Online Second-Hand Bookstores’ Strategic Decisions: A Theoretical Perspective

Abstract

The development of second-hand bookstores has received ample attention in the book industry. However, research on their operational strategies is still in its infancy, especially concerning how they compete with new bookstores. We used the Hotelling model to investigate second-hand bookstores’ differentiation strategy. We analyzed how product mismatch problems and consumer preferences affect bookstores’ strategies and profits. We came to three major conclusions. First, second-hand bookstores can effectively compete with new bookstores using a product differentiation strategy. Moreover, it is better to set high prices for special old books. Second, a higher breadth and depth of mismatching will increase revenue for second-hand bookstores and lower revenue for new bookstores. Third, new bookstores’ profit decreases with consumers’ preferences for special old books, while the influence of such preferences on the second-hand bookstores’ profits depends on the transportation cost. These findings provide valuable insights for second-hand bookstores to compete with new bookstores. An important implication of this study is that using a differentiation strategy for second-hand bookstores needs to consider the mismatch problems and consumers’ preferences.

1. Introduction

The current linear production and consumption dominate society, greatly increasing the burden of natural resource conservation and environmental protection [1]. At present, the research on circular economy and sustainable development has attracted significant attention from academia [2]. How to transition to a circular economy and rationally utilize those products at the end of their life cycles have become key factors in improving resource utilization and social welfare [3]. Product reuse is a win-win opportunity in this case [2]. Second-hand products are goods that are acquired after being used by another consumer [4]. This trend towards second-hand consumption has spread into the book industry. Recently, the scale of China’s paper book market has continued to expand at a rate of more than 10% every year [5]. Thus, the number of books stored by users is growing rapidly. However, the reuse rate of many books is low, and discarding books or selling them as waste products is too wasteful, so the second-hand book trade has begun to be welcomed by the market worldwide and has developed rapidly [6].

Moreover, the internet revolution has changed our lives and influenced the used book industry [7]. An important feature of traditional offline used book transactions is that buyers and sellers can negotiate prices [8]. Generally, second-hand books in brick-and-mortar bookstores or street markets are sold at a price set by the bookseller, and consumers can only negotiate within a specific range on this basis [9]. Booksellers tend to set prices arbitrarily, and there is a considerable gap between the price and value of second-hand books [10]. In this case, information asymmetry between consumers and booksellers is significant. Thus, unreasonable transactions often cause unfairness to buyers [7]. Nowadays, with the development of e-commerce, online second-hand book sales platforms have emerged, such as Duozhuayu and Confucius Second hand Books.com in China. Online used bookstores can set prices openly, making book conditions and prices transparent [11]. The openness of the online platform allows buyers to compare the prices of different booksellers quickly. Thus, online platforms have created a fair environment for used book transactions and promoted the development of the used book industry [5].

In this context, the second-hand market must compete with the new book market because of the extended life cycle of books. Moreover, the consumption of second-hand books will attract consumers who might otherwise not have bought books at all. Thus, managers in the second-hand book industry need to consider their business strategies to gain a competitive advantage against the new book market. Understanding the purchase behavior of consumers and targeting strategies is crucial. To examine competition between two firms, the Hotelling method develops a location model to demonstrate the relationship between location and behavior of firms and is proved applicable and robust in economic study. In the existing studies, the model framework has been applied to various contexts, such as the oil market [12], transportation market [13], and advertising market [14]. Still, it misses the intersection of the second-hand and new book industries.

Compared with new bookselling platforms, second-hand bookstores have the following characteristics. First, certain second-hand books in second-hand bookstores can be substitutes for new books. Online transactions promote the circulation of second-hand books, and researchers can quickly obtain research materials through the Internet. Therefore, an advantage of second-hand bookstores over new bookstores is that the books they sell have a differentiated advantage. More specifically, in this paper, we consider unique characteristics of the second-hand book industry using a Hotelling model. Herein, the following research questions are pursued:

• (1). Can second-hand bookstores implement a product differentiation strategy to increase their profits?

• (2). How can second-hand bookstores gain an advantage in the competition with new bookstores?

• (3). What price strategy should second-hand bookstores use to compete with new bookstores?

• (4). How does the mismatch problem affect the new and second-hand bookstores’ pricing, profit, and demand?

• (5). How do consumers’ preferences affect the new and second-hand bookstores’ pricing strategies, revenue, and market demand?

The overarching aim of this study is to expand the understanding of online second-hand bookstores’ strategic decisions and their consumers’ behaviors. First, our work theoretically examines how second-hand bookstores and new bookstores compete in a platform market. Second, we complement existing theoretical studies within the book industry research by using a Hotelling model. Third, we examine equilibrium when the second-hand bookstore implements a differentiation strategy, followed by considerations of the mismatch problem and heterogeneous consumer preferences.

The rest of this study is structured as follows. Section 2 reviews relevant literature. In Section 3, we construct the model framework and investigate the case that the second-hand bookstore does not adopt any strategy, considering both Cournot and Stackelberg’s competition. Section 4 studies the effectiveness of the differentiation strategy adopted by the second-hand bookstore, and we also examine the mismatch problem in the second-hand bookstore. Section 5 extends the model to include consumers’ preferences, and we analyze its effects on the equilibrium. Conclusions are presented in Section 6.

2. Literature Review

We studied second-hand bookstores’ strategic decisions in the platform market, especially when competing with new bookstores. Thus, our research is related to the literature on the second-hand book industry and the relationship between the new and second-hand goods in the market. We discuss these streams of studies below.

2.1. Second-Hand Book Industry

In the book market, second-hand books refer to books that were previously owned by other people [15]. Second-hand bookstores acquire and sell second-hand books with collectible value. Usually, the acquisition price is relatively low, and the selling price is relatively high; second-hand bookstores can earn a considerable price difference and profit from it [15]. The motivations for customers to purchase second-hand books include economic and ideological motivations [16]. Economic motivation refers to the consumer’s desire to buy products at a lower price. Furthermore, ideological motivation is the consumer’s nostalgic pleasure in buying used goods and the urge to display uniqueness.

