Analysis on the influence of retailer's introduction of store brand under manufacturer's product line strategy

Yuanyuan Jiao; Yuhang Li; Jun Du

doi:10.3934/jimo.2021220

Article Contents

2023, Volume 19, Issue 2: 1084-1106. Doi: 10.3934/jimo.2021220

This issue Previous Article Distributionally robust multi-period portfolio selection subject to bankruptcy constraints Next Article Selling by clicks or leasing by bricks? A dynamic game for pricing durable products in a dual-channel supply chain

Analysis on the influence of retailer's introduction of store brand under manufacturer's product line strategy

1.
Business School, Nankai University, Tianjin 300071, China
2.
College of Management and Economics, Tianjin University, Tianjin 300072, China

^* Corresponding author: Yuhang Li
^* Corresponding author: Yuhang Li

Received: February 28, 2021

Revised: July 31, 2021

Early access: December 2021

Published: February 2023

This work was supported by the National Natural Science Foundation of China [grant numbers 71672095

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Abstract

As the introduction of store brand (SB) by retailers becomes more common, national brand manufacturers (NBM) in the market often respond with a product line strategy. Then how should retailers decide their product lines under the situation that NBM adopt product line strategy? Under the manufacturer's product line strategy, this paper compares and analyzes three competitive strategies for retailers to introduce high-end SB, low-end SB and SB product line. The results show that: First, no matter what kind of competition strategy the retailer adopts, (1) the introduction of SB occupies the market share of NB; (2) the retailer gains the highest unit income from the high-end products of NB manufacturers. Second, the relationship between production cost and encroachment effects directly affects the retailer's choice of optimal competitive strategy and the manufacturer's preference for its competitive strategy choice. Finally, from the perspective of profit maximization, the optimal choice for retailers is the product line competition strategy, but this strategy is not obvious compared with the high-end competition strategy. Therefore, when retailers introduce product lines, if they are effectively threatened by national brand manufacturers (for example, NBM refuses to provide national brands to retailers), retailers may choose the suboptimal strategy, that is, to introduce store brand high-end products. When the high-end products of store brands are better than the low-end products of national brands and the production cost of products is relatively high, this strategy just meets the expectations of national brand manufacturers for retailers to introduce strategies.

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Mathematics Subject Classification: Primary: 90B06, 91-10.

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Figure 1. Encroachment of demand under high-end competition strategy $ \left(\sigma>\theta\right) $

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Figure 2. Encroachment of demand under high-end competition strategy $ \left(\sigma<\theta\right) $

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Figure 3. Encroachment of demand under low-end competition strategy

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Figure 4. Encroachment of demand under product line competition strategy$ \left(\sigma>\theta\right) $

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Figure 5. Encroachment of demand under product line competition strategy$ \left(\sigma<\theta\right) $

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Figure 6. Profit comparison of retailer under different competitive strategies$ \left(\sigma>\theta\right) $

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Figure 7. Profit comparison of retailer under different competitive strategies$ \left(\sigma<\theta\right) $

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Figure 8. Profit comparison of NBM under different competitive strategies$ \left(\sigma>\theta\right) $

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Figure 9. Profit comparison of NBM under different competitive strategies$ \left(\sigma<\theta\right) $

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Table 1. The notations used in this paper

Notations	Explanations
NB; SB	National Brand; Store Brand
NH; NL;SH; SL	NB high-end product; NB Low-end product; SB high-end product; SB Low-end product
$ u_{ij} $	The utility of consumers' purchasing $ j $ products of brand $ i $, where $ i={N,S};j={H,L} $
$ \theta,\sigma,\upsilon $	brand internal encroachment effect; brand external encroachment effect; Consumers' evaluation of NH
$ w_{ij},p_{ij} $	the wholesale price of $ j $ product of brand $ i $; the retail price of $ j $ product of brand $ i $
$ c_{ij},q_{ij} $	The production cost of $ j $ products of brand $ i $; The demand of $ j $ products of brand $ i $
$ \pi_{M}^{k},\pi_{R}^{k} $	Manufacturer's profit; retailer's profit; where $ k = 0,1,2,3 $ respectively indicate that there is no SB, SH competition strategy, SL competition strategy, product line competition strategy
$ \Delta_{H}^{k},\Delta_{L}^{k} $	The difference between the retailer's profit from unit SH and SL and the profit from the benchmark model
$ (x,y) $	The corresponding profit expressions for retailers under different introduction strategies, $ k=0,1,2,3 $ corresponding to $ (x, y)={(0,0),(1,0),(0,1),(1,1)} $

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Table 2. Changes in market share of national brands under different competitive strategies

strategy type	Encroachment effect relationship	NH's market share has encroached	NL's market share has encroached	Market Expansion
High-end competition strategy (Introduction of SH)	$ \sigma >\theta $	A1	B1	/
High-end competition strategy (Introduction of SH)	$ \sigma <\theta $	/	A2	B2
Low-end competition strategy(Introduction of SL)	$ \theta\sigma <\theta $	/	A3	B3
Competitive strategy of product line (Introduction of SH and SL)	$ \sigma >\theta $	A4	B4+A5	B5
	$ \sigma <\theta $	/	A6	B6+A7
Notes: The values of A1-A7 and B1-B6 are shown in Figure 1 to Figure 5.

