Optimal Pricing Strategy of New Products and Remanufactured Products Considering Consumers’ Switching Purchase Behavior

Abstract

Due to income constraints, increased awareness of environmental protection and preference for new products, consumers generate switching purchases between new and remanufactured products, which often lead to a “cannibalization effect” in the market, and make sellers fall into a vicious circle of price reduction. Considering consumers’ switching purchase behavior, this study examines the pricing problem of new products and remanufactured products in the competitive market environment. Based on two-period duopoly asymmetric price game models, there has been less research on the effectiveness of the price matching strategy and the traditional dynamic pricing strategy, which is the issue that this paper is dedicated to discussing. This study analyzes the equilibrium profits and their influencing factors under the dynamic pricing and price matching strategies of sellers, and discusses the simplified solution of the model. The results show that consumer learning costs, initial consumers and product differences can affect the sellers’ pricing decisions. Consumers’ learning costs of products reduces the equilibrium profit of the manufacturer and increases that of the remanufacturer. Initial consumers are not always advantageous for sellers’ profitability. Product differences affect the determination of the seller’s equilibrium strategy. In the optimal strategy, the remanufacturer should insist on price matching, while the manufacturer should choose dynamic pricing or price matching according to the product differences. This study provides sellers with insights to choose appropriate and custom pricing strategies to maximize profit as well as prevent the majority of consumers switching purchase.

1. Introduction

With the rapid development of science and technology, the speed of product renewal has become faster. According to the data analysis of the Global New Products Database (GNPD), products that have been launched in Europe, the Middle East and Africa accelerate significantly in the first half of 2022.The industries involved extend from emerging electronic products, cosmetics and fashion to traditional fields such as home furnishing and automobiles. At the same time, with the advent of the global era of green environmental protection, the recycling and remanufacturing of old products has received more and more attention. Remanufactured products are obtained by renovating or remanufacturing recycled old products (or parts), which are not significantly different from new products in function and appearance. Apple’s official website shows that the products sold in the online store include manufactured mobile phones, and that each Apple-certified remanufactured product has undergone a rigorous refurbishment process, including comprehensive testing according to the same rigorous functional standards as new Apple products.

In order to meet the needs of consumers at different levels, manufacturers need to produce and sell new products and remanufactured products at the same time, which leads to the generation of consumer switching purchase behavior. For example, consumers who prefer remanufactured products at the beginning may delay their purchase and opt for the new product because of a strong desire to try new functions or new materials. However, due to income constraints, increased awareness of environmental protection and so on, consumers who want to buy new products will choose to compromise and finally buy remanufactured products. Considering these switching purchase behavior of consumers and the competition between new products and remanufactured products, manufacturers and remanufacturers face great challenges in product sales. First of all, new products and remanufactured products often lead to a “cannibalization effect” in the market, which greatly reduces the success rate of new products on the market. Secondly, consumers’ switching purchase between new products and remanufactured products makes sellers fall into a vicious circle of price reduction in order to retain the original demand. The current research has generally concluded that consumers’ switching purchase behavior has a negative impact on sellers’ profits. Research shows that if the manufacturer or remanufacturer ignores some of the above-mentioned strategic behavior in the pricing decision, both of them may incur about 20% less profits [1]. Li et al. [2] have only analyzed equilibrium profits under dynamic pricing, and found that dynamic pricing leads to a decline in corporate equilibrium profits and consumer rationality when firms sharply discount prices. When introducing the price matching strategy to optimize the seller’s profit, scholars have drawn different conclusions. For example, Anne and Shaffer [3] show that when both asymmetric product substitutability and shelf space availability are considered, the price under PMG may even fall below the competitive level, while Chen et al. [4] argue that price-match guarantees can generate a competition-enhancing effect. Additionally, in the actual business process, the Chairman of Best Buy, has pointed out that consumer switching purchase behavior made the company “face very difficult challenges, and the only effective way to break the bottom line of sales profit of products is to re-plan the pricing strategy of the sales market”.

Some manufacturers try to apply dynamic pricing as a reactive response to dispose of products, and they dynamically adjust prices over time. However, one of the drawbacks of dynamic pricing strategy is that the manufacturers and remanufacturers cannot segment consumers completely because the strategy often prompts consumers to postpone and switch purchases between new products and remanufactured products. In general, scholars find it difficult to eliminate or weaken the negative impact of consumer behavior by using dynamic pricing. Given that consumers’ perception of quality is positively correlated with product price, some manufacturers use price matching guarantee (PMG) to discourage consumers from postponing and switching purchases. Price matching guarantee (PMG) is a policy and practice that quickly matches another seller’s quotation for the same or similar goods, or refunds the difference in price within a short time after the sale. This policy and practice has attracted increasing interest from the academics and practitioners [5]. McWilliams and Gerstner prove that the PMG strategy can be used to prevent consumers from switching purchases because it reduces the dissatisfaction of consumers who find a lower price elsewhere after purchase [6]. In fact, manufacturers and remanufacturers are trying to adopt price matching. For example, Huawei relaunched the P40 Pro 5G mobile phone in 2022 and supported the sale of new and remanufactured products at the same price. Additionally, some scholars have questioned whether when there are too many consumers, PMGs are unprofitable, and they believe that sellers’ equilibrium price matching strategies depend on the relative importance of the demand-expansion and competition-intensifying effects [7]. Based on these points, this paper will analyze the effectiveness of price matching. We mainly study the competitive equilibrium of dynamic pricing and matching pricing when new products and remanufactured products are sold at the same time, and solve the following problems:

(1)

Under the dynamic pricing strategy, does consumers’ switching purchase behavior between new products and manufactured products have a negative impact on manufacturers and remanufacturers? How should this impact be described?

(2)

Can the price matching strategy mitigate the effects of consumers’ switching purchase behavior when new products and remanufactured products are sold at the same time?

To address these questions, we first present the traditional dynamic pricing strategy. Then, we introduce three types of price matching strategies. These strategies are as follows: first, only the manufacturer implements price matching and uses the price of the remanufactured product in the next period as their own pricing (BMP). Second, only the remanufacturer implements price matching and uses the price of the new product in the next period as their own pricing (OMP). Third, the manufacturer and the remanufacturer match the pricing of a product from their respective competitors in the next period, which suggests that the seller wants to retain their consumers and encourage them to buy products in the current period (RMP). The third matching strategy combines the two previous matching strategies to make the seller’s sales exclusive, with little possibility for consumers to shift. Finally, we obtain the relevant factors that affect the profits of manufacturers and remanufacturers and the applicable conditions of the price matching strategy through the equilibrium analysis.

On the basis of the existing research, this paper discusses the following three innovative points: (1) The form of consumers’ switching buying behavior; (2) The simplification of the model; (3) The comparison of optimal strategies. First, we introduce consumers’ switching purchase behavior into the pricing decision of sellers’ products in two periods. Specifically, we consider that consumers’ switching behavior occurs between new products and remanufactured products in different periods and divide them into two categories. One is that consumers who view remanufactured products at the beginning may delay their purchase and turn to buy new products; the other is that consumers who view new products will switch to buying remanufactured products. Second, we simplify the solution process of the equilibrium profits of sellers under different strategies, which reduces the space for optimal decision-making of sellers. Third, we analyze the effectiveness of the price matching strategy compared with the traditional dynamic pricing strategy. The price matching strategy takes a competitor’s future price as a seller’s current price, which helps to explain the connotation of price matching. Considering consumers’ switching purchase behavior, the only choice for the remanufacturer is to take the future price of the competitor’s new products as the current product price, while the manufacturer can judge whether to match the future price of competitors according to the differences between new products and remanufactured products. This provides some guidance for the pricing of sellers.

The remainder of this paper is organized as follows. Section 2 briefly reviews the literature, while Section 3 introduces the basic model in detail, including the model framework, assumptions, and method. Section 4 analyses the dynamic pricing strategy equilibrium and the price matching strategies equilibrium. Section 5 conducts a numerical analysis on the impact of the related factors influencing the sellers’ profits and their optimal strategy. Section 6 presents the conclusions and directions for future research. All relevant proofs appear in Appendix A.

2. Literature Review

This study is related to two research streams: the pricing strategy in the case of simultaneous sales of multiple products, and the pricing strategy selection problem considering consumers’ switching purchase behavior.

