Contract design considering data driven marketing: with and without the cap and trade regulation

Abstract

Due to the natural advantage of data acquisition, data driven marketing (DDM) has been widely adopted online. The major concern of managers, especially when facing severe environmental issues, is to adjust proper measures regarding DDM promotions (including consumers’ DDM preference and offline free riding), so as to ensure efficient implementation of the cap and trade regulation (CTR). We developed two models: No-CTR and CTR under centralized and decentralized scenarios, and then proposed corresponding contracts to achieve Pareto improvement. The validity of both coordination schemes has been demonstrated. Contrary to conventional wisdom, the results indicate that encouraging DDM promotions and consumers’ environmental awareness are not always beneficial to the manufacturer or retailers. Certain conditions should be met based on promotions and wholesale price. Secondly, comparing No-CTR with CTR, promotions should be set for different carbon quotas to ensure all members prefer CTR. Especially, when carbon quota is moderate, below a certain level can also be optimal. Furthermore, numerical analysis suggests that with the increase of carbon trading price, the online quantity will evolve from downward to upward, as will both retailers’ profit. Entrepreneurs should be mindful of balancing the trade-offs contingent on DDM promotion effect because it plays an essential role in CTR implementation efficiency. In our model, stochastic demand is also considered.

Appendices

Appendix A

1.1 Proof of Theorem 1

Function (13) is concave and has the maximum value with respect to $C_o,C_r,B,e$ if the Hessian matrix is negative definite.

$$\begin{aligned} H(\pi _{sc}^N)= \begin{vmatrix} -p_rf(C_r)&0&0&0\\0&-p_of(C_o)&0&0\\0&0&-k_m&0\\0&0&0&-k_o\\ \end{vmatrix} \end{aligned}$$

$D_1=-p_rf(C_r)<0, D_2=p_rp_of(C_r)f(C_o)>0, D_3=-k_mD_2<0,D_4=-k_oD_3>0,$ which means this matrix is strictly negative, and $\pi _{sc}^N$ is a strictly concave function with unique optimal reaction.

1.2 Proof of Theorem 3

The Hessian matrix about Function (24):

$$\begin{aligned} H(\pi _{sc}^{B*})= \begin{vmatrix} \frac{-p_r}{A}&0&0&\lambda p_c\\0&\frac{-p_o}{A}&0&\lambda p_c\\0&0&-k_o&\lambda p_c(\eta +\gamma \eta )\\\lambda p_c&\lambda p_c&\lambda p_c(\eta +\gamma \eta )&4\lambda p_c\beta -k_m\\ \end{vmatrix} \end{aligned}$$

$D_1=-p_r/A<0; D_2=p_rp_o/A>0; D_3=-k_op_rp_o/A<0; D_4=(-Ak_op_c^2\lambda ^2(p_o+p_r)+k_mk_op_op_r- \eta ^2p_c^2p_op_r\lambda ^2(1+r)^2 - 4k_o\beta p_cp_op_r\lambda )/A^2>0$ When $k_mk_op_op_r>\eta ^2p_c^2p_op_r\lambda ^2(1+r)^2+4k_o\beta p_cp_op_r\lambda + Ak_op_c^2\lambda ^2(p_o+p_r)$, the condition of negative matrix is satisfied.

Specifically:

$$\begin{aligned} X_1&=A\lambda ^2k_op_o+A\lambda ^2k_op_r+\left( 1+\gamma \right) ^2\eta ^2\lambda ^2p_op_r\\ X_2&=A\lambda ^2k_o+2\beta \lambda ^2k_o+\lambda ^2A_ok_o+\lambda ^2A_rk_o +\left( 1+\gamma \right) \eta ^2\lambda ^2\left( p_o-p_r\right) \\ X_3&=-k_mk_op_op_r+pc^2X1+4pc\beta \lambda k_op_op_r\\ X_4&=X_2p_r-A\lambda ^2k_op_o-\left( 1+\gamma \right) ^2\eta ^2\lambda ^2\left( p_o-p_r\right) p_r\\ X_5&=\lambda k_mk_op_r-\beta \lambda k_op_r^2+3\beta \lambda k_op_op_r\\ X_6&=A\lambda ^2p_o^2-A\gamma \lambda ^2p_op_r+(-\gamma -1)\lambda ^2A_op_op_r +(-\gamma -1)\lambda ^2A_rp_op_r\\ {}&-A\lambda ^2p_op_r +A\gamma \lambda ^2p_r^2-2\beta \gamma \lambda ^2p_op_r-2\beta \lambda ^2p_op_r\\ X_7&=\gamma \lambda k_mp_op_r+\lambda k_mp_op_r-\beta (\gamma -3)\lambda p_o^2p_r +3\beta \gamma \lambda p_op_r^2-\beta \lambda p_op_r^2\\&-\eta \left( p_o+\gamma p_r\right) \left( k_mp_op_r\right) \\ X_8&=2\beta \lambda k_op_op_r+\lambda p_op_r\left( -\left( 2A+A_o+A_r\right) k_o -\left( 1+\gamma \right) \eta ^2\left( p_o+\gamma p_r\right) \right) \end{aligned}$$

