{[4(a - c)meC - (2m - \theta^{2} )C^{2} ]} {(4me^{2} )}}} \right. \kern-0pt} {(4me^{2} )}}.$

Proof of Theorem 8

We use a contract $(\hat{w}^{c\psi *} ,\hat{s}^{c\psi *} )$ to explore the supply chain coordination, where $\hat{s}^{*} = {{\theta (a - c - be)} \mathord{\left/ {\vphantom {{\theta (a - c - be)} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}$ when $0 < C < C_{6}$ and $\hat{s}^{*} = {{C\theta } \mathord{\left/ {\vphantom {{C\theta } {(2me)}}} \right. \kern-0pt} {(2me)}}$ when $C \ge C_{6} .$ Similar to the analysis of Theorem 1, we obtain the response function $\hat{q}_{1}^{\psi } = - {{[\hat{q}_{2}^{\psi } r + (1 + 2r)(w - a - \hat{s}^{*} \theta )]} \mathord{\left/ {\vphantom {{[\hat{q}_{2}^{\psi } r + (1 + 2r)(w - a - \hat{s}^{*} \theta )]} {[2(1 + r)]}}} \right. \kern-0pt} {[2(1 + r)]}}$ in the decentralized supply chain. In order to let $\hat{q}_{1}^{\psi } = \hat{q}_{1}^{*}$ and $\hat{q}_{2}^{\psi } = \hat{q}_{2}^{*} ,$ we derive that $\hat{w}^{c\psi *} = c + be + {{(a - c - be)mr} \mathord{\left/ {\vphantom {{(a - c - be)mr} {[(1 + 2r)(2m - \theta^{2} )]}}} \right. \kern-0pt} {[(1 + 2r)(2m - \theta^{2} )]}}$ when $0 < C < C_{6} ;$ $\hat{w}^{c\psi *} = [(1 + 2r){{(2ame + C\theta^{2} ) - mC(2 + 3r)]} \mathord{\left/ {\vphantom {{(2ame + C\theta^{2} ) - mC(2 + 3r)]} {[2(1 + 2r)me]}}} \right. \kern-0pt} {[2(1 + 2r)me]}}$ when $C \ge C_{6} .$ Thus, the profits of the two retailers and manufacturer are $\hat{\pi }_{{R_{1} }}^{c\psi *} = \hat{\pi }_{{R_{2} }}^{c\psi *} = m(a - c - be)^{2} [2m{{(1 + r) - (1 + 2r)(1 - \psi )\theta^{2} ]} \mathord{\left/ {\vphantom {{(1 + r) - (1 + 2r)(1 - \psi )\theta^{2} ]} {[2(1 + 2r)(2m - \theta^{2} )^{2} ]}}} \right. \kern-0pt} {[2(1 + 2r)(2m - \theta^{2} )^{2} ]}}$ and $\hat{\pi }_{M}^{c\psi *} = m(a - c - be)^{2} [{{2mr - (1 + 2r)\theta^{2} \psi ]} \mathord{\left/ {\vphantom {{2mr - (1 + 2r)\theta^{2} \psi ]} {[(1 + 2r)(2m - \theta^{2} )^{2} ] + bC}}} \right. \kern-0pt} {[(1 + 2r)(2m - \theta^{2} )^{2} ] + bC}}$ when $0 < C < C_{6} ;$ $\hat{\pi }_{{R_{1} }}^{c\varphi *} = \hat{\pi }_{{R_{2} }}^{c\varphi *} = C^{2} [{{2m(1 + r) - (1 + 2r)\theta^{2} (1 - \psi )]} \mathord{\left/ {\vphantom {{2m(1 + r) - (1 + 2r)\theta^{2} (1 - \psi )]} {[8e^{2} (1 + 2r)m]}}} \right. \kern-0pt} {[8e^{2} (1 + 2r)m]}}$ and $\hat{\pi }_{M}^{c\psi *} = C[{{4em(1 + 2r)(a - c) - 2Cm(2 + 3r) + C(1 + 2r)\theta^{2} (2 - \psi )]} \mathord{\left/ {\vphantom {{4em(1 + 2r)(a - c) - 2Cm(2 + 3r) + C(1 + 2r)\theta^{2} (2 - \psi )]} {[4e^{2} (1 + 2r)m]}}} \right. \kern-0pt} {[4e^{2} (1 + 2r)m]}}$ when $C \ge C_{6} .$ When $C \ge C_{6} ,$ we derive that, if $C_{6} \le C \le C_{7} ,$ $\hat{w}^{c\psi *} \ge c;$ if $C > C_{7} ,$ $\hat{w}^{c\psi *} < c,$ where $C_{7} = {{2(a - c)me} \mathord{\left/ {\vphantom {{2(a - c)me} {[{{2m(2 + 3r)} \mathord{\left/ {\vphantom {{2m(2 + 3r)} {(2 + 4r) - \theta^{2} }}} \right. \kern-0pt} {(2 + 4r) - \theta^{2} }}]}}} \right. \kern-0pt} {[{{2m(2 + 3r)} \mathord{\left/ {\vphantom {{2m(2 + 3r)} {(2 + 4r) - \theta^{2} }}} \right. \kern-0pt} {(2 + 4r) - \theta^{2} }}]}}.$ Thus, when $C > C_{7} ,$ the supply chain is non-coordinated.

