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Price discrimination based on purchase behavior and service cost in competitive channels

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Abstract

With the recent emergence of cloud computing and big data technologies, collection of consumers’ information is widespread. Retailers use consumers’ purchase history and consumptive habits data to price discriminate between current and new, high-cost and low-cost consumers. We investigate behavior-based pricing (BBP) and consumers cost-based pricing (CCP) simultaneously in a competitive two-period market in which bricks and clicks retailers sell products to high-cost-type and low-cost-type consumers during two periods. We examine how the price discrimination affects the channel members’ prices, market shares and profits. We find that dual price discriminations (BBP and CCP) decrease the service cost advantage retailer’s profit, but increase the service cost disadvantage retailer’s profit if the consumer’s travel cost is low. Compared the market shares of retailers, it is interesting that a cost advantage retailer serves more type-H consumers under the case of BBP and CCP than other cases. In addition, our results illustrate that cost disadvantage retailers prefer to reward the current consumers in the second period. Additionally, we find that consumers may benefit from price discrimination that they face a lower price in the case of BBP and CCP than other cases under certain conditions. Even more egregious, the current type-H consumers served by the cost advantage retailer enjoy a lower price than the type-L consumers served by the competitor.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 71771164 and 71702129), Humanity and Social Science Youth Foundation of Ministry of Education of China (No. 17YJC630232).

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Authors and Affiliations

  1. College of Management and Economics, Tianjin University, Tianjin, 300072, China

    Man Xu & Wansheng Tang

  2. School of Management, Tianjin University of Technology, Tianjin, 300384, China

    Chi Zhou

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  1. Man Xu
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Correspondence to Wansheng Tang.

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Appendix

Appendix

Proof of Proposition 1

In the case of no price discrimination, retailers’ profits in two periods are identical. Thus, they can simply maximize their per period profits:

$$\begin{aligned} \pi _{A1}^\mathrm{{ND}}= & {} (p_{A1}^\mathrm{{ND}}-s_{A}^{H})x_{1}^\mathrm{{ND}}+p_{A1}^\mathrm{{ND}}x_{1}^\mathrm{{ND}},\\ \pi _{B1}^\mathrm{{ND}}= & {} (p_{B1}^\mathrm{{ND}}-s_{B}^{H})(1-x_{1}^\mathrm{{ND}})+p_{B1}^\mathrm{{ND}}(1-x_{1}^\mathrm{{ND}}), \end{aligned}$$

where \(x_{1}^\mathrm{{ND}}=\frac{t+p_{B1}^\mathrm{{ND}}-p_{A1}^\mathrm{{ND}}}{2t}\).

Solving the first-order optimality conditions:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial \pi _{A1}^\mathrm{{ND}}}{\partial p_{A1}^\mathrm{{ND}}}=\frac{2t+2p_{B1}^\mathrm{{ND}}-4p_{A1}^\mathrm{{ND}}+s_{A}^{H}}{2t}=0,\\&\frac{\partial \pi _{B1}^\mathrm{{ND}}}{\partial p_{B1}^\mathrm{{ND}}}=\frac{2t-4p_{B1}^\mathrm{{ND}}+2p_{A1}^\mathrm{{ND}}+s_{B}^{H}}{2t}=0, \end{aligned} \right. \end{aligned}$$

we obtain \({p_{A1}^\mathrm{{ND}}}^{\!*}=t+\frac{s_{A}^{H}}{3}+\frac{s_{B}^{H}}{6}\) and \({p_{B1}^\mathrm{{ND}}}^{\!*}=t+\frac{s_{A}^{H}}{6}+\frac{s_{B}^{H}}{3}\).

Correspondingly, substituting \({p_{A1}^\mathrm{{ND}}}^{\!*}\) and \({p_{B1}^\mathrm{{ND}}}^{\!*}\) to the retailers A’s and B’s market demands, we obtain \({D_{A1}^\mathrm{{ND}}}^{\!*}=x_{1}^\mathrm{{ND}}=1-\frac{s_{A}^{H}-s_{B}^{H}}{6t}\) and \({D_{B1}^\mathrm{{ND}}}^{\!*}=1-x_{1}^\mathrm{{ND}}=1+\frac{s_{A}^{H}-s_{B}^{H}}{6t}\), to the profits, we obtain \({\pi _{A}^\mathrm{{ND}}}^{\!*}=\frac{(-6t+s_{A}^{H}-s_{B}^{H})^{2}}{9t}\), and \({\pi _{B}^\mathrm{{ND}}}^{\!*}=\frac{(6t+s_{A}^{H}-s_{B}^{H})^{2}}{9t}\). \(\square \)

Proof of Lemma 1

In the case of single price discrimination (BBP), we solve the two periods backward, solving first for the second-period equilibrium strategies.