An emerging body of the literature provides empirical studies related to the book industry. Chen et al. [17] exploited a natural experiment to investigate the impact of e-bookselling on print sales. They found that delaying e-book availability has no increasing effect on print book sales. Moreover, there is a weak relationship between the sales of e-books and print books in the short term. Wang and Miller [18] investigated book publishers’ product portfolios in e-books and printed books. Their results show that large publishers protect the printed book ecosystem more. Moreover, considering the second-hand book market, Ghose et al. [19] used a unique dataset from Amazon.com’s new and used book marketplaces to measure the degree to which second-hand books cannibalize new book sales. They also used these results to measure the publisher’s welfare and consumer surplus changes. Their analysis shows that second-hand books are poor substitutes for new books for most of Amazon’s consumers, and selling second-hand books increases consumers’ surplus and publishers’ welfare. Ellison and Ellison [20] show that some kinds of used books are in limited supply and would be highly valued only by a small fraction of people. Improved search technologies will increase customer flow, and sellers will subsequently increase prices for used books. As a result, the increased equilibrium match quality would increase the overall social welfare.

Consistent with these studies, we examine the second-hand book industry, especially regarding the price strategy and competition between old and new bookselling platforms. Taking this further, we deal with the mismatch problem and consumer preferences. This study has implications for bookstore platforms’ choices regarding new and used books.

2.2. The Relationship between the New and Used Goods

Another research stream focuses on theoretical modeling to study the interactions between new and used products. Some studies consider the reuse of used products from the perspective of the supply chain. Feng et al. [21] built a Stackelberg game to consider a reverse supply chain with dual-recycling channels. They argue that the online channel can improve the recycling price in the traditional channel. Moreover, the overall profit of the whole supply chain system is improved after introducing the online channel. Liu et al. [22] developed a price competition model for the waste electrical and electronic equipment market within both formal and informal sectors. They found that both the formal and informal sectors prefer high-quality products. In contrast, the informal sector can capture a larger market share of used products because of a better acquisition price. Refaie et al. [23] designed a multi-echelon closed-loop supply chain model for durable products. They show that reusing durable products can result in a reduction in purchasing costs and an increase in annual savings and revenue. Xu et al. [24] developed game-theoretic models to study the different operation modes of used trading platforms that affect supply chain members’ profits, environment, and social welfare. The results suggest that the resale mode can improve profits for the online platform and the used product supplier. All these theoretical studies focus on the firm strategies regarding the characteristics of the used products, given various assumptions on consumer behavior. Our paper contributes to this stream of literature by considering the mismatch problem between customers and used products in the context of the second-hand book industry. Furthermore, this theoretical study also highlights the role of customers’ preferences in influencing bookstores’ strategies.

Empirical research on the used-products market is limited. Using consumer-level data on an online platform, Bloemhof and Corbett [25] empirically investigated the factors affecting the price difference between used, remanufactured, and new goods in the iPod industry. They explain their findings through the lenses of information asymmetry and adverse selection. The results show that consumers’ willingness to pay for the used products is subtly related to the nature of that product. Taking this study further, we focus on consumers’ preference for second-hand products. We analyze the influence of consumers’ preferences on firms’ strategies.

2.3. Literature Review Summary

In the above-reviewed studies, some scholars have examined the market of used and new products in supply chains. However, most studies do not consider the competition among sellers of used and new products or online scenarios. To fill the gap in the platform economy, our research has contributed to the existing studies on the competitive market of used and new products, especially in the book industry. We developed a price strategy for second-hand bookstores to compete with new book sellers. Then, we propose a product differentiation strategy that second-hand bookstores can use when competing with new book sellers. It presents that second-hand bookstores can effectively compete with new bookstores using a product differentiation strategy. Based on this context, we investigate the influence of the problem of product mismatch on second-hand bookstores. Then, the model is extended by considering consumers’ preferences, and we analyze its effects on second-hand bookstores’ strategy. Finally, to assist second-hand bookstores in implementing these strategies effectively, we propose specific measures for product promotion.

3. Assumption, Notations, and Models

This section presents Cournot and Stackelberg’s competition models that do not consider the mismatch problem and consumers’ preferences. We provide the basic assumptions of the models. Then, we analyze the models’ equilibrium results.

3.1. Basic Assumptions

We developed a game-theoretical model that analyzes the competition between an online second-hand bookstore and an online new bookstore. The books offered by these two different suppliers (used bookstore and new bookstore) vary in some respects, and therefore, consumers are attracted.

Assumption 1. 

The consumers can choose to purchase a book from a second-hand or a new bookstore, and each consumer can buy at most one book.

Assumption 2. 

Two types of bookstores are both rational economic agents, and the pursuit of profit maximization is their final aim. New bookstores only sell new common books, while second-hand bookstores sell both old common books and old special books.

Assumption 2 of rational economic agents has been widely used in the marketing literature [26,27,28]. In addition, to move away from other issues and focus solely on the substitute effects and competition between a new bookstore and second-hand bookstore, we assume new bookstores sell one type of book (new common books), and second-hand bookstores sell two types of books (old common books and special books).

Assumption 3. 

New common books and second-hand common books can substitute for each other.

The conditions of the used book include “As new”, “Fine”, “Very good”, “Good”, “Fair”, “Poor”, “Ex-library”, “Book club”, and “Binding copy” [29,30]. For a used book with the condition “As New”, the implication is that it is a perfect copy. That is to say, an ordinary used book in perfect condition can be an ideal substitute for a new book. Thus, for common books, the used and the new can substitute for each other.

Assumption 4. 

Any new books cannot substitute for an old special book.

In Assumption 4, we show that second-hand books include not only common books but also rare old books and autographed books of famous people, which are often valuable for research and collection [10]. Special old books are rare books out of print or autographed books of famous people. These old books are often valuable for specific customers and cannot be bought in new bookstores; thus, they cannot be replaced by new books [31]. Thus, a new book is not a substitute for a special old book.

Assumption 5. 

The selling price of the book in a new bookstore is$P_n$. The book’s selling price in a second-hand bookstore is for ordinary and special second-hand books.

Assumption 6. 

The direct utility of purchasing one item of the used book or new book is the same for a consumer, denoted by $U$.

Assumption 7. 

The number of two groups of consumers who buy a new or used book is normalized to 1, and the consumers are uniformly distributed on a line of 0 to 1. Two types of bookstores are exogenously located at the ends of the line (the new bookstore at 0 and the second-hand bookstore at 1, respectively).

Assumption 8. 

We assume the index $x$shows the position of consumers on the line, representing customers’ heterogeneous preferences for bookstores.