Summarizing the above analysis, we can get Theorem 8.

Proof of Theorem 9

Following a similar process in proving Theorem 8, we can get Theorem 9.

Proof of Theorem 10

Following a similar process in proving Theorem 1, we can get Theorem 10.

Proof of Theorem 11

We use a contract $(\overline{w}_{1}^{c\psi *} ,\overline{w}_{2}^{c\psi *} ,\overline{s}^{c\psi *} )$ to explore the supply chain coordination, where $\overline{s}^{*} = {{\theta (a - c - be)} \mathord{\left/ {\vphantom {{\theta (a - c - be)} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}$ when $0 < C_{{M_{1} }} + C_{{M_{2} }} < C_{9}$ $\overline{s}^{*} = {{(C_{{M_{1} }} + C_{{M_{2} }} )\theta } \mathord{\left/ {\vphantom {{(C_{{M_{1} }} + C_{{M_{2} }} )\theta } {(2me)}}} \right. \kern-0pt} {(2me)}}$ when $C_{{M_{1} }} + C_{{M_{2} }} \ge C_{9} .$ We obtain the response functions $\overline{q}_{1}^{\psi } = - {{[2\overline{q}_{2}^{\psi } r + (1 + 2r)(\overline{w}_{1} - a - \overline{s}^{*} \theta )]} \mathord{\left/ {\vphantom {{[2\overline{q}_{2}^{\psi } r + (1 + 2r)(\overline{w}_{1} - a - \overline{s}^{*} \theta )]} {[2(1 + r)]}}} \right. \kern-0pt} {[2(1 + r)]}}$ and $\overline{q}_{2}^{\psi } = - {{[2\overline{q}_{1}^{\psi } r + (1 + 2r)(\overline{w}_{2} - a - \overline{s}^{*} \theta )]} \mathord{\left/ {\vphantom {{[2\overline{q}_{1}^{\psi } r + (1 + 2r)(\overline{w}_{2} - a - \overline{s}^{*} \theta )]} {[2(1 + r)]}}} \right. \kern-0pt} {[2(1 + r)]}}$ in the decentralized supply chain. In order to let $\overline{q}_{1}^{\psi } = \overline{q}_{1}^{*}$ and $\overline{q}_{2}^{\psi } = \overline{q}_{2}^{*} ,$ we derive that $\overline{w}_{1}^{c\psi *} = \overline{w}_{2}^{c\psi *} = c + be$ when $0 < C_{{M_{1} }} + C_{{M_{2} }} < C_{9} ;$ $\overline{w}_{1}^{c\psi *} = \overline{w}_{2}^{c\psi *} = [{{2ame + (C_{{M_{1} }} + C_{{M_{2} }} )(\theta^{2} - 2m)]} \mathord{\left/ {\vphantom {{2ame + (C_{{M_{1} }} + C_{{M_{2} }} )(\theta^{2} - 2m)]} {(2me)}}} \right. \kern-0pt} {(2me)}}$ when $C_{{M_{1} }} + C_{{M_{2} }} \ge C_{9} .