Revenue management originates from the US airline industry in the end of 1970s, for a long time, dynamic pricing had been widely used by sellers to sell perishable products such as air tickets, whose product prices change with time. The extant literature has reviewed dynamic pricing from different perspectives. Levin and McGill [8] found that in a multi-period dynamic game model, sellers can increase revenue by implementing dynamic pricing strategies when consumers have price comparison behavior. Dasu and Tong [9] and Chew et al. [10] reach the same conclusion. Prasad et al. [11] find that the number of non-comparable consumers at a certain threshold can improve sellers’ revenues based on a two-stage dynamic pricing model. Liu and Zhang [12] highlight that companies with low product quality have greater profit losses. Li et al. [2] show that dynamic pricing leads to a decline in corporate equilibrium profits and consumer rationality when firms sharply discount prices. The new products and remanufactured products we studied have similar characteristics to perishable products such as air tickets. In the analysis of the dynamic pricing model, scholars have mentioned the strategic behavior of consumers and their preference for products, but there is no detailed description of consumers’ switching purchase behavior. In the era of advocating for green development and consumer personalized demand, it is particularly critical to choose the perspective from which to analyze the reasonable pricing of new products and remanufactured products. Our problem is essentially a more complex dynamic program that considers the competitive environment and consumers’ behavior.

Different products are sold at the same time, which means fierce competition for profits among the sellers behind their respective products. Therefore, sellers have to grasp the reaction of competitors when setting their pricing strategies. The phenomenon of the duopoly competition has been a topic of discussion since the seminal papers of early researchers. Moorthy [13] examines how consumer preferences, costs, and price competition affect a firm’s competitive product strategy under a duopoly competition. Li et al. [14] demonstrate that on the competitive pricing of duopoly, increasing the differentiation between sellers could reverse the unfavorable situation in the competition. The price competition between the new products and remanufactured products discussed in this paper is essentially the competitive problem of duopoly and multi-products. Actually, the competition between new manufactured products and remanufactured products within and across competitive channels significantly affects the pricing decisions of manufacturers. Wen et al. [15] found that when there is new product competition between the remanufacturer and the manufacturer, adding direct channels can improve the remanufacturer’s ability to resist the uncertainty of consumer behavior. Similarly, Liu et al. [16] proposed that if manufacturers decide to produce remanufactured products, they can also sell remanufactured products in their online direct channels. Niu et al. [17] considered sellers who sell new products and remanufactured products at the same time, and found that retailers can eliminate their yield uncertainty by producing remanufactured products themselves or relying on third-party remanufacturers. Qiao and Su [18] built two-period game models to address the choice issues of the original equipment manufacturer’s licensing strategy and the independent remanufacturer’s distribution channel, and found that the original equipment manufacturer’s licensing strategy choice depends on the sizes of the fixed licensing fee and the obtained willingness-to-pay. For the analysis of consumer behavior, Ma et al. [19] found that when only the reference quality effect behavior is considered, higher unit remanufacturing cost, lower remanufacturing rates, and higher customers’ discount rates can offset some of the negative impacts of reference quality effects. Baghdadi et al. [20] believe that customers are often skeptical about the quality and durability of remanufactured products, they develop a Stackelberg game model to optimize the pricing decisions for remanufactured products. Cai et al. [21] used reverse induction to obtain the equilibrium profits under the strategy the supplier actively shares to demand information from the retailer. Through the Stackelberg game analysis, Han et al. [22] found that under decentralized decision-making, the optimal profit of construction and demolition waste resources with government subsidies using the supply chain is higher. Yang et al. [23] showed that when the consumers’ valuation difference for remanufactured products is small, the monopolistic original equipment manufacturer can increase their profit by reminding consumers to consider the effects of anticipated regret. Among many consumer behaviors, transfer purchase behavior is one of the most common behaviors [24]. Therefore, a large number of scholars have discussed the product pricing decisions under the transfer purchase behavior of consumers. Consumer switching purchase behavior is a multi-party price comparison behavior based on utility maximization, which often occurs between different products or different channels of the same product. Shin [25] confirmed that differentiated services will ease the price competition between sellers, resulting in higher profits for both sides of the competition. Liu et al. [26] found that prices and profits of sellers declined when consumers switch channels to buy products. Different from the above research literature, we mainly focus on the problem of duopoly competition between the manufacturer and the remanufacturer, meanwhile, analyzing consumers’ switching purchase behavior based on a two-period dynamic game model.

Previous studies have shown that the emergence of consumer switch purchase behavior intensifies market competition, so scholars have proposed many pricing optimization methods. Marketing researchers have paid close attention to PMGs and documented their impact on price competition. The price matching strategy requires a seller to quickly match the price of the same or similar goods of another seller, or refund the price difference within a short time after the sale, so as to achieve the goal of consistent pricing of products in different sales channels. Hviid and Shaffer [27] found that an infinitesimally small hassle cost could attenuate the ability of PMG to mitigate price competition. Anne and Shaffer [3] showed that when both asymmetric product substitutability and shelf space availability are considered, the price under PMG may even fall below the competitive level. Hess and Gerstner [28] also agree with the above views. In contrast, Chen et al. [4] argued that price-match guarantees can generate a competition-enhancing effect, considering that price-match encourages consumers’ price search behavior, and thus, exaggerates price competition. In addition, when consumers are heterogeneous, PMG can be used to price discriminate one consumer type over the others. Janssen and Parakhonyak [29] found that PMG increases consumers’ option value of purchase and raises the prices of products. Xing and Liu [30] point out that the establishment of selective refund contracts based on price matching can improve the service level of retailers, even if some consumers switch to purchase. Chen and Chen [31] found that the seller with experience advantages can implement price matching strategies according to the shopping costs of consumers, which can help reduce the negative impact of consumer switching purchase on the seller’s income. Mehra et al. [32] proposed price matching as a short-term strategy and an exclusive product assortment as a long-term strategy. Zhao et al. [33] found that a price matching strategy is more effective than discrete-time dynamic pricing in the presence of strategic purchasing behavior. According to the definition of price matching strategy in the existing research, we discuss the competitive equilibrium between a manufacturer and a remanufacturer when the current price of a seller is equal to the future price of the competitor. Moreover, scholars have not reached an agreement on the effectiveness of the price matching strategy, such as Anne and Shaffer [3], who have obtained that the price matching strategy would have a positive impact on the seller, while Chen et al. [4] argued that price-match guarantees can generate a negative impact. Therefore, it is necessary for us to further discuss the effectiveness of the sellers’ price matching strategy, especially in the competitive environment of manufacturers and remanufactures.

To highlight the differences between new products and remanufactured products, we reflect the product difference and consumer learning cost into the utility function of consumers. To the best of our knowledge, few studies have simultaneously analyzed the impact of different consumer switching purchase behaviors on the competition between new and remanufactured products, and so we consider different consumers in a sales cycle: some consumers may first visit the manufacturer but then switch to purchasing remanufactured products; some consumers may first visit the remanufacturer but switch to purchasing new products. Then, we try to judge the effectiveness of the price matching strategy through equilibrium analysis between new products and remanufactured products.

3. Hypotheses and Methods

In a common market, a manufacturer (denoted with the subscript $r$) and a remanufacturer (denoted with the subscript $o$) compete to sell new products and remanufactured products to consumers. Each seller releases $N$ products to the spot market. At the end of the sales period, the residual value of the product is zero. We refer to the manufacturer or the remanufacturer who encounters the consumer in the first period as the seller $i$($i=r,o$); if no purchase has occurred before the second period, the consumer visits the seller $j$, $j=r,o$ and $j\ne i$. When a seller encounters the consumer in the second period, they will be aware that the consumer has visited the competitor in the previous period. In period $t$, the price of a seller posted is denoted by $p_{ti}$, where $t=1,2$, $i=r,o$. We assume that both sellers are risk-neutral, aiming at the maximization of profits and denoting the seller $i$’s expected profit-to-go function in period $t$ as ${\pi }_{ti}$.

Hypothesis 1:

In the two periods, the manufacturer only sells new products and the remanufacturer only sells remanufactured products.

Hypothesis 2:

Manufacturers and remanufacturers price products for the purpose of maximizing profits.