Assumptions 5–8 are consistent with the literature stream of the Hotelling model [32]. The model and its variant we built are mainly based on the work of Hotelling [19], which is used to characterize the differentiation of the market. In the model, products are offered by different suppliers. These suppliers are different in some characteristics and thus attract different purchasers. Therefore, we employ the Hotelling model to describe the competition between the used and new bookstores. Sources for such preferences in Assumption 8 vary, such as the horizontal difference in books (the books’ content), the customers’ physical locations, and familiarity with the two bookstores.

Assumption 9. 

We used$t$to measure the linear transportation cost a consumer incurs to travel to a new bookstore.

Assumption 9 is a key assumption in the Hotelling model and allows the model to capture the intuition behind it. An assumption of non-linear cost can only algebraically change the model’s result [20]. Transportation costs can be interpreted as costs associated with searching for a book on the new book website, reading users’ book reviews, and previewing parts of the book. For example, on JD.com’s bookselling website, users can preview the first 50 or 100 pages of a book. After this, consumers decide their preferences for the book and whether they need to buy it. They are the cost related to transportation and are not included in the selling price. Therefore, a consumer whose position is $x$ and purchases at the new bookstore will pay a cost $t\cdot x$. On the other hand, the transportation cost of a consumer whose position is $x$ and purchases at the second-hand bookstore is $t\cdot \left(1-x\right)$. These two kinds of transportation costs in the Hotelling model mainly show the different preferences of consumers for the new bookstore and used bookstore.

Assumption 10. 

Production costs related to bookstores are normalized to zero.

This assumption is for simplicity. Similar assumptions can be found in much operational literature [33,34].

Based on the above assumptions, in the two subsections, we develop our benchmark model of two types of market structures where online bookstores compete, including the Cournot case and the Stackelberg case. Both stores sell ordinary books in these cases, and we will not consider selling special old books in the old bookstore. (The case of selling special old books in a second-hand bookstore is analyzed in the next section.) The difference between used and new books is that they have different conditions and degrees of newness. In the Cournot case, two types of online bookstores have equivalent market power, and they set prices simultaneously. In the Stackelberg case, however, the new bookstore has more market power and thus acts as the leader and sets a price first; the second-hand bookstore follows. Figure 1 shows the model structure of this study. All parameters’ notations are summarized and shown in Table 1.

3.2. Cournot Equilibrium

In this case, we assume second-hand and new bookstores have the same market potential and share. Thus, two types of online stores tend to set the selling price at the same time. A consumer’s net direct utility from buying one unit of the product from a new bookstore at a distance $x$ is $U_n=U-P_n-t\cdot x$, where $P_n$ is the price charged by the new bookstore and $t\cdot x$ is the total transportation cost. In contrast, the utility a consumer derives from a used bookstore is $U_o=U-P_o-t\cdot x-t\cdot \left(1-x\right)$, where $P_0$ is the price charged by the used bookstore and $t\cdot x+t\cdot \left(1-x\right)$ is the total cost of transportation. (This setting is based on our observations of consumers’ buying behavior. We conducted an online questionnaire survey. Among the 80 valid questionnaires returned, we found that 75% of consumers would check the relevant information of books on the online new book platform before buying used books on the corresponding used book platform. Among users with such a behavior, more than 90% of consumers discover and determine the books they want to buy during the process of browsing the new book platform. These users mainly examine the sales, word-of-mouth from book reviews, and the degree of popularity from books displayed on the new book platform. After that, they will choose, compare, and buy the substitutes of new book on the used book platform.) Generally, to buy a used book online, consumers tend to find relevant information about the version of the book on a new bookselling platform, such as book reviews, and preview parts of the book. Then, consumers will switch to a used bookselling platform to find suitable second-hand books to buy, such as by checking customer reviews and ratings of the store and selecting second-hand books according to their quality and conditions. It is equivalent to spending time browsing new and used bookselling platforms.

Consistent with the previous study [35,36], the new and used bookstores are independent and make their own decisions to maximize their profits. Therefore, we applied the Nash game model. Consumers are indifferent in choosing between the new bookstore and the second-hand bookstore if $U_n=U_o$ and solving for $x_a$. Here, we obtain $x_a=1-\left(P_n-P_o\right)/t$. Thus, the demand for the new bookstore is $D_n=x_a=1-(P_n-P_o)/t$, and that for the second-hand bookstore is $D_o=1-x_a=(P_n-P_o)/t$.

The profit of a new bookstore is ${\pi }_n=D_n\cdot P_n=P_n\cdot (1-(P_n-P_o)/t)$, and that for the second-hand bookstore is ${\pi }_o=D_o\cdot P_o=P_o\cdot (P_n-P_o)/t$. Taking the first-order derivative of ${\pi }_n$ concerning $P_n$ and ${\pi }_o$ for $P_o$, respectively, and letting them be equal to zero, we obtain the optimal selling price $P_n^a$ and $P_o^a$. Here, we have $P_n^a=2t/3$, $P_o^a=t/3$. Substituting $P_n^a$ and $P_o^a$ into equations of profits and demand, we obtain the new bookstore and second-hand bookstores’ market demand and gains, respectively, as displayed in Table 2.

3.3. Stackelberg Equilibrium

There are more discounts and promotions at new online bookstores than used ones. For example, we can see many promotions on the Dangdang platform in China (Dangdang is now a major player in China’s e-commerce market focusing on new bookselling), especially during the Double 11 shopping festival (Double 11 shopping festival is the most popular shopping festival in China which began on 20 October 2021; it is similar to Black Friday or Cyber Monday) or other major festivals. However, there is less publicity associated with used book websites. In addition, the market share of the new book market is much larger than that of the used book market. Thus, we assume that the new bookstores may have more pricing power than the used bookstores. In such a setting, the new bookstore is a Stackelberg leader who sets the price of the books first, and then the second-hand bookstore decides its price for the second-hand books. The second-hand bookstore must change its optimal price to deal with the new bookstore’s price adjustment. Therefore, the new bookstore is the price leader and determines its optimal price by considering second-hand bookstore prices. That is to say, the new bookstore needs to insert the second-hand bookstore’s profit function into its profit function and then determine the optimal price.