$ Thus, the profits of the retailer and two manufacturers are $\overline{\pi }_{R}^{c\psi *} = m(a - c - be)^{2} [2m{{ - (1 - \psi )\theta^{2} ]} \mathord{\left/ {\vphantom {{ - (1 - \psi )\theta^{2} ]} {(2m - \theta^{2} )^{2} }}} \right. \kern-0pt} {(2m - \theta^{2} )^{2} }},$ $\overline{\pi }_{{M_{1} }}^{c\psi *} = bC_{{M_{1} }} - m(a - c - be)^{2} {{\theta^{2} \psi } \mathord{\left/ {\vphantom {{\theta^{2} \psi } {[2(2m - \theta^{2} )^{2} ]}}} \right. \kern-0pt} {[2(2m - \theta^{2} )^{2} ]}}$ and $\overline{\pi }_{{M_{2} }}^{c\psi *} = bC_{{M_{2} }} - m(a - c - be)^{2} {{\theta^{2} \psi } \mathord{\left/ {\vphantom {{\theta^{2} \psi } {[2(2m - \theta^{2} )^{2} ]}}} \right. \kern-0pt} {[2(2m - \theta^{2} )^{2} ]}}$ when $0 < C_{{M_{1} }} + C_{{M_{2} }} < C_{9} ;$ $\overline{\pi }_{R}^{c\psi *} = (C_{{M_{1} }} + C_{{M_{2} }} )^{2} [{{2m - \theta^{2} (1 - \psi )]} \mathord{\left/ {\vphantom {{2m - \theta^{2} (1 - \psi )]} {(4e^{2} m)}}} \right. \kern-0pt} {(4e^{2} m)}},$ $\bar{\pi }_{{M_{1} }}^{{c\psi *}} = \left\{ {4b\left( {C_{{M_{1} }} - C_{{M_{2} }} } \right)e^{2} m - 4\left( {C_{{M_{1} }} + C_{{M_{2} }} } \right)m\left[ {C_{{M_{1} }} + C_{{M_{2} }} + e(c - a)} \right] + (2 - \psi )(C_{{M_{1} }} + C_{{M_{2} }} )^{2} \theta ^{2} } \right\}/\left( {8e^{2} m} \right)$ and $\bar{\pi }_{{M_{2} }}^{{c\psi *}} = {{\{ 4b(C_{{M_{2} }} - C_{{M_{1} }} )e^{2} m - 4(C_{{M_{1} }} + C_{{M_{2} }} )m[C_{{M_{1} }} + C_{{M_{2} }} + e(c - a)] + (2 - \psi )(C_{{M_{1} }} + C_{{M_{2} }} )^{2} \theta ^{2} \} } \mathord{\left/ {\vphantom {{\{ 4b(C_{{M_{2} }} - C_{{M_{1} }} )e^{2} m - 4(C_{{M_{1} }} + C_{{M_{2} }} )m[C_{{M_{1} }} + C_{{M_{2} }} + e(c - a)] + (2 - \psi )(C_{{M_{1} }} + C_{{M_{2} }} )^{2} \theta ^{2} \} } {(8e^{2} m)}}} \right. \kern-\nulldelimiterspace} {(8e^{2} m)}}$ when $C_{{M_{1} }} + C_{{M_{2} }} \ge C_{9} .$ When $C_{{M_{1} }} + C_{{M_{2} }} \ge C_{9} ,$ we derive that, if $C_{9} \le C_{{M_{1} }} + C_{{M_{2} }} \le C_{10} ,$ $\overline{w}_{1}^{c\psi *} = \overline{w}_{2}^{c\psi *} \ge c;$ if $C_{{M_{1} }} + C_{{M_{2} }} > C_{10} ,$ $\overline{w}_{1}^{c\psi *} = \overline{w}_{2}^{c\psi *} < c,$ where $C_{10} = {{2(a - c)me} \mathord{\left/ {\vphantom {{2(a - c)me} {(2m - \theta^{2} )}}} \right. \kern-0pt} {(2m - \theta^{2} )}}.$ Summarizing the above analysis, we can get Theorem 11.

Proof of Theorem 12

Following a similar process in proving Theorem 11, we can get Theorem 12.