There are $N$ units similar consumers, whose valuations for the product are drawn independently from a uniform distribution on [0,1]. Additionally, $N_i$ consumers visit the seller $i$($i=r,o$) in the first period, with $N_r+N_o=N$. In the second period, it is possible for each consumer to visit another seller, which means that consumers either buy from the manufacturer or the remanufacturer. The number of consumers buying from seller $i$ in period $t$ is denoted as $n_{ti}$($t=1,2$,$i=r,o$). Each consumer’s reservation price for an ideal new product purchased from the manufacturer is $v$, and for the remanufactured product sold by the remanufacturer is $\theta v$. $\theta \in (0,1]$ represents the difference between new products and remanufactured products, which is reflected in the differences in raw materials, manufacturing processes, functions, etc. Unlike purchasing remanufactured products, consumers can obtain the freshness and real experience of product updates through new marketing forms, such as a new product release and free trial service, which helps to improve the valuation of new products. $f\left(v\right)$ denotes the density of $v$ and the distribution function is $F\left(v\right)$, which are well-known to sellers. Since new products have just entered the market, consumers do not know their functional performance, and need to spend time and energy to learn. In contrast, consumers are familiar with remanufactured products because they have been used or sold in the market before. We assume that the learning cost of each consumer for an ideal new product from the manufacturer is $s$ ($s\in [0,1]$). Moreover, the product cost information and use report of remanufactured products in the market are relatively complete; we ignore the consumer learning cost for an ideal remanufactured product purchased from the remanufacturer. In daily life, fast fashion clothing brands, such as Everlane, HoneyBy and Story Mfg, will actively disclose their production costs to consumers. Research institutions, such as IHS Markit, will release various product analysis reports, so $s$ represents the timeliness of consumers’ access to authoritative information and the difficulty of the learning product. The utility function of each consumer who purchases the new product is $U_{tr}=v-p_{tr}-s$; the utility function of each consumer who purchases from the remanufactured product is $U_{to}=\theta v-p_{to}$ and $t=\left(1,2\right)$. When $U_{2i}>U_{1j}$ ($i=r,o$, $j=r,o$, $i\ne j$), a consumer’s switching purchase behavior occurs. Thus, let $q_{\left(n_{1i},N_i\right)}$ denote the probability of $N_i$ consumers who visit seller $i$ to buy $n_{1i}$ products in the first period, where $n_{1i}=0.1\dots N_i$, $i=r,o$. Let $q_{\left(n_{2j},N_i-n_{1i}\right)}$ denote the probability of $N_i-n_{1i}$ consumers visiting the seller $j$ to buy in the second period, where $n_{2j}=0.1\dots \left(N_i-n_{1i}\right)$, $j=r,o$, $i\ne j$.

Hypothesis 3:

A large number of consumers visit sellers at random, and each consumer can only know the pricing of their corresponding products after visiting the seller.

Hypothesis 4:

Consumers make purchase decisions with the goal of maximizing their personal utility. Some consumers visit the manufacturer first to learn about the information of new products. Consumers who have purchased new products withdraw from the market and consumers who have not purchased products will turn to visit the remanufacturer to make purchase decisions. The other consumers visit the remanufacturer first to learn about the information of remanufactured products. Consumers who have purchased remanufactured products withdraw from the market and consumers who have not purchased products will turn to visit the remanufacturer to make purchase decisions.

The parameters involved in the paper and their meanings are shown in

Table 1. The parameters involved in the paper and their meanings.

Symbol	Meanings
r	The subscript of the manufacturer
o	The subscript of the remanufacturer
t	Sales period
i	A seller in the first period
j	A seller in the second period
$N$	Number of consumers initially entering the market
$n$	Number of consumers purchasing products
q	The probability of consumers purchasing products
v	Each consumer’s reservation price for an ideal new product purchased from the manufacturer
$\theta$	Difference between new products and remanufactured products
s	Consumer learning costs
U	Consumer utility
p	The product pricing
$\pi$	The profits of a seller

.

This study first presents the traditional dynamic pricing strategy (TDP) as a benchmark, and then introduces and analyses three price matching strategies (BMP, OMP and RMP). Among them, the three price matching situations are as follows: the situation when only the manufacturer implements price matching is called the BMP strategy; the situation when only the remanufacturer implement price matching is called the OMP strategy; the situation when the manufacturer and the remanufacturer match the pricing of a product from their respective competitors is called RMP. From the above analysis, the decision-making process of two sellers can be obtained, as shown in Figure 1.

The decision-making process of the traditional dynamic pricing strategy is as follows: Step 1, in the first period, $N_i$, consumers visits the two sellers, and each seller announces the product pricing, $p_{1i}$, $i=r,o$; Step 2, consumers decide whether to buy or not; Step 3, in the second period, $\left(N_i-n_{1i}\right)$, consumers switch to another seller for comparison; Step 4, each seller announces the product pricing, $p_{2i}$, $i=r,o$; Step 5, consumers who have not purchased products before make purchase decisions in the second period. If the BMP or OMP strategy is implemented, the seller will announce $p_{1i}^{BMP}$ ($p_{1i}^{OMP}$) in Step 1, and $p_{1i}^{BMP}=p_{2j}^{BMP}$ ($p_{1i}^{\mathit{OMP}}=p_{2j}^{OMP}$), $i,j=r,o$, $i\ne j$. If the RMP strategy is implemented, the sellers will announce $p_{1r}^{RMP}$ and $p_{1o}^{RMP}$ in Step 1, then set $p_{1r}^{RMP}=p_{2o}^{RMP}$ and $p_{1o}^{RMP}=p_{2r}^{RMP}$. Next, we will first analyze the TDP strategy and then establish three models of price matching strategies and discuss them in combination with practical cases.

4. Model Analysis and Results

4.1. Traditional Dynamic Pricing Strategy Equilibrium Analysis

TDP strategies are widely used by sellers. Consider the switching buying behavior of consumers across periods and products; if $U_{2o}>U_{1r}$, that is $\theta v-p_{2o}^{TDP}>v-p_{1r}^{TDP}-s$, the probability that a consumer visits the manufacturer and then buys the remanufactured product is $F\left(\frac{p_{1r}^{TDP}-p_{2o}^{TDP}+s}{1-\theta }\right)$, and the probability of $N_r$ consumers buying $n_{1r}$ products from the manufacturer in the first period is $q_{\left(n_{1r},N_r\right)}^{TDP}=\frac{N_r!}{n_{1r}!(N_r-n_{1r})!}{\left[1-F\left(\frac{p_{1r}^{TDP}-p_{2o}^{TDP}+s}{1-\theta }\right)\right]}^{n_{1r}}F{\left(\frac{p_{1r}^{TDP}-p_{2o}^{TDP}+s}{1-\theta }\right)}^{N_r-n_{1r}}$. If $U_{2r}>U_{1o}$, that is $v-p_{2r}-s>\theta v-p_{1o}$, the probability that a consumer visits the remanufacturer and then buys the new products is $1-F\left(\frac{p_{2r}^{TDP}-p_{1o}^{TDP}+s}{1-\theta }\right)$, and the probability of $N_o$ consumers buying $n_{1o}$ products from the remanufacturer in the first period is $q_{\left(n_{1o},N_o\right)}^{TDP}=\frac{N_o!}{n_{1o}!(N_o-n_{1o})!}F{\left(\frac{p_{2r}^{TDP}-p_{1o}^{TDP}+s}{1-\theta }\right)}^{n_{1o}}{\left[1-F\left(\frac{p_{2r}^{TDP}-p_{1o}^{TDP}+s}{1-\theta }\right)\right]}^{N_o-n_{1o}}$. Then, we analyze the probability of consumers switching in the second period. Consumers waiting for the second period will buy the new products when $U_{2i}>0$ ($i=r,o$), and so the probability of $N_r-n_{1r}$ consumers buying $n_{2o}$ remanufactured products is $q_{\left(n_{2o},N_r-n_{1r}\right)}^{TDP}=\frac{\left(N_r-n_{1r}\right)!}{n_{2o}!(N_r-n_{1r}-n_{2o})!}{\left[1-F\left(\frac{p_{2o}^{TDP}}{\theta }\right)\right]}^{n_{2\mathrm{o}}}F{\left(\frac{p_{2o}^{TDP}}{\theta }\right)}^{N_r-n_{1r}-n_{2o}}$. Similarly, the probability of $N_o-n_{1o}$ consumers buying $n_{2r}$ new products is $q_{\left(n_{2r},N_o-n_{1o}\right)}^{TDP}=\frac{\left(N_o-n_{1o}\right)!}{n_{2r}!(N_o-n_{1o}-n_{2r})!}{\left[1-F\left(p_{2r}^{TDP}+s\right)\right]}^{n_{2\mathrm{r}}}F{\left(p_{2r}^{TDP}+s\right)}^{N_o-n_{1o}-n_{2r}}$.