1.3 Proof of Corollary 1

$$\begin{aligned} e^{CD*}-e^{ND*}=\frac{2\beta w-2\beta \lambda p_c}{k_m-4\beta \lambda p_c}-\frac{2w\beta }{k_m}+\frac{\overline{Y}p_c\lambda }{k_m-4\beta \lambda p_c} \end{aligned}$$

After simplification, we found properties of positive and negative depends on $8w\beta ^2-2\beta k_m+k_m\overline{Y}$. When $8w\beta ^2+\left( \overline{Y}-2\beta \right) k_m>0$, $e^{CD*}-e^{ND*}>0$. Since $\partial d_o/\partial e=\beta>0,\partial d_o/\partial B=\eta >0$, $d_o$ is increase in e, B. Since $B^{CD*}=B^{ND*}$, when $e^{CD*}-e^{ND*}>0$, $d_o^{CD*}>d_o^{ND*}$. Since $C_o^{ND*}=C_o^{CD*},q_o=C_o+d_o$, we can concluded $q_o^{CD*}>q_o^{ND*}$, so is $q_r$.

1.4 Proof of Corollary 2

$$\begin{aligned}\begin{aligned} \frac{\partial e^{CD*}}{\partial \eta }&=\frac{2\eta p_c\lambda (p_o-w)(\gamma +1)}{k_o(k_m-4\beta p_c\lambda )}>0, \frac{\partial e^{ND*}}{\partial \eta }=0, \frac{\partial B^{CD*}}{\partial \eta }=\frac{\partial B^{ND*}}{\partial \eta }=\frac{p_o-w}{k_0}>0\\ \frac{\partial q_o^{CD*}}{\partial \eta }&= \frac{2\eta (p_o-w)(k_m-3\beta p_c\lambda +\beta p_c \lambda \gamma )}{k_o(k_m-4\beta p_c\lambda )}>\frac{\partial q_o^{ND*}}{\partial \eta }= \frac{2\eta (p_o-w)}{k_o}\\ \frac{\partial q_r^{CD*}}{\partial \eta }&=\frac{2\eta (p_o-w)(k_m\gamma -3\beta p_c\lambda \gamma +\beta p_c \lambda )}{k_o(k_m-4\beta p_c\lambda )} = \frac{2\eta \gamma (p_o-w)(k_m-3\beta p_c\lambda +\frac{\beta p_c \lambda }{\gamma })}{k_o(k_m-4\beta p_c\lambda )}\\ {}&>\frac{\partial q_r^{ND*}}{\partial \eta }=\frac{2\eta \gamma (p_o-w)}{k_o}\\ \frac{\partial q_o^{CD*}}{\partial \eta }&-\frac{\partial q_r^{CD*}}{\partial \eta }=\frac{2\eta (p_o-w)(1-\gamma )}{k_o}>0; \frac{\partial q_o^{ND*}}{\partial \eta }-\frac{\partial q_r^{ND*}}{\partial \eta } = \frac{2\eta (p_o-w)(1-\gamma )}{k_o}>0 \end{aligned} \end{aligned}$$

1.5 Proof of Corollary 6

$\pi _m^{CD*}-\pi _m^{ND*}=p_cX_{10}/\left( 2\left( k_m-4p_c\beta \lambda \right) k_m\right)$, its ± (positive or negative) depends on $X_{10}$, which is a quadratic convex function about $\overline{Y}$. Through the classical root formula, we can get restrictions for w and $\overline{Y}$ to ensure: $\pi _m^{CD*}-\pi _m^{ND*}>0$.

If $w<p_c\lambda$: $\overline{E_2}<E_g,\overline{E_2}<\overline{E_4}, \left\{ \begin{array}{ll} \overline{E_3}<\overline{E_2},if w>w_1\\ \overline{E_3}>\overline{E_2},if w<w_1 \end{array} \right.$

If $p_c\lambda<w<\left( -2p_c\beta \lambda +k_m\right) /2\beta :$ $\overline{E_1}<E_g,\overline{E_2}<\overline{E_3}<\overline{E_4}$

Specifically:

$$\begin{aligned} \begin{aligned} Y_3&=\frac{-2w\beta -2p_c\beta \lambda +k_m}{p_c\lambda } -\frac{\sqrt{\left( -4p_c\beta \lambda +k_m\right) \left( 4w^2\beta ^2+k_m\left( -2E_gp_c-4w\beta +k_m\right) \right) }}{p_c\lambda \sqrt{k_m}}\\ Y_4&=\frac{-2w\beta -2p_c\beta \lambda +k_m}{p_c\lambda } +\frac{\sqrt{\left( -4p_c\beta \lambda +k_m\right) \left( 4w^2\beta ^2+k_m\left( -2E_gp_c-4w\beta +k_m\right) \right) }}{p_c\lambda \sqrt{k_m}}\\ \overline{E_3}&=\frac{8w^2\beta ^3\lambda -4w\beta ^2\lambda k_m+2p_c\beta ^2\lambda ^2k_m}{4p_c\beta \lambda k_m-k_m^2}\\ \overline{E_4}&=\frac{4w^2\beta ^2-4w\beta k_m+k_m^2}{2p_ck_m}\\ w_1&=\frac{p_c\lambda \left( 2p_c\beta \lambda -k_m\right) \left( 2p_c\beta \lambda k_m -k_m^2+2p_c\beta \lambda \sqrt{k_m\left( -4p_c\beta \lambda +k_m\right) }\right) }{16{p_c}^3\beta ^3\lambda ^3-4p_c\beta \lambda k_m^2+k_m^3}\\ X_{10}&=16w^2\beta ^3\lambda +\lambda (-8E_gp_c\beta +4w(\overline{Y}-2\beta )\beta +p_c(\overline{Y}+2\beta )^2\lambda )k_m + 2(E_g-\overline{Y}\lambda )k_m^2 \end{aligned} \end{aligned}$$

