The Implications of Supplier Learning on Retailer’s Private Label Encroachment

Abstract

This study examines retailers’ decisions of whether and when to introduce private labels considering the supplier learning effect. We construct a two-period framework including a supplier and a retailer with national brand and private label divisions and elucidate equilibrium results under the decentralized structure. Our analysis shows that the retailer’s optimal choices of private label encroachment are related to the supplier learning rate and product competition intensity. Specifically, the retailer chooses the first-period encroachment of private labels when product competition is weak; when encountering fierce product competition, the retailer’s choices transition from no encroachment to the second-period encroachment and, ultimately, to the first-period encroachment of private labels as the supplier learning effect elevates. Moreover, the retailer’s first-period encroachment of private labels benefits the supplier, consumers and society, but its second-period encroachment and even no encroachment of private labels would be harmful to those firms, especially when product competition intensity is high and the supplier learning rate is relatively low. Our study offers implications for industry participants to make choices of private label encroachment and understand how it influences the supply chain ecosystem in the presence of the supplier learning effect.

1. Introduction

Online retailing has become an essential part of global e-commerce trade in the internet era. Amid the relentless digitalization of modern life, consumers enjoy the inherent benefits including the convenience and the accessibility from online retailing. According to a report from Statista, global online retailing sales approximated 6 trillion U.S. dollars in 2024. This key indicator is projected to follow a 31% growth over ensuing years, with the online retailing market size expected to reach 8 trillion U.S. dollars by 2028 [1]. There are an estimated 2.77 billion individuals engaging in consumption activities from retailers [2].

Beyond serving as distribution channels regarding suppliers’ national brands, retailers introduce private labels, like Amazon Basics, Costco’s Kirkland, and JD’s Jingzao, to leverage consumer behaviors and enhance profitability [3,4]. The Private Label Manufacturers Association shows that aggregate market sales hit 271 billion U.S. dollars in 2024, within a 23.6% increase in annual sales over the prior four years [5]. Meanwhile, in the context of selling national brands, retailers are always confronted with the decision of whether and when to introduce private labels: first-period encroachment versus second-period encroachment. The first-period encroachment refers to retailers introducing private labels for concurrent sales with national brands, while the second-period encroachment refers to retailers doing so for subsequent sales to national brands. In business operations, both encroachment options are separately adopted by retailers. For example, Amazon offers its private label Amazon Basics batteries for sale alongside national brand supplier’s Duracell batteries on its site, while JD’s private label Jingzao introduces its proprietary bicycles only after national brand supplier Fujita wholesales bicycles to JD for resale.

For upstream suppliers, the market-derived supplier learning effect represents a pivotal determinant in production and operations management. This phenomenon signifies the decline in unit cost accompanying cumulative production, owing to the enhanced experience and operational efficiency [6,7,8]. Its underlying drivers encompass improved labor and managerial familiarity, manufacturability optimization, technology advancement, etc., in various production and manufacturing industries, like electronics, daily necessities and apparel. Initial products generally involve poorly defined processes, which affords substantial improvement potential through applicable methods such as process standardization and technical empowerment. Moreover, based on cumulative production, the supplier’s existing production experience also significantly influences cost efficiency in subsequent periods [9,10]. For example, Unilever produces consumer packaged goods based on the established manufacturing system; it has attained 5% labor productivity growth and 8% cost reduction because of the supplier learning effect [11].

The issue of private label encroachment, also defined as private label introduction, has received extensive attention in recent literature [12,13,14,15]. Most of the existing studies focus on investigating whether retailers should introduce private labels. To our knowledge, there are few academic works considering the retailer’s decisions of private label encroachment (first-period encroachment versus second-period encroachment), although this is conspicuous in practical online retailing. Several scholars investigate the impacts of product quality, outsourcing, and producer or pricing leadership on the retailer’s decision of whether to introduce private labels [4,16,17]. Thus, the current theory fails to elucidate how the supplier learning effect shapes the retailer’s strategic decision regarding whether and when to introduce private labels. Meanwhile, retailer JD decentralizes its private label and national brand divisions independently with individual order authority; thus, it is not clear how the decentralized division structure influences the retailer’s choices of private label encroachment. Based on the aforementioned motivation examples and research gaps, we construct theoretical models to explore the following questions: (1) What conditions does the retailer choose to introduce private labels considering the supplier learning effect? (2) Is the retailer more likely to choose the first-period encroachment with the supplier learning effect due to the second-period production cost advantage? (3) What are the implications of the retailer’s private label encroachment on supply chain stakeholders (i.e., the supplier, consumers and society)? For these purposes, we construct a stylized supply chain model to study strategic interactions with a supplier and a retailer and aim to examine the implications of the supplier learning effect on the retailer’s decisions of private label encroachment. We analyze three operation strategies (no encroachment, first-period encroachment, and second-period encroachment) with/without the supplier learning effect. There are three main results derived from our analysis.

Regarding the retailer’s decisions of private label encroachment, the retailer should balance the interaction between product competition and the supplier learning effect. When product competition intensity is low, the retailer always chooses the first-period encroachment of private labels to capitalize on the production cost reduction. When product competition is high, the retailer would choose from no encroachment to second-period encroachment and then to first-period encroachment as the supplier learning effect becomes more significant. Moreover, the intuition that the retailer should introduce private labels as early as possible to leverage the production cost reduction advantage with the supplier learning effect only holds when product competition intensity is low. Specifically, in contexts of relatively fierce product competition, as the supplier learning effect prevails, the retailer’s choices shift from no encroachment to the second-period encroachment. Only when the supplier learning effect arrives at a significant level does its choice switch back to the first-period encroachment. Finally, the retailer’s optimal choices regarding private label encroachment are not necessarily aligned with the supplier’s profit, supply chain profit, consumer surplus and social welfare. In particular, consumers and society prefer the first-period encroachment of private labels, whereas in equilibrium, the retailer delays or even avoids its encroachment when product competition is fierce and the supplier learning effect is not significant.

Our study provides the following contributions. First, earlier literature focuses on whether retailers introduce private labels within a single period, whereas we incorporate the possibility of multiple periods. Thus, our study can provide guidance for retailers on whether to introduce private labels and when to do so. Second, we encompass operations dimensions explicitly in the supplier learning effect and product competition and highlight how these market factors impact the retailer’s decisions of private label encroachment. As noticed earlier, private label business continues to increase, and it is vital for retailers to adopt the optimal private label encroachment options to elevate revenues under the specific business environment. Finally, our study indicates that, in a competitive market, retailers are not always better off introducing private labels as early as possible. Under some market contexts, retailers could benefit from postponing or even deterring private label encroachment when considering the supplier learning effect.

The remainder of our study is as follows. Section 2 summarizes the literature. Section 3 formulates the model. Section 4 examines equilibrium results with/without the supplier learning effect. We further study the retailer’s encroachment decisions and their impacts on the supply chain ecosystem in Section 5. We conclude with our results in Section 6. All proofs are summarized in Appendix A.

2. Literature Review

Our paper relates to the literature on private label encroachment. The extensive studies focus on private label encroachment from market factors including outsourcing producer [17,18,19], product quality [12,16], brand differentiation [20,21,22], channel encroachment [23,24], etc. Kumar et al. [25] show that the retailer’s sourcing strategies for private labels depend on consumer types, consumer valuations and marginal requirement. Fang et al. [26] indicate that retailers choose to introduce private labels if national brands have a unit quality cost disadvantage relative to private labels. Liao et al. [17] point out that, when sourcing from the national brand supplier, the retailer introduces private labels with relatively lower quality in comparison to scenarios of the independent supplier and the insourcing. Zhong and Huo [27] show that the government subsidy makes the retailer have more incentives to introduce private labels, which improves environmental performance but harms the supplier. Xiao et al. [14] demonstrate that the retailer opts to introduce private labels when the private label exhibits relatively high quality, whereas it does not do so when its own leadership is dominant. Jin et al. [28] investigate that the retailer has a greater (lesser) tendency to introduce private labels in a dual-channel model within the scenario of different (unified) wholesale price. Li et al. [4] find that the retailer often introduces private labels under the wholesale structure, while it is less likely to do that under the platform structure. However, our paper discusses whether and when the retailer should introduce private labels (first-period encroachment versus second-period encroachment) under the decentralized division structure where the retailer delegates order authority to managers of individual brand divisions. Moreover, we analyze the role of the supplier learning effect, a market factor characterized by its scarcity in private label literature but pervasiveness in practical retailing operations.