$${\pi }_r^{TDP}\left(N_r,N_o\right)=\underset{p_{tr}^{TDP}}{Max}{\displaystyle \sum _{n_{1r}=0}^{N_r}{q}_{\left(n_{1r},N_r\right)}^{TDP}}\left[p_{1r}^{TDP}·n_{1r}+{\displaystyle \sum _{n_{1o}=0}^{N_o}{q}_{\left(n_{1o},N_o\right)}^{TDP}}·p_{2r}^{TDP}·{\displaystyle \sum _{n_{2r}=0}^{N_o-n_{1o}}\left(n_{2r}·q_{\left(n_{2r},N_o-n_{1o}\right)}^{TDP}\right)}\right]$$

(1)

$${\pi }_o^{TDP}\left(N_r,N_o\right)=\underset{p_{t_o}^{TDP}}{Max}{\displaystyle \sum _{n_{1o}=0}^{N_o}{q}_{\left(n_{1o},N_o\right)}^{TDP}}\left[p_{1_o}^{TDP}·n_{1r}+{\displaystyle \sum _{n_{1r}=0}^{N_r}{q}_{\left(n_{1r},N_r\right)}^{TDP}}·p_{2_o}^{TDP}·{\displaystyle \sum _{n_{2o}=0}^{N_r-n_{1r}}\left(n_{2o}·q_{\left(n_{2o},N_r-n_{1r}\right)}^{TDP}\right)}\right]$$

(2)

From Equations (1) and (2), the quantity of products sold in each period is uncertain, and it is difficult to determine the equilibrium price and profits of the manufacturer and the remanufacturer. Therefore, we consider simplifying the solution process and discuss the equilibrium decisions of the two sellers when $N=1$.

When a consumer first visits the manufacturer, the probability that he/she makes a purchase in the first period is $1-F\left(\frac{p_{1r}^{TDP}-p_{2o}^{TDP}+s}{1-\theta }\right)$, and the expected profits of the manufacturer in the first period can be simplified as

$$\begin{array}{l}{\pi }_{1r}^{TDP}\left(1,0\right)=\underset{p_{1r}^{TDP}}{\mathit{max}} p_{1r}^{TDP}·\left[1-F\left(\frac{p_{1r}^{TDP}-p_{2o}^{TDP}+s}{1-\theta }\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}v-p_{1r}^{TDP}-s\ge 0\end{array}$$

(3)

The expected profits of the remanufacturer in the second period can be simplified as

$$\begin{array}{l}{\pi }_{2o}^{TDP}\left(1,0\right)=\underset{p_{2O}^{TDP}}{\mathit{max}} p_{2o}^{TDP}·F\left(\frac{p_{1r}^{TDP}-p_{2o}^{TDP}+s}{1-\theta }\right)·\left[1-(\frac{p_{2o}^{TDP}}{\theta })\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}\theta v-p_{2o}^{TDP}>0\end{array}$$

(4)

When a consumer first visits the remanufacturer, the expected profits of the remanufacturer in the first period can be simplified as

$$\begin{array}{l}{\pi }_{1o}^{TDP}\left(0,1\right)=\underset{p_{1o}^{TDP}}{\mathit{max}} p_{1o}^{TDP}·F\left(\frac{p_{2r}^{TDP}-p_{1o}^{TDP}+s}{1-\theta }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}\theta v-p_{1o}^{TDP}>0\end{array}$$

(5)

The expected profits of the manufacturer in the second period can be simplified as

$$\begin{array}{l}{\pi }_{2r}^{TDP}\left(0,1\right)=\underset{p_{2r}^{TDP}}{\mathit{max}} p_{2r}^{TDP}\left[1-F\left(\frac{p_{2r}^{TDP}-p_{1o}^{TDP}+s}{1-\theta }\right)\right]·\left[1-F\left(p_{2r}^{TDP}+s\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t\hspace{1em}v-p_{2_{\mathrm{r}}}^{TDP}-s>0\end{array}$$

(6)

Based on the analysis above, the two sellers’ equilibrium profits can be calculated as shown in Theorem 1. To simplify the presentation, we define

$$\begin{array}{l}\lambda =\sqrt{4s^2-12s\theta +17{\theta }^2+8s-12\theta +4}\\ \mu =\sqrt{9s^2+8s\theta +16{\theta }^2-22s-24\theta +17}\end{array}$$

(7)

Theorem 1.

Under the TDP strategy, when there is only one consumer in the market, the optimal expected profits of sellers are as follows:

$$\begin{gathered} {\pi }_{1r}^{TDP*}\left(1,0\right)=\frac{{\left(10-6s-7\theta -\lambda \right)}^2}{256\left(1-\theta \right)},{\pi }_{2o}^{TDP*}\left(1,0\right)=\frac{{\left(3s-4\theta +7-\mu \right)}^2}{256\left(1-\theta \right)}, \\ {\pi }_{2r}^{TDP*}\left(0,1\right)=\frac{\left(7-5\mathrm{s}+4\theta -\mu \right)\left(9-3s-12\theta +\mu \right)\left(1-3s+4\theta +\mu \right)}{1024\theta (1-\theta )}, \\ {\pi }_{1o}^{TDP*}\left(0,1\right)=\frac{\left(2+\theta +2s-\lambda \right)\left(6+6s-9\theta +\lambda \right)\left(7\theta -2s-2+\lambda \right)}{1024\theta (1-\theta )}. \end{gathered}$$

The proof of Theorem 1 is provided in Appendix A. Theorem 1 shows that when there is only one consumer in the market, the sellers’ equilibrium profits are affected by the service difference between two sellers and the learning cost. According to Equations (1) and (2), we obtain Theorem 2 after a series of calculations.

Theorem 2.

Under the TDP strategy, when there are N consumers in the market, the expected profits of the two sellers are as follows:

$${\pi }_r^{TDP*}\left(N_r,N_o\right)=N_r·{\pi }_{1r}^{TDP*}\left(1,0\right)+N_o·{\pi }_{2r}^{TDP*}\left(0,1\right)$$

(8)

$${\pi }_o^{TDP*}\left(N_r,N_o\right)=N_o·{\pi }_{1o}^{TDP*}\left(1,0\right)+N_r·{\pi }_{2o}^{TDP*}\left(0,1\right)$$

(9)

The proof of Theorem 2 is provided in Appendix A. According to Theorem 2, when there are multiple consumers, the sellers’ optimal profits can be transformed into the expected profits of the sellers when there is only one consumer in the market multiplied by the number of consumers who initially arrive at the corresponding seller. This helps us analyze the equilibrium profit of sellers. According to the above simplified procedure, $\theta \in (0,1]$, in order to ensure that the product price and the seller’s income are not negative, we randomly take the value of $s$, and assume that $N_R=N_O=50$. Taking “$s=0.1$” as an example, the changes in sellers’ profits with $\theta$ are shown in Figure 2. In fact, taking other values of $s$ can result in similar changes in sellers’ profits.

As shown in Figure 2, with the increase in $\theta$, the profits of the manufacturer decreases, and the profits of the remanufacturer increases first and then decreases. It can be seen that it is difficult for sellers to eliminate the adverse effects of consumers’ switching purchase behavior, which leads to a decrease in profits. Therefore, this study will introduce some new pricing strategies to improve sellers’ profits.

4.2. Equilibrium Analysis for Price Matching Strategies

4.2.1. BMP Strategy Equilibrium Analysis

Under the BMP strategy, the manufacturer first wants to increase their profits by increasing the number of targeted consumers who prefer the new products when $p_{1r}^{BMP}=p_{2o}^{BMP}$, so that the consumer will not be tempted by the low price of the remanufacturer and switch to them. The remanufacturer, meanwhile, does not implement matching pricing but sets a new price for each period. Similar to the TDP strategy analysis, the profits of sellers are as follows:

$${\pi }_r^{BMP}\left(N_r,N_o\right)=\underset{p_{tr}^{BMP}}{Max}{\displaystyle \sum _{n_{1r}=0}^{N_r}{q}_{\left(n_{1r},N_r\right)}^{BMP}}\left[p_{1r}^{BMP}·n_{1r}+{\displaystyle \sum _{n_{1o}=0}^{N_o}{q}_{\left(n_{1o},N_o\right)}^{BMP}}·p_{2r}^{BMP}·{\displaystyle \sum _{n_{2r}=0}^{N_o-n_{1o}}\left(n_{2r}·q_{\left(n_{2r},N_o-n_{1o}\right)}^{BMP}\right)}\right]$$