1.6 Proof of Corollary 8

$$\begin{aligned} \begin{aligned}&\pi _{m}^{NRS}-\pi _{m}^{ND}=\frac{1}{2{{k}_{m}}{{k}_{o}}{{p}_{o}}{{p}_{r}}}\left( {{\beta }^{2}}{{k}_{o}}{{p}_{o}}{{p}_{r}}{{B}_{1}}+{{B}_{4}} \right) \\&\pi _{r}^{NRS}-\pi _{r}^{ND}=\frac{1}{2{{k}_{m}}{{k}_{o}}{{p}_{r}}}\left( {{\beta }^{2}}{{k}_{o}}{{p}_{r}}{{B}_{2}}+{{B}_{5}} \right) \\&\pi _{o}^{NRS}-\pi _{o}^{ND}=\frac{1}{2{{k}_{m}}{{k}_{o}}{{p}_{r}}}\left( 2{{\beta }^{2}}{{k}_{o}}{{p}_{o}}{{B}_{3}}+{{B}_{6}} \right) \\ \end{aligned} \end{aligned}$$

We observe that $\pi _{i}^{NRS}-\pi _{i}^{ND}$ is convex in $\beta$ if $x_{1}^{N}$ and $x_{2}^{N}$ satisfying:

$$\begin{aligned}{} & {} \frac{4w\left( -2w+{{p}_{o}}+{{p}_{r}} \right) }{{{\left( {{p}_{o}}+{{p}_{r}} \right) }^{2}}}<x_{2}^{N}<1-\frac{4{{w}^{2}}}{{{\left( {{p}_{o}}+{{p}_{r}} \right) }^{2}}}\\{} & {} \quad \frac{4w\left( -w+{{p}_{r}} \right) -\left( {{p}_{o}}+{{p}_{r}} \right) \left( \left( -2+x_{2}^{N} \right) {{p}_{o}}+\left( x_{2}^{N}-2\gamma \right) {{p}_{r}} \right) }{2\left( {{p}_{o}}+{{p}_{r}} \right) \left( {{p}_{o}}+\gamma {{p}_{r}} \right) }\\{} & {} \quad<x_{1}^{N}<\frac{2{{w}^{2}}+p_{o}^{2}+\gamma p_{r}^{2}+{{p}_{o}}\left( -2w+\left( 1+\gamma \right) {{p}_{r}} \right) }{\left( {{p}_{o}}+{{p}_{r}} \right) \left( {{p}_{o}}+\gamma {{p}_{r}} \right) } \end{aligned}$$

Then, there exists an unique $\beta =\beta _i$ satisfying $\pi _{i}^{NRS}>\pi _{i}^{ND}$ in the interval $\beta _i<\beta <1$ if $B_j<0$, or $\pi _{i}^{NRS}>\pi _{i}^{ND}$ holds true if $B_j>0$, $(i,j)\in \left\{ (m,4),(r,5),(o,6)\right\}$. Therefore, there must be an efficient intersection of $\beta$ supporting above inequality and one of the examples is given below: $\max \left\{ {{\beta }_{m}},{{\beta }_{r}},{{\beta }_{o}} \right\}<\beta <1$.