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Table 3. The equilibrium price of each product under different introduction strategies of store brands

Strategy type	encroachment effect relationship	$ w_{NH} $	$ w_{NL} $	$ p_{NH} $	$ p_{NL} $	$ p_{SH} $	$ p_{SL} $
Benchmark Model	/	$ \frac{1+c_{NH}}{2} $	$ \frac{\theta+c_{NL}}{2} $	$ \frac{1+c_{NH}}{2} $	$ \frac{3\theta+c_{NL}}{4} $	/	/
High-end competition strategy (Introduction of SH)	$ \sigma >\theta $	$ \frac{1-\sigma+c_{NH}+c_{SH}}{2} $	$ \frac{\sigma c_{NL}+\theta c_{SH}}{2} $	$ \frac{3-\sigma+c_{NH}+c_{SH}}{4} $	$ \frac{2\theta\sigma+\sigma c_{NL}+\theta c_{SH}}{4\sigma} $	$ \frac{\sigma+c_{SH}}{2} $	/
High-end competition strategy (Introduction of SH)	$ \sigma <\theta $	$ \frac{1-\sigma+c_{NH}+c_{SH}}{2} $	$ \frac{\theta-\sigma+c_{NL}+c_{SH}}{2} $	$ \frac{3-\sigma+c_{NH}+c_{SH}}{4} $	$ \frac{3\theta-\theta\sigma+c_{NL}+c_{SH}}{4} $	$ \frac{\sigma+c_{SH}}{2} $	/
Low-end competition strategy (Introduction of SL)	$ \theta\sigma <\theta $	$ \frac{1-\theta\sigma+c_{NH}+c_{SL}}{2} $	$ \frac{\theta-\theta\sigma+c_{NL}+c_{SL}}{2} $	$ \frac{3-\theta\sigma+c_{NH}+c_{SL}}{4} $	$ \frac{3\theta-\theta\sigma+c_{NL}+c_{SL}}{4} $	/	$ \frac{\theta\sigma+c_{SL}}{2} $
Competitive strategy of product line (Introduction of SH and SL)	$ \sigma >\theta $	$ \frac{1-\sigma+c_{NH}+c_{SH}}{2} $	$ \frac{\left( \sigma-\theta\right)c_{SL}}{2\sigma\left(1-\theta\right)}+\frac{\theta\left( 1-\sigma\right) c_{SH}}{2\sigma\left( 1-\theta\right)} +\frac{c_{NL}}{2} $	$ \frac{3-\sigma+c_{NH}+c_{SH}}{4} $	$ \frac{\left( \sigma-\theta\right)c_{SL}}{4\sigma\left(1-\theta\right)}+\frac{\theta\left( 1-\sigma\right) c_{SH}}{4\sigma\left( 1-\theta\right)} +\frac{c_{NL}+2\theta}{4} $	$ \frac{\sigma+c_{SH}}{2} $	$ \frac{\theta\sigma+c_{SL}}{2} $
	$ \sigma <\theta $	$ \frac{1-\sigma+c_{NH}+c_{SH}}{2} $	$ \frac{\theta-\sigma+c_{NL}+c_{SH}}{2} $	$ \frac{3-\sigma+c_{NH}+c_{SH}}{4} $	$ \frac{3\theta-\theta\sigma+c_{NL}+c_{SH}}{4} $	$ \frac{\sigma+c_{SH}}{2} $	$ \frac{\theta\sigma+c_{SL}}{2} $

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Table 4. The equilibrium price of each product under different introduction strategies of store brands

Strategy type	$ \sigma >\theta $		$ \sigma <\theta $
High-end competition strategy (Introduction of SH)	$ \Delta_{H}^{1}=\frac{\sigma-\sigma^2 c}{4} $	$ \Delta_{L}^{1}=\frac{\theta-\theta\sigma c}{4} $	$ \Delta_{H}^{1}=\frac{\sigma-\sigma^2 c}{4} $	$ \Delta_{H}^{1}=\frac{\sigma-\sigma^2 c}{4} $
Low-end competition strategy (Introduction of SL)	$ \Delta_{H}^{2}=\frac{\theta\sigma-\sigma^2 c}{4} $	$ \Delta_{L}^{2}=\frac{\theta\sigma+\theta^2\sigma^2 c}{4} $	$ \Delta_{H}^{2}=\frac{\theta\sigma-\sigma^2 c}{4} $	$ \Delta_{L}^{2}=\frac{\theta\sigma+\theta^2\sigma^2 c}{4} $
Competitive strategy of product line (Introduction of SH and SL)	$ \Delta_{H}^{3}=\frac{\sigma-\sigma^2 c}{4} $	$ \Delta_{L}^{3}=\frac{\theta-\theta c\left[ \sigma\left( 1-\sigma\right) +\theta\sigma\right]}{4} $	$ \Delta_{H}^{3}=\frac{\sigma-\sigma^2 c}{4} $	$ \Delta_{L}^{3}=\frac{\sigma+\sigma^2 c}{4} $