(10)

$${\pi }_o^{BMP}\left(N_r,N_o\right)=\underset{p_{t_o}^{BMP}}{Max}{\displaystyle \sum _{n_{1o}=0}^{N_o}{q}_{\left(n_{1o},N_o\right)}^{BMP}}\left[p_{1_o}^{BMP}·n_{1o}+{\displaystyle \sum _{n_{1r}=0}^{N_r}{q}_{\left(n_{1r},N_r\right)}^{BMP}}·p_{2_O}^{BMP}·{\displaystyle \sum _{n_{2o}=0}^{N_r-n_{1r}}\left(n_{2o}·q_{\left(n_{2o},N_r-n_{1r}\right)}^{BMP}\right)}\right]$$

(11)

where the probability of $N_r$ consumers buying $n_{1r}$ products from the manufacturer is $q_{\left(n_{1r},N_r\right)}^{BMP}=\frac{N_r!}{n_{1r}!(N_r-n_{1r})!}{\left[1-F\left(\frac{s}{1-\theta }\right)\right]}^{n_{1r}}F{\left(\frac{p_{1_r}^{BMP}-p_{2_o}^{BMP}+s}{1-\theta }\right)}^{N_r-n_{1r}}$; the probability of $N_o$ consumers buying $n_{1o}$ products from the remanufacturer is $q_{\left(n_{1o},N_o\right)}^{BMP}=\frac{N_o!}{n_{1o}!(N_o-n_{1o})!}{\left[F\left(\frac{p_{1_o}^{BMP}-p_{2_r}^{BMP}-s}{1-\theta }\right)\right]}^{n_{1o}}{\left[1-F\left(\frac{p_{1_o}^{BMP}-p_{2_r}^{BMP}-s}{1-\theta }\right)\right]}^{N_o-n_{1o}}$; the probability of $N_r-n_{1r}$ consumers buying $n_{2o}$ products from the remanufacturer is $q_{\left(n_{2o},N_r-n_{1r}\right)}^{BMP}=\frac{\left(N_r-n_{1r}\right)!}{n_{2o}!(N_r-n_{1r}-n_{2o})!}{\left[1-F\left(\frac{p_{2_o}^{BMP}}{\theta }\right)\right]}^{n_{2o}}F{\left(\frac{p_{2_o}^{BMP}}{\theta }\right)}^{N_r-n_{1r}-n_{2o}}$; and the probability of $N_o-n_{1o}$ consumers buying $n_{2r}$ products from the remanufacturer is $q_{\left(n_{2r},N_o-n_{1o}\right)}^{BMP}=\frac{\left(N_o-n_{1o}\right)!}{n_{2r}!(N_o-n_{1o}-n_{2\mathrm{r}})!}{\left[1-F\left(p_{2r}^{BMP}+s\right)\right]}^{n_{2\mathrm{r}}}{\left[F\left(p_{2r}^{BMP}+s\right)\right]}^{N_o-n_{1o}-n_{2\mathrm{r}}}$.

When $N=1$, the expected profits of the manufacturer in the first period can be written as

$$\begin{array}{l}{\pi }_{1r}^{BMP}\left(1,0\right)=p_{1r}^{BMP}·\left[1-F\left(\frac{s}{1-\theta }\right)\right]=\mathit{max}{p}_{2o}^{BMP}·\left[1-F\left(\frac{s}{1-\theta }\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}v-p_{1r}^{BMP}-s>0\end{array}$$

(12)

The expected profits of the remanufacturer in the second period can be written as

$$\begin{array}{l}{\pi }_{2o}^{BMP}\left(1,0\right)=\underset{p_{2o}^{BMP}}{\mathit{max}} p_{2o}^{BMP}·F\left(\frac{s}{1-\theta }\right)·\left[1-(\frac{p_{2_O}^{BMP}}{\theta })\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}\theta v-p_{2o}^{BMP}>0\end{array}$$

(13)

The expected profits of the remanufacturer in the first period are as follows:

$$\begin{array}{l}{\pi }_{1o}^{BMP}\left(0,1\right)=\underset{p_{1o}^{BMP}}{\mathit{max}} p_{1o}^{BMP}·F\left(\frac{p_{2r}^{BMP}-p_{1o}^{BMP}+s}{1-\theta }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}\theta v-p_{1o}^{BMP}>0\end{array}$$

(14)

The expected profits of the manufacturer in the second period are as follows:

$$\begin{array}{l}{\pi }_{2r}^{BMP}\left(0,1\right)=\underset{p_{2r}^{BMP}}{\mathit{max}} p_{2r}^{BMP}\left[1-F\left(\frac{p_{2r}^{BMP}-p_{1o}^{BMP}+s}{1-\theta }\right)\right]·\left[1-F\left(p_{2r}^{BMP}+s\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}v-p_{2\mathrm{r}}^{BMP}-s>0\end{array}$$

(15)

Theorem 3.

Under the BMP strategy, when there is only one consumer in the market, the optimal profits of the two sellers are as follows:

$$\begin{array}{l}{\pi }_{1r}^{BMP*}\left(1,0\right)=\frac{\theta \left(1-\theta -s\right)}{2\left(1-\theta \right)},{\pi }_{2o}^{BMP*}\left(1,0\right)=\frac{\theta s}{4(1-\theta )},{\pi }_{1o}^{BMP*}\left(0,1\right)=\frac{{\left(7+3s-4\theta +\mu \left(\theta \right)\right)}^2}{256\left(1-\theta \right)},\\ {\pi }_{2r}^{BMP*}\left(0,1\right)=\frac{\left(7-5s-4\theta +\mu \left(\theta \right)\right)\left(3s+12\theta -9+\mu \left(\theta \right)\right)\left(3s-1-4\theta +\mu \left(\theta \right)\right)}{1024(1-\theta )}\end{array}$$

The proof of Theorem 3 is in Appendix A.

Theorem 4.

Under the BMP strategy, when there are N consumers in the market, the expected profits of the two sellers are as follows:

$${\pi }_r^{BMP*}\left(N_r,N_o\right)=N_r·{\pi }_{1r}^{BMP*}\left(1,0\right)+N_o·{\pi }_{2r}^{BMP*}\left(0,1\right)$$

(16)

$${\pi }_o^{BMP*}\left(N_r,N_o\right)=N_o·{\pi }_{1o}^{BMP*}\left(1,0\right)+N_r·{\pi }_{2o}^{BMP*}\left(0,1\right)$$

(17)

The proof of Theorem 4 is provided in Appendix A. According to Theorem 4, only the manufacturer can match the price, and the two sellers’ optimal profit functions can also be simplified.

4.2.2. OMP Strategy Equilibrium Analysis

Using OMP strategy, the remanufacturer first wants to increase their profits by increasing the number of targeted consumers who prefer purchasing remanufactured products. When $p_{1o}^{OMP}=p_{2r}^{OMP}$, the remanufacturer can improve the probability of target consumers to buy products in the current period. Similar to Section 4.2.1, the profits of the manufacturer (${\pi }_r^{OMP}$) and the remanufacturer (${\pi }_o^{OMP}$) are

$${\pi }_r^{OMP}\left(N_r,N_o\right)=\underset{p_{tr}^{OMP}}{Max}{\displaystyle \sum _{n_{1r}=0}^{N_r}{q}_{\left(n_{1r},N_r\right)}^{OMP}}\left[p_{1r}^{OMP}·n_{1r}+{\displaystyle \sum _{n_{1o}=0}^{N_o}{q}_{\left(n_{1o},N_o\right)}^{OMP}}·p_{2r}^{OMP}·{\displaystyle \sum _{n_{2r}=0}^{N_o-n_{1o}}\left(n_{2r}·q_{\left(n_{2r},N_o-n_{1o}\right)}^{OMP}\right)}\right]$$

(18)

$${\pi }_o^{OMP}\left(N_r,N_o\right)=\underset{p_{t_o}^{OMP}}{Max}{\displaystyle \sum _{n_{1o}=0}^{N_o}{q}_{\left(n_{1o},N_o\right)}^{OMP}}\left[p_{1o}^{OMP}·n_{1o}+{\displaystyle \sum _{n_{1r}=0}^{N_r}{q}_{\left(n_{1r},N_r\right)}^{OMP}}·p_{2o}^{OMP}·{\displaystyle \sum _{n_{2o}=0}^{N_r-n_{1r}}\left(n_{2o}·q_{\left(n_{2o},N_r-n_{1r}\right)}^{OMP}\right)}\right]$$