Specifically,

$$\begin{aligned}&{{B}_{1}}=\left( -4{{w}^{2}}+\left( 1-x_{2}^{N} \right) {{\left( {{p}_{o}}+{{p}_{r}} \right) }^{2}} \right) \\&{{B}_{2}}=4w\left( w-{{p}_{r}} \right) +\left( {{p}_{o}}+{{p}_{r}} \right) \left( \left( -2+2x_{1}^{N}+x_{2}^{N} \right) {{p}_{o}}+\left( x_{2}^{N}+2\left( -1+x_{1}^{N} \right) \gamma \right) {{p}_{r}} \right) \\&{{B}_{3}}=2{{w}^{2}}+\left( 1-x_{1}^{N} \right) p_{o}^{2}+\gamma \left( 1-x_{1}^{N} \right) p_{r}^{2}+{{p}_{o}}\left( -2w+\left( 1-x_{1}^{N} \right) \left( 1+\gamma \right) {{p}_{r}} \right) \\&{{B}_{4}}={{k}_{m}}({{\eta }^{2}}{{p}_{o}}{{p}_{r}}(2{{w}^{2}}\left( 1+\gamma \right) +\left( x_{1}^{N}-2\left( -1+x_{1}^{N}+x_{2}^{N} \right) \gamma \right) p_{o}^{2}\\&\quad + {{\gamma }^{2}}(x_{1}^{N}-2x_{2}^{N}+2\gamma -2x_{1}^{N}\gamma )p_{r}^{2}-2{{p}_{o}}(w\left( 1+\gamma \right) \\&\quad +\gamma (-x_{1}^{N}+x_{2}^{N}+\left( -2+2x_{1}^{N}+x_{2}^{N} \right) \gamma ){{p}_{r}})) \\&\quad +{{k}_{o}}({{p}_{o}}(2A{{w}^{2}}-2w\left( 2A+{{A}_{o}}+{{A}_{r}} \right) {{p}_{r}}+(2\left( -1+x_{1}^{N} \right) \gamma \left( {{A}_{o}}-{{A}_{r}} \right) \\&\quad -\left( -1+x_{2}^{N} \right) \left( A+2{{A}_{r}} \right) )p_{r}^{2})\\&\quad +2A{{w}^{2}}{{p}_{r}}+(A-Ax_{2}^{N}+2x_{1}^{N}{{A}_{o}}-2\left( -1+x_{1}^{N}+x_{2}^{N} \right) {{A}_{r}})p_{o}^{2}{{p}_{r}})) \\&{{B}_{5}}={{k}_{m}}(2\gamma {{\eta }^{2}}{{p}_{r}}(-{{w}^{2}}+{{p}_{o}}\left( w+\left( -1+x_{1}^{N}+x_{2}^{N} \right) {{p}_{o}} \right) \\&\quad +(w+(-1+x_{2}^{N}+\left( -2+2x_{1}^{N}+x_{2}^{N} \right) \gamma ){{p}_{o}}){{p}_{r}}+\gamma \left( x_{2}^{N}+\left( -1+x_{1}^{N} \right) \gamma \right) p_{r}^{2}) \\&\quad +{{k}_{o}}(-A{{w}^{2}}+(2w\left( A+{{A}_{r}} \right) +\left( -1+x_{1}^{N}+x_{2}^{N} \right) \left( A+2{{A}_{r}} \right) {{p}_{o}}){{p}_{r}}\\&\quad +(-1+x_{2}^{N}+\left( -1+x_{1}^{N} \right) \gamma )\left( A+2{{A}_{r}} \right) p_{r}^{2})) \\&{{B}_{6}}={{k}_{m}}(-{{\eta }^{2}}{{p}_{o}}({{w}^{2}}+x_{1}^{N}p_{o}^{2}+\left( -1+x_{1}^{N} \right) {{\gamma }^{2}}p_{r}^{2}-2{{p}_{o}}(w-\left( -1+x_{1}^{N} \right) \gamma {{p}_{r}}))\\&\quad -{{k}_{o}}(A{{w}^{2}}+x_{1}^{N}\left( A+2{{A}_{o}} \right) p_{o}^{2}+{{p}_{o}}(-2w\left( A+{{A}_{o}} \right) +\left( -1+xa1 \right) \gamma \left( A+2{{A}_{o}} \right) {{p}_{r}}))) \\ \end{aligned}$$

1.7 Proof of Theorem 6

$$\begin{aligned}\begin{aligned} C_1&=p_op_r\left( \left( 2\left( A+\beta \right) +A_o+A_r\right) k_o +\left( 1+\gamma \right) \eta ^2\left( p_o+\gamma p_r\right) \right) \\ C_2&=p_op_rk_o\left( -k_m+\beta \left( p_o+p_r\right) \right) \\ C_3&=-\left( 1+\gamma \right) \eta ^2p_o\left( p_o-p_r\right) -k_o\left( \left( A+2\beta +A_o+A_r\right) p_o-Ap_r\right) \\ C_4&=k_op_o\left( k_m-\beta p_o+3\beta p_r\right) \\ C_5&=-\left( 1+\gamma \right) \left( A+2\beta +A_o+A_r\right) p_op_r+Ap_o^2+A\gamma p_r^2\\ C_6&=p_op_r\left( \left( 1+\gamma \right) k_m -\beta \left( \left( -3+\gamma \right) p_o+\left( 1-3\gamma \right) p_r\right) \right) \\ C_7&=\left( 1-\gamma ^2\right) \eta ^2p_o\left( p_o-p_r\right) p_r +k_o\left( 2\left( A+2\beta +A_o+A_r\right) p_op_r-Ap_o^2-Ap_r^2\right) \\ C_8&=k_op_op_r\left( k_m+\beta \left( p_o+p_r\right) \right) \\ C_9&=\left( 1+\gamma \right) ^2\eta ^2p_op_r+Ak_o\left( p_o+p_r\right) \\ C_{10}&=p_op_r\left( \left( 2A-2\beta +A_o+A_r\right) k_o +\left( 1+\gamma \right) \eta ^2\left( p_o+\gamma p_r\right) \right) \end{aligned} \end{aligned}$$