Here, we solve this profit maximization problem by inserting $P_o$ and $P_n$ into the profit function of the second-hand bookstore, then take the first-order derivative and let it equal zero. We can obtain the response function of the second-hand bookstore to the new bookstore as $P_o=P_n/2$. Then, inserting this response function into the profit function of the new bookstore and taking the first-order derivative of ${\pi }_n$ to $P_n$, we obtain $P_n=t$, which is the price the new bookstore charges, and $P_o=t/2$ is the price when the used bookstore can observe and respond. Substituting the value of prices into equations of profits and demand, we obtain the demand and profits of the new bookstore and second-hand bookstores, as shown in Table 2. By comparing the equilibrium results between the case of Cournot and Stackelberg, we can establish the following proposition.

Proposition 1.

The later new bookstores enter the market, the smaller the selling price of new books, the larger the number of sales, and the smaller profits they gain. On the other hand, the earlier second-hand bookstores enter the market, the smaller the selling price of used books, the smaller the number of sales, and the smaller profits they gain.

We can know from Table 2 that $P_n^a<P_n^b,P_o^a<P_o^b;D_n^a>D_n^b,D_o^a<D_o^b;{\pi }_n^a<{\pi }_n^b,{\pi }_o^a<{\pi }_o^b$. These results are in line with the common knowledge of competition theory. Compared with the Stackelberg competition, in the case of the Cournot competition, the competition is fiercer. Thus, the equilibrium prices and profits are more extensive. On the other hand, for the number of sales, the comparative results are different. The new bookstore will achieve a higher selling number in the Cournot case, while the second-hand bookstore will achieve a higher selling number in the Stackelberg case. That is because customers are less sensitive to the price of used books but more sensitive to the price of new books. This shows that the new bookstore can have the first-move advantage over the competition in the Stackelberg case, while the second-hand bookstore can benefit from entering the market later. The reasons are mainly twofold. First, the first entrant usually bears the uncertainty of the market environment, which is determined by consumers’ preferences and tastes. The first entrant might have a mistaken estimation of the market and thus makes strategic mistakes. The second entrant can avoid such mistakes by observing the actions of the first mover and making higher profits. Thus, the price follower can gain a competitive advantage. Second, the second-hand bookstore will free-ride the new bookstore. That is to say, the information provided by the new bookstore platform has a positive external effect on the second-hand bookstore. Part of the customer traffic of second-hand bookstores comes from new bookstore platforms. Many consumers plan to buy these books just because they found a popular book or a book with a good reputation on the new book platform. Therefore, from the perspective of second-hand bookstores, they should follow the new bookstores to obtain higher profits.

4. The Second-Hand Bookstore Implements a Differentiation Strategy

In this section, we consider the case after second-hand bookstores apply a differentiation strategy and analyze the market equilibrium results. We assume that second-hand bookstores sell special second-hand books in this case. The differentiation strategy for second-hand bookstores includes promoting and emphasizing the uniqueness and irreplaceability of their second-hand books through advertising or other means of publicity. Thus, second-hand bookstores sell two types of books: ordinary second-hand books and special second-hand books. The new bookstore still sells only new books. Ordinary second-hand books in good condition are similar to the new books and they can substitute each other.

Consumers will choose only one product unit from the new or second-hand bookstore. We assume the proportion of $\lambda (0<\lambda <1)$ consumers whose best-matching book is offered by the used bookstore. (Some subjects are not written and published often, or out of fashion. People can only find that subject on some used books.) Thus, the proposition of users whose best-matching book is offered by the new bookstore is $1-\lambda$. The utility of choice other than the best match is $U-\epsilon$ and $0<\epsilon <U$.

For a consumer whose best match is a special used book, he/she buys the special used book at a second-hand bookstore; then, he/she will obtain utility $U_r=U-P_r-t\cdot x-t\cdot \left(1-x\right)$. Another consumer with the same preference for a special used book who purchases the ordinary used book at the same bookstore obtains utility $U_o=U-P_o-t\cdot x-t\cdot \left(1-x\right)-\epsilon$. We assume that $U_r>U_o$ always holds, which means for the consumers whose best match is a special used book, if they buy at a second-hand bookstore, they will always choose to buy the special used book. Moreover, a consumer who changes to a new bookstore to buy a new book will incur other costs. The utility, in this case, will be $U_n=U-t\cdot x-P_n-\epsilon$. In the setting, $U_r=U_n$ we will derive the indifference point of $x_b=-\frac{P_n+\epsilon -t-P_r}{t}$. Therefore, we may conclude that $\lambda \left(1-x_b\right)$ of consumers will buy the special used book at the second-hand bookstore, and $\lambda x_b$ of consumers will change channels and buy a new book at the new bookstore.

On the other hand, a consumer whose best matching book is an ordinary used book and who buys an ordinary used book at a second-hand bookstore derives the utility $U_o=U-P_o-t\cdot x-t\cdot \left(1-x\right)$. A consumer with the same ordinary used book preference purchases the special used book at the same store and obtains the utility $U_r=U-P_r-t\cdot \left(1-x\right)-t\cdot x-\epsilon$. We assume that $U_o>U_r$ always holds, which means for the consumers whose best match is an ordinary used book, if they buy at a second-hand bookstore, they will always choose to buy the ordinary used book. However, a consumer who buys an ordinary book at a new bookstore derives the utility $U_n=U-P_n-t\cdot x$. Solving for $U_o=U_n$, we can obtain the indifference point $x_a=-\frac{P_n-t-P_o}{t}$. Therefore, we may conclude that $\left(1-\lambda \right)\left(1-x_a\right)$ of consumers will purchase the ordinary used book at the second-hand bookstore while $\left(1-\lambda \right)x_a$ of consumers will switch channels to buy at the new bookstore. The decision-making process of consumers is presented in Figure 1.

The demand for the new book and used books at the new bookstore and used bookstore is as follows:

Special used book demand at second-hand bookstores: $D_r=\lambda \left(1-x_b\right)$

Ordinary used book demand at second-hand bookstores: $D_o=\left(1-\lambda \right)\left(1-x_a\right)$

Ordinary new book demand at new bookstores: $D_n=\lambda x_b+\left(1-\lambda \right)x_a$

Therefore, profits of the second-hand bookstore and new bookstore can be expressed as follows:

Second-hand bookstore’s profit: ${\pi }_s=P_r\cdot D_r+P_o\cdot D_o$

New bookstore’s profit: ${\pi }_n=P_n\cdot D_n$

To find the equilibrium price that maximizes the profit of both bookstores, we take the first-order derivative of ${\pi }_s$ and ${\pi }_N$ to $P_o$, $P_r$ and $P_n$, respectively, and let the derivatives equal zero; we obtain $\partial {\pi }_s/\partial P_o=0;\partial {\pi }_s/\partial P_r=0;\partial {\pi }_n/\partial P_n=0$. Therefore, from the above equations, we obtain the optimal selling price $P_o^c,P_r^c,P_n^c$. Moreover, we can also derive the used bookstore and new bookstore’s demand and profit by substituting $P_o^c,P_r^c,P_n^c$ into the equations above. We summarize the equilibrium results in the fourth column of Table 2.