(19)

where the probability of consumers buying products from the remanufacturer in the second period is $q_{\left(n_{2o},N_r-n_{1r}\right)}^{OMP}=\frac{\left(N_r-n_{1r}\right)!}{n_{2o}!(N_r-n_{1r}-n_{2o})!}{\left[1-F\left(\frac{p_{2_o}^{OMP}}{\theta }\right)\right]}^{n_{2\mathrm{o}}}F{\left(\frac{p_{2_o}^{OMP}}{\theta }\right)}^{N_r-n_{1r}-n_{2o}}$; the probability of consumers buying products from the manufacturer in the first period is $q_{\left(n_{1r},N_r\right)}^{OMP}=\frac{N_r!}{n_{1r}!(N_r-n_{1r})!}{\left[1-F\left(\frac{p_{1_r}^{OMP}-p_{2_o}^{OMP}+s}{1-\theta }\right)\right]}^{n_{1r}}F{\left(\frac{p_{1_r}^{OMP}-p_{2_o}^{OMP}+s}{1-\theta }\right)}^{N_r-n_{1r}}$ the probability of consumers buying products from the remanufacturer in the first period is $q_{\left(n_{1o},N_o\right)}^{OMP}=\frac{N_o!}{n_{1o}!(N_o-n_{1o})!}{\left[F\left(\frac{s}{1-\theta }\right)\right]}^{n_{1o}}{\left[1-F\left(\frac{s}{1-\theta }\right)\right]}^{N_o-n_{1o}}$ and the probability of consumers buying products from the manufacturer in the second period is $q_{\left(n_{2r},N_o-n_{1o}\right)}^{OMP}=\frac{\left(N_o-n_{1o}\right)!}{n_{2\mathrm{r}}!(N_o-n_{1o}-n_{2\mathrm{r}})!}{\left[1-F\left(p_{2r}^{OMP}+s\right)\right]}^{n_{2\mathrm{r}}}{\left[F\left(p_{2r}^{OMP}+s\right)\right]}^{N_o-n_{1o}-n_{2\mathrm{r}}}$.

Similar to the previous analysis, when there is only one consumer who visits the manufacturer in the market, the expected profits of the two sellers are simplified as

$$\begin{array}{l}{\pi }_{1r}^{OMP}\left(1,0\right)=\underset{p_{1r}^{OMP}}{\mathit{max}} p_{1r}^{OMP}·\left[1-F\left(\frac{p_{1r}^{OMP}-p_{2o}^{OMP}+s}{1-\theta }\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}v-p_{1r}^{OMP}-s>0\end{array}$$

(20)

$$\begin{array}{l}{\pi }_{2o}^{OMP}\left(1,0\right)=\underset{p_{2o}^{OMP}}{\mathit{max}} p_{2o}^{OMP}·F\left(\frac{p_{1r}^{OMP}-p_{2o}^{OMP}+s}{1-\theta }\right)·\left[1-(\frac{p_{2o}^{OMP}}{\theta })\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}\theta v-p_{2o}^{OMP}>0\end{array}$$

(21)

When there is only one consumer who visits the remanufacturer in the market, the expected profits of the two sellers are

$$\begin{array}{l}{\pi }_{1o}^{OMP}\left(0,1\right)=\mathit{max} p_{1o}^{OMP}·F\left(\frac{s}{1-\theta }\right)=p_{2r}^{OMP}·F\left(\frac{s}{1-\theta }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}\theta v-p_{1o}^{OMP}>0\end{array}$$

(22)

$$\begin{array}{l}{\pi }_{2r}^{OMP}\left(0,1\right)=\underset{p_{2r}^{OMP}}{\mathit{max}} p_{2r}^{OMP}\left[1-F\left(\frac{s}{1-\theta }\right)\right]·\left[1-F\left(p_{2r}^{OMP}+s\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}v-p_{2\mathrm{r}}^{OMP}-s>0\end{array}$$

(23)

The equilibrium prices of the OMP strategy are shown in Theorem 5.

Theorem 5.

Under the OMP strategy, when there is only one consumer in the market, the optimal profits of the two sellers are as follows:

$$\begin{array}{l}{\pi }_{1r}^{OMP*}\left(1,0\right)=\frac{{\left(10-6s-7\theta +\lambda \left(\theta \right)\right)}^2}{256\left(1-\theta \right)},\\ {\pi }_{2o}^{OMP*}\left(1,0\right)=\frac{\left(2s+\theta +2+\lambda \left(\theta \right)\right)\left(9\theta -6s-6+\lambda \left(\theta \right)\right)\left(2s+2-7\theta +\lambda \left(\theta \right)\right)}{1024\theta (1-\theta )},\\ {\pi }_{2r}^{OMP*}\left(0,1\right)=\frac{{\left(1-s\right)}^2\left(1-\theta -s\right)}{4(1-\theta )},{\pi }_{1o}^{OMP*}\left(0,1\right)=\frac{\left(1-s\right)s}{2\left(1-\theta \right)}.\end{array}$$

The proof of Theorem 5 is in Appendix A.

Theorem 6.

Under the OMP strategy, when there are N consumers in the market, the expected profits of the two sellers are as follows:

$${\pi }_r^{OMP*}\left(N_r,N_o\right)=N_r·{\pi }_{1r}^{OMP*}\left(1,0\right)+N_o·{\pi }_{2r}^{OMP*}\left(0,1\right)$$

(24)

$${\pi }_o^{OMP*}\left(N_r,N_o\right)=N_o·{\pi }_{1o}^{OMP*}\left(1,0\right)+N_r·{\pi }_{2o}^{OMP*}\left(0,1\right)$$

(25)

The proof of Theorem 6 is provided in Appendix A. According to Theorem 6, only the remanufacturer can match the price and the sellers’ optimal profits can also be simplified.

4.2.3. RMP Strategy Equilibrium Analysis

Under the RMP strategy, both sellers in the first period match competitors’ second prices, which means $p_{1r}^{RMP}=p_{2o}^{RMP}$ and $p_{1r}^{RMP}=p_{2o}^{RMP}$; then, the probability of $N_r$ consumers buying $n_{1r}$ products from the manufacturer in the first period is $q_{\left(n_{1r},N_r\right)}^{RMP}=\frac{N_r!}{n_{1r}!(N_r-n_{1r})!}{\left[1-F\left(\frac{s}{1-\theta }\right)\right]}^{n_{1r}}F{\left(\frac{s}{1-\theta }\right)}^{N_r-n_{1r}}$ and the probability of $N_o$ consumers buying $n_{1o}$ products from the remanufacturer in the first period is $q_{\left(n_{1o},N_o\right)}^{RMP}=\frac{N_o!}{n_{1o}!(N_o-n_{1o})!}F{\left(\frac{s}{1-\theta }\right)}^{n_{1o}}{\left[1-F\left(\frac{s}{1-\theta }\right)\right]}^{N_o-n_{1o}}$. Similar to Section 4.2.1, the profits of the sellers are as follows:

$${\pi }_r^{RMP}\left(N_r,N_o\right)=\underset{p_{tr}^{RMP}}{Max}{\displaystyle \sum _{n_{1r}=0}^{N_r}{q}_{\left(n_{1r},N_r\right)}^{RMP}}\left[p_{1r}^{RMP}·n_{1r}+{\displaystyle \sum _{n_{1o}=0}^{N_o}{q}_{\left(n_{1o},N_o\right)}^{RMP}}·p_{2r}^{RMP}·{\displaystyle \sum _{n_{2r}=0}^{N_o-n_{1o}}\left(n_{2r}·q_{\left(n_{2r},N_o-n_{1o}\right)}^{RMP}\right)}\right]$$

(26)

$${\pi }_o^{RMP}\left(N_r,N_o\right)=\underset{p_{t_o}^{RMP}}{Max}{\displaystyle \sum _{n_{1o}=0}^{N_o}{q}_{\left(n_{1o},N_o\right)}^{RMP}}\left[p_{1_o}^{RMP}·n_{1o}+{\displaystyle \sum _{n_{1r}=0}^{N_r}{q}_{\left(n_{1r},N_r\right)}^{RMP}}·p_{2_O}^{RMP}·{\displaystyle \sum _{n_{2o}=0}^{N_r-n_{1r}}\left(n_{2o}·q_{\left(n_{2o},N_r-n_{1r}\right)}^{RMP}\right)}\right]$$