The retailer A aims to decide the optimal prices \(p_{A2}^\mathrm{{O\!-\!SD}}\) and \(p_{A2}^\mathrm{{N\!-\!SD}}\) to maximize the second-period profit

$$\begin{aligned} \begin{aligned} \pi _{A2}^{\mathrm{{SD}}}&= (p_{A2}^\mathrm{{O\!-\!SD}}-s_{A}^{H})x_{A}^\mathrm{{H\!-\!SD}}+p_{A2}^\mathrm{{O\!-\!SD}}x_{A}^\mathrm{{L\!-\!SD}}\\&+\,(p_{A2}^\mathrm{{N\!-\!SD}}-s_{A}^{H})(x_{B}^\mathrm{{H\!-\!SD}}-x_{1}^\mathrm{{SD}})\\&+\,p_{A2}^\mathrm{{N\!-\!SD}}(x_{B}^\mathrm{{L\!-\!SD}}\!-x_{1}^\mathrm{{SD}}), \end{aligned} \end{aligned}$$
(14)

and retailer B aims to decide the optimal prices \(p_{B2}^\mathrm{{O\!-\!SD}}\) and \(p_{B2}^\mathrm{{N\!-\!SD}}\) to maximize the second-period profit

$$\begin{aligned} \begin{aligned} \pi _{B2}^{\mathrm{{SD}}}&=\, (p_{B2}^\mathrm{{O\!-\!SD}}-s_{B}^{H})(1-x_{B}^\mathrm{{H\!-\!SD}})\\&+\,p_{B2}^\mathrm{{O\!-\!SD}}(1-x_{B}^\mathrm{{L\!-\!SD}})\\&+\,(p_{B2}^\mathrm{{N\!-\!SD}}-s_{B}^{H})(x_{1}^\mathrm{{SD}}-x_{A}^\mathrm{{H\!-\!SD}})+p_{B2}^\mathrm{{N\!-\!SD}}(x_{1}^\mathrm{{SD}}-x_{A}^\mathrm{{L\!-\!SD}}), \end{aligned} \end{aligned}$$
(15)

where \(x_{A}^\mathrm{{H\!-\!SD}}=x_{A}^\mathrm{{L\!-\!SD}}=\frac{t+p_{B2}^\mathrm{{N\!-\!SD}}-p_{A2}^\mathrm{{O\!-\!SD}}}{2t}\), \(x_{B}^\mathrm{{H\!-\!SD}}=x_{B}^\mathrm{{L\!-\!SD}}=\frac{t+p_{B2}^\mathrm{{O\!-\!SD}}-p_{A2}^\mathrm{{N\!-\!SD}}}{2t}.\)

The Hessian for the retailers A’s and B’s profits is given by

$$\begin{aligned} H_{A}=H_{B}=\left[ \begin{array}{cc} -\frac{2}{t}&{} 0\\ 0 &{} -\frac{2}{t}\\ \end{array} \right] , \end{aligned}$$

which can deduce that the retailers’ profits are concavity.

Then, solving the first-order optimality conditions:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial \pi _{A2}^\mathrm{{SD}}}{\partial p_{A2}^\mathrm{{N\!-\!SD}}}=\frac{(2-4x_{1}^\mathrm{{SD}})t+2p_{B2}^\mathrm{{O\!-\!SD}}+s_{A}^{H}-4p_{A2}^\mathrm{{N\!-\!SD}}}{2t}=0,\\&\frac{\partial \pi _{A2}^\mathrm{{SD}}}{\partial p_{A2}^\mathrm{{O\!-\!SD}}}=\frac{2t+2p_{B2}^\mathrm{{N\!-\!SD}}-4p_{A2}^\mathrm{{O\!-\!SD}}+s_{A}^{H}}{2t}=0,\\&\frac{\partial \pi _{B2}^\mathrm{{SD}}}{\partial p_{B2}^\mathrm{{N\!-\!SD}}}=\frac{(4x_{1}^\mathrm{{SD}}-2)t+2p_{A2}^\mathrm{{O\!-\!SD}}+s_{B}^{H}-4p_{B2}^\mathrm{{N\!-\!SD}}}{2t}=0,\\&\frac{\partial \pi _{B2}^\mathrm{{SD}}}{\partial p_{B2}^\mathrm{{O\!-\!SD}}}=\frac{2t+2p_{A2}^\mathrm{{N\!-\!SD}}-4p_{B2}^\mathrm{{O\!-\!SD}}+s_{B}^{H}}{2t}=0,\\ \end{aligned} \right. \end{aligned}$$

we can obtain the results as shown in Lemma 1. \(\square \)

Proof of Proposition 2

The retailer A’s first-period profit in the case of BBP

$$\begin{aligned} \pi _{A1}^\mathrm{{SD}}=(p_{A1}^\mathrm{{SD}}-s_{A}^{H})x_{1}^\mathrm{{SD}}+p_{A1}^\mathrm{{SD}}x_{1}^\mathrm{{SD}}, \end{aligned}$$

and the retailer B’s first-period profit in the case of BBP

$$\begin{aligned} \pi _{B1}^\mathrm{{SD}}=(p_{B1}^\mathrm{{SD}}-s_{B}^{H})(1-x_{1}^\mathrm{{SD}})+p_{B1}^\mathrm{{SD}}(1-x_{1}^\mathrm{{SD}}), \end{aligned}$$

where \(x_{1}^\mathrm{{SD}}=\frac{8t-6p_{A1}+6p_{B1}+s_{A}^{H}-s_{B}^{H}}{16t}\).

Similar to the proof of Proposition 1, by substituting Lemma 1’s results to retailers A’s and B’s profits and solving the first-order conditions, we obtain the retailers A’s and B’s optimal first-period prices as shown in Proposition 2.

In addition, by substituting Lemma 1’s results to the retailer A’s first-market demand \(D_{A1}^\mathrm{{SD}}=2x_{1}^\mathrm{{SD}}\) and the second market demand \(D_{A2}^\mathrm{{SD}}=x_{A}^\mathrm{{H\!-\!SD}}+x_{A}^\mathrm{{L\!-\!SD}}+x_{B}^\mathrm{{H\!-\!SD}}+x_{B}^\mathrm{{L\!-\!SD}}-2x_{1}^\mathrm{{SD}}\), retailer B’s first-market demand \(D_{B1}^\mathrm{{SD}}=2(1-x_{1}^\mathrm{{SD}})\) and the second-market demand \(D_{B2}^\mathrm{{SD}}=2-x_{B}^\mathrm{{H\!-\!SD}}-x_{B}^\mathrm{{L\!-\!SD}}-x_{A}^\mathrm{{H\!-\!SD}}-x_{A}^\mathrm{{L\!-\!SD}}+2x_{1}^\mathrm{{SD}}\), we can obtain the results as shown in Proposition 2.