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Table 5. The impact of price encroachment under different competitive strategy

Strategy type	Encroachment effect relationship	$ w_{NH},p_{NH} $	$ w_{NL},p_{NL} $	$ p_{SH} $	$ p_{sL} $
High-end competition strategy (Introduction of SH)	$ \sigma >\theta $	$ \sigma <\frac{1}{2c}\left(\uparrow \right) $; $ \sigma >\frac{1}{2c}\left( \downarrow\right) $	$ \uparrow $	$ \uparrow $	/
High-end competition strategy (Introduction of SH)	$ \sigma <\theta $	$ \sigma <\frac{1}{2c}\left(\uparrow \right) $; $ \sigma >\frac{1}{2c}\left( \downarrow\right) $	$ \sigma <\frac{1}{2c}\left(\uparrow \right) $; $ \sigma >\frac{1}{2c}\left( \downarrow\right) $	$ \uparrow $	/
Low-end competition strategy (Introduction of SL)	$ \theta\sigma <\theta $	$ \theta\sigma <\frac{1}{2c}\left(\uparrow \right) $; $ \theta\sigma >\frac{1}{2c}\left( \downarrow\right) $	$ \theta\sigma <\frac{1}{2c}\left(\uparrow \right) $; $ \theta\sigma >\frac{1}{2c}\left( \downarrow\right) $	/	$ \uparrow $
Competitive strategy of product line (Introduction of SH and SL)	$ \sigma >\theta $	$ \sigma <\frac{1}{2c}\left(\uparrow \right) $; $ \sigma >\frac{1}{2c}\left( \downarrow\right) $	$ \sigma <\frac{1+\theta}{2}\left(\uparrow \right) $; $ \sigma >\frac{1+\theta}{2c}\left( \downarrow\right) $	$ \uparrow $	$ \uparrow $
	$ \sigma <\theta $	$ \sigma <\frac{1}{2c}\left(\uparrow \right) $; $ \sigma >\frac{1}{2c}\left( \downarrow\right) $	$ \sigma <\frac{1}{2c}\left(\uparrow \right) $; $ \sigma >\frac{1}{2c}\left( \downarrow\right) $	$ \uparrow $	$ \uparrow $
Notes:"$\uparrow$" indicates that the price increases with the increase of encroachment effect, that is, the positive effect; "$\downarrow$" indicates that the price decreases with the increase of encroachment effect, that is, negative influence; "/" means not affected by encroachment effect.

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Table 6. The order of prices under different competition strategies

Stragegy type	$ c >\frac{1}{\sigma\left( 1+\theta\right) } $	$ c <\frac{1}{\sigma\left( 1+\theta\right) } $	$ \frac{\left( 1-\sigma\right) }{\sigma\left( 1-\theta\sigma\right) } <c <\frac{1}{\sigma\left( 1+\theta\right) } $	$ c >\frac{1}{\sigma\left( 1+\theta\right) } $	$ c <\frac{\left( 1-\sigma\right) }{\sigma\left( 1-\theta\sigma\right) } $
$ \sigma >\theta $	$ w_{NH}^{0} >w_{NH}^{1}=w_{NH}^{3} >w_{NH}^{2} $	$ w_{NH}^{0} >w_{NH}^{2} >w_{NH}^{1}=w_{NH}^{3} $	$ w_{NL}^{0} >w_{NL}^{1} >w_{NL}^{2} >w_{NL}^{3} $	$ w_{NL}^{0} >w_{NL}^{1} >w_{NL}^{3} >w_{NL}^{2} $	$ w_{NL}^{0} >w_{NL}^{2} >w_{NL}^{1} >w_{NL}^{3} $
$ \sigma >\theta $	$ p_{NH}^{0} >p_{NH}^{1}=p_{NH}^{3} >p_{NH}^{2} $	$ p_{NH}^{0} >p_{NH}^{2} >p_{NH}^{1}=p_{NH}^{3} $	$ p_{NL}^{0} >p_{NL}^{1} >p_{NL}^{2} >p_{NL}^{3} $	$ p_{NL}^{0} >p_{NL}^{1} >p_{NL}^{3} >p_{NL}^{2} $	$ p_{NL}^{0} >p_{NL}^{2} >p_{NL}^{1} >p_{NL}^{3} $
Stragegy type	$ c >\frac{1}{\sigma\left( 1+\theta\right) } $	$ c <\frac{1}{\sigma\left( 1+\theta\right) } $	$ c >\frac{1}{\sigma\left( 1+\theta\right) } $	$ c <\frac{1}{\sigma\left( 1+\theta\right) } $	$ {} $
$ \sigma <\theta $	$ w_{NH}^{0} >w_{NH}^{1}=w_{NH}^{3} >w_{NH}^{2} $	$ w_{NH}^{0} >w_{NH}^{2} >w_{NH}^{1}=w_{NH}^{3} $	$ w_{NL}^{0} >w_{NL}^{1}+w_{NL}^{3} >w_{NL}^{2} $	$ w_{NL}^{0} >w_{NL}^{2} >w_{NL}^{1}+w_{NL}^{3} $	$ {} $
$ \sigma <\theta $	$ p_{NH}^{0} >p_{NH}^{1}=p_{NH}^{3} >p_{NH}^{2} $	$ p_{NH}^{0} >p_{NH}^{2} >p_{NH}^{1}=p_{NH}^{3} $	$ p_{NL}^{0} >p_{NL}^{1}=p_{NL}^{3} >p_{NL}^{2} $	$ p_{NL}^{0} >p_{NL}^{2} >p_{NL}^{1}=p_{NL}^{3} $	$ {} $

