Group buying decisions of competing retailers with emergency procurement

Abstract

Group buying has been in vogue for many years, particularly for short-life-cycle products. This paper investigates the effects of pricing and ordering in group buying, and considers the joint decision on pricing and ordering of short-life-cycle products in competing markets if retailers could provide the emergency procurement to meet all stochastic demands. It is shown that retailers always prefer to launch group buying in a single channel, which is also observed in a dual channel for competing retailers due to emergency procurement. These findings differ from those presented in the related literatures.

Appendix

Proof of Lemma 1

Introducing the transformation of variable [$z^{G0}=Q^{G0}-y(p)$], we can get the new expected profit as follows.

$$\begin{aligned} \Pi ^{G0}(p,Q)= & {} \Pi ^{G0}(p,z) \\= & {} py(p)+p\mu +E\left[ v(z-x)^{+}-g(x-z)^{+}\right] -c(z+y(p)) \\= & {} (p-c)(A-\beta p)+(p-g)\mu +(g-c)z-(g-v)\int _{-\infty }^z {(z-x)} f(x)dx \end{aligned}$$

for which the Hessian matrix is negative due to $\frac{\partial ^{2}\Pi ^{G0}(p,z)}{\partial p^{2}}=-2\beta <0$ and $\frac{\partial ^{2}\Pi ^{G0}(p,z)}{\partial p^{2}}\frac{\partial ^{2}\Pi ^{G0}(p,z)}{\partial z^{2}}-\frac{\partial ^{2}\Pi ^{G0}(p,z)}{\partial p\partial z}\frac{\partial ^{2}\Pi ^{G0}(p,z)}{\partial z\partial p}=2\beta (g-v)f(z)>0$ for any $\beta \in (0,1]$, satisfying the second order condition for a maximum. Therefore, $\Pi ^{G0}(p,Q)$ is joint concave on (p, Q).

Hence, solving for $\frac{\partial \Pi ^{G0}(p,z)}{\partial p}=0,\frac{\partial \Pi ^{G0}(p,z)}{\partial z}=0$ simultaneously, we obtain $p^{G0^{*}}$ and $z^{G0^{*}}$,

$$\begin{aligned} p^{G0^{*}}=\frac{A+\beta c+\mu }{2\beta },\quad z^{G0^{*}}=F^{-1}\left( \frac{g-c}{g-v}\right) . \end{aligned}$$

and then the optimal ordering quantity is

$$\begin{aligned} Q^{G0^{*}}=y\left( p^{G0^{*}}\right) +z^{s0*}=\frac{A-\beta c-\mu }{2}+F^{-1}\left( \frac{g-c}{g-v}\right) . \end{aligned}$$

Certainly, the nonnegative retail prices and ordering quantity require: $A-\beta c-\mu >0$, then it will be satisfied easily as long as A (i.e., the base demand) is not too small. For the rest of this paper we will assume that condition to be true.

Proof of Lemma 2

Defining $z^{G1}=Q^{G1}-y(p,q)$, we can write

$$\begin{aligned} \Pi ^{G1}(p,Q,q)= & {} \Pi ^{G1}(p,z,q) \\= & {} py(p,\;q)+p\mu -(1-\gamma )pq+E\left[ v(z-x)^{+}-g(x-z)^{+}\right] \\&-\,c(z+y(p,\,q))-c_0 q \\= & {} (p-c)(A-\beta p+q)-(\lambda q+c_0 )q+(p-g)\mu +(g-c)z\\&-\,(g-v)\int _{-\infty }^z {(z-x)} f(x)dx \end{aligned}$$

It is easily established that the third order Hessian matrix of this problem is negative definite due to $\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial p^{2}}=-2\beta <0$, $\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial p^{2}}\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial z^{2}}-\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial p\partial z}\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial z\partial p}=2\beta (g-v)f(z)>0$ and

$$\begin{aligned} \left( {{\begin{array}{ccc} {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial p^{2}}}&{}\quad {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial p\partial z}}&{}\quad {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial p\partial q}} \\ {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial z\partial p}}&{}\quad {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial z^{2}}}&{}\quad {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial z\partial q}} \\ {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial q\partial p}}&{}\quad {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial q\partial z}}&{}\quad {\frac{\partial ^{2}\Pi ^{G1}(p,z,q)}{\partial q^{2}}} \\ \end{array} }} \right) \end{aligned}$$

equals $(1-4\lambda \beta )(g-v)f(z)<0$ for any $\lambda >\frac{1}{4\beta }$, implying $\Pi ^{G1}(p,Q,q)$ is joint concave on (p, Q, q).