Finally, we get the retailers A’s and B’s optimal profits \(\pi _{A}^\mathrm{{SD}}\) and \(\pi _{B}^\mathrm{{SD}}\) as shown in Proposition 2. \(\square \)

Proof of Corollary 1

Comparing the retailers’ prices over two periods and based on the assumption \(t>\frac{5}{16}|s_{A}^{H}-s_{B}^{H}|\), we obtain \({p_{A1}^\mathrm{{SD}}}^{\!*}-{p_{A2}^\mathrm{{N\!-\!SD}}}^{\!*}=t>0\), \({p_{B1}^\mathrm{{SD}}}^{\!*}-{p_{B2}^\mathrm{{N\!-\!SD}}}^{\!*}=t>0\), \({p_{A1}^\mathrm{{SD}}}^{\!*}-{p_{A2}^\mathrm{{O\!-\!SD}}}^{\!*}=\frac{2}{3}t+\frac{1}{16}(s_{A}^{H}-s_{B}^{H})>0\), \({p_{B1}^\mathrm{{SD}}}^{\!*}-{p_{B2}^\mathrm{{O\!-\!SD}}}^{\!*}=\frac{2}{3}t+\frac{1}{16}(s_{B}^{H}-s_{A}^{H})>0\), \({p_{A2}^\mathrm{{N\!-\!SD}}}^{\!*}-{p_{A2}^\mathrm{{O\!-\!SD}}}^{\!*}=-\frac{1}{3}t+\frac{1}{16}(s_{A}^{H}-s_{B}^{H})<0\), \({p_{B2}^\mathrm{{N\!-\!SD}}}^{\!*}-{p_{A2}^\mathrm{{O\!-\!SD}}}^{\!*}=-\frac{1}{3}t-\frac{1}{16}(s_{A}^{H}+s_{B}^{H})<0\), which can deduce the results as shown in Corollary 1. \(\square \)

Proof of Lemma 2

In the case of dual price discrimination (BBP and CCP), we solve the two periods backward, solving first for the second-period equilibrium strategies.

The retailer A aims to decide the optimal prices \(p_{A2}^\mathrm{{H\!-\!D\!D}}\), \(p_{A2}^\mathrm{{L\!-\!D\!D}}\) and \(p_{A2}^\mathrm{{N\!-\!SD}}\) to maximize the second-period profit

$$\begin{aligned} \begin{aligned} \pi _{A2}^{\mathrm{{D\!D}}}=&(p_{A2}^\mathrm{{H\!-\!D\!D}}-s_{A}^{H})x_{A}^\mathrm{{H\!-\!D\!D}}+p_{A2}^\mathrm{{L\!-\!D\!D}}x_{A}^\mathrm{{L\!-\!D\!D}}+(p_{A2}^\mathrm{{N\!-\!D\!D}}-s_{A}^{H})\\&\times \,(x_{B}^\mathrm{{H\!-\!D\!D}}-x_{1}^\mathrm{{D\!D}})+p_{A2}^\mathrm{{N\!-\!D\!D}}(x_{B}^\mathrm{{L\!-\!D\!D}}-x_{1}^\mathrm{{D\!D}}),\\ \end{aligned} \end{aligned}$$

and retailer B aims to decide the optimal prices \(p_{B2}^\mathrm{{H\!-\!D\!D}}\), \(p_{B2}^\mathrm{{L\!-\!D\!D}}\) and \(p_{B2}^\mathrm{{N\!-\!SD}}\) to maximize the second-period profit

$$\begin{aligned} \begin{aligned} \pi _{B2}^{\mathrm{{SD}}}=&(p_{B2}^\mathrm{{H\!-\!D\!D}}-s_{B}^{H})(1-x_{B}^\mathrm{{H\!-\!D\!D}}+p_{B2}^\mathrm{{L\!-\!D\!D}}(1-x_{B}^\mathrm{{L\!-\!D\!D}})\\&+(p_{B2}^\mathrm{{N\!-\!D\!D}}-s_{B}^{H})(x_{1}^\mathrm{{D\!D}}-x_{A}^\mathrm{{H\!-\!D\!D}})+p_{B2}^\mathrm{{N\!-\!D\!D}}(x_{1}^\mathrm{{D\!D}}-x_{A}^\mathrm{{L\!-\!D\!D}}),\\ \end{aligned} \end{aligned}$$

where \(x_{A}^\mathrm{{i\!-\!D\!D}}=\frac{t+p_{B2}^\mathrm{{N\!-\!D\!D}}-p_{A2}^\mathrm{{i\!-\!D\!D}}}{2t}\) and \(x_{B}^\mathrm{{i\!-\!D\!D}}=\frac{t+p_{B2}^\mathrm{{i\!-\!D\!D}}-p_{A2}^\mathrm{{N\!-\!D\!D}}}{2t}\), \(i\in \{H,L\}\).

The Hessian for the retailers A’s and B’s profits is given by

$$\begin{aligned} H_{A}^\mathrm{{\!D\!D}}=H_{B}^\mathrm{{\!D\!D}}=\left[ \begin{array}{ccc} -\frac{2}{t}&{} 0 &{} 0\\ 0 &{} -\frac{1}{t}&{} 0\\ 0 &{} 0 &{} -\frac{1}{t}\\ \end{array} \right] , \end{aligned}$$

which can deduce that the retailers’ profits are concavity.