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Table 7. Optimal profit for manufacturers and retailers under different introduction strategies

Benchmark Model	/	$ \left\{ \begin{array}{l} \pi_{M}^{0}=\frac{1-2c+c^2 E}{8}\\ \pi_{R}^{0}=\frac{1-2c+c^2 E}{16}\end{array} \right. $
High-end competition strategy (Introduction of SH)	$ \sigma >\theta $	$ \left\{ \begin{array}{l} \pi_{M}^{1}=\frac{A+c^2 B}{8} \\ \pi_{R}^{1}=\frac{D+c^2 \left( B+4\Sigma ^3\right) }{16}\end{array} \right. $
High-end competition strategy (Introduction of SH)	$ \sigma <\theta $	$ \left\{ \begin{array}{l} \pi_{M}^{1}=\frac{A+\left( 1+\sigma\right) c^2 \left[ E+\sigma\left( 1+\theta+\sigma\right) \right] }{8}\\ \pi_{R}^{1}=\frac{D+c^2 \left[ E+\sigma\left( \theta^2 -\theta\sigma+3\sigma^2\right) \right] }{16}\end{array} \right. $
Low-end competition strategy (Introduction of SL)	$ \theta\sigma <\theta $	$ \left\{ \begin{array}{l} \pi_{M}^{2}=\frac{F+\left( 1-\theta\sigma\right) c^2 \left[ E+\theta\sigma\left( 1+\theta\sigma+\theta^2\right) \right] }{8} \\ \pi_{R}^{2}=\frac{G+c^2 \left[ E+\theta^3\sigma\left( 1 -\sigma+3\sigma^2\right) \right] }{16}\end{array} \right. $
Competitive strategy of product line (Introduction of SH and SL)	$ \sigma >\theta $	$ \left\{ \begin{array}{l} \pi_{M}^{3}=\frac{F+\left( 1-\sigma\right) c^2 \left[ 1+\sigma\left( 2-\theta^2+\theta^3\right) +E \sigma^2\right] }{8} \\ \pi_{R}^{3}=\frac{D+c^2 \left[ B+\theta^3\left( 1-\sigma\right) +\sigma^3\left( 1+3E\right) \right] }{16}\end{array} \right. $
	$ \sigma <\theta $	$ \left\{ \begin{array}{l} \pi_{M}^{3}=\frac{A+\left( 1-\sigma\right) c^2 \left[ E+\sigma\left( 1+\theta+\sigma\right) \right] }{8} \\ \pi_{R}^{3}=\frac{D+c^2 \left[ E+\theta^3\left( 1-2\theta\right) \left( 3+2\theta\right) -\theta\sigma\left( \sigma-\theta\right) \right] }{16}\end{array} \right. $
Remark	/	$ A=1-\sigma-2c\left( 1-2\sigma^2\right);B=1+\sigma\left( 1-\theta^2\right) -\sigma^2\left( 1-\theta\right) - \\ \sigma^3;D=1+3\sigma-2c\left( 1+3\sigma^2\right) ;E=1+\theta-\theta^2;F= \\ \left( 1-\theta\sigma\right) -2c\left( 1-\theta^2\sigma^2\right) ;G=1+3\theta\sigma-2c\left( 1+3\theta^3\sigma^3\right) $

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