Our study also relates to the stream on the supplier learning effect, where the supplier’s per-unit production cost declines with order quantity due to the accumulation of knowledge base and manufacturing experience [8,29]. Wright [6] proposes initially that the labor cost decreases effectively as the production quantity elevates. Then, most literature focuses on this effect on the firms’ manufacturing and operations management [10,30,31,32,33]. Shum et al. [34] investigate how supplier learning influences its pricing decision when interacting with strategic consumers and find that the cost reduction may be harmful to the supplier. Feng and Chan [35] explore how the supplier decides optimal lot-sizing and pricing strategies when considering the supplier learning effect. Hong et al. [8] study how the supplier learning effect shapes its procurement choices and show that decentralized procurement benefits the supplier when both product competition and the supplier learning effect are high. Gao et al. [10] show that the supplier learning effect benefits or harms the supplier depending on the cost-reduction efficiency and the information asymmetry friction in a procurement context. Gupta et al. [36] find that supplier entry is not an optimal option if the supplier learning effect is pronounced and the operation cost of the direct channel is moderate. Different from the mentioned studies in terms of product pricing, outsourcing, procurement and channel encroachment, we build a stylized theoretical model to examine how the supplier learning effect shapes the retailer’s choices of private label encroachment. Accordingly, we construct a two-period framework incorporating the supplier learning effect and counterintuitively find that the retailer may choose the second-period encroachment and even no encroachment of private labels although the cost reduction is considerable in the first-period encroachment scenario.

3. Model Setting

This study constructs a supply chain where a supplier (referred as $s$) produces and sells national brand products (referred as $n$) to a retailer (referred as $r$) at the wholesale price $w_n$, and the retailer sells products to consumers at the retail price $p_n$. Besides national brand products, the retailer assesses the introduction of private label products (referred to as $p$), which are offered by the supplier with the wholesale price $w_p$ and subsequently sold by its private label division at the retail price $p_p$. For example, JD has outsourced its private label product “Jingantu” lubricant to a leading lubricant supplier FUCHS, who also wholesales the same type of national brand products as JD. As mentioned in the Introduction, the supplier learning effect plays a role in large-scale production industries, such as consumer electronics, consumer-packaged goods and domestic appliances. For example, Anker, who is a dominant electronics supplier, distributes portable charger products to Amazon and also produces similar items for Amazon’s private label—Amazon Basics. It can achieve a considerable production cost reduction via offering Amazon a larger order quantity. Given the common production line and product technics for two products (as shown in products of Anker and Amazon Basics), the initial unit production cost of both products is assumed to be identical and denoted as $c$, which further serves as that in the first period, that is, $c_1=c.$ This ensures that our results are driven exclusively by the trade-off between the supplier learning effect and product competition rather than other factors such as asymmetric production cost, aligning with the motivation of our study and rendering the research question interesting; otherwise, we achieve an intuitive outcome that the retailer introduces private labels within an advantageous production cost efficiency over national brands and vice versa. Following the supplier learning effect model of Gupta et al. [36] and Gao et al. [10], the unit production cost in the second period exhibits a linear decrease with respect to the order quantity in the first period, i.e., $c_2=c-\phi \sum q_{1i}$, where $\phi \in \left[0,1\right]$ represents the supplier learning rate in which it produces the unit product and $q_{1i}$ represents the first-period order quantity of product $i$ ($i=\left\{n,p\right\}$). It implies that the higher the quantity the retailer’s two divisions order in the first period, the higher the cost reduction they may obtain in the second period.

We strive to examine the retailer’s optimal choices of private label encroachment under the decentralized division structure in which the retailer’s national brand and private label divisions operate with independent authority over product procurement and order quantity decisions and divisional managers are authorized and incentivized to maximize the profitability of their own respective units, while the retailer strives to choose the optimal private label encroachment strategy based on the sum of divisional profits. Moreover, there are no internal transfer prices or coordination mechanisms between the two divisions, which operate with a high degree of autonomy and maintain separate profit and loss (P&L) statements. This decentralized setting is common in retail practice (i.e., JD’s division operations) and is also universally employed in the existing literature [8,37]. The inverse demand function is shown as $p_{ti}=a-q_{ti}-\theta q_{tj}$ for product $i$ in period $t$, where $a$ shows the maximum market price to pay for two products and $\theta \in \left[0,1\right]$ shows the degree of homogenization between two products and indicates product competition intensity of two products ($j=\left\{n,p\right\}$, $t=\left\{\mathrm{1,2}\right\}$). When $\theta =0$, two products are independent; when $\theta =1$, both are perfectly homogeneous. Similarly, when the retailer chooses not to introduce private labels and the retailer only sells national brand products, then the demand function for national brand products in period $t$ is $p_{tn}=a-q_{tn}$. The inverse demand function is universally adopted in existing studies [38,39,40,41,42]. When the retailer chooses the first-period (no or second-period) encroachment of private labels, the supplier provides order quantity $Q_1=\sum q_{1i}$ ($Q_1=q_{1n}$) for the retailer in the first period, and the unit production cost in the second period is $c_2=c-\phi Q_1$. To ensure our model yields interior solutions where second-period production cost, all order quantities, wholesale/retail prices, and margin profits are strictly non-negative, we assume that the initial unit production cost is relatively high, namely, ${\displaystyle \frac{\left[2{\left(2+\theta \right)}^2+\left(5+2\theta \right)\phi \right]\phi a}{2\left(2+\theta +\phi \right){\left(2+\theta \right)}^2-{\phi }^2}}\le c\le a$. This constraint ensures that the cost reduction achieved through the supplier learning effect does not drive the production cost to an infeasible negative value in the second period, thus preserving the economic validity of our analysis. The notations and definitions are shown in

Table 1. Summary of notations.

Notation	Definition
a	The maximum market price to pay for two products
c	The initial unit production cost of two products
$\theta$	Product competition intensity of two products
$\phi$	Supplier learning rate
$p_{ti}$	Retail price for product i in period t, where $t=\left\{1,2\right\}$$, i=\left\{n,p\right\}$
${\pi }_f$	$\mathrm{Profit} \mathrm{of} \mathrm{firm} f, \mathrm{where} f=\left\{s,r\right\}$
$\pi$	Supply chain profit
CS	Consumer surplus
SW	Social welfare
Decision variables
$w_{ti}$	Wholesale price for product i in period t
$q_{ti}$	Order quantity for product i in period t

.

The game sequence is illustrated in Figure 1. First, the retailer decides whether and when to introduce private labels, including no encroachment, first-period encroachment, and second-period encroachment, which can be seen as a long-term decision. Then, based on the retailer’s encroachment choices, the sequences of each period are shown as follows:

(1) If the retailer chooses not to introduce private labels, the supplier sets the wholesale price $w_{tn}$; then, the retailer’s national brand division sets the order quantity $q_{tn}$.

(2) If the retailer chooses to introduce private labels in the first period, the supplier sets wholesale prices $w_{tn}$ and $w_{tp}$; then, the retailer’s national brand and private label divisions set order quantities $q_{tn}$ and $q_{tp}$, respectively.

(3) If the retailer chooses to introduce private labels in the second period, the supplier sets the wholesale price $w_{1n}$; then, the retailer’s national brand division sets the order quantity $q_{1n}$ in the first period; in the second period, the supplier sets wholesale prices $w_{2n}$ and $w_{2p}$; then, the retailer’s two divisions set order quantities $q_{2n}$ and $q_{2p}$, respectively.

4. Equilibrium Analysis

4.1. Without the Supplier Learning Effect

In this section, we explore three scenarios of private label encroachment decisions for the retailer without the supplier learning effect, where the supplier’s unit production costs of both periods are identical. We obtain equilibrium results related with no encroachment, first-period encroachment, and second-period encroachment, which are represented by superscripts “ON”, “OF”, and “OS”, respectively.

4.1.1. No Encroachment of Private Label (Strategy ON)

Under the premise of the no encroachment scenario, the retailer only sells the supplier’s national brand; then, the supplier’s and retailer’s profit functions in the second period are

$${\pi }_{s2}^{ON}=\left(w_{2n}^{ON}-c\right)q_{2n}^{ON},$$

(1)

$${\pi }_{r2}^{ON}=\left(p_{2n}^{ON}-w_{2n}^{ON}\right)q_{2n}^{ON}.$$

(2)

The total profit functions of the two firms during the two periods are

$${\pi }_s^{ON}=\left(w_{1n}^{ON}-c\right)q_{1n}^{ON}+{\pi }_{s2}^{ON},$$

(3)

$${\pi }_r^{ON}{=\left(p_{1n}^{ON}-w_{1n}^{ON}\right)q_{1n}^{ON}+\pi }_{r2}^{ON}.$$

(4)

Using backward induction, the equilibrium results under the two periods are summarized in Lemma 1.