1.8 Proof of Theorem 8

Firstly:

$$\begin{aligned} \Delta \pi _m=\pi _m^{CFC*}-\pi _m^{CD*}=\pi _m^{CCS*}+T-\pi _m^{CD*}=-T_1+T \end{aligned}$$

(1)

$$\begin{aligned} \begin{aligned} \Delta \pi _{r+o}&=\pi _r^{CFC*}+\pi _o^{CFC*}-\pi _r^{CD*}-\pi _o^{CD*}\\&=\pi _r^{CCS*}-\varphi T+\pi _o^{CCS*}-(1-\varphi )T-\pi _r^{CD*}-\pi _o^{CD*}=T_2-T \end{aligned} \end{aligned}$$

(2)

According to the literature, Nash negotiation agreement (negotiation solution) can be seen as the maximum point of Nash product in feasible region. Therefore, the Nash negotiation agreement can be transformed into optimization problem as:

$$\begin{aligned} \mathop {\max }\limits _{(\pi _o^{CFC*}+\pi _r^{CFC*},\pi _m^{CFC*})\in \Omega } =\left( \Delta \pi _m\right) ^u\left( \Delta \pi _{r+o}\right) ^\tau s.t.\Delta \pi _m+\Delta \pi _{r+o}=\Delta \pi _{sc} \end{aligned}$$

$$\begin{aligned} \Delta \pi _m=\frac{u}{u+\tau }\Delta \pi _{sc}; \Delta \pi _{r+o}=\frac{\tau }{u+\tau }\Delta \pi _{sc} \end{aligned}$$

(3)

Put formula (1) and (2) together:

$$\begin{aligned} \Delta \pi _{sc}=T_2-T_1 \end{aligned}$$

(4)

Put formula (1)–(4) together, Eq. (41) can be proved.

Secondly:

$$\begin{aligned} \begin{aligned} \Delta \pi _r&=\pi _r^{CFC*}-\pi _r^{CD*}=\pi _r^{CCS*}-\varphi T-\pi _r^{CD*}=\varphi _2T-\varphi T \\ \Delta \pi _o&=\pi _o^{CFC*}-\pi _o^{CD*}=\pi _o^{CCS*}-(1-\varphi )T-\pi _o^{CD*}=\varphi T-\varphi _1T \end{aligned} \end{aligned}$$

(5)

In the same way, it can be translated into

$$\begin{aligned} \mathop {\max }\limits _{(\pi _o^{CFC*},\pi _r^{CFC*})\in \Omega } =\left( \Delta \pi _o\right) ^{v_o}\left( \Delta \pi _r\right) ^{v_r}s.t.\Delta \pi _o+\Delta \pi _r=\Delta \pi _{r+o} \end{aligned}$$

(6)

$$\begin{aligned} \Delta \pi _o=\frac{v_o}{v_o+v_r}\Delta \pi _{r+o}; \Delta \pi _r=\frac{v_r}{v_o+v_r}\Delta \pi _{r+o} \end{aligned}$$

(7)

Put formula (5)–(7) together, Eq. (42) can be proved.

Appendix B

We will give the derivation process with cost c, h, s, since they are considered in Extension (Sect. 7). Let costs be 0, we can obtain the corresponding result of benchmark models.

$$\begin{aligned} \begin{aligned} \max {\pi _r}(q_r)&= \left\{ \begin{array}{ll} \overline{E_3}<\overline{E_2},if w>w_1\\ \overline{E_3}>\overline{E_2},if w<w_1 \end{array} \right. \\&= \left\{ \begin{array}{ll} p_r(d_r+\varepsilon _r)-w(d_r+C_r)-h(C_r-\varepsilon _r),\varepsilon _r\le C_r\\ p_r(d_r+C_r)-w(d_r+C_r)-s(\varepsilon _r-C_r),\varepsilon _r>C_r \end{array} \right. \\&=\int _{0}^{C_r}\left( p_r(d_r+\varepsilon _r) -w(d_r+C_r)-h(C_r-\varepsilon _r)\right) f(\varepsilon _r)d\varepsilon _r \\ {}&\quad + \int _{C_r}^{\infty }\left( p_r(d_r+C_r)-w(d_r+C_r) -s(\varepsilon _r-C_r)\right) f(\varepsilon _r)d\varepsilon _r \\&= \left( p_rd_r-w(d_r+C_r)-hC_r\right) F(C_r) +\int _{0}^{C_r}{\left( p_r+h\right) \varepsilon _rf(\varepsilon _r)d\varepsilon _r} \\&\quad + \int _{C_r}^{\infty }\left( (p_r-w)(d_r+C_r)+sC_r\right) f(\varepsilon _r)d\varepsilon _r -s\int _{C_r}^{\infty }\varepsilon _rf(\varepsilon _r)d\varepsilon _r \end{aligned} \end{aligned}$$

(1)