Then, similar to the proof of Lemma 1, solving the first-order optimality conditions:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial \pi _{A2}^\mathrm{{D\!D}}}{\partial p_{A2}^\mathrm{{N\!-\!D\!D}}}=\frac{4p_{B2}^\mathrm{{\!H\!-\!D\!D}}-16p_{A2}^\mathrm{{\!N\!-\!D\!D}}+6p_{A1}^\mathrm{{D\!D}}-6p_{B1}^\mathrm{{D\!D}}+3s_{A}^{H}+s_{B}^{H}+4p_{B2}^{\!L\!-\!D\!D}}{8t}=0,\\&\frac{\partial \pi _{A2}^\mathrm{{D\!D}}}{\partial p_{A2}^\mathrm{{H\!-\!D\!D}}}=\frac{t+p_{B2}^\mathrm{{N\!-\!D\!D}}-2p_{A2}^\mathrm{{H\!-\!D\!D}}+s_{A}^{H}}{2t}=0,\\&\frac{\partial \pi _{A2}^\mathrm{{D\!D}}}{\partial p_{A2}^\mathrm{{L\!-\!D\!D}}}=\frac{t+p_{B2}^\mathrm{{N\!-\!D\!D}}-2p_{A2}^\mathrm{{L\!-\!D\!D}}}{2t}=0,\\&\frac{\partial \pi _{B2}^\mathrm{{D\!D}}}{\partial p_{B2}^\mathrm{{N\!-\!D\!D}}}=\frac{4p_{A2}^\mathrm{{\!H\!-\!D\!D}}-16p_{B2}^\mathrm{{\!N\!-\!D\!D}}+6p_{B1}^\mathrm{{D\!D}}-6p_{A1}^\mathrm{{D\!D}}+3s_{B}^{H}+s_{A}^{H}+4p_{A2}^{\!L\!-\!D\!D}}{8t}=0,\\&\frac{\partial \pi _{B2}^\mathrm{{D\!D}}}{\partial p_{B2}^\mathrm{{H\!-\!D\!D}}}=\frac{t+p_{A2}^\mathrm{{N\!-\!D\!D}}-2p_{B2}^\mathrm{{H\!-\!D\!D}}+s_{B}^{H}}{2t}=0,\\&\frac{\partial \pi _{B2}^\mathrm{{D\!D}}}{\partial p_{B2}^\mathrm{{L\!-\!D\!D}}}=\frac{t+p_{A2}^\mathrm{{N\!-\!D\!D}}-2p_{B2}^\mathrm{{L\!-\!D\!D}}}{2t}=0,\\ \end{aligned} \right. \end{aligned}$$

we can obtain the results as shown in Lemma 2. \(\square \)

Proof of Proposition 3

The proof is similar to Proposition 2. \(\square \)

Proof of Corollary 2

First, comparing the second-period prices between the current type-H consumers and new consumers in the case of BBP and CCP, we obtain: \({p_{A2}^\mathrm{{H-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{N-\!D\!D}}}^{\!*}=\frac{1}{16}s_{B}^{H}+\frac{3}{16}s_{A}^{H}+\frac{1}{3}t>0\), \({p_{B2}^\mathrm{{H-\!D\!D}}}^{\!*}-{p_{B2}^\mathrm{{N-\!D\!D}}}^{\!*}=\frac{1}{16}s_{A}^{H}+\frac{3}{16}s_{B}^{H}+\frac{1}{3}t>0\), which can deduce the results as shown in Corollary 2 (i).

Second, comparing the second-period prices between the current type-L consumers and new consumers in the case of BBP and CCP, we can write that

$$\begin{aligned} {p_{A2}^\mathrm{{N-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{L-\!D\!D}}}^{\!*}=-\frac{1}{3}t+\frac{5}{16}s_{A}^{H}-\frac{1}{16}s_{B}^{H}, \end{aligned}$$
(16)

solving \({p_{A2}^\mathrm{{N-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{L-\!D\!D}}}^{\!*}>0\), we obtain \(t<\frac{15}{16}s_{A}^{H}-\frac{3}{16}s_{B}^{H}\).

And then, based on the assumption that \(t>\frac{5}{16}|s_{A}^{H}-s_{B}^{H}|\), we can obtain that the condition \(t<\frac{15}{16}s_{A}^{H}-\frac{3}{16}s_{B}^{H}\) is satisfied when \(s_{A}^{H}>\frac{2}{5}s_{B}^{H}\) .

Similar to the above analysis, solving \({p_{B2}^\mathrm{{N-\!D\!D}}}^{\!*}-{p_{B2}^\mathrm{{L-\!D\!D}}}^{\!*}>0\) based on the assumption, we can obtain the result shown in Corollary 2 (iii). \(\square \)

Proof of Corollary 3

Comparing the retailer’s second-period demand of the type-H consumers to the first period, we can write that \({D_{A2}^\mathrm{{H\!-\!D\!D}}}^{\!*}-{D_{A1}^\mathrm{{H\!-\!D\!D}}}^{\!*}=x_{A}^\mathrm{{H\!-\!D\!D}}+x_{B}^\mathrm{{H\!-\!D\!D}}-2x_{1}^\mathrm{{D\!D}}=\frac{s_{B}^{H}-s_{A}^{H}}{4t}\).