Lemma 1.

Without the supplier learning effect, the equilibrium results under no encroachment of private labels are as follows: (1) Under two periods:$w_{tn}^{ON\ast }={\displaystyle \frac{a+c}{2}}$, $q_{tn}^{ON\ast }={\displaystyle \frac{a-c}{4}}$and $p_{tn}^{ON\ast }={\displaystyle \frac{3a+c}{4}}$; (2) The two-period total profits of two firms: ${\pi }_s^{ON\ast }={\displaystyle \frac{{(a-c)}^2}{4}}$, ${\pi }_r^{ON\ast }={\displaystyle \frac{{(a-c)}^2}{8}}$.

4.1.2. First-Period Encroachment of Private Label (Strategy OF)

We now consider the scenario where the retailer introduces private labels in the first period. Similar to Section 4.1.1, the second-period profit functions of the supplier’s and retailer’s divisions are

$${\pi }_{s2}^{OF}=\left(w_{2n}^{OF}-c\right)q_{2n}^{OF}+\left(w_{2p}^{OF}-c\right)q_{2p}^{OF},$$

(5)

$${\pi }_{r2i}^{OF}=\left(p_{2i}^{OF}-w_{2i}^{OF}\right)q_{2i}^{OF}.$$

(6)

Given the second-period equilibrium decisions, the total profit functions of the supplier’s and retailer’s divisions are

$${\pi }_s^{OF}=\left(w_{1n}^{OF}-c\right)q_{1n}^{OF}+\left(w_{1p}^{OF}-c\right)q_{1p}^{OF}+{\pi }_{s2}^{OF},$$

(7)

$${\pi }_{ri}^{OF}=\left(p_{1i}^{OF}-w_{1i}^{OF}\right)q_{1i}^{OF}+{\pi }_{r2i}^{OF}.$$

(8)

We utilize backward induction to obtain the equilibrium results, as Lemma 2 shows.

Lemma 2.

Without the supplier learning effect, the equilibrium results under the first-period private label encroachment are as follows: (1) Under two periods: $w_{ti}^{OF\ast }={\displaystyle \frac{a+c}{2}}$, $q_{ti}^{OF\ast }={\displaystyle \frac{a-c}{2\left(2+\theta \right)}}$, $p_{ti}^{OF\ast }={\displaystyle \frac{3a+c+a\theta +c\theta }{2(2+\theta )}}$; (2) The two-period total profits of two firms: ${\pi }_s^{OF\ast }={\displaystyle \frac{{(a-c)}^2}{2+\theta }}$and ${\pi }_r^{OF\ast }={\displaystyle \frac{{(a-c)}^2}{{\left(2+\theta \right)}^2}}$.

4.1.3. Second-Period Encroachment of Private Label (Strategy OS)

We now consider the scenario where the retailer introduces private labels in the second period. Similarly, the second-period profit functions of the supplier’s and retailer’s divisions are

$${\pi }_{s2}^{OS}=\left(w_{2n}^{OS}-c\right)q_{2n}^{OS}+\left(w_{2p}^{OS}-c\right)q_{2p}^{OS},$$

(9)

$${\pi }_{r2i}^{OS}=\left(p_{2i}^{OS}-w_{2i}^{OS}\right)q_{2i}^{OS}.$$

(10)

Given the equilibrium decisions in the second period, the total profit functions of the supplier’s and retailer’s national brand division are

$${\pi }_s^{OS}=\left(w_{1n}^{OS}-c\right)q_{1n}^{OS}+{\pi }_{s2}^{OS},$$

(11)

$${\pi }_{rn}^{OS}=\left(p_{1n}^{OS}-w_{1n}^{OS}\right)q_{1n}^{OS}+{\pi }_{r2n}^{OS}.$$

(12)

We use backward induction to obtain the equilibrium results, as Lemma 3 shows.

Lemma 3.

Without the supplier learning effect, the equilibrium results under the second-period private label encroachment are as follows: (1) Under two periods: $w_{1n}^{OS\ast }=w_{2i}^{OS\ast }={\displaystyle \frac{a+c}{2}}$, $q_{1n}^{OS\ast }={\displaystyle \frac{a-c}{4}}$, $p_{1n}^{OS\ast }={\displaystyle \frac{3a+c}{4}}$, $q_{2i}^{OS\ast }={\displaystyle \frac{a-c}{2\left(2+\theta \right)}}$, $p_{2i}^{OS\ast }={\displaystyle \frac{3a+c+a\theta +c\theta }{2(2+\theta )}}$; (2) The two-period total profits of two firms: ${\pi }_s^{OS\ast }={\displaystyle \frac{\left(6+\theta \right){(a-c)}^2}{8\left(2+\theta \right)}}$and ${\pi }_r^{OS\ast }={\displaystyle \frac{\left(12+4\theta +{\theta }^2\right){(a-c)}^2}{16{\left(2+\theta \right)}^2}}$.

4.2. With the Supplier Learning Effect

In this section, we examine three scenarios of private label encroachment decisions for the retailer with the supplier learning effect, where the supplier’s unit production cost in the second period is linearly decreased with the first-period order quantity. We obtain equilibrium results related to no encroachment, first-period encroachment, and second-period encroachment, which are represented by superscripts “WN”, “WF”, and “WS”, respectively.

4.2.1. No Encroachment of Private Label (Strategy WN)

We analyze the benchmark case under no introduction, where the supplier only sells national brand products via the retailer. We start with the second period of the game model. Based on the order quantity $q_{1n}^{WN}$, the supplier’s second-period unit production cost is $c_2^{WN}=c-\phi q_{1n}^{WN}$. Then, the supplier’s and the retailer’s second-period profit functions are

$${\pi }_{s2}^{WN}=\left(w_{2n}^{WN}-c_2^{WN}\right)q_{2n}^{WN},$$

(13)

$${\pi }_{r2}^{WN}=\left(p_{2n}^{WN}-w_{2n}^{WN}\right)q_{2n}^{WN}.$$

(14)

By using backward induction, the total profit functions of the two firms are

$${\pi }_s^{WN}=\left(w_{1n}^{WN}-c\right)q_{1n}^{WN}+{\pi }_{s2}^{WN},$$

(15)

$${\pi }_r^{WN}=\left(p_{1n}^{WN}-w_{1n}^{WN}\right)q_{1n}^{WN}+{\pi }_{r2}^{WN}.$$

(16)

Similarly, we obtain equilibrium results with the supplier learning effect under the no encroachment scenario, as Lemma 4 shows.

Lemma 4.

With the supplier learning effect, the equilibrium results under no encroachment of private label are as follows:

(1) In the first period: $w_{1n}^{WN\ast }={\displaystyle \frac{128\left(a+c\right)-16\left(a-c\right)\phi -8\left(3a+c\right){\phi }^2-(a-c){\phi }^3}{32(8-{\phi }^2)}}$, $q_{1n}^{WN\ast }={\displaystyle \frac{\left(a-c\right)(8+3\phi )}{4(8-{\phi }^2)}}$, $p_{1n}^{WN\ast }={\displaystyle \frac{8\left(3a+c\right)-3\left(a-c\right)\phi -4a{\phi }^2}{4(8-{\phi }^2)}}$;

(2) In the second period: $w_{2n}^{WN\ast }={\displaystyle \frac{32\left(a+c\right)-8\left(a-c\right)\phi +\left(7a+c\right){\phi }^2}{8(8-{\phi }^2)}}$, $q_{2n}^{WN\ast }={\displaystyle \frac{\left(a-c\right)(32+8\phi -{\phi }^2)}{16(8-{\phi }^2)}}$, $p_{2n}^{WN\ast }={\displaystyle \frac{32\left(3a+c\right)-8\left(a-c\right)\phi -\left(15a+c\right){\phi }^2}{16(8-{\phi }^2)}}$;

(3) The two-period total profits of two firms: ${\pi }_s^{WN\ast }={\displaystyle \frac{(128+48\phi +{\phi }^2){(a-c)}^2}{64(8-{\phi }^2)}}$, ${\pi }_r^{WN\ast }={\displaystyle \frac{(2048+768\phi -176{\phi }^2-48{\phi }^3+7{\phi }^4){(a-c)}^2}{256{(8-{\phi }^2)}^2}}$.