Noted:

$$\begin{aligned} \begin{aligned}&\int _{0}^{C_r}{\left( p_r+h\right) \varepsilon _rf(\varepsilon _r)d\varepsilon _r} =(p+h)\left( C_rF(C_r)-\int _{0}^{C_r}{F(\varepsilon _r)d\varepsilon _r}\right) \\&s\int _{C_r}^{\infty }\varepsilon _rf(\varepsilon _r)d\varepsilon _r =s\left( \int _{0}^{\infty }\varepsilon _rf(\varepsilon _r)d\varepsilon _r -\int _{0}^{C_r}\varepsilon _rf(\varepsilon _r)d\varepsilon _r\right) \\&= s\left( E(\varepsilon _r)-C_rf(C_r)+\int _{0}^{C_r}{F(\varepsilon _r)d\varepsilon _r}\right) \\&=\int _{C_r}^{\infty }\left( (p_r-w)(d_r+C_r)+sC_r\right) f(\varepsilon _r)d\varepsilon _r \\&= (p_r-w)(d_r+C_r)+sC_r-\left( (p_r-w)(d_r+C_r)+sC_r\right) F(C_r) \end{aligned} \end{aligned}$$

(2)

Substitute formula (2) into formula (1)

$$\begin{aligned}\begin{aligned}&\pi _r(q_r)=\pi _r(C_r)=(p_r-w)(d_r+C_r)+s\left( C_r-E(\varepsilon _r)\right) -(p_r+s+h)\int _{0}^{C_r}{F(\varepsilon _r)d\varepsilon _r} \\&=(p_r-w)(d_r+C_r)+s\left( C_r-E(\varepsilon _r)\right) -(p_r+s+h)\int _{0}^{C_r}{F(x)dx} \end{aligned} \end{aligned}$$

Similarly, the expression of $\pi _o(C_o,B)$ can also be demonstrated.

Appendix C

$\pi _m^{CD*}=X_9/\left( 2k_m-8p_c\beta \lambda \right)$, its ± (positive or negative) depends on $X_9$, which is a quadratic convex function about $\overline{Y}$. Through the classical root formula, we can find some restrictions for w and $\overline{Y}$ to ensure $\pi _m^{CD*}>0$:

when,$w<p_c\lambda$    $\left\{ \begin{array}{lll} \overline{Y}>Y_2,if\ E_g<\overline{E_1}\\ \overline{Y}<Y_1 \ or \ \overline{Y}>Y_2,if\ \overline{E_1}<E_g<\overline{E_2}\\ \overline{Y}\in {\mathbb {R}}^+,if\ \overline{E_2}<E_g \end{array} \right.$

when,$w>p_c\lambda$    $\left\{ \begin{array}{ll} \overline{Y}>Y_2,if\ E_g<\overline{E_1}\\ \overline{Y}\in {\mathbb {R}}^+,if\ \overline{E_1}<E_g \end{array} \right.$

In this paper, we only consider the region $\overline{Y}\in {\mathbb {R}}^+$ for the simplicity.Therefore, Assuming:

$$\begin{aligned} \left\{ \begin{array}{ll} \overline{E_2}<E_g,ifw<p_c\lambda \\ \overline{E_1}<E_g,ifw>p_c\lambda \end{array} \right. \\ \end{aligned}$$

Specifically:

$$\begin{aligned} \begin{aligned} Y_1&=\frac{2p_cw\beta \lambda -2{p_c}^2\beta \lambda ^2-wk_m+p_c\lambda k_m}{{p_c}^2\lambda ^2}-\frac{\sqrt{\left( 4p_c\beta \lambda -k_m\right) \left( 2E_g{p_c}^3\lambda ^2-\left( w-p_c\lambda \right) ^2k_m\right) }}{{p_c}^2\lambda ^2}\\ Y_2&=\frac{2p_cw\beta \lambda -2{p_c}^2\beta \lambda ^2 -wk_m+p_c\lambda k_m}{{p_c}^2\lambda ^2} +\frac{\sqrt{\left( 4p_c\beta \lambda -k_m\right) \left( 2E_g{p_c}^3\lambda ^2-\left( w-p_c\lambda \right) ^2k_m\right) }}{{p_c}^2\lambda ^2}\\ \overline{E_1}&=\frac{2w^2\beta ^2-4p_cw\beta ^2\lambda +2{p_c}^2\beta ^2\lambda ^2}{4{p_c}^2\beta \lambda -p_ck_m}\\ \overline{E_2}&=\frac{w^2k_m-2p_cw\lambda k_m +{p_c}^2\lambda ^2k_m}{2{p_c}^3\lambda ^2}\\ X_9&={p_c}^2\lambda ^2{\overline{Y}}^2+\overline{Y} \left( -4p_cw\beta \lambda +4{p_c}^2\beta \lambda ^2+2\left( w-p_c\lambda \right) k_m\right) \\&+4w^2\beta ^2-8E_g{p_c}^2\beta \lambda -8p_cw\beta ^2\lambda +4pc^2\beta ^2\lambda ^2+2E_gp_ck_m \end{aligned} \end{aligned}$$

Appendix D

Without CTR:

$$\begin{aligned} E\left[ \pi _{sc}^N(C_o,C_r,B,e)\right] =\sum _{i=o,r}\left( p_i(C_i+d_i) -p_i\int _{0}^{C_i}{F(x)dx}\right) -\frac{1}{2}k_oB^2-\frac{1}{2}k_me^2 \end{aligned}$$