Thus, if \(s_{B}^{H}>s_{A}^{H}\), then \({D_{A2}^\mathrm{{H\!-\!D\!D}}}^{\!*}-{D_{A1}^\mathrm{{H\!-\!D\!D}}}^{\!*}>0\), otherwise, \({D_{A2}^\mathrm{{H\!-\!D\!D}}}^{\!*}-{D_{A1}^\mathrm{{H\!-\!D\!D}}}^{\!*}<0\). \(\square \)

Proof of Proposition 4

Through Propositions 2 and 3, we obtain that the first-period prices in the case of single price discrimination and dual price discrimination are equal, i.e., \({p_{A1}^\mathrm{{\!D\!D}}}^{\!*}={p_{A1}^\mathrm{{SD}}}^{\!*}\) and \({p_{B1}^\mathrm{{\!D\!D}}}^{\!*}={p_{B1}^\mathrm{{SD}}}^{\!*}\).

In addition, comparing the retailer’s first-period prices in the environment of price discrimination and no price discrimination, we can obtain: \({p_{A1}^\mathrm{{\!D\!D}}}^{\!*}-{p_{A1}^\mathrm{{ND}}}^{\!*}=\frac{1}{3}t+\frac{1}{24}(s_{A}^{H}-s_{B}^{H})\), and \({p_{B1}^\mathrm{{\!D\!D}}}^{\!*}-{p_{B1}^\mathrm{{ND}}}^{\!*}=\frac{1}{3}t+\frac{1}{24}(s_{B}^{H}-s_{A}^{H})\).

Because t satisfies the assumption \(t>\frac{5}{16}|s_{A}^{H}-s_{B}^{H}|\), which can deduce that \(\frac{1}{3}t>\frac{5}{48}|s_{A}^{H}-s_{B}^{H}|>\frac{1}{24}|s_{A}^{H}-s_{B}^{H}|\), thus \({p_{A1}^\mathrm{{\!D\!D}}}^{\!*}-{p_{A1}^\mathrm{{ND}}}^{\!*}>0\) and \({p_{B1}^\mathrm{{\!D\!D}}}^{\!*}-{p_{B1}^\mathrm{{ND}}}^{\!*}>0\). \(\square \)

Proof of Proposition 5

Through Propositions 2 and 3, we obtain that the second-period prices for new consumers in the case of single price discrimination and dual price discrimination are equal, i.e., \({p_{A2}^\mathrm{{N\!-\!D\!D}}}^{\!*}={p_{A2}^\mathrm{{N\!-\!SD}}}^{\!*}\) and \({p_{B2}^\mathrm{{N\!-\!D\!D}}}^{\!*}={p_{B2}^\mathrm{{N\!-\!SD}}}^{\!*}\).

In addition, comparing the retailer’s second-period price for new consumers in the case of price discrimination and the second-period price in the case of no price discrimination, we obtain: \({p_{A2}^\mathrm{{N\!-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{ND}}}^{\!*}=-\frac{2}{3}t+\frac{1}{24}(s_{A}^{H}-s_{B}^{H})\), \({p_{B2}^\mathrm{{N\!-\!D\!D}}}^{\!*}-{p_{B2}^\mathrm{{ND}}}^{\!*}=-\frac{2}{3}t+\frac{1}{24}(s_{B}^{H}-s_{A}^{H})\).

Based on the assumption, we can deduce that \(-\frac{2}{3}t<-\frac{5}{24}|s_{A}^{H}-s_{B}^{H}|<-\frac{1}{24}|s_{A}^{H}-s_{B}^{H}|\). Thus, we can obtain the results as shown in Proposition 5. \(\square \)

Proof of Proposition 6

In the case of BBP and no price discrimination, the prices of the current type-H consumers in the second period are \({p_{A2}^\mathrm{{O\!-\!SD}}}^{\!*}\) and \({p_{A2}^\mathrm{{\!ND}}}^{\!*}\).

First, comparing the current type-H consumers’ second-period prices in the case of BBP with that in other cases, we can obtain: \({p_{A2}^\mathrm{{O\!-\!SD}}}^{\!*}-{p_{A2}^\mathrm{{H\!-\!D\!D}}}^{\!*}=-\frac{1}{4}s_{A}^{H}<0\), \({p_{B2}^\mathrm{{O\!-\!SD}}}^{\!*}-{p_{B2}^\mathrm{{H\!-\!D\!D}}}^{\!*}=-\frac{1}{4}s_{B}^{H}<0\), \({p_{A2}^\mathrm{{O\!-\!SD}}}^{\!*}-{p_{A2}^\mathrm{{\!ND}}}^{\!*}=-\frac{1}{3}t-\frac{1}{48}s_{A}^{H}-\frac{1}{48}s_{B}^{H}<0\), \({p_{B2}^\mathrm{{O\!-\!SD}}}^{\!*}-{p_{B2}^\mathrm{{\!ND}}}^{\!*}=-\frac{1}{3}t+\frac{1}{48}s_{A}^{H}-\frac{1}{48}s_{B}^{H}<0\), which can deduce the results as shown in Proposition 6 (i).

Next, comparing the second-period prices for the current type-H consumers in the case of dual price discrimination and no price discrimination, we can obtain: \({p_{A2}^\mathrm{{H\!-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{\!ND}}}^{\!*}=-\frac{1}{3}t+\frac{11}{48}s_{A}^{H}+\frac{1}{48}s_{B}^{H}\) and \({p_{B2}^\mathrm{{H\!-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{\!ND}}}^{\!*}=-\frac{1}{3}t+\frac{1}{48}s_{A}^{H}+\frac{11}{48}s_{B}^{H}\).