4.2.2. First-Period Encroachment of Private Label (Strategy WF)

Similar to Section 4.2.1, we start with the second period. Based on order quantities of the retailer’s division $q_{1n}^{WF}$ and $q_{1p}^{WF}$, the supplier’s second-period unit production cost is $c_2^{WF}=c-\phi \left(q_{1n}^{WF}+q_{1p}^{WF}\right)$. Then, the second-period profit functions of the supplier’s and retailer’s divisions are

$${\pi }_{s2}^{WF}=\left(w_{2n}^{WF}-c_2^{WF}\right)q_{2n}^{WF}+\left(w_{2p}^{WF}-c_2^{WF}\right)q_{2p}^{WF},$$

(17)

$${\pi }_{r2i}^{WF}=\left(p_{2i}^{WF}-w_{2i}^{WF}\right)q_{2i}^{WF}.$$

(18)

Given the second-period equilibrium decisions via backward induction, the total profit functions of the supplier’s and retailer’s divisions are

$${\pi }_s^{WF}=\left(w_{1n}^{WF}-c\right)q_{1n}^{WF}+\left(w_{1p}^{WF}-c\right)q_{1p}^{WF}+{\pi }_{s2}^{WF},$$

(19)

$${\pi }_{ri}^{WF}=\left(p_{1i}^{WF}-w_{1i}^{WF}\right)q_{1i}^{WF}+{\pi }_{r2i}^{WF}.$$

(20)

We obtain the equilibrium results, as Lemma 5 shows.

Lemma 5.

With the supplier learning effect, the equilibrium results under the first-period encroachment of private labels are as follows:

(1) In the first period: $w_{1i}^{WF\ast }={\displaystyle \frac{\begin{array}{c}2{\left(2+\theta \right)}^5\left(a+c\right)-\left(3+2\theta \right){\left(2+\theta \right)}^3\left(a-c\right)\phi \\ -2{\left(2+\theta \right)}^2\left(5a+c+2a\theta \right){\phi }^2-\left(a-c\right){\phi }^3\end{array}}{4{\left(2+\theta \right)}^2\left[{\left(2+\theta \right)}^3-\left(3+\theta \right){\phi }^2\right]}}$, $q_{1i}^{WF\ast }={\displaystyle \frac{\left[2{\left(2+\theta \right)}^2+(5+2\theta )\phi \right]\left(a-c\right)}{4\left[{(2+\theta )}^3-\left(3+\theta \right){\phi }^2\right]}}$, $p_{1i}^{WF\ast }={\displaystyle \frac{2{\left(2+\theta \right)}^2\left(3a+c+a\theta +c\theta \right)-\left(5+2\theta \right)\left(1+\theta \right)\left(a-c\right)\phi -4a\left(3+\theta \right){\phi }^2}{4\left[{(2+\theta )}^3-\left(3+\theta \right){\phi }^2\right]}}$;

(2) In the second period: $w_{2i}^{WF\ast }={\displaystyle \frac{2{\left(2+\theta \right)}^3\left(a+c\right)-2{\left(2+\theta \right)}^2\left(a-c\right)\phi -\left(11a+c+4a\theta \right){\phi }^2}{4\left[{(2+\theta )}^3-\left(3+\theta \right){\phi }^2\right]}}$, $q_{2i}^{WF\ast }={\displaystyle \frac{\left[2{\left(2+\theta \right)}^3+2{\left(2+\theta \right)}^2\phi -\phi \right]\left(a-c\right)}{4\left(2+\theta \right)\left[{(2+\theta )}^3-\left(3+\theta \right){\phi }^2\right]}}$, $p_{2i}^{WF\ast }={\displaystyle \frac{\begin{array}{c}2{\left(2+\theta \right)}^3\left(3a+c+a\theta +c\theta \right)-2\left(1+\theta \right){\left(2+\theta \right)}^2\left(a-c\right)\phi \\ -\left(23a+c+19a\theta +c\theta +4a{\theta }^2\right){\phi }^2\end{array}}{4\left(2+\theta \right)\left[{(2+\theta )}^3-\left(3+\theta \right){\phi }^2\right]}}$;

(3) The two-period total profits of two firms: ${\pi }_s^{WF\ast }={\displaystyle \frac{\left[8{\left(2+\theta \right)}^4+4\left(5+2\theta \right){\left(2+\theta \right)}^2\phi +{\phi }^2\right]{(a-c)}^2}{8{\left(2+\theta \right)}^2\left[{(2+\theta )}^3-\left(3+\theta \right){\phi }^2\right]}}$, ${\pi }_r^{WF\ast }={\displaystyle \frac{\left[8{\left(2+\theta \right)}^6+4\left(7+3\theta \right){\left(2+\theta \right)}^4\phi -\left(3+2\theta \right){\left(2+\theta \right)}^2{\phi }^2-4\left(3+\theta \right){\left(2+\theta \right)}^2{\phi }^3+2\left(3+\theta \right){\phi }^4\right]{(a-c)}^2}{8{\left(2+\theta \right)}^2{\left[{(2+\theta )}^3-\left(3+\theta \right){\phi }^2\right]}^2}}$.

4.2.3. Second-Period Encroachment of Private Label (Strategy WS)

Similarly, given order quantity $q_{1n}^{WS}$, the supplier’s second-period unit production cost is $c_2^{WS}=c-\phi q_{1n}^{WS}$. Then, the profit functions of the supplier’s and retailer’s divisions in the second period are

$${\pi }_{s2}^{WS}=\left(w_{2n}^{WS}-c_2^{WS}\right)q_{2n}^{WS}+\left(w_{2p}^{WS}-c_2^{WS}\right)q_{2p}^{WS},$$

(21)

$${\pi }_{r2i}^{WS}=\left(p_{2i}^{WS}-w_{2i}^{WS}\right)q_{2i}^{WS}.$$

(22)

Given the equilibrium decisions in the second period via backward induction, the total profit functions of the supplier’s and retailer’s national brand divisions are

$${\pi }_s^{WS}=\left(w_{1n}^{WS}-c\right)q_{1n}^{WS}+{\pi }_{s2}^{WS},$$

(23)

$${\pi }_{rn}^{WS}=\left(p_{1n}^{WS}-w_{1n}^{WS}\right)q_{1n}^{WS}+{\pi }_{r2n}^{WS}.$$

(24)

Similarly, we also obtain the equilibrium results, as Lemma 6 shows.

Lemma 6.

With the supplier learning effect, the equilibrium results under the second-period encroachment of private labels are as follows:

(1) In the first period: $w_{1n}^{WS\ast }={\displaystyle \frac{\begin{array}{c}8{\left(2+\theta \right)}^4\left(a+c\right)-4\left(3+2\theta \right){\left(2+\theta \right)}^2\left(a-c\right)\phi \\ -2{\left(2+\theta \right)}^2\left(5a+c+2a\theta \right){\phi }^2-(a-c){\phi }^3\end{array}}{4{\left(2+\theta \right)}^2\left[4{\left(2+\theta \right)}^2-(3+\theta ){\phi }^2\right]}}$, $q_{1n}^{WS\ast }={\displaystyle \frac{\left[2{\left(2+\theta \right)}^2+(5+2\theta )\phi \right]\left(a-c\right)}{2\left[4{\left(2+\theta \right)}^2-(3+\theta ){\phi }^2\right]}}$, $p_{1n}^{WS\ast }={\displaystyle \frac{2{\left(2+\theta \right)}^2\left(3a+c\right)-(5+2\theta )\left(a-c\right)\phi -2a(3+\theta ){\phi }^2}{2\left[4{\left(2+\theta \right)}^2-(3+\theta ){\phi }^2\right]}}$;