(1)

Substituting centralized decisions into (1), gives optimal total profit.

$$\begin{aligned} \pi _{sc}^{NC*}=\frac{1}{2}\left( \left( A+2A_o\right) p_o+\left( A+2A_r\right) p_r +\frac{\beta ^2\left( p_o+p_r\right) ^2}{k_m}+\frac{\eta ^2\left( p_o+\gamma p_r\right) ^2}{k_o}\right) \end{aligned}$$

(2)

Substituting decentralized decisions into (1), gives optimal total profit.

$$\begin{aligned} \pi _{sc}^{ND*}= \frac{1}{2}\left( \begin{aligned}&-\frac{Aw^2}{p_o}+\left( A+2A_o\right) p_o-\frac{Aw^2}{p_r}+\left( A+2A_r\right) p_r \\&+\frac{4w\beta ^2\left( -w+p_o+p_r\right) }{k_m}-\frac{\eta ^2\left( p_o-w\right) \left( w+p_o+2\gamma p_r\right) }{k_o} \end{aligned} \right) \end{aligned}$$

(3)

Subtracting Formula (1) and Formula (2) gives

$$\begin{aligned} \pi _{sc}^{NC*}-\pi _{sc}^{ND*}=\frac{Aw^2}{2}\left( \frac{1}{p_o} +\frac{1}{p_r}\right) +\frac{\beta ^2\left( p_o+p_r-2w\right) ^2}{2k_m}+\frac{\eta ^2\left( w+\gamma p_r\right) ^2}{2k_o} \end{aligned}$$

It can be easily seen that the outcome is positive. So, we can conclude that the total profit of the supply chain under centralized decision-making is greater than that under decentralized decision-making without CTR.

With CTR:

From numerical illustrations, we can find $\pi _{sc}^{CC*}-\pi _{sc}^{CD*}$ is increase in $\gamma$ and $\eta$. When $\gamma =\eta =0, \pi _{sc}^{CC*}-\pi _{sc}^{CD*}>0$. Then, centralized profits dominate decentralized profits to a certain degree.

Fig. 8

Comparison of equilibrium supply chain profit with CTR

It is well known that double marginalization contributes to the profit gap between a centralized system and a decentralized system. Spengler (1950) was the first to identify the problem of “double marginalization”, holding that decentralized decisions in general are inefficient and lead to inferior performance.

A large body of supply chain contract literature [see Cachon (2003) for a comprehensive survey] suggests that the wholesale price contract cannot coordinate a supply chain, because there are different margins and neither firm considers the entire supply chain’s margin when making a decision. The standard supply chain coordination mechanisms are generally aimed at eliminating the decision inefficiency (or the double marginalization effect), and achieving the same efficiency level as a centralized system (Peng et al., 2018; Xu et al., 2018; Hosseini-Motlagh et al., 2019; Xu et al., 2023).

Appendix E

The following analysis proves the robustness of the results.

The expected profits of the manufacturer and retailers are listed respectively:

$$\begin{aligned}\begin{aligned} \mathop {\max }{\hat{\pi }}_m^N(e)&=(w-c)\left( q_o+q_r\right) -\frac{1}{2}k_me^2\\ \mathop {\max }{\hat{\pi }}_m^C(e)&=(w-c)\left( q_o+q_r\right) -\frac{1}{2}k_me^2-p_c\left( \lambda \left( 1-e\right) q-E_g\right) \\ {\mathop {\max }{\hat{\pi }}}_r(q_r)&=p_r\min {(}q_r,D_r)-wq_r-h\max {(}q_r-D_r,0)-s\max {(D_r-q_r,0)}\\ {\max {\hat{\pi }}}_o(q_o,B)&=p_o\min {(}q_o,D_o) -wq_o-\frac{1}{2}k_0B^2-h\max {(}q_o-D_o,0)\\ {}&-s\max {(D_o-q_o,0)} \end{aligned} \end{aligned}$$

Through simplification, these functions take the following form. See Appendix B for the demonstration.

$$\begin{aligned}{} & {} {\max }E\left[ {\hat{\pi }}_m^N(e)\right] =(w-c)\left( C_o+d_o+C_r+d_r\right) -\frac{1}{2}k_me^2 \underset{{}}{\mathop {\max }}\,E\left[ \widehat{\pi }_{m}^{B}(e) \right] \\{} & {} \quad =(w-c)\left( {{C}_{o}}+{{d}_{o}}+{{C}_{r}}+{{d}_{r}} \right) -\frac{1}{2}{{k}_{m}}{{e}^{2}}\\{} & {} \qquad -{{p}_{c}}\left( \lambda \left( 1-e \right) \left( {{C}_{o}}+{{d}_{o}}\ +{{C}_{r}}+{{d}_{r}} \right) -{{E}_{g}} \right) \mathop {\max }\limits _{C_r}E\left[ {\hat{\pi }}_r^N(C_r)\right] \\{} & {} \quad =(p_r-w)\left( C_r+d_r\right) +s(C_r-\mu )\\{} & {} \quad -\left( p_r+s+h\right) \int _{0}^{C_r}{F(x)dx} \mathop {\max }\limits _{C_o,B}E\left[ {\hat{\pi }}_o^N(C_o,B)\right] =(p_o-w)\left( C_o+d_o\right) +s(C_o-\mu )\\{} & {} \quad -\left( p_o+s+h\right) \int _{0}^{C_0}{F(x)dx}-\frac{1}{2}k_oB^2 \end{aligned}$$