Solving \({p_{A2}^\mathrm{{H\!-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{\!ND}}}^{\!*}<0\) for t, we obtain \(t>\frac{1}{16}s_{B}^{H}+\frac{11}{16}s_{A}^{H}\). Combined with the assumptions, it can deduce that if \(s_{A}^{H}<\frac{1}{4}s_{B}^{H}\), \({p_{A2}^\mathrm{{H\!-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{\!ND}}}^{\!*}<0\) holds for all t, while if \(s_{A}^{H}>\frac{1}{4}s_{B}^{H}\), \({p_{A2}^\mathrm{{H\!-\!D\!D}}}^{\!*}-{p_{A2}^\mathrm{{\!ND}}}^{\!*}<0\) holds for \(t>\frac{1}{16}s_{B}^{H}+\frac{11}{16}s_{A}^{H}\). \(\square \)

Proof of Proposition 7

The proof is similar to Proposition 6. \(\square \)

Proof of Proposition 8

Through the comparison between \({p_{i2}^\mathrm{{H\!-\!D\!D}}}^{\!*}\) and \({p_{i2}^\mathrm{{L\!-\!D\!D}}}^{\!*}\), \({p_{i2}^\mathrm{{H\!-\!D\!D}}}^{\!*}\) and \({p_{i2}^\mathrm{{N\!-\!D\!D}}}^{\!*}\), it can easily obtain the results shown in Proposition 8 (i).

Because the current type-H and type-L consumers in the same channel face the same price under the cases of BBP and non-price discrimination, and comparing different types of consumers between different channels, we obtain: \({p_{A2}^\mathrm{{O\!-\!SD}}}^{\!*}-{p_{B2}^\mathrm{{O\!-\!SD}}}^{\!*}=\frac{1}{8}(s_{A}^{H}-s_{B}^{H})\). It can deduce that \({{p_{A2}^\mathrm{{O\!-\!SD}}}^{\!*}}^{\!*}-{{p_{B2}^\mathrm{{O\!-\!SD}}}^{\!*}}^{\!*}>0\) holds if \(s_{A}^{H}>s_{B}^{H}\). Similarly, solving \({p_{A2}^\mathrm{{ND}}}^{\!*}-{p_{B2}^\mathrm{{ND}}}^{\!*}<0\), we obtain \(s_{A}^{H}>s_{B}^{H}\). \(\square \)

Proof of Proposition 9

Given the retailers’ first-period demands: \({D_{A1}^\mathrm{{\!ND}}}^{\!*}=1-\frac{s_{A}^{H}-s_{B}^{H}}{6t}\), \({D_{A1}^\mathrm{{\!SD}}}^{\!*}=1-\frac{s_{A}^{H}-s_{B}^{H}}{16t}\), \({D_{A1}^\mathrm{{\!D\!D}}}^{\!*}=1-\frac{s_{A}^{H}-s_{B}^{H}}{16t}\), \({D_{B1}^\mathrm{{\!ND}}}^{\!*}=1+\frac{s_{A}^{H}-s_{B}^{H}}{6t}\), \({D_{B1}^\mathrm{{\!SD}}}^{\!*}=1+\frac{s_{A}^{H}-s_{B}^{H}}{16t}\), \({D_{B1}^\mathrm{{\!D\!D}}}^{\!*}=1+\frac{s_{A}^{H}-s_{B}^{H}}{16t}\), we can obtain the results shown in Proposition 9 (i) through simple comparisons.

Based on the previous analysis, we obtain if \(s_{A}^{H}>s_{B}^{H}\), then \(D_{A1}^{j*}<D_{B1}^{j*}\), otherwise \(D_{A1}^{j*}>D_{B1}^{j*}\), \(j\in \{\mathrm{ND}, \mathrm{SD}, \mathrm{DD}\}\). According to the definition \(\varDelta _{A1-B1}^\mathrm{{D\!D}}=|{D_{A1}^\mathrm{{D\!D}}}^{\!*}-{D_{B1}^\mathrm{{D\!D}}}^{\!*}|=|1-\frac{s_{A}^{H}-s_{B}^{H}}{8t}|\), \(\varDelta _{A1-B1}^\mathrm{{SD}}=|{D_{A1}^\mathrm{{SD}}}^{\!*}-{D_{B1}^\mathrm{{SD}}}^{\!*}|=|1-\frac{s_{A}^{H}-s_{B}^{H}}{8t}|\), \(\varDelta _{A1-B1}^\mathrm{{ND}}=|{D_{A1}^\mathrm{{ND}}}^{\!*}-{D_{B1}^\mathrm{{ND}}}^{\!*}|=|1-\frac{s_{A}^{H}-s_{B}^{H}}{3t}|\), we obtain that if \(s_{A}^{H}>s_{B}^{H}\), \(\varDelta _{A1-B1}^\mathrm{{D\!D}}=\varDelta _{A1-B1}^\mathrm{{SD}}=\frac{s_{A}^{H}-s_{B}^{H}}{8t}-1\), \(\varDelta _{A1-B1}^\mathrm{{ND}}=\frac{s_{A}^{H}-s_{B}^{H}}{3t}-1\), which can deduce \(\varDelta _{A1-B1}^\mathrm{{D\!D}}=\varDelta _{A1-B1}^\mathrm{{SD}}<\varDelta _{A1-B1}^\mathrm{{ND}}\). And if \(s_{A}^{H}<s_{B}^{H}\), \(\varDelta _{A1-B1}^\mathrm{{D\!D}}=\varDelta _{A1-B1}^\mathrm{{SD}}=1-\frac{s_{A}^{H}-s_{B}^{H}}{8t}\), \(\varDelta _{A1-B1}^\mathrm{{ND}}=\frac{1-s_{A}^{H}-s_{B}^{H}}{3t}\), which can also deduce \(\varDelta _{A1-B1}^\mathrm{{D\!D}}=\varDelta _{A1-B1}^\mathrm{{SD}}<\varDelta _{A1-B1}^\mathrm{{ND}}\). \(\square \)