(27)

where the probability of consumers buying products from the manufacturer in the first period is $q_{\left(n_{1r},N_r\right)}^{RMP}=\frac{N_r!}{n_{1r}!(N_r-n_{1r})!}{\left[1-F\left(\frac{s}{1-\theta }\right)\right]}^{n_{1r}}F{\left(\frac{s}{1-\theta }\right)}^{N_r-n_{1r}}$; the probability of consumers buying products from the remanufacturer in the second period is $q_{\left(n_{2o},N_r-n_{1r}\right)}^{RMP}=\frac{\left(N_r-n_{1r}\right)!}{n_{2o}!(N_r-n_{1r}-n_{2o})!}{\left[1-F\left(\frac{p_{2_o}^{RMP}}{\theta }\right)\right]}^{n_{2\mathrm{o}}}F{\left(\frac{p_{2_o}^{RMP}}{\theta }\right)}^{N_r-n_{1r}-n_{2o}}$; the probability of consumers buying products from the remanufacturer in the first period is $q_{\left(n_{1o},N_o\right)}^{RMP}=\frac{N_o!}{n_{1o}!(N_o-n_{1o})!}{\left[F\left(\frac{s}{1-\theta }\right)\right]}^{n_{1o}}{\left[1-F\left(\frac{s}{1-\theta }\right)\right]}^{N_o-n_{1o}}$ and the probability of consumers buying products from the manufacturer in the second period is $q_{\left(n_{2r},N_o-n_{1o}\right)}^{RMP}=\frac{\left(N_o-n_{1o}\right)!}{n_{2\mathrm{r}}!(N_o-n_{1o}-n_{2\mathrm{r}})!}{\left[1-F\left(p_{2r}^{RMP}+s\right)\right]}^{n_{2\mathrm{r}}}{\left[F\left(p_{2r}^{RMP}+s\right)\right]}^{N_o-n_{1o}-n_{2\mathrm{r}}}$.

We also discuss the equilibrium decision of the two sellers when $N=1$. When there is only one consumer in the market, the expected profits of the two sellers are:

$$\begin{array}{l}{\pi }_{1r}^{RMP}\left(1,0\right)=\mathit{max} p_{1r}^{RMP}·\left[1-F\left(\frac{s}{1-\theta }\right)\right]=p_{2o}^{RMP}·\left[1-F\left(\frac{s}{1-\theta }\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}v-p_{1r}^{RMP}-s>0\end{array}$$

(28)

$$\begin{array}{l}{\pi }_{2o}^{RMP}\left(1,0\right)=\underset{p_{2o}^{RMP}}{\mathit{max}} p_{2o}^{RMP}·F\left(\frac{s}{1-\theta }\right)·\left[1-(\frac{p_{2o}^{RMP}}{\theta })\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}\theta v-p_{2o}^{RMP}>0\end{array}$$

(29)

$$\begin{array}{l}{\pi }_{1o}^{RMP}\left(0,1\right)=\mathit{max} p_{1o}^{RMP}·F\left(\frac{s}{1-\theta }\right)=p_{2r}^{RMP}·F\left(\frac{s}{1-\theta }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}\theta v-p_{1o}^{RMP}>0\end{array}$$

(30)

$$\begin{array}{l}{\pi }_{2r}^{RMP}\left(0,1\right)=\underset{p_{2r}^{RMP}}{\mathit{max}} p_{2r}^{RMP}\left[1-F\left(\frac{s}{1-\theta }\right)\right]·\left[1-F\left(p_{2r}^{RMP}+s\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.\hspace{1em}v-p_{2\mathrm{r}}^{RMP}-s>0\end{array}$$

(31)

Theorem 7.

Under the RMP strategy, when there is only one consumer in the market, the optimal profits of the two sellers are as follows:

$$\begin{array}{l}{\pi }_{1r}^{RMP*}\left(1,0\right)={\pi }_{1r}^{BMP*}\left(1,0\right)=\frac{\theta \left(1-\theta -s\right)}{2\left(1-\theta \right)},{\pi }_{1o}^{RMP*}\left(1,0\right)={\pi }_{1o}^{OMP*}\left(1,0\right)=\frac{\left(1-s\right)s}{2\left(1-\theta \right)},\\ {\pi }_{2r}^{RMP*}\left(0,1\right)={\pi }_{2r}^{OMP*}\left(0,1\right)=\frac{{\left(1-s\right)}^2\left(1-\theta -s\right)}{4(1-\theta )},{\pi }_{2o}^{RMP*}\left(0,1\right)={\pi }_{2o}^{BMP*}\left(0,1\right)=\frac{\theta s}{4(1-\theta )}\end{array}$$

The proof of Theorem 7 is in Appendix A.

Theorem 8.

Under the RMP strategy, when there are N consumers in the market, the expected profits of the two sellers are as follows:

$${\pi }_r^{RMP*}\left(N_r,N_o\right)=N_r·{\pi }_{1r}^{RMP*}\left(1,0\right)+N_o·{\pi }_{2r}^{RMP*}\left(0,1\right)$$

(32)

$${\pi }_o^{RMP*}\left(N_r,N_o\right)=N_o·{\pi }_{1o}^{RMP*}\left(1,0\right)+N_r·{\pi }_{2o}^{RMP*}\left(0,1\right)$$

(33)

The proof of Theorem 8 is in Appendix A.

5. Numerical Examples

5.1. The Impact of Product Differences between New Products and Remanufactured Products and Consumers Learning Costs on Profits

As previously defined, as $\theta$ becomes smaller, the differences between the two sellers is greater, and with the increase in $s$, consumers do not receive authoritative information on time, and it is more difficult to learn new product information. In order to analyze the impact of a single parameter on the sellers’ pricing decision and highlight the disadvantages of the two sellers as much as possible, we assume that $s=0$, $N_R=N_O=50$, and the difference between the two sellers is only reflected in product differences. The impact of product differences between new products and remanufactured products on sellers’ profits under the four strategies is shown in Figure 3.

Figure 3a implies that with the increase in $\theta$, the profits of the manufacturer using the TDP as well as OMP strategies decrease, while the profits of the manufacturer using the RMP as well as BMP strategies increase. Figure 3b shows that the increase in $\theta$ may be detrimental for the remanufacturer’s profit, especially when the profits fall to zero under the RMP strategy. With different strategies, the influence of $\theta$ on profits varies widely owing to the consumer’s switching behavior. Using the TDP strategy, with $\theta$ increases, consumers have an incentive to switch simply because of the low price of the second period, and the two sellers gradually form a symmetrical equilibrium in the competition. However, using the OMP strategy, sellers have the same price ($p_{1o}^{OMP}=p_{2r}^{OMP}$), and consumers visiting the remanufacturer have little incentive to buy the new products; thus, the profits of the manufacturer have declined. Using the BMP strategy, the manufacturer loses some consumers, which leads to a decline in profits.

In addition, $s$ also reflects the consumers’ learning ability, which has an impact on the consumers’ switching purchase behavior. $s=0$ means that there is no difference in the ability of consumers to learn about new products and remanufactured products, and $s>0$ means that it is more difficult for consumers to learn about new products. In general, in order to introduce the impact of $s$ on the seller, under the condition that the product pricing and the sellers’ profit are not negative, it can be assumed that s = 0.1. The impact of search cost on sellers’ profits is shown in Figure 4. ($N_R=N_O=50$).

Owing to the presence of $s$, the two sellers’ profit curves have changed compared to Figure 3, which means that the cost of learning for consumers prevents manufacturers from increasing profits, but helps the remanufacturer to gain more profits. Figure 4 shows that with the increase in $\theta$, the manufacturer’s profit curves using the TDP and RMP strategies become two parabolas. The remanufacturer uses the price matching strategy to increase their profit. Considering the influence of $s$ and $\theta$ on profits, also reflects the strength of the characteristics of the two sellers in the competition. When $s$ is small, the advantage of remanufactured products is weak. Consumers who prefer high-tech or novel products can easily switch to manufacturers. When $\theta$ is large, the advantage of the new product is weak. Thus, consumers who seek low prices are more inclined to switch to the remanufacturer; the two sellers may achieve a win–win situation in competition.

5.2. Optimal Strategy for the Manufacturer and the Remanufacturer

The size of the initial consumers may vary due to the difference in sellers. To observe the change trend of the sellers’ profits there are three conditions: (1) The manufacturer has more initial consumers; (2) The manufacturers and the remanufacturer have the same number of initial consumers; (3) The remanufacturers have more initial consumers, we list three typical cases: $N_r:N_o=10:90$, $N_r:N_o=50:50$, and $N_r:N_o=90:10$. The profits (${\pi }_r,{\pi }_o$) are shown in

Table 2. The equilibrium profits of the manufacturer under a different strategy ($s=0.1$).