Hence, solving for $\frac{\partial \Pi ^{G1}(p,z,q)}{\partial p}=0$, $\frac{\partial \Pi ^{G1}(p,z,q)}{\partial z}=0$, $\frac{\partial \Pi ^{G1}(p,z,q)}{\partial q}=0$ simultaneously, we can obtain p, q and $z^{G1^{*}}$:

$$\begin{aligned} p=\frac{A+q+\beta c+\mu }{2\beta },\quad q=\frac{p-c-c_0 }{2\lambda },\quad z^{G1^{*}}=F^{-1}\left( \frac{g-c}{g-v}\right) . \end{aligned}$$

Solving the reaction function results the following optimal price, the optimal group buying quantity and the optimal ordering quantity

$$\begin{aligned} p^{G1^{*}}= & {} \frac{2\lambda (A+\beta c+\mu )-c-c_0 }{4\lambda \beta -1},\quad q^{G1^{*}}=\frac{A-\beta c-2\beta c_0 +\mu }{4\lambda \beta -1}\\ Q^{G1^{*}}= & {} y\left( p^{G1^{*}},q^{G1^{*}}\right) +z^{G1^{*}}=\frac{2\lambda \beta \left( {A-\beta c-\mu } \right) +\mu -\beta c_0 }{4\lambda \beta -1}+F^{-1}\left( \frac{g-c}{g-v}\right) \end{aligned}$$

Proof of Proposition 1

Due to similarity, here we just investigate the profit/price/ordering quantity comparison of G0 and G1.

$$\begin{aligned} \Pi ^{G1^{*}}-\Pi ^{G0^{*}}= & {} \frac{(A-\beta c-2\beta c_0 +\mu )^{2}}{4\beta (4\lambda \beta -1)}>0;\\ p^{G1^{*}}-p^{G0^{*}}= & {} \frac{A-\beta c-2\beta c_0 +\mu }{2\beta (4\lambda \beta -1)}=\frac{q^{G1^{*}}}{2\beta }>0;\\ Q^{G1^{*}}-Q^{G0^{*}}= & {} \frac{A-\beta c-2\beta c_0 +\mu }{2(4\lambda \beta -1)}=\frac{q^{G1^{*}}}{2}>0. \end{aligned}$$

Thus, the retailer always prefers G1 to G0.

Proof of Lemma 3

By substituting $z_i^{G00} =Q_i^{G00} -y(p_i ,p_{3-i} )$, the expected profit can be written conveniently as:

$$\begin{aligned} \Pi _i^{G00}(p_i,Q_i)= & {} \Pi _i^{G00}(p_i,z_i)\\= & {} p_i y(p_i ,\;p_{3-i} )+p_i \mu +E\left[ v(z-x)^{+}-g(x-z)^{+}\right] \\= & {} p_i(A_i-\beta _i p_i+\theta p_{3-i})+(p_i-g)\mu +gz_i -(g-v)\int _{-\infty }^{z_i } {(z_i -x)} f(x)dx \end{aligned}$$

It can be easily seem that the second order Hessian matrix is negative definite because $\frac{\partial ^{2}\Pi _i^{G00} (p_i ,z_i )}{\partial p_i^2 }=-2\beta _i <0$ and $\frac{\partial ^{2}\Pi _i^{G00} (p_i ,z_i )}{\partial p_i^2 }\frac{\partial ^{2}\Pi _i^{G00} (p_i ,z_i )}{\partial z_i^2 }-\frac{\partial ^{2}\Pi _i^{G00} (p_i ,z_i )}{\partial p_i \partial z_i }\frac{\partial ^{2}\Pi _i^{G00} (p_i ,z_i )}{\partial z_i \partial p_i }=2\beta _i (g-v)f(z_i )>0$ for any $\beta \in (0,1]$, implying $\Pi _i^{G00} (p_i ,z_i)$ is joint concave on $(p_i,z_i)$.

Solving for $\frac{\partial \Pi _i^{G00} (p_i ,z_i )}{\partial p_i }=0$, $\frac{\partial \Pi _i^{G00}(p_i,z_i)}{\partial z_i }=0$ simultaneously, we can obtain $p_i$ and $z_i^{G00^{*}}$:

$$\begin{aligned} p_i =\frac{A_i +\theta p_{3-i} +\mu }{2\beta _i },\quad z_i^{G00^{*}} =F^{-1}\left( \frac{g}{g-v}\right) . \end{aligned}$$