(2) In the second period: $w_{2i}^{WS\ast }={\displaystyle \frac{8{\left(2+\theta \right)}^2\left(a+c\right)-2{\left(2+\theta \right)}^2\left(a-c\right)\phi -\left(11a+c+4a\theta \right){\phi }^2}{4\left[4{\left(2+\theta \right)}^2-(3+\theta ){\phi }^2\right]}}$, $q_{2i}^{WS\ast }={\displaystyle \frac{\left[8{\left(2+\theta \right)}^2+2{\left(2+\theta \right)}^2\phi -{\phi }^2\right]\left(a-c\right)}{4\left(2+\theta \right)\left[4{\left(2+\theta \right)}^2-(3+\theta ){\phi }^2\right]}}$, $p_{2i}^{WS\ast }={\displaystyle \frac{\begin{array}{c}8{\left(2+\theta \right)}^2\left(3a+\theta a+c+\theta c\right)-2\left(1+\theta \right){\left(2+\theta \right)}^2\left(a-c\right)\phi \\ -\left[\left(23+19\theta +4{\theta }^2\right)a+\left(1+\theta \right)c\right]{\phi }^2\end{array}}{4\left(2+\theta \right)\left[4{\left(2+\theta \right)}^2-(3+\theta ){\phi }^2\right]}}$;

(3) The two-period total profits of two firms: ${\pi }_s^{WS\ast }={\displaystyle \frac{\left[\begin{array}{c}4\left(6+\theta \right){\left(2+\theta \right)}^3\\ +4\left(5+2\theta \right){\left(2+\theta \right)}^2\phi +{\phi }^2\end{array}\right]{(a-c)}^2}{8{\left(2+\theta \right)}^2\left[4{\left(2+\theta \right)}^2-(3+\theta ){\phi }^2\right]}}$, ${\pi }_r^{WS\ast }={\displaystyle \frac{\left[\begin{array}{c}4{\left(2+\theta \right)}^4\left(12+4\theta +{\theta }^2\right)+4\left(7+2\theta \right){\left(2+\theta \right)}^4\phi -{\left(2+\theta \right)}^2\left(3-12\theta -4{\theta }^2\right){\phi }^2\\ -2\left(3+\theta \right){{\left(2+\theta \right)}^2\phi }^3+\left(3+\theta \right){\phi }^4\end{array}\right]{(a-c)}^2}{4{\left(2+\theta \right)}^2{\left[4{\left(2+\theta \right)}^2-(3+\theta ){\phi }^2\right]}^2}}$.

4.3. Comparison of Equilibrium Results

To distill the underlying mechanisms of the retailer’s private label encroachment considering the supplier learning effect, we contrast the equilibrium results (i.e., wholesale prices, retail prices, order quantities and corresponding margin profits) with/without the supplier learning effect, as the following shows.

4.3.1. Comparison Outcomes Without the Supplier Learning Effect

We compare the equilibrium results for three scenarios without the supplier learning effect: no encroachment, first-period encroachment, and second-period encroachment. The analytical results are summarized in Proposition 1.

Proposition 1.

(1) $w_{1n}^{ON\ast }=w_{1n}^{OS\ast }=w_{1n}^{OF\ast }$, $w_{2n}^{ON\ast }=w_{2n}^{OS\ast }=w_{2n}^{OF\ast }$, $w_{2p}^{OS\ast }=w_{2p}^{OF\ast }$;

(2) $q_{1n}^{ON\ast }=q_{1n}^{OS\ast }\ge q_{1n}^{OF\ast }$ ${q_{2n}^{ON\ast }\ge q_{2n}^{OS\ast }=q}_{2n}^{OF\ast }$, $q_{2p}^{OF\ast }=q_{2p}^{OS\ast }$;

(3) $p_{1n}^{ON\ast }=p_{1n}^{OS\ast }\ge p_{1n}^{OF\ast }$, $p_{2n}^{ON\ast }\ge p_{2n}^{OS\ast }=p_{2n}^{OF\ast }$, $p_{2p}^{OF\ast }=p_{2p}^{OS\ast }$;

(4) $\left(w_{1n}^{ON\ast }-c\right)=\left(w_{1n}^{OS\ast }-c\right)=\left(w_{1n}^{OF\ast }-c\right)$, $\left(w_{2n}^{ON\ast }-c\right)=\left(w_{2n}^{OS\ast }-c\right)=\left(w_{2n}^{OF\ast }-c\right)$, $\left(w_{2p}^{OF\ast }-c\right)=\left(w_{2p}^{OS\ast }-c\right)$;

(5) $\left(p_{1n}^{OS\ast }-w_{1n}^{OS\ast }\right)=\left(p_{1n}^{ON\ast }-w_{1n}^{ON\ast }\right)\ge \left(p_{1n}^{OF\ast }-w_{1n}^{OF\ast }\right)$, $\left(p_{2n}^{ON\ast }-w_{2n}^{ON\ast }\right)\ge \left(p_{2n}^{OF\ast }-w_{2n}^{OF\ast }\right)=\left(p_{2n}^{OS\ast }-w_{2n}^{OS\ast }\right)$, $\left(p_{2p}^{OF\ast }-w_{2p}^{OF\ast }\right)=\left(p_{2p}^{OS\ast }-w_{2p}^{OS\ast }\right)$.

Proposition 1 indicates that the private label encroachment in a period has no impact on the supplier’s wholesale price without the supplier learning effect, that is, the supplier sets the equal wholesale price in the first/second period for the national brand and the private label, respectively, leading to the supplier obtaining an equal margin profit eventually (i.e., $w_{ti}^{X\ast }={\displaystyle \frac{a+c}{2}}$, $w_{ti}^{X\ast }-c={\displaystyle \frac{a-c}{2}}$, where $X=\left\{ON,OF,OS\right\}$). Moreover, compared with the no encroachment scenario, the retailer’s private label encroachment in a period triggers product competition, which decreases the national brand’s order quantity, retail price, and marginal profit. Regarding the retailer’s private label division, it sets the same order quantity (retail price) for the first-period and second-period encroachment scenarios, ultimately achieving an identical marginal profit. Proposition 1 shows that, while the private label encroachment could entice the retailer to obtain revenues from the private label, it weakens the national brand division via product competition, and contemporaneous decisions depend on product competition.

We further analyze the impacts of market parameters (i.e., initial unit production cost $c$ and product competition intensity $\theta$) on the supplier’s and retailer’s profits without the supplier learning effect, as shown in Proposition 2.

Proposition 2.

The effects of market parameters on the supplier’s and retailer’s profits without the supplier learning effect are shown in

Table 2. Sensitivity analysis of firms’ profits without the supplier learning effect.

Scenarios	$(\partial {\mathit{\pi }}_{\mathit{s}}^{\mathit{X}\mathbf{\ast }}/\partial \mathit{c}$$, \partial {\mathit{\pi }}_{\mathit{s}}^{\mathit{X}\mathbf{\ast }}/\partial \mathit{\theta }$)	$(\partial {\mathit{\pi }}_{\mathit{r}}^{\mathit{Y}\mathbf{\ast }}/\partial \mathit{c}$$, \partial {\mathit{\pi }}_{\mathit{r}}^{\mathit{Y}\mathbf{\ast }}/\partial \mathit{\theta }$)
Strategy ON	(−, none)	(−, none)
Strategy OF	(−,−)	(−,−)
Strategy OS	(−,−)	(−,−)

Note: “−” and “none” represent “negative” and “no” impacts, respectively; $X=\left\{ON,OF,OS\right\}$.

.

Proposition 2 shows that an increase in the initial unit production cost exerts a negative impact on the supplier’s and retailer’s profits among the three scenarios (i.e., strategies ON, OF, and OS) and an increase in product competition intensity also reduces the two firms’ profits when the retailer chooses to introduce private labels (i.e., strategies OF and OS). This is because the increased production cost causes the supplier and the retailer to elevate the wholesale price and the retail price, respectively. This dual price increment leads to a reduction in overall order quantities, resulting in the shrinkage of profits for both the supplier and the retailer. Moreover, the elevated product competition intensifies price pressure in the end market, and then, the two divisions of the retailer are compelled to reduce order quantities and retail prices; thus, both firms are worse off from the heightened product competition.

4.3.2. Comparison Outcomes with the Supplier Learning Effect

Similarly, we compare the equilibrium results for the three scenarios with the supplier learning effect and obtain analytical results as shown in Proposition 3.

Proposition 3.