Proof of Proposition 10

Given the retailers’ second-period demands: \({D_{A2}^\mathrm{{\!ND}}}^{\!*}=1-\frac{s_{A}^{H}-s_{B}^{H}}{6t}\), \({D_{A2}^\mathrm{{\!SD}}}^{\!*}=1-\frac{5(s_{A}^{H}-s_{B}^{H})}{16t}\), \({D_{A2}^\mathrm{{\!D\!D}}}^{\!*}=1-\frac{5(s_{A}^{H}-s_{B}^{H})}{16t}\), \({D_{B2}^\mathrm{{\!ND}}}^{\!*}=1+\frac{s_{A}^{H}-s_{B}^{H}}{6t}\), \({D_{B2}^\mathrm{{\!SD}}}^{\!*}=1+\frac{5(s_{A}^{H}-s_{B}^{H})}{16t}\), \({D_{B2}^\mathrm{{\!D\!D}}}^{\!*}=1+\frac{5(s_{A}^{H}-s_{B}^{H})}{16t}\), the following proof is similar to Proposition 9. \(\square \)

Proof of Proposition 11

Comparing the two-period demands as given in the proof of Propositions 9 and 10 under different pricing discriminations, we can obtain: \(\varTheta _{A2-A1}^\mathrm{{\!D\!D}}={D_{A2}^\mathrm{{\!D\!D}}}^{\!*}-{D_{A1}^\mathrm{{\!D\!D}}}^{\!*}=\frac{s_{B}^{H}-s_{A}^{H}}{4t}\), \(\varTheta _{A2-A1}^\mathrm{{\!SD}}={D_{A2}^\mathrm{{\!SD}}}^{\!*}-{D_{A1}^\mathrm{{\!SD}}}^{\!*}=\frac{s_{B}^{H}-s_{A}^{H}}{4t}\), \(\varTheta _{B2-B1}^\mathrm{{\!D\!D}}={D_{B2}^\mathrm{{\!D\!D}}}^{\!*}-{D_{B1}^\mathrm{{\!D\!D}}}^{\!*}=\frac{s_{A}^{H}-s_{B}^{H}}{4t}\), \(\varTheta _{B2-B1}^\mathrm{{\!SD}}={D_{B2}^\mathrm{{\!SD}}}^{\!*}-{D_{B1}^\mathrm{{\!SD}}}^{\!*}=\frac{s_{A}^{H}-s_{B}^{H}}{4t}\). Therefore, Proposition 11 can be deduced naturally. \(\square \)

Proof of Proposition 12

According to the results given in the paper, retailers’ profits in the first period are: \({\pi _{A1}^\mathrm{{\!ND}}}^{\!*}=\frac{(s_{A}^{H}-s_{B}^{H}-6t)^2}{36t}\), \({\pi _{A1}^\mathrm{{SD}}}^{\!*}=\frac{(s_{A}^{H}-s_{B}^{H}-16t)(3s_{A}^{H}-3s_{B}^{H}-32t)}{384t}\), \({\pi _{A1}^\mathrm{{D\!D}}}^{\!*}=\frac{(s_{A}^{H}-s_{B}^{H}-16t)(3s_{A}^{H}-3s_{B}^{H}-32t)}{384t}\).

Comparing \({\pi _{A1}^\mathrm{{\!ND}}}^{\!*}\) and \({\pi _{A1}^\mathrm{{\!SD}}}^{\!*}\), we obtain: \({\pi _{A1}^\mathrm{{\!ND}}}^{\!*}-{\pi _{A1}^\mathrm{{\!SD}}}^{\!*}=\frac{-384t^2-144(s_{A}^{H}-s_{B}^{H})t+23(s_{A}^{H}-s_{B}^{H})^2}{1152t}\). Solving \({\pi _{A1}^\mathrm{{\!ND}}}^{\!*}-{\pi _{A1}^\mathrm{{\!SD}}}^{\!*}=0\) at t, we obtain \({\hat{t}}_{1}=\frac{1}{8}(\frac{\sqrt{219}}{6}-\frac{3}{2})(s_{A}^{H}-s_{B}^{H})\) and \({\hat{t}}_{2}=\frac{1}{8}(-\frac{\sqrt{219}}{6}-\frac{3}{2})(s_{A}^{H}-s_{B}^{H})\).

Obviously, if \(s_{A}^{H}-s_{B}^{H}>0\) and \(t<{\hat{t}}_{1}\), or \(s_{A}^{H}-s_{B}^{H}<0\) and \(t<{\hat{t}}_{2}\), we obtain \({\pi _{A1}^\mathrm{{\!ND}}}^{\!*}>{\pi _{A1}^\mathrm{{\!SD}}}^{\!*}\). However, based on the assumption that \(t>\frac{5}{16}(s_{A}^{H}-s_{B}^{H})\), we find that if \(s_{A}^{H}-s_{B}^{H}>0\), then \(t>{\hat{t}}_{1}\); thus, the cost disadvantage retailer A’s first-period profits satisfies \({\pi _{A1}^\mathrm{{\!ND}}}^{\!*}<{\pi _{A1}^\mathrm{{\!SD}}}^{\!*}={\pi _{A1}^\mathrm{{\!D\!D}}}^{\!*}\).