Solving the reaction function $p_i$ results into the following price

$$\begin{aligned} p_i^{G00^{*}} =\frac{2\beta _{3-i} A_i +\theta A_{3-i} +(2\beta _{3-i} +\theta )\mu }{4\beta _i \beta _{3-i} -\theta ^{2}}, \end{aligned}$$

and the corresponding ordering quality is

$$\begin{aligned} Q_i^{G00^{*}}= & {} y\left( p_i^{G00^{*}} ,\;p_{3-i}^{G00^{*}}\right) +z_i^{G00^{*}} =\frac{\beta _i (2\hbox {A}_i \beta _{3-i} +\theta A_{3-i} )+(\theta ^{2}+\beta _i \theta -2\beta _i \beta _{3-i})\mu }{4\beta _i \beta _{3-i} -\theta ^{2}}\\&+\,F^{-1}\left( \frac{g}{g-v}\right) . \end{aligned}$$

Proof of Lemma 4

We define $z_i^{G11} =Q_i^{G11} -y(p_i ,q_i )$, the expected profit simplifies to

$$\begin{aligned} \Pi _i^{G11}(p_i,Q_i,q_i)= & {} \Pi _i^{G11} (p_i ,z_i ,q_i ) \\= & {} p_i y(p_i ,\;q_i )+p_i \mu -(1-\gamma )p_i q_i +E\left[ v(z_i -x)^{+}-g(x-z_i )^{+}\right] \\= & {} p_i (A_i -\beta _i p_i +q_i +\theta p_{3-i} )-\lambda q_i^2 +(p_i -g)\mu +gz_i \\&-\,(g-v)\int _{-\infty }^{z_i } {(z_i -x)} f(x)dx \end{aligned}$$

for which the Hessian matrix is negative due to $\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial p_i^2 }=-2\beta _i <0$, $\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial p_i^2 }\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial z_i^2 }-\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial p_i \partial z_i }\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial z_i \partial p_i }=2\beta _i (g-v)f(z_i )>0$ and

$$\begin{aligned} \left( {{\begin{array}{ccc} {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial p_i^2 }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial p_i \partial z_i }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial p_i \partial q_i }} \\ {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial z_i \partial p_i }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial z_i^2 }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial z_i \partial q_i }} \\ {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial q_i \partial p_i }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial q_i \partial z_i }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial q_i^2 }} \\ \end{array} }} \right) \end{aligned}$$

equals $(1-4\lambda \beta _i )(g-v)f(z_i )<0$ for any $\lambda >\frac{1}{4\beta _i }$, satisfying the third order condition for a maximum and implying $\Pi _i^{G11} (p_i ,z_i ,q_i )$ is joint concave on $(p_i,z_i,q_i)$.

Solving for $\frac{\partial \Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial p_i }=0$, $\frac{\partial \Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial z_i }=0$, $\frac{\partial \Pi _i^{G11} (p_i ,z_i ,q_i )}{\partial q_i }=0$ simultaneously, we can obtain $p_i$, $z_i^{G11^{*}}$ and $q_i$:

$$\begin{aligned} p_i =\frac{A_i +q_i +-\theta p_{3-i} +\mu }{2\beta _i },\quad z_i^{G11^{*}}=F^{-1}\left( \frac{g}{g-v}\right) ,\quad q_i =\frac{p_i }{2\lambda }. \end{aligned}$$

Solving the reaction function $p_i$, we can get the optimal price

$$\begin{aligned} p_i^{G11^{*}} =\frac{2\lambda ((4\lambda \beta _{3-i} -1)A_i +2\lambda \theta A_{3-i} +(4\lambda \beta _{3-i} -1+2\lambda \theta )\mu )}{(4\lambda \beta _i -1)(4\lambda \beta _{3-i} -1)-4\lambda ^{2}\theta ^{2}}, \end{aligned}$$

and the corresponding Group Buying level is

$$\begin{aligned} q_i^{G11^{*}} =\frac{(4\lambda \beta _{3-i} -1)A_i +2\lambda \theta A_{3-i} +(4\lambda \beta _{3-i} -1+2\lambda \theta )\mu }{(4\lambda \beta _i -1)(4\lambda \beta _{3-i} -1)-4\lambda ^{2}\theta ^{2}}. \end{aligned}$$

Furthermore, the optimal ordering quality is

$$\begin{aligned} Q_i^{G11^{*}}= & {} y\left( p_i^{G11^{*}} ,q_i^{G11^{*}}\right) +z_i^{G11^{*}} \\= & {} \frac{2\lambda \beta _i (4\lambda \beta _{3-i} -1)A_i +4\lambda ^{2}\theta \beta _i A_{3-i} +(4\lambda \beta _{3-i} -1+4\lambda ^{2}\theta ^{2}-2\lambda \beta _i (4\lambda \beta _{3-i} -1-2\lambda \theta ))\mu }{(4\lambda \beta _i -1)(4\lambda \beta _{3-i} -1)-4\lambda ^{2}\theta ^{2}}\\&+\,F^{-1}\left( \frac{g}{g-v}\right) \end{aligned}$$