Without CTR:

$$\begin{aligned} {\hat{C}}_o^{ND*}=\frac{(p_o-w+s)A}{p_o+s+h};{\hat{B}}^{ND*}=\frac{\eta (p_o-w)}{k_o}\\ {\hat{C}}_r^{ND*}=\frac{(p_r-w+s)A}{p_r+s+h};{\hat{e}}^{ND*}=\frac{2(w-c)\beta }{k_m} \end{aligned}$$

Then,

$$\begin{aligned} {\hat{q}}_r^{ND*}=A_r-\frac{2\beta ^2(c-w)}{k_m}+\frac{\eta ^2\gamma (p_o-w)}{k_o}+\frac{A(p_r+s-w)}{h+p_r+s} \end{aligned}$$

With CTR:

$$\begin{aligned}\begin{aligned} {\hat{C}}_o^{ND*}&=\frac{(p_o-w+s)A}{p_o+s+h};{\hat{B}}^{ND*}=\frac{\eta (p_o-w)}{k_o}\\ {\hat{C}}_r^{ND*}&=\frac{(p_r-w+s)A}{p_r+s+h};{\hat{e}}^{CD*}=\frac{2\left( w-c\right) \beta -\lambda p_c\left( 2\beta - \widehat{Y}\right) }{k_m-4p_c\beta \lambda }\\ \widehat{Y}&=A_o+A_r-\frac{\eta ^2\left( w-p_o\right) }{k_o} -\frac{\gamma \eta ^2\left( w-p_o\right) }{k_o}+\frac{A\left( s-w+p_o\right) }{h+s+p_o}+\frac{A\left( s-w+p_r\right) }{h+s+p_r} \end{aligned} \end{aligned}$$

Then,

$$\begin{aligned}\begin{aligned} {\hat{q}}_o^{CD*}&=A_o+ \frac{\eta ^2(p_o-w)}{k_o}+\frac{A(p_o+s-w)}{h+p_o+s} + \frac{2(w-c)\beta ^2-\lambda p_c\beta (2\beta -\widehat{Y})}{k_m-4p_c\beta \lambda }\\ {\hat{q}}_r^{CD*}&=A_r+ \frac{\eta ^2\gamma (p_o-w)}{k_o}+\frac{A(p_r+s-w)}{h+p_r+s} +\frac{2(w-c)\beta ^2-\lambda p_c\beta (2\beta -\widehat{Y})}{k_m-4p_c\beta \lambda } \end{aligned} \end{aligned}$$

Through Comparison:

$$\begin{aligned}\begin{aligned}&{\hat{B}}^{ND*}={\hat{B}}^{CD*}\\&{\hat{e}}^{ND*}<{\hat{e}}^{CD*},{\hat{q}}_o^{ND*}<{\hat{q}}_o^{CD*},{\hat{q}}_r^{ND*}<{\hat{q}}_r^{CD*}\\&if\ \widehat{Y}>\frac{8c\beta ^2-8w\beta ^2+2\beta k_m}{k_m}\\&{\hat{\pi }}_o^{ND*}<{\hat{\pi }}_o^{CD*},if\ \widehat{Y}>\frac{8c\beta ^2-8w\beta ^2+2\beta k_m}{k_m}\\&{\hat{\pi }}_r^{ND*}<{\hat{\pi }}_r^{CD*},if\ \widehat{Y}>\frac{8c\beta ^2-8w\beta ^2+2\beta k_m}{k_m} \end{aligned} \end{aligned}$$

The value of ${\hat{\pi }}_m^{CD*}-{\hat{\pi }}_m^{ND*}$ depends on:

$$\begin{aligned}\begin{aligned}&16\left( c-w\right) ^2\beta ^3\lambda -\lambda k_m\left( 8E_gp_c\beta -4\left( w-c\right) \left( \widehat{Y}-2\beta \right) \beta -p_c\left( \widehat{Y}+2\beta \right) ^2\lambda \right) \\&\quad +2\left( E_g-\widehat{Y}\lambda \right) k_m^2 \end{aligned} \end{aligned}$$

Looking back, refer to Proof of Corollary 6, the value of $\pi _m^{CD*}-\pi _m^{ND*}$ depends on:

$$\begin{aligned} \left( 16w^2\beta ^3\lambda -\lambda k_m\left( 8E_gp_c\beta -4w\left( \overline{Y}-2\beta \right) \beta -p_c\left( \overline{Y}+2\beta \right) ^2\lambda \right) +2\left( E_g-\overline{Y}\lambda \right) k_m^2\right) \end{aligned}$$

It can be concluded that s, h, c do not change the essence of the problem, so our mechanism still works.