Proposition 2.

Implementing a differentiation strategy will increase demand and profit for second-hand bookstores. The differentiation strategy is an effective tool to compete with new bookstores.

See the Appendix A for the proof.

As the second-hand bookstore adopts a differentiation strategy, demand for the ordinary book decreases in both bookstores, reducing the new bookstore’s profit. However, for the used bookstore, the increased demand for special second-hand books offsets the decreased demand for ordinary second-hand books, leading to the growth of the second-hand bookstore’s profit. Thus, implementing a differentiation strategy will increase the second-hand bookstore’s demand and profit, and the differentiation strategy may be a powerful tool to compete with the new bookstore. This conclusion is also verified in Figure 2. When consumers realize the existence of special second-hand books, larger values of $\lambda$ and $\epsilon$ will make second-hand bookstores obtain higher profits than those without a differentiation strategy.

Proposition 3.

When second-hand bookstores consider implementing a differentiation strategy to compete with new bookstores, it may be better to raise the price of special second-hand books.

Proposition 3 shows that the retail price of the special book is higher than that of the ordinary book $P_r^c>P_o^c$ in the second-hand bookstore. Its comparison relationship is not affected by $\lambda$. This may be because when the value of $\lambda$ is larger, consumers prefer special second-hand books, and the demand for special second-hand books in the second-hand bookstore is greater for ordinary second-hand books. Therefore, the second-hand bookstore will set a higher price for special second-hand books to maximize its profit. On the other hand, when the value of $\lambda$ is small, users prefer ordinary used books, and the demand for special second-hand books in second-hand bookstores is smaller than that for ordinary used books. Therefore, the profit of the used bookstore mainly depends on the sales of ordinary second-hand books. A higher price will be set for special second-hand books to avoid price competition between the two kinds of books.

Effects of the Mismatch Problem

Because the information is opaque, different merchants have differences in the status of their products, and there is no uniform standard. There may be cases where consumers’ expectations do not match the products they receive. The bookstore’s managers need to know how the book mismatch problem will affect price strategies, consumer demands, and suppliers’ profits. In line with the previous study by Liu et al. [37], we consider two dimensions of the mismatch problem: the breadth and depth of mismatching consumers. The breadth of mismatching consumers is denoted as $\lambda$ the proportion of users with mismatch problems. The depth of mismatching consumers is denoted as $\epsilon$, which indicates the degree of the mismatch problem. The mathematical derivation process is displayed in the Appendix.

Proposition 4.

From the above results, we can deduce that the effect of mismatch on prices is:

•  (i) 

  $\partial P_n^c/\partial \lambda <0;\partial P_o^c/\partial \lambda <0;\partial P_r^c/\partial \lambda <0$.

•  (ii) 

  $\partial P_n^c/\partial \epsilon <0;\partial P_o^c/\partial \epsilon <0;\partial P_r^c/\partial \epsilon >0$.

Proposition 4(i) shows that the breadth of mismatching ($\lambda$) from consumers is negatively associated with the prices of new books in a new bookstore and second-hand books in a second-hand bookstore. It makes sense that the larger the amounts of consumers who have a mismatch problem, the more likely the second-hand bookstore will set a lower price for the second-hand books to retain consumers. The new bookstore may also lower the new book’s price to compete with second-hand bookstores. Proposition 4(ii) indicates that the depth of mismatching ($\epsilon$) is positively related to the price of special old books in a second-hand bookstore and negatively associated with the price of books in a new bookstore and ordinary books in a second-hand bookstore. As the depth of the mismatch problem becomes more severe, consumers will be more likely to purchase special old books; thus, their selling prices will increase. Moreover, the new and second-hand bookstore will decrease their prices for ordinary books to attract users.

Proposition 5.

From the above results, we can deduce that the effect of mismatch on demands is:

•  (i) 

  $\partial D_n^c/\partial \lambda <0;\partial D_o^c/\partial \lambda <0;\partial D_r^c/\partial \lambda >0;\partial D_s^c/\partial \lambda >0$;

•  (ii) 

  $\partial D_n^c/\partial \epsilon <0;\partial D_o^c/\partial \epsilon <0;\partial D_r^c/\partial \epsilon >0;\partial D_s^c/\partial \epsilon >0$.

Proposition 5(i) indicates that the demand for special old books will increase with $\lambda$, while the demand for ordinary books at either the new or second-hand bookstore decreases as $\lambda$ grows larger. More consumers will change to a second-hand bookstore to buy special old books when the breadth of mismatch problems increases. As a result, demand for special books will increase in a used bookstore, and demand for ordinary second-hand books will decrease. This phenomenon will also hurt the demand for new books in a new bookstore. Moreover, Proposition 5(i) shows that the total demand for second-hand bookstores increased $\lambda$. Although demand for ordinary books in second-hand bookstores decreases, increasing demand for special books will offset the loss and bring higher total market demand.

Proposition 5(ii) states that the demand for ordinary books in second-hand bookstores and new books in new bookstores decreases $\epsilon$. In contrast, demand for special old books at second-hand bookstores is positively associated with $\epsilon$. Consumers tend to buy the special used book when the depth of mismatch problems increases. In this case, the demand for special books will increase, while demand for ordinary books will decrease. Moreover, the relationship between demand for all kinds of books in second-hand bookstores and $\epsilon$ is positively related. The reason for these results is similar to the explanation in Proposition 5(i). As the depth of the mismatch problem increases, more consumers will switch to a second-hand bookstore to buy special old books.

Proposition 6.