Proof of Lemma 5

First of all, we define $z_i^{G10} =Q_i^{G10} -y(p_i ,q_i )$, the expected profit of Retailer i can be written as:

$$\begin{aligned} \Pi _i^{G10} (p_i ,Q_i ,q_i)= & {} \Pi _i^{G10} (p_i ,z_i ,q_i ) \\= & {} p_i y(p_i ,\;q_i )+p_i \mu -(1-\gamma )p_i q_i +E\left[ v(z_i -x)^{+}-g(x-z_i )^{+}\right] \\= & {} p_i (A_i -\beta _i p_i +q_i +\theta p_{3-i} )-\lambda q_i^2 +(p_i -g)\mu +gz_i \\&-\,(g-v)\int _{-\infty }^{z_i } {(z_i -x)} f(x)dx. \end{aligned}$$

for which the Hessian matrix is negative due to $\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial p_i^2 }=-2\beta _i <0$, $\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial p_i^2 }\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial z_i^2 }-\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial p_i \partial z_i }\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial z_i \partial p_i }=2\beta _i (g-v)f(z_i )>0$ and

$$\begin{aligned} \left( {{\begin{array}{ccc} {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial p_i^2 }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial p_i \partial z_i }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial p_i \partial q_i }} \\ {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial z_i \partial p_i }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial z_i^2 }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial z_i \partial q_i }} \\ {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial q_i \partial p_i }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial q_i \partial z_i }}&{}\quad {\frac{\partial ^{2}\Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial q_i^2 }} \\ \end{array} }} \right) \end{aligned}$$

equals $(1-4\lambda \beta _i )(g-v)f(z_i )<0$ for any $\lambda >\frac{1}{4\beta _i }$, satisfying the third order condition for a maximum and implying $\Pi _i^{G10} (p_i ,z_i ,q_i )$ is joint concave on $(p_i ,z_i ,q_i)$.

Hence, solving for $\frac{\partial \Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial p_i }=0$, $\frac{\partial \Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial z_i }=0$, $\frac{\partial \Pi _i^{G10} (p_i ,z_i ,q_i )}{\partial q_i }=0$ simultaneously, we can obtain $p_i$, $z_i^{G10*}$ and $q_i$:

$$\begin{aligned} p_i =\frac{A_i +q_i +\theta p_{3-i} +\mu }{2\beta _i },\quad z_i^{G10*} =F^{-1}\left( \frac{g}{g-v}\right) ,\quad q_i =\frac{p_i }{2\lambda }. \end{aligned}$$

Secondly, defining $z_{3-i}^{G10} =Q_{3-i}^{G10} -y(p_{3-i})$, the corresponding expected profit of Retailer (3-i) simplifies to

$$\begin{aligned} \Pi _{3-i}^{G10} (p_{3-i} ,Q_{3-i})= & {} \Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} ) \\= & {} p_{3-i} y(p_{3-i} )+p_{3-i} \mu +E\left[ v(z_{3-i} -x)^{+}-g(x-z_{3-i})^{+}\right] \\= & {} p_{3-i} (A_{3-i} -\beta _{3-i} p_{3-i} +\theta p_i )+(p_{3-i} -g)\mu +gz_{3-i} \\&-\,(g-v)\int _{-\infty }^{z_{3-i} } {(z_{3-i} -x)}f(x)dx \end{aligned}$$

for which the Hessian matrix is negative due to $\frac{\partial ^{2}\Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} )}{\partial p_{3-i}^2 }=-2\beta _{3-i} <0$ and $\frac{\partial ^{2}\Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} ,q_{3-i} )}{\partial p_{3-i}^2 }\frac{\partial ^{2}\Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} ,q_{3-i} )}{\partial z_{3-i}^2 }-\frac{\partial ^{2}\Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} ,q_{3-i} )}{\partial p_{3-i} \partial z_{3-i} }\frac{\partial ^{2}\Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} ,q_{3-i} )}{\partial z_{3-i} \partial p_{3-i} }=2\beta _{3-i} (g-v)f(z_{3-i} )>0$ for any $\beta \in (0,1]$, satisfying the second order condition for a maximum and implying $\Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} ,q_{3-i})$ is joint concave on $(p_{3-i} ,z_{3-i})$.