(1) $w_{1n}^{WN\ast }\ge w_{1n}^{WS\ast }\ge w_{1n}^{WF\ast }$; when $\theta \le {\theta }_1$or $\theta >{\theta }_1$and $\phi \le {\phi }_1$, $w_{2n}^{WN\ast }\ge w_{2n}^{WS\ast }\ge w_{2n}^{WF\ast }$, otherwise, $w_{2n}^{WS\ast }\ge w_{2n}^{WN\ast }\ge w_{2n}^{WF\ast }$; $w_{2p}^{WS\ast }\ge w_{2p}^{WF\ast }$;

(2) When $\theta \le {\theta }_2$and $\phi \ge {\phi }_2$, $q_{1n}^{WF\ast }{\ge q_{1n}^{WS\ast }\ge q}_{1n}^{WN\ast }$; when $\theta >{\theta }_1$and $\phi \ge {\phi }_1$, ${q_{1n}^{WN\ast }\ge q}_{1n}^{WS\ast }\ge q_{1n}^{WF\ast }$, otherwise, $q_{1n}^{WS\ast }\ge max\left\{q_{1n}^{WF\ast },q_{1n}^{WN\ast }\right\}$; when $\theta \le {\theta }_3$and $\phi \ge {\phi }_3$, $q_{2n}^{WF\ast }\ge max\left\{q_{2n}^{WS\ast },q_{2n}^{WN\ast }\right\}$, otherwise, $q_{2n}^{WN\ast }\ge q_{2n}^{WF\ast }\ge q_{2n}^{WS\ast }$; $q_{2p}^{WF\ast }\ge q_{2p}^{WS\ast }$;

(3) When $\theta \le {\theta }_1$or $\theta >{\theta }_1$and $\phi \le {\phi }_1$, $p_{1n}^{WN\ast }\ge p_{1n}^{WS\ast }\ge p_{1n}^{WF\ast }$, otherwise, $p_{1n}^{WS\ast }\ge p_{1n}^{WN\ast }\ge p_{1n}^{WF\ast }$; $p_{2n}^{WN\ast }\ge p_{2n}^{WS\ast }\ge p_{2n}^{WF\ast }$; $p_{2p}^{WS\ast }\ge p_{2p}^{WF\ast }$;

(4) $\left(w_{1n}^{WN\ast }-c\right)\ge \left(w_{1n}^{WS\ast }-c\right)\ge \left(w_{1n}^{WF\ast }-c\right)$, $\left(w_{2n}^{WF\ast }-c_2^{WF}\right)\ge \left(w_{2n}^{WS\ast }-c_2^{WS}\right)\ge \left(w_{2n}^{WN\ast }-c_2^{WN}\right)$, $\left(w_{2p}^{WF\ast }-c_2^{WF}\right)\ge \left(w_{2p}^{WS\ast }-c_2^{WS}\right)$;

(5) When $\theta \le {\theta }_4$and $\phi \ge {\phi }_4$, $\left(p_{1n}^{WF\ast }-w_{1n}^{WF\ast }\right)\ge \left(p_{1n}^{WS\ast }-w_{1n}^{WS\ast }\right)\ge \left(p_{1n}^{WN\ast }-w_{1n}^{WN\ast }\right)$, otherwise, $\left(p_{1n}^{WS\ast }-w_{1n}^{WS\ast }\right)\ge max\left\{\left(p_{1n}^{WF\ast }-w_{1n}^{WF\ast }\right), \left(p_{1n}^{WN\ast }-w_{1n}^{WN\ast }\right)\right\}$; when $\theta \le {\theta }_3$and $\phi \ge {\phi }_3$, $\left(p_{2n}^{WF\ast }-w_{2n}^{WF\ast }\right)\ge max\left\{\left(p_{2n}^{WS\ast }-w_{2n}^{WS\ast }\right), \left(p_{2n}^{WN\ast }-w_{2n}^{WN\ast }\right)\right\}$, otherwise, $\left(p_{2n}^{WN\ast }-w_{2n}^{WN\ast }\right)\ge \left(p_{2n}^{WF\ast }-w_{2n}^{WF\ast }\right)\ge \left(p_{2n}^{WS\ast }-w_{2n}^{WS\ast }\right)$; $\left(p_{2p}^{WF\ast }-w_{2p}^{WF\ast }\right)\ge \left(p_{2p}^{WS\ast }-w_{2p}^{WS\ast }\right)$.

Proposition 3 indicates the dynamics of the supplier’s and retailer’s two divisions in pricing, ordering, and margin profit across two periods. Proposition 3(1) shows that the supplier decides the lowest first-period wholesale price in the first-period encroachment case to incentivize more orders and then leverages the second-period cost advantage due to the cost learning effect, leading to the lowest wholesale prices for both products in the second period. With considerable cost learning efficiency and intense competition, the supplier charges the highest wholesale price in the second-period encroachment case to offset profit losses from the private label competition. Proposition 3(2) indicates that, with a high supplier learning rate, the resulting cost advantage allows the national brand division to order more across both periods in the first-period encroachment case when the learning effect dominates. As competition intensity rises and the competition effect becomes dominant, the division instead places more first-period orders in the no encroachment case. Driven by wholesale price differences, the private label division places more second-period orders in the first-period encroachment case. Proposition 3(3) finds that the retailer’s two divisions charge the lowest retail prices across two periods in the first-period encroachment case owing to the supplier learning effect and product competition. With weak product competition or fierce product competition and the inferior supplier learning effect, the national brand division could obtain more monopolistic profits via charging the higher retail price in the no encroachment case. Proposition 3(4) shows that the supplier achieves the highest first-period margin profit for the national brand in the no encroachment case due to the monopolistic pricing, while the highest second-period margin profit for both products is achieved in the first-period encroachment case due to the production cost advantage. Proposition 3(5) examines that, with a high learning rate, the national brand division acquires the highest margin profit for two periods in the first-period encroachment case due to the dominant supplier learning effect; as product competition intensifies, it does so for the first-period in the second-period encroachment case, considering the dominant competition effect.

We further analyze the impacts of market parameters (i.e., initial unit production cost $c$, product competition intensity $\theta$, and supplier learning rate $\phi$) on the supplier’s and retailer’s profits with the supplier learning effect, which are shown in Proposition 4.

Proposition 4.

The effects of market parameters on the supplier’s and retailer’s profits with the supplier learning effect are shown in

Table 3. Sensitivity analysis of firms’ profits with the supplier learning effect.

Scenarios	$(\partial {\mathit{\pi }}_{\mathit{s}}^{\mathit{Y}\mathbf{\ast }}/\partial \mathit{c}$$, \partial {\mathit{\pi }}_{\mathit{s}}^{\mathit{Y}\mathbf{\ast }}/\partial \mathit{\theta }$$, \partial {\mathit{\pi }}_{\mathit{s}}^{\mathit{Y}\mathbf{\ast }}/\partial \mathit{\phi }$)	$(\partial {\mathit{\pi }}_{\mathit{r}}^{\mathit{Y}\mathbf{\ast }}/\partial \mathit{c}$$, \partial {\mathit{\pi }}_{\mathit{r}}^{\mathit{Y}\mathbf{\ast }}/\partial \mathit{\theta }$$, \partial {\mathit{\pi }}_{\mathit{r}}^{\mathit{Y}\mathbf{\ast }}/\partial \mathit{\phi }$)
Strategy WN	(−, none, +)	(−, none, +)
Strategy WF	(−,−, +)	(−,−, +)
Strategy WS	(−,−, +)	(−,−, +)

Note: “+”, “−”, and “none” represent “positive”, “negative”, and “no” impacts, respectively; $Y=\left\{WN, WF,WS\right\}$.

.

Proposition 4 examines the impact of market parameters (i.e., initial unit production cost $c$, product competition intensity $\theta$, and supplier learning rate $\phi$) on profits of the supplier and the retailer in the presence of the supplier learning effect. Our findings indicate that an increase in the initial unit production cost or product competition intensity exerts negative impacts on both firms’ profits, which are consistent with the scenario without the supplier learning effect. However, an increase in the supplier learning rate effectively boosts the profits of both the supplier and the retailer. The rationale is that a higher learning rate enables the supplier to reduce its production cost in the second period, and the reduction in the production cost incentivizes the supplier to lower its wholesale price. It, in turn, motivates both divisions of the retailer to increase their order quantities. Subsequently, both the supplier and the retailer achieve higher overall profits from the elevated supplier learning rate.

5. Retailer’s Encroachment Decisions and Their Impacts

In this section, we firstly examine the retailer’s decisions of private label encroachment with/without the supplier learning effect. Then, we study whether the retailer is more likely to choose the first-period encroachment of private labels considering the supplier learning effect. Finally, we investigate the impacts of the retailer’s decisions of private label encroachment on supply chain stakeholders (i.e., the supplier, consumers and society).