From the above results, we can deduce that the effect of mismatch on profits is:

•  (i) 

  $\partial {\pi }_n^c/\partial \lambda <0;\partial {\pi }_s^c/\partial \lambda >0$;

•  (ii) 

  $\partial {\pi }_n^c/\partial \epsilon <0;\partial {\pi }_s^c/\partial \epsilon >0$.

Proposition 6(i) shows that the revenue for the used bookstore increases with the growth of $\lambda$, while the profit of the new bookstore decreases $\lambda$. Concerning the profit of the second-hand bookstore, although the demand and price for ordinary second-hand books decrease, the demand for special old books increases, which means the total profit will increase. This result is consistent with the previous analysis for the new bookstore. The total profit will decrease with the growth of $\lambda$, the demand, and the price for new books. Proposition 6(ii) shows that a larger depth of mismatch will result in a larger profit for the second-hand bookstore and a smaller profit for the new bookstore. From this proposition, we can find that a second-hand bookstore may employ a differentiation strategy and provide different books from the new bookstore. Thus, the depth of mismatch will increase, and second-hand bookstores can make greater profits.

These results indicate that the mismatch’s depth and breadth affect the second-hand bookstore’s performance. Suppose more customers prefer special old books, which will benefit second-hand bookstores and is not suitable for new bookstores. From the perspective of second-hand bookstores, they can adopt a differentiation strategy to enhance their profit. Through market research and surveys, they can explore market potential and sell second-hand books that meet consumer preferences. Moreover, second-hand bookstores can provide satisfactory customer service by improving product delivery and return speed. Thus, they can increase customer satisfaction and gain competitiveness.

5. Considering the Preferences of Consumers

In this case, we introduce $\tau$ to describe the proportion of consumers who have preferences for special old books and are familiar with special old books. Thus, they can be potential consumers of special old books. These consumers will purchase special or ordinary books, as in the previous case. However, a proportion of $1-\tau$ consumers who have no preferences for special old books may choose to buy either ordinary used books in a second-hand bookstore or new books in a new bookstore. Therefore, there will be only two choices made by these consumers. For the consumers who buy an ordinary used book directly at the second-hand bookstore, their utility will be $U_o=U-P_o-t\cdot x-t\cdot \left(1-x\right)$. If they switch channels and decide to buy the new book at the new bookstore, their utility will be $U_n=U-P_n-t\cdot x$. In the setting $U_n=U_o$, we obtain the indifference point $x_c=-\frac{P_n-t-P_o}{t}$. For the proportion of consumers $1-\tau$, we derive that $\left(1-\tau \right)\left(1-x_c\right)$ consumers will purchase the special used book from the used bookstore, while $\left(1-\tau \right)x_c$ of consumers will switch channels to buy the new book at a new bookstore. Taking the first-order derivative of ${\pi }_s$ to $P_o$ and $P_r$, ${\pi }_n$ to $P_n$, respectively, and letting them be equal to zero, we obtain the optimal selling price $P_o^d,P_r^d,P_n^d$. Substituting $P_o^d,P_r^d,P_n^d$ into equations of profits and demands, we obtain the used bookstore and new bookstore’s demand and revenue, as presented in Table 2.

The following part explores the influence of consumers’ preferences on the used bookstore and new bookstore’s pricing decisions, demands, and profits.

Proposition 7.

From the above results, we can deduce the effect of consumers’ preferences for special old books on price, demand, and profits:

•  (i) 

  $\partial P_n^d/\partial \tau <0;\partial P_o^d/\partial \tau <0;\partial P_r^d/\partial \tau <0$.

•  (ii) 

  $\partial D_o^d/\partial \tau <0$if$t>\left(2\epsilon \lambda \tau -\epsilon \right)/2,$and vice versa.$\partial D_r^d/\partial \tau >0$if$t>\left(2\epsilon \lambda \tau -3\epsilon \right)/2$, and vice versa.$\partial D_n^d/\partial \tau <0$.

•  (iii) 

  $\partial {\pi }_n^d/\partial \tau <0;$ $\partial {\pi }_s^d/\partial \tau >0$if$t>\left(10\epsilon \lambda \tau -9\epsilon \right)/8$, and vice versa.

Proposition 7(i) shows that as consumers have a stronger preference for special old books, the prices for ordinary books at both new and second-hand bookstores will decrease, and the price of the used book in the second-hand bookstore will also decrease. Therefore, customers’ preferences for special old books are negatively related to book prices, including new and used books. Proposition 7(ii) indicates that the demand for ordinary books at new bookstores decreases when the preference for special old books grows. It is intuitive. However, the impact of customers’ preference for special old books on the demand for special books and ordinary books at the second-hand bookstore is related to the value of the transportation cost $t$. If $t$ is relatively large, in a second-hand bookstore, the demand for special old books always increases with customers’ preference for special old books. In contrast, the demand for ordinary second-hand books will decrease with customers’ preference for special old books.

Proposition 7(iii) states that the new bookstore’s revenue will decrease with $\tau$, and the influence of $\tau$ on second-hand bookstore’s profit also depends on the transportation cost $t$. If the transportation cost is high, the profit of the second-hand bookstore will always increase with $\tau$. Otherwise, if the transportation is small, the profit of the second-hand bookstore will decrease. This shows that the growth of customers’ preference for special old books will not necessarily improve earnings for second-hand bookstores. The reason for this might be that when the transportation cost factor $t$ is small, the difference in utility a consumer has between purchasing a used book or a new book becomes smaller; thus, the competition between these two entities becomes fiercer. The second-hand bookstore’s profit thus decreases. However, as the customers’ preference for special old books reaches a high level, new bookstores will continue to lower prices to retain consumers. Thus, second-hand bookstores will have to reduce the prices of both new and used books to gain a competitive advantage. In this case, although the price for second-hand books has decreased, the increased sale of second-hand and new books may lead to an increase in the overall profit of the second-hand bookstore.

In summary, a strategy of enhancing customers’ preference for special old books can be used for second-hand bookstores. Raising consumers’ preference for special old books will increase second-hand bookstores’ profits if the transportation cost factor is relatively large. Advertising and membership systems are some regular measures to improve consumers’ preference for special old books. On the other hand, if second-hand bookstores keep promoting second-hand books’ specialty when the transportation cost is relatively low, this may cause some fierce competition with new bookstores and damage the overall profits of the second-hand bookstores. Thus, for second-hand bookstores to take measures to attract customers, they should consider the value of the transportation cost factor. As mentioned in Section 2, the transportation cost factor is related to searching for a book and making the purchase decision on online platforms, including reading users’ book reviews and previewing parts of the book. When the transportation cost factor is low, it is relatively easy for customers to find a book and make purchase decisions on online platforms. In this circumstance, second-hand bookstores should not overly promote the specialty of second-hand books to avoid price competition with new bookstores. When the transportation cost factor is large, second-hand bookstores can obtain larger profits by promoting the specialty of their products.

