Group buying decisions of competing retailers with emergency procurement

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.

The authors gratefully acknowledge support from the National Natural Science Foundation of China through Grant U1204701, 71301045 and 71511117, and the Innovation Ability Construction Projects for Shanghai University through Grant 15590501800.

Authors and Affiliations

1. Research Center of Logistics, Shanghai Maritime University, Shanghai, 201306, China

   Rong Zhang

2. School of Economics and Management, Shanghai Maritime University, Shanghai, 201306, China

   Bin Liu

Corresponding author

Correspondence to Bin Liu.

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.

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 \(z^{G0^{*}}\),

and then the optimal ordering quantity is

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

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

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 \(z^{G1^{*}}\):

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

Proof of Proposition 1

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

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:

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 \(z_i^{G00^{*}}\):

Solving the reaction function \(p_i\) results into the following price

and the corresponding ordering quality is

Proof of Lemma 4

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

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

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 \(q_i\):

Solving the reaction function \(p_i\), we can get the optimal price

and the corresponding Group Buying level is

Furthermore, the optimal ordering quality is

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:

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

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 \(q_i\):

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

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.

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

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

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

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

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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):

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):

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

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

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Proof of Proposition 2

The Proposition 2 is a direct result from Lemma 6.

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