5.1. Retailer’s Encroachment Decisions Without the Supplier Learning Effect

Based on the equilibrium results in Lemmas 1–3, we elucidate the retailer’s optimal decisions of private label encroachment, which are shown in Proposition 5.

Proposition 5.

Without the supplier learning effect,

(1) When  $\theta <{\theta }_5$,  ${\pi }_r^{OF\ast }\ge {\pi }_r^{OS\ast }\ge {\pi }_r^{ON\ast }$ and  $\left({\pi }_r^{OF\ast }-{\pi }_r^{OS\ast }\right)$ decreases in  $\theta$;

(2) When $\theta \ge {\theta }_5$, ${\pi }_r^{ON\ast }\ge {\pi }_r^{OS\ast }\ge {\pi }_r^{OF\ast }$, and $\left({\pi }_r^{ON\ast }-{\pi }_r^{OS\ast }\right)$increases in $\theta$.

Proposition 5 shows that, without the supplier learning effect, the choice of first-period encroachment of private labels is not always a beneficial option for the retailer. Based on the equilibrium outcomes in Proposition 1, even though introducing private labels could achieve demand expansion, it would also decrease the national brand division’s order quantity, retail price, and marginal profit. When product competition intensity is low, the retailer benefits from the private label’s demand increment, and the positive effect of the private label’s demand expansion dominates the negative effect of the national brand division’s demand shrinkage; thus, the retailer tends to introduce private labels in the first period. With the increase in product competition, the negative effect of the national brand division’s demand shrinkage intensifies, and the retailer would reduce the incentive to choose the first-period encroachment of private labels. However, when product competition intensity is high, the retailer chooses not to introduce private labels to avoid excessive losses because of intense product competition, and it is more willing to choose no encroachment of private labels with the increase in product competition. The managerial implications of Proposition 5 show that, without the supplier learning effect, when the retailer decides to introduce private label products as early as possible, it should adopt a differentiated positioning strategy to avoid falling into a homogenized competition with its own national brand divisions, leading to obtaining more profits in the first-period encroachment of private labels under lower product competition.

5.2. Retailer’s Encroachment Decisions with the Supplier Learning Effect

Based on the equilibrium results in Lemmas 4–6, we examine the retailer’s optimal decisions of private label encroachment with the supplier learning effect, which are shown in Proposition 6.

Proposition 6.

With the supplier learning effect,

(1) When $\theta <{\theta }_5$, ${\pi }_r^{WF\ast }\ge {\pi }_r^{WS\ast }\ge {\pi }_r^{WN\ast }$;

(2) When $\theta \ge {\theta }_5$, (i) If $\phi \le {\phi }_1$, ${\pi }_r^{WN\ast }\ge {\pi }_r^{WS\ast }\ge {\pi }_r^{WF\ast }$; (ii) If ${\phi }_1<\phi \le {\phi }_2$, ${\pi }_r^{WS\ast }\ge max\left\{{\pi }_r^{WF\ast }, {\pi }_r^{WN\ast }\right\}$; (iii) If $\phi >{\phi }_2$, ${\pi }_r^{WF\ast }\ge {\pi }_r^{WS\ast }\ge {\pi }_r^{WN\ast }$.

Proposition 6 shows a clear transition in the retailer’s choices of private label encroachment. Some may intuitively think that the retailer has a greater propensity to introduce private labels in the first period due to the production cost reduction advantage. However, our analysis shows that it is not always immutable. When product competition intensity is low, the retailer has an incentive to choose the first-period encroachment of private labels due to the dominant cost reduction advantage and demand expansion. Specifically, given that the retailer’s two divisions procure products from the supplier individually, the retailer expects that the two divisions compete for consumer demands via a lower retail price when product competition intensity is high, so that it would carefully consider the interplay between the advantage of production cost reduction and the weakness of retail price decline. Based on intense product competition, if the supplier learning rate is high, the retailer still chooses the first-period encroachment of private labels due to the dominant production cost reduction advantage. However, if the supplier learning rate is low, the positive role of the production cost reduction from the first-period encroachment is limited, and the retailer avoids fierce product competition and chooses not to introduce private labels. As the supplier learning rate elevates, the supplier learning effect becomes increasingly considerable, and the supplier reduces the wholesale price to entice the retailer’s divisions to order more in the first period; thus, retail prices decline accordingly. While retail prices of the two divisions decrease, margin profits of the two divisions increase with the supplier learning rate in the private label encroachment, such that the retailer chooses to do so in the second period. The analytical insight of Proposition 6 shows that retailers need to consider market factors (i.e., product competition and the supplier learning effect) in their decisions on whether and when to introduce private labels, especially under high product competition in the presence of the supplier learning effect. Since a high supplier learning rate that facilitates product cost reduction is vital for retailers to achieve cost edges in fierce product competition, retailers ought to proactively build deep partnerships with suppliers endowed with strong capabilities in cost learning.

5.3. Effects of Supplier Learning on Retailer’s Encroachment Decisions

Given the results of Propositions 5 and 6, we examine the interaction mechanisms between the private label encroachment scenarios (i.e., no encroachment, first-period encroachment, and second-period encroachment) and the supplier learning effect as follows.

Proposition 7.

The comparisons of the retailer’s profits under the three scenarios with/without the supplier learning effect feature an interval characteristic as follows.

(1) When  $\theta <{\theta }_5$,  ${\pi }_r^{OF\ast }\ge {\pi }_r^{OS\ast }\ge {\pi }_r^{ON\ast }$,  ${\pi }_r^{WF\ast }\ge {\pi }_r^{WS\ast }\ge {\pi }_r^{WN\ast }$;

(2) When  $\theta \ge {\theta }_5$,

    (a) if  $\phi \le {\phi }_1$,  ${\pi }_r^{ON\ast }\ge {\pi }_r^{OF\ast }\ge {\pi }_r^{OS\ast }$,  ${\pi }_r^{WN\ast }\ge {\pi }_r^{WS\ast }\ge {\pi }_r^{WF\ast }$;

    (b) if  ${\phi }_1<\phi \le {\phi }_2$,  ${\pi }_r^{ON\ast }\ge {\pi }_r^{OF\ast }\ge {\pi }_r^{OS\ast }$,  ${\pi }_r^{WS\ast }\ge max\left\{{\pi }_r^{WF\ast }, {\pi }_r^{WN\ast }\right\}$;

    (c) if $\phi >{\phi }_2$, ${\pi }_r^{ON\ast }\ge {\pi }_r^{OF\ast }\ge {\pi }_r^{OS\ast }$, ${\pi }_r^{WF\ast }\ge {\pi }_r^{WS\ast }\ge {\pi }_r^{WN\ast }$.

Figure 2 depicts the effects of supplier learning on the retailer’s encroachment decisions. Conventional wisdom thinks that, compared to the analysis without the supplier learning effect, the retailer is more willing to choose the first-period encroachment of private labels with the supplier learning effect due to the production cost advantage and market expansion. However, Proposition 7 shows that it is not always invariable. When product competition is low (i.e., $\theta <{\theta }_5$), the retailer indeed always chooses the first-period encroachment of private labels regardless of the supplier learning effect. Nevertheless, when product competition is high (i.e., $\theta \ge {\theta }_5$), the retailer has a higher propensity to introduce private labels in a period with the supplier learning effect compared to scenarios without the supplier learning effect. The rationale is that the retailer avoids excessive losses given fierce product competition and thus chooses no encroachment of private labels without the supplier learning effect. With the supplier learning effect, the retailer could enjoy the production cost reduction advantage in the second period due to the first-period order quantity, and it benefits more from the higher order quantity as the supplier learning effect becomes more considerable. Therefore, the retailer’s private label decisions switch from no encroachment to the second-period encroachment and, finally, to the first-period encroachment as the supplier learning rate increases. The implications of Proposition 7 indicate that the supplier learning effect could incur retailers to have more incentives to introduce private labels, especially when product competition is fierce, which provide guidelines for retailers choosing the private label under intense competition with the supplier learning effect.