6. Conclusions

6.1. Theoretical Implications

We built a Hotelling model consisting of an online second-hand bookstore and an online new bookstore. We analyzed the equilibrium price, demand, and profits for both the Cournot case and Stackelberg case retailers based on our assumptions. Additionally, we examined the effectiveness of the used bookstore’s differentiation strategy and the mismatch problem’s influence on retailers’ strategies, including breadth and depth. Furthermore, we extended the model by considering consumers’ preferences and proposed measures for second-hand bookstores to implement differentiation strategies more effectively. Thus, our research provides guidance for second-hand bookstores to ensure that they can attract more consumers and remain profitable.

Our theoretical analysis shows that, for a second-hand bookstore, it is an effective measure to implement a differentiation strategy. The second-hand bookstore can earn a larger profit by introducing and promoting special old books. Moreover, the second-hand bookstore’s profit increases with the value of depth and breadth of book mismatch, while the new bookstore’s profit decreases. The results also illustrate that second-hand bookstores can adopt a differentiation strategy to enhance their profit. Finally, the influence of customers’ preferences on the revenue of a second-hand bookstore is related to the transportation cost. When the transportation cost factor is relatively small, the profit of second-hand bookstores will decrease with consumers’ preferences for special old books and vice versa. Thus, expanding the consumers’ preferences for special old books will not necessarily increase profits for second-hand bookstores.

The overall contribution of this study is threefold. First, we investigated an industry mostly ignored by extant literature in the second-hand product market. Although prior research has acknowledged the increasing importance of the second-hand book market, most book industry studies focus on new books and lack investigation of the characteristics of second-hand books. Second, although the operation problem of second-hand products has been well studied in the literature, little is known about how the second-hand bookstore’s differentiation strategy will influence the book market’s competition and equilibrium. Specifically, we investigate this issue in the context of the platform economy. Third, on a broader level, this study contributes to extending our knowledge about expanding the second-hand book market to enhance social welfare. Several factors must be considered to successfully operate second-hand bookstores, such as the mismatch problem and consumers’ purchasing preferences.

6.2. Practical Implications

Some managerial implications are revealed in this study as a suggestion for managers to focus on the economic aspects of second-hand books. Since second-hand bookstores attract customers by selling special used or cheap ordinary books, the competition between new bookstores and second-hand bookstores is fierce. Managers need to attract customers. Uniqueness, low price, and nostalgic preferences are characteristics of second-hand books. These traits make second-hand bookstores different from new bookstores. However, the used book platforms largely ignore these measures, and there is especially an issue with trusting descriptions. Therefore, managers of second-hand bookstores can provide better services to attract customers, such as providing truthful descriptions of books’ conditions and traits, customers’ reviews, and ratings of the bookstore, as well as quality assurances and easy return policies. Furthermore, certain customers are interested in buying special old books, despite their high prices and poor conditions. These books are used for the hobby of collection. Our study implies that second-hand booksellers can innovatively change their price policies to obtain perceived value from customers. For example, platforms can provide an environment for bargaining and auctioning these special books.

Regarding new bookstore platforms, they should also implement strategies to retain their customers who are likely to switch to second-hand bookstores. New bookstores can offer discounts or coupons to promote their products. Moreover, sophisticated customer services will be helpful to retain customers, such as providing enriched and transparent book reviews and introductions, attractive return policies, and quick delivery. These measures can enhance customer relationships and encourage their purchase intentions in the future.

6.3. Limitations

Although this study contributes to the growing platform competition literature [38,39,40], there are several limitations that open avenues for future research. First, this model omits many detailed features of second-hand bookstores and new bookstores, such as their relationships with book suppliers [6], after-sales services [41], and spillover effects between different forms of stores [42]. Future researchers can consider networks of the whole supply chain [43,44,45,46,47,48]. Second, our research only examines the behavior of consumers and bookstores from a theoretical model. We may draw more interesting conclusions if we can collect relevant data on consumers and retailers on e-commerce platforms and conduct empirical analysis. Last, we only discussed the equilibrium conditions for maximizing retailer profits. We may yield different results if we consider maximizing social welfare [49,50]. In such an extension, the strategy of used bookstores would be more complex. All these avenues are promising and deserve future research.

Author Contributions

Y.W., A.M., Z.H., J.P. and J.O. contributed equally to the content of the study. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no competing interests.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. The model structures.

Figure 2. Comparison between [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] with different values of [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] ([Forumla omitted. See PDF.]).

Table 1

Parameters and variables.

Parameters Notation	Definition
$P_o,P_r,P_n$	The selling price of ordinary old books, special old books, and new books, respectively
$D_o,D_r,D_n$	The demand for ordinary old books, special old books, and new books, respectively
${\pi }_s$	Profit of the second-hand bookstore
${\pi }_n$	Profit of the new bookstore
$\tau$	The proportion of consumers who prefer special second-hand books
$\epsilon$	The depth of the mismatching
$\lambda$	The breadth of mismatching
$o$	The subscript for an ordinary old book
$r$	The subscript for a special old book
$n$	The subscript for a new book
$a$	The superscript for “Cournot model”
$b$	The superscript for “Stackelberg model”
$c$	The superscript for the model considering the mismatch
$d$	The superscript for the model considering consumers’ preferences

Table 2

Equilibrium outcome summary.