Thus, solving for $\frac{\partial \Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} ,q_{3-i} )}{\partial p_{3-i} }=0$ and $\frac{\partial \Pi _{3-i}^{G10} (p_{3-i} ,z_{3-i} ,q_{3-i} )}{\partial z_{3-i} }=0$ simultaneously, we can obtain $p_{3-i}$ and $z_{3-i}^{G10*}$ as follows.

$$\begin{aligned} p_{3-i} =\frac{A_{3-i} +\theta p_i +\mu }{2\beta _{3-i} },\quad z_i^{G10*}=F^{-1}\left( \frac{g}{g-v}\right) . \end{aligned}$$

Thirdly, solving for the reaction function $p_i$, $q_i$ and $p_{3-i}$ simultaneously, we can get

$$\begin{aligned} p_i^{G10*}= & {} \frac{\lambda (2\beta _{3-i} A_i +\theta A_{3-i} +(2\beta _{3-i} +\theta )\mu )}{4\lambda \beta _i \beta _{3-i} -\beta _{3-i}-\lambda \theta ^{2}},\\ q_i^{G10*}= & {} \frac{2\beta _{3-i} A_i +\theta A_{3-i} +(2\beta _{3-i} +\theta )\mu }{2(4\lambda \beta _i \beta _{3-i} -\beta _{3-i} \hbox {-}\lambda \theta ^{2})},\\ p_{3-i}^{G10*}= & {} \frac{2\lambda \theta A_i +(4\lambda \beta _i -1)A_{3-i} +(2\lambda \theta +4\lambda \beta _i -1)\mu )}{2(4\lambda \beta _i \beta _{3-i} -\beta _{3-i} -\lambda \theta ^{2})}. \end{aligned}$$

Then, the optimal stocking of the two retailers can be derived:

$$\begin{aligned} Q_i^{G10*}= & {} \frac{2\lambda \beta _i \beta _{3-i} A_i +\lambda \theta \beta _i A_{3-i} -\left( 2\lambda \beta _i \beta _{3-i} -\lambda \theta \beta _i -\beta _{3-i} -\lambda \theta ^{2}\right) \mu }{4\lambda \beta _i \beta _{3-i} -\beta _{3-i} -\lambda \theta ^{2}}\\&+\,F^{-1}\left( \frac{g}{g-v}\right) ,\\ Q_{3-i}^{G10*}= & {} \frac{2\lambda \theta \beta _{3-i} A_i +(4\lambda \beta _i -1)\beta _{3-i} A_{3-i} +\left( 2\lambda \theta \beta _{3-i} -4\lambda \beta _i \beta _{3-i} +\beta _{3-i} +2\lambda \theta ^{2}\right) \mu }{2\left( 4\lambda \beta _i \beta _{3-i} -\beta _{3-i}-\lambda \theta ^{2}\right) }\\&+\,F^{-1}\left( \frac{g}{g-v}\right) . \end{aligned}$$

Fig. 1

Comparison of the optimal retail price $p_i^*$ ($\mu =0.5$) among G11, G10, G01 and G00

Fig. 2

Comparison of the optimal stocking quantity $Q_i^*$ ($\mu =0.5$) among G11, G10, G01 and G00

Proof of Lemma 6

It is straightforward to get these results from Table 1. Here we omit these processes, but they are illustrated by Figs. 1, 2 and 3.

Firstly, we compare the retail prices under different scenarios. The retail price is the highest in the G11 followed by G10, G01 and then G00 (shown in Fig. 1):

$$\begin{aligned} p_{i}^{{G11*}} \ge p_{i}^{{G10*}} \ge p_{i}^{{G01*}} \ge p_{i}^{{G00*}} \end{aligned}$$

Secondly, we compare the deterministic portion of the stocking quantity under the four scenarios. The result is similar to the price (shown in Fig. 2):

$$\begin{aligned} Q_{i}^{{G11*}} \ge Q_{i}^{{G10*}} \ge Q_{i}^{{G01*}} \ge Q_{i}^{{G00*}} \end{aligned}$$

Finally, we compare the profits. A retailer makes the largest profit in G11, and the smallest in G00 (shown in Fig. 3) as expected:

$$\begin{aligned} \Pi _{i}^{{G11*}} \ge \Pi _{i}^{{G10*}} \ge \Pi _{i}^{{G01*}} \ge \Pi _{i}^{{G00*}} \end{aligned}$$

Fig. 3

Comparison of the optimal retail profit $\Pi _i^*$ ($\mu =0.5$) among G11, G10, G01 and G00

Proof of Proposition 2

The Proposition 2 is a direct result from Lemma 6.