5.4. Impacts of Retailer’s Private Label Encroachment Decisions on Supply Chain Stakeholders

In this subsection, we analyze the impacts of the retailer’s private label encroachment on the supplier’s profit, supply chain profit, consumer surplus (CS) and social welfare (SW) with/without the supplier learning effect. Notably, consumer surplus refers to the net surplus difference between the consumption valuation consumers obtain from the product transaction and the total amount they actually need to pay; social welfare refers to the sum of consumer surplus and the supply chain firms’ profits, which represents the total economic efficiency to society from the product transaction. Following the standard approach in the literature [21,43], under the setting of linear demand functions (i.e., $p_{tn}=a-q_{tn}$ for a single-product case, and $p_{ti}=a-q_{ti}-\theta q_{tj}$ for a two-product case), the consumer surplus formulas in period t for these two cases are ${CS}_t=a{q}_{tn}-{\displaystyle \frac{1}{2}}{q_{tn}}^2-p_{tn}{q}_{tn}$ and ${CS}_t=a\left(q_{tn}+q_{tp}\right)-{\displaystyle \frac{1}{2}}\left({q_{tn}}^2+2\theta q_{tn}{q}_{tp}+{q_{tp}}^2\right)-\left(p_{tn}{q}_{tn}+p_{tp}{q}_{tp}\right)$, respectively, and total consumer surplus across two periods is $CS={CS}_1+{CS}_2$, where $i,j=\left\{n,p\right\}$, $t=\left\{\mathrm{1,2}\right\}$. Furthermore, following Hong et al. [8] and Jin et al. [44], the total social welfare formula is $SW=CS+{\pi }_s+{\pi }_r$, which is aggregated equally across two periods with no intertemporal discount. The analytical results are shown in the following propositions.

Proposition 8.

Without the supplier learning effect,

(1)  ${\pi }_s^{OF\ast }\ge {\pi }_s^{OS\ast }\ge {\pi }_s^{ON\ast }$,  ${\pi }^{OF\ast }\ge {\pi }^{OS\ast }\ge {\pi }^{ON\ast }$, both  $\left({\pi }_s^{OF\ast }-{\pi }_s^{OS\ast }\right)$ and  $\left({\pi }^{OF\ast }-{\pi }^{OS\ast }\right)$ decrease in  $\theta$;

(2) ${CS}^{OF\ast }\ge {CS}^{OS\ast }\ge {CS}^{ON\ast }$, ${SW}^{OF\ast }\ge {SW}^{OS\ast }\ge {SW}^{ON\ast }$, both $\left({CS}^{OF\ast }-{CS}^{OS\ast }\right)$and $\left({SW}^{OF\ast }-{SW}^{OS\ast }\right)$decrease in $\theta$.

Combined with the analytical results of Proposition 5, Proposition 8 indicates that the supplier always benefits from the first-period encroachment of private labels due to the increment of total order quantities (i.e., $Q^{OF\ast }\ge Q^{OS\ast }\ge Q^{ON\ast }$), ultimately maximizing the supply chain profit. Moreover, consumers could obtain a lower retail price in two periods due to the increased order quantity from the first-period encroachment of private labels, which also increases social welfare eventually. With the increase in product competition, the positive effect of demand expansion in the first period declines; thus, both the profit and the welfare advantages become less predominant in the first-period encroachment scenario. The implication of Proposition 8 shows that the optimal decision for the retailer in introducing private labels benefits all supply chain stakeholders when product competition is relatively not intense; otherwise, the retailer’s choice of not introducing private labels would harm these stakeholders.

We further analyze the preferences of stakeholders with the supplier learning effect. The analytical results are summarized in Proposition 9.

Proposition 9.

With the supplier learning effect,

(1)  ${\pi }_s^{WF\ast }\ge {\pi }_s^{WS\ast }\ge {\pi }_s^{WN\ast }$,  ${\pi }^{WF\ast }\ge {\pi }^{WS\ast }\ge {\pi }^{WN\ast }$, both  $\left({\pi }_s^{WF\ast }-{\pi }_s^{WS\ast }\right)$ and  $\left({\pi }^{WF\ast }-{\pi }^{WS\ast }\right)$ increase in  $\phi$ while decrease in  $\theta$;

(2) ${CS}^{WF\ast }\ge {CS}^{WS\ast }\ge {CS}^{WN\ast }$, ${SW}^{WF\ast }\ge {SW}^{WS\ast }\ge {SW}^{WN\ast }$, both $\left({CS}^{WF\ast }-{CS}^{WS\ast }\right)$and $\left({SW}^{WF\ast }-{SW}^{WS\ast }\right)$increase in $\phi$while decrease in $\theta$.

Proposition 9 shows that the supplier, supply chain, consumers and society always prefer the retailer to choose the first-period encroachment of private labels, that is, the first-period encroachment of private labels is more beneficial for the supplier’s profit, supply chain profit, consumer surplus and social welfare compared to no encroachment and the second-period encroachment. The reasoning is that the supplier benefits more from the considerable margin profit from the first-period encroachment (i.e., both $\left(w_{2n}^{WF\ast }-c_2^{WF}\right)$ and $\left(w_{2p}^{WF\ast }-c_2^{WF}\right)$ are the highest), thus improving supply chain performance. As the retailer’s two divisions compete for consumer demands via a lower retail price, the first-period encroachment scenario achieves this for two products in both periods, which ultimately improves consumer surplus and social welfare. Moreover, both the profit and welfare advantages from the first-period encroachment of private labels become more significant as the supplier learning rate increases but more contracted as product competition intensity elevates. The analytical insights of Proposition 9 indicate that the first-period encroachment of private labels achieves a multi-win situation for firms and welfare increment, but no encroachment of private labels would harm firms and welfare.

6. Conclusions

In this study, we build a two-period framework with a supplier and a retailer and explore the latter’s optimal choices of private label encroachment (first-period versus second-period encroachment) under the decentralized division structure, where the retailer delegates the order authority to managers of individual divisions. Unlike prior studies that overlook the supplier learning effect, we incorporate the dynamic change in the supplier’s second-period production cost driven by the supplier learning effect and identify a new operational perspective for private label encroachment decisions.

This study examines that, under the decentralized division structure, the retailer’s optimal decisions of private label encroachment need to balance the supplier learning effect against product competition. That is, when product competition intensity is low, the retailer chooses the first-period encroachment of private labels considering the dominant supplier learning effect; when product competition intensity is high, the retailer’s choices shift from no encroachment to the second-period encroachment and, eventually, to the first-period encroachment as the supplier learning effect becomes more significant. Without the supplier learning effect, heightened product competition makes the retailer deter the private label encroachment. Moreover, the supplier learning effect could incur retailers to have more incentives to introduce private labels, especially when product competition is fierce. Concerning the impacts of the retailer’s optimal choices on supply chain stakeholders, our study shows that misalignment may occur, especially when the supplier learning effect is low and product competition is high. While the supplier, consumers and society uniformly favor the first-period encroachment of private labels, the retailer opts for the second-period encroachment or even no encroachment of private labels in equilibrium.

Our study provides several managerial insights regarding the strategic encroachment of private labels between suppliers and retailers. First, from the standpoint of retailers, they should strategically align with the trade-off between the supplier learning effect and product competition when deciding to choose private label encroachment. Specifically, retailers should differentiate product offerings between two divisions to reduce product homogeneity and competitive pressure, collaborate with suppliers to strengthen the cost learning efficiency, and prioritize first-period private label encroachment to fully leverage dynamic cost advantages. From the standpoint of suppliers, when facing retailers’ private label encroachment decisions, they should evaluate product competition status in advance. Through upgrading production systems to improve cost learning efficiency, suppliers enable retailers to employ the first-period encroachment strategy, ultimately leading to a win–win equilibrium across the supply chain. From the standpoint of social planners, they could establish a supply chain collaboration network and enable suppliers and retailers to jointly evaluate product competition and the cost learning efficiency. This action actively prevents the abandonment of private label encroachment in the highly competitive market where suppliers exhibit strong cost learning efficiency. Through implementing and enabling supports to improve the suppliers’ cost learning efficiency, social planners could incentivize retailers to choose the first-period encroachment of private labels, which generates collective benefits for all stakeholders in the supply chain ecosystem eventually.

This study concludes with modeling limitations and directions for future research. First, this study is only conducted on the deterministic demand setting; future work could examine retailers’ decisions on private label encroachment in the stochastic demand situation. Moreover, we construct the reselling relationship between suppliers and retailers, but more retailers play the marketplace role; it would be valuable to explore retailers’ decisions in platform operations. Finally, we only consider the scenario in which retailers source private label products from national brand suppliers; it is tempting to examine the case involving third-party suppliers or in-house private label production.