	Cournot	Stackelberg	Considering Mismatch	Considering ConsumerPreferences
Price	$P_n^a=2t/3$	$P_n^b=t$	$P_n^c=2t/3-\lambda \epsilon /3$	$P_n^d=-1/3\epsilon \lambda \tau +2/3t$
	$P_o^a=t/3$	$P_o^b=t/2$	$P_o^c=t/3-\lambda \epsilon /6$	$P_o^d=-1/6\epsilon \lambda \tau +t/3$
	/	/	$P_r^c=\epsilon /2+t/3-\lambda \epsilon /6$	$P_r^d=-1/6\epsilon \lambda \tau +t/3+\epsilon /2$
Sales	$D_n^a=2/3$	$D_n^b=1/2$	$D_n^c=\left(2t-\lambda \epsilon \right)/3t$	$D_n^d=\left(2t-\epsilon \lambda \tau \right)/3t$
	$D_o^a=1/3$	$D_o^b=1/2$	$D_o^c=\left(\left(-1+\lambda \right)\left(\lambda \epsilon -2t\right)\right)/6t$	$D_o^d=\left(\epsilon \lambda \tau -2t\right)\left(\lambda \tau -1\right)/6t$
	/	/	$D_r^c=\left(\left(3\epsilon +2t-\lambda \epsilon \right)\lambda \right)/6t$	$D_r^d=-\left(\lambda \left(\epsilon \lambda \tau -3\epsilon -2t\right)\tau \right)/6t$
Profit	${\pi }_n^a=4t/9$	${\pi }_n^b=t/2$	${\pi }_n^c=A$	${\pi }_n^d=C$
	${\pi }_o^a=t/9$	${\pi }_o^b=t/4$	${\pi }_s^c=B$	${\pi }_s^d=D$

Note: wherein $A=1/9\frac{{\left(\epsilon \lambda -2t\right)}^2}{t}$; $B=-1/36\frac{5{\epsilon }^2{\lambda }^2-9{\epsilon }^2\lambda -8\epsilon \lambda t-4t^2}{t}$; $C={\left(\epsilon \lambda \tau -2t\right)}^2/9t$; $D=-1/36\frac{5{\epsilon }^2{\lambda }^2{\tau }^2-9{\epsilon }^2\lambda \tau -8\lambda \epsilon t\tau -4t^2}{t}$.

Appendix A

Proof of Proposition 2.

$$\begin{gathered} D_n^c-D_n^a<0 \\ {D}_o^c-D_o^a=1/6\frac{\left(-1+\lambda \right)\left(\epsilon \lambda -2t\right)}{t}-1/3=1/6\frac{\lambda \left(\epsilon \lambda -\epsilon -2t\right)}{t}<0 \\ {\pi }_n^c-{\pi }_n^a<0 \\ {\pi }_s^c-{\pi }_o^a=-1/36\frac{\lambda \epsilon \left(5\lambda \epsilon -9\epsilon -8t\right)}{t} \end{gathered}$$

We assume $t=1/5,t=1/2,t=1$ and draw a two-dimensional graph, as shown in Figure A1. The two-dimensional planes are all above the 0 plane, so it is clear that ${\pi }_s^c-{\pi }_o^a>0$.□

Figure A1. Values of [Forumla omitted. See PDF.].

Proof of Proposition 4.

$$\begin{gathered} \partial P_n^c/\partial \lambda =-\epsilon /3<0;\partial P_o^c/\partial \lambda =\partial P_r^c/\partial \lambda =-\epsilon /6<0 \\ \partial P_o^c/\partial \epsilon =-\lambda /6<0;\partial P_r^c/\partial \epsilon =-\lambda /6+1/2>0;\partial P_n^c/\partial \epsilon =-\lambda /3<0 \end{gathered}$$

□

Proof of Proposition 5.

$$\begin{gathered} \partial D_n^c/\partial \lambda =-\frac{\epsilon }{3t}<0;\partial D_o^c/\partial \lambda =1/6\frac{2\lambda \epsilon -\epsilon -2t}{t}<0 \\ \partial D_r^c/\partial \lambda =-1/6\frac{2\lambda \epsilon -3\epsilon -2t}{t}>0;\partial D_s^c/\partial \lambda =1/3\frac{\epsilon }{t}>0 \\ \partial D_n^c/\partial \epsilon =-1/3\frac{\lambda }{t}<0;\partial D_o^c/\partial \epsilon =1/6\frac{\left(-1+\lambda \right)\lambda }{t}<0 \\ \partial D_r^c/\partial \epsilon =-1/6\frac{\lambda \left(\lambda -3\right)}{t}>0;\partial D_s^c/\partial \epsilon =1/3\frac{\lambda }{t}>0 \end{gathered}$$

□

Proof of Proposition 6.

$$\begin{gathered} \partial {\pi }_n^c/\partial \lambda =2/9\frac{\left(\lambda \epsilon -2t\right)\epsilon }{t}<0;\partial {\pi }_s^c/\partial \lambda =-1/36\frac{\epsilon \left(10\lambda \epsilon -9\epsilon -8t\right)}{t}>0 \\ \partial {\pi }_n^c/\partial \epsilon =2/9\frac{\left(\lambda \epsilon -2t\right)\lambda }{t}<0;\partial {\pi }_s^c/\partial \epsilon =-1/18\frac{\lambda \left(5\lambda \epsilon -9\epsilon -4t\right)}{t}>0 \end{gathered}$$

□

Proof of Proposition 7.

$$\begin{gathered} \partial P_n^d/\partial \tau =-1/3\lambda \epsilon <0;\partial P_o^d/\partial \tau =\partial P_r^d/\partial \tau =-1/6\lambda \epsilon <0 \\ \partial D_n^d/\partial \tau =-1/3\frac{\lambda \epsilon }{t}<0 \end{gathered}$$

$\partial D_o^d/\partial \tau =1/6\frac{\lambda \left(2\epsilon \lambda \tau -\epsilon -2t\right)}{t}$. If $t>\left(2\epsilon \lambda \tau -\epsilon \right)/2,$ then $\partial D_o^d/\partial \tau <0$, and vice versa.

$\partial D_r^d/\partial \tau =-1/6\frac{\lambda \left(2\epsilon \lambda \tau -3\epsilon -2t\right)}{t}$. If $t>\left(2\epsilon \lambda \tau -3\epsilon \right)/2$, then $\partial D_r^d/\partial \tau >0$, and vice versa.

$\partial {\pi }_s^d/\partial \tau =-1/36\frac{\lambda \epsilon \left(10\epsilon \lambda \tau -9\epsilon -8t\right)}{t}$. If $t>\left(10\epsilon \lambda \tau -9\epsilon \right)/8$, then $\partial {\pi }_s^d/\partial \tau >0$$$\partial {\pi }_n^d/\partial \tau =2/9\frac{\left(\epsilon \lambda \tau -2t\right)\epsilon \lambda }{t}<0$$

□

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