θ	$${\mathit{N}}_{\mathit{r}}:{\mathit{N}}_{\mathit{o}}$$	Strategy
		TDP	BMP	OMP	RMP
0.05	10:90	15.1	13.3	18.3	16.5
	50:50	17.3	8.4	19.1	13.2
	90:10	19.6	3.5	19.9	10.2
0.35	10:90	12.7	12.4	17.2	16.4
	50:50	14.9	13.5	17.4	16.0
	90:10	17.2	14.5	17.7	15.0
0.75	10:90	5.3	7.0	11.6	13.2
	50:50	5.8	13.9	9.3	17.3
	90:10	6.3	20.8	7.7	21.5

and

Table 3. The equilibrium profits of the remanufacturer under a different strategy ($s=0.1$).

θ	$${\mathit{N}}_{\mathit{r}}:{\mathit{N}}_{\mathit{o}}$$	Strategy
		TDP	BMP	OMP	RMP
0.05	10:90	4.3	4.3	4.3	4.3
	50:50	2.7	2.4	2.7	2.4
	90:10	1.1	0.6	1.1	0.6
0.35	10:90	5.1	4.8	6.6	6.4
	50:50	4.6	3.3	5.4	4.1
	90:10	4.0	1.7	4.2	1.9
0.75	10:90	4.8	5.1	16.6	17.0
	50:50	4.6	5.2	11.2	12.8
	90:10	4.4	7.2	5.8	8.6

.

Table 2 and Table 3 show that it is not always advantageous for sellers to have a large number of initial consumers. Especially when $\theta$ is low, the manufacturer’s profits decrease with the increase in the initial consumer scale under the BMP and RMP strategies; when $\theta$ is moderate, the manufacturer’s profits decrease with the increase in the initial consumer scale under the RMP strategy; when $\theta$ is high, the manufacturer’s profits decrease with the increase in the initial consumer scales under the OMP strategy, and the remanufacturer’s profits decrease with the increase in the initial consumer scale under the BMP strategy. Taking the BMP strategy as an example, the decrease in seller’s profits can be explained as the manufacturer implementing the price matching in the first period; when the difference between new products and remanufactured products becomes less obvious, the manufacturer will lose the advantage of charging higher prices. As the initial consumers increase, the consumers’ influence on the remanufacturer’s purchase is greater, which leads to the decline in the manufacturer’s profits because the low price becomes the most important factor in attracting consumers. With the increase in initial consumers, the greater the negative impact of consumers purchasing new products, the greater the decrease in the remanufacturers’ profits. The change in sellers’ profits in the case of other strategies can also be explained by the above analysis method. Therefore, sellers should maintain a certain range of consumers and adopt reasonable price matching strategies to increase profits.

The change in the initial consumer size will not affect the choice of sellers’ equilibrium strategy. When $\theta$ is small or moderate, the OMP strategy is the most common choice of the manufacturer and the remanufacturer; when $\theta$ is large, the RMP strategy is the most common choice of sellers. This means that, regardless of the product differences between new products and remanufactured products, the remanufacturer should choose price matching, and the manufacturer should choose dynamic pricing or price matching according to the product differences (when $\theta$ is small or moderate, the manufacturer should choose dynamic pricing; when $\theta$ is large, the manufacturer should choose price matching). Next, we will analyze the reasons why the remanufacturer is unable to flexibly adjust their pricing strategies. When new products and remanufactured products are becoming more and more similar, the remanufacturer will try to meet the consumers’ demand for novelty to narrow the service gap. However, in the actual operation process, it is difficult for the remanufacturers to invest a lot of money, time and innovation again to promote products, and consumers are likely to have aesthetic and technical fatigue. Therefore, the remanufacturer, who has lost service advantages, may further exert their low price advantages, turn passivity into initiative, and promote the manufacturer to match their price, which achieves a Pareto improvement.

In general, considering consumers’ switching purchase behavior, the only choice for the remanufacturer is price matching, while the manufacturer can choose dynamic pricing or price matching according to product differences. In actual production, the original equipment manufacturer (OEM) usually lacks sufficient funds, specialized equipment and the technical level to make profits from remanufacturing. For example, the Ford Company tried to enter the field of remanufacturing abandoned vehicles, but failed. Unlike OEM, the third-party remanufacturer (TPR) not only has advanced remanufacturing technology, but also tends to form economies of scale for remanufacturing the used products of many brands. “Remanufacturing” recycling can optimize the performance of waste products and minimize the consumption of resources. Meanwhile, remanufactured products have a large space for value preservation and appreciation. Therefore, TPR becomes the main competitor of OEM. Huawei’s P40 Pro 5G mobile phone (Huawei, Shenzhen, China) was successfully re-launched at its original price, providing a new way for the competition between OME and TRI.

6. Conclusions

Consumers’ switching purchase behavior often occurs in the actual sales activities of new and remanufactured products. Due to the expectation of price reduction in remanufactured products or the uncertainty of the performance of new products, consumers often choose to postpone the purchase of products. In the case that manufactured products and remanufactured products are sold at the same time, some consumers who pursue novelty may delay their purchase and opt for new products from the manufacturer, while some consumers who prefer practicality and low prices, may switch to remanufactured products. To maximize profits, sellers should consider switching purchase behaviors; they should also make use of their own characteristics and advantages to choose appropriate pricing strategies. Given this background, this study analyzes the factors that affect sellers’ equilibrium profits, and discusses the optimal pricing strategy for sellers.

The key point of the price matching strategy, compared with traditional dynamic pricing, is that it uses the price of a competitor’s product as a reference. Our conclusions are divided into two categories; one is the impact of various parameters on the sellers, and the other is the sellers’ optimal pricing strategy. Considering the influence of parameters, we find that consumer learning costs, initial consumers and product differences can affect the sellers’ pricing decisions. Specifically, consumer learning costs reduce the profits of the manufacturer but increases that of the remanufacturer. The large number of initial consumers is not always advantageous for sellers to make more profits. Product differences affect the determination of the seller’s equilibrium strategy, when the differences between new products and remanufactured products are obvious ($\theta$ is small or moderate), OMP is the equilibrium strategy for sellers and when the differences between new products and remanufactured products are not obvious ($\theta$ is large), the RMP strategy is the equilibrium strategy for sellers. In the optimal strategy, the remanufacturer should insist on price matching, while the manufacturer should choose dynamic pricing or price matching according to the product differences.

The impact of consumer learning costs on the manufacturer and the remanufacturer are obvious. Learning about new products is not easy for consumers, so the remanufacturer can take advantage of this to gain more profits. At the beginning, sellers face more consumers, and in the future, they may encounter more consumers’ switching purchase, and their profits will be damaged. In short, considering consumers’ switching purchase behavior, the remanufacturer should choose price matching and the manufacturer should choose dynamic pricing or price matching based on differences between new products and remanufactured products. In fact, whether sellers sell new products or remanufactured products in a competitive environment, the purpose is to eliminate consumers’ uncertainty and encourage them to buy products. Therefore, sellers need to choose product pricing according to their own sales characteristics. For example, manufacturers can provide live demonstrations and free trial services of different series of products for consumers who buy TVs, mobile phones and other electronic products, adapt to the functions of new products. Traditionally, compared with remanufactured products, the learning process of new products can make consumers feel the performance of these products more intuitively, and attract many consumers to purchase products. At this time, the market price of remanufactured products is more competitive than that of new products, which can attract price-sensitive consumers to buy and create considerable profit space for enterprises. Some remanufacturers are also using other technologies to improve the consumer perception of remanufactured products. For example, Dell uses the diversified marketing methods of third-party platforms to sell remanufactured products. At this time, the differences between new products and remanufactured products are no longer obvious, and the two sellers return to a new round of price competition, which is the RMP strategy mentioned in this paper. Of course, with the improvement of the accuracy of price forecasting in the future, price matching can not only be limited to this, but also can be set in a certain range in combination with the sensitivity of consumers to prices, which is likely to extend the concept of price matching.

In this research, we only consider the duopoly case. It would be interesting to investigate the pricing of new products and remanufactured products by considering different channel structures. Now, many sellers have begun to implement “online and offline” sales patterns, and the case of a seller considering adding another channel might be particularly interesting. It has to be said that the psychology of consumers is quite complex, and the final behavior is not completely rational; thus, there are some limitations in our interpretation of consumers. In the future, we can combine some irrational factors to further explain the problem.