The proofs of retailer’s B first-period profits and retailers’ second-period profits under different pricing strategies are similar. \(\square \)

Proof of Proposition 13

Comparing \({\pi _{A2}^\mathrm{{\!SD}}}^{\!*}\) and \({\pi _{A1}^\mathrm{{\!SD}}}^{\!*}\), \({\pi _{A2}^\mathrm{{\!D\!D}}}^{\!*}\) and \({\pi _{A1}^\mathrm{{\!D\!D}}}^{\!*}\), we obtain:

$$\begin{aligned}&{\pi _{A2}^\mathrm{{\!SD}}}^{\!*}\!-\!{\pi _{A1}^\mathrm{{\!SD}}}^{\!*}\\&\quad =\frac{-1792t^2-288(s_{A}^{H}-s_{B}^{H})t+99{(s_{A}^{H}-s_{B}^{H})}^2}{2304t},\\&{\pi _{A2}^\mathrm{{\!D\!D}}}^{\!*}\!-\!{\pi _{A1}^\mathrm{{\!D\!D}}}^{\!*}\\&\quad =\frac{-1792t^2\!-\!288(s_{A}^{H}-s_{B}^{H})t\!+\!99(s_{A}^{H}-s_{B}^{H})^2+144{s_{A}^{H}}(s_{A}^{H}-2s_{B}^{H})}{2304t}, \end{aligned}$$

thus \({\pi _{A2}^\mathrm{{\!D\!D}}}^{\!*}-{\pi _{A1}^\mathrm{{\!D\!D}}}^{\!*}\!-\!({\pi _{A2}^\mathrm{{\!SD}}}^{\!*}\!-\!{\pi _{A1}^\mathrm{{\!SD}}}^{\!*})\!=\!\frac{144{s_{A}^{H}}(s_{A}^{H}\!-\!2s_{B}^{H})}{2304t}.\)

Clearly, when \(s_{A}^{H}-2s_{B}^{H}>0\), \({\pi _{A2}^\mathrm{{\!D\!D}}}^{\!*}-{\pi _{A1}^\mathrm{{\!D\!D}}}^{\!*}>{\pi _{A2}^\mathrm{{\!SD}}}^{\!*}-{\pi _{A1}^\mathrm{{\!SD}}}^{\!*}\). Similarly, we can obtain the results of retailer B’s profits. \(\square \)

Proof of Proposition 14

According to the results obtained in Propositions 1, 2, and 3, we compare the retailer A’s profits \({\pi _{A}^\mathrm{{ND}}}^{\!*}\), \({\pi _{A}^\mathrm{{SD}}}^{\!*}\) and \({\pi _{A}^\mathrm{{D\!D}}}^{\!*}\) in pairs as follows:

$$\begin{aligned} \begin{aligned}&{\pi _{A}^\mathrm{{D\!D}}}^{\!*}\!-\!{\pi _{A}^\mathrm{{ND}}}^{\!*}\\&\quad =\frac{-256t^2-288(s_{A}^{H}-s_{B}^{H})t\!+\!7(s_{A}^{H}-s_{B}^{H})^2\!+\!144{s_{A}^{H}}(s_{A}^{H}-2s_{B}^{H})}{2304t},\\&{\pi _{A}^\mathrm{{D\!D}}}^{\!*}-{\pi _{A}^\mathrm{{SD}}}^{\!*}=\frac{s_{A}^{H}(s_{A}^{H}-2s_{B}^{H})}{16t},\\&{\pi _{A}^\mathrm{{SD}}}^{\!*}-{\pi _{A}^\mathrm{{ND}}}^{\!*}=\frac{-256t^2-288(s_{A}^{H}-s_{B}^{H})t+7(s_{A}^{H}-s_{B}^{H})^2}{2304t}.\\ \end{aligned} \end{aligned}$$

Solving \({\pi _{A}^\mathrm{{D\!D}}}^{\!*}-{\pi _{A}^\mathrm{{ND}}}^{\!*}>0\) and \({\pi _{A}^\mathrm{{D\!D}}}^{\!*}-{\pi _{A}^\mathrm{{SD}}}^{\!*}>0\), we obtain

$$\begin{aligned}&t{<}\frac{9}{16}(s_{A}^{H}-s_{B}^{H}){+}\frac{1}{8} \sqrt{22(s_{A}^{H}-s_{B}^{H})^2+36s_{A}^{H}(s_{A}^{H}-2s_{B}^{H})}\\&\quad \mathrm {and} \ s_{A}^{H}>2s_{B}^{H}. \end{aligned}$$

In combination with assumption, we can deduce the result in Proposition 14 (i).

Solving \({\pi _{A}^\mathrm{{SD}}}^{\!*}-{\pi _{A}^\mathrm{{D\!D}}}^{\!*}>0\) and \({\pi _{A}^\mathrm{{SD}}}^{\!*}-{\pi _{A}^\mathrm{{ND}}}^{\!*}>0\), we obtain

$$\begin{aligned} t<\frac{1}{16}(9+2\sqrt{22})(s_{A}^{H}-s_{B}^{H}) \ \ \mathrm {and} \ s_{B}^{H}<s_{A}^{H}<2s_{B}^{H}. \end{aligned}$$

In combination with assumption, we can deduce the result in Proposition 14 (ii).

In other situations, Proposition 14 (iii) is established.

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Xu, M., Tang, W. & Zhou, C. Price discrimination based on purchase behavior and service cost in competitive channels. Soft Comput 24, 2567–2588 (2020). https://doi.org/10.1007/s00500-019-03760-7

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