# Dynamic Pricing Strategy for Two-Generation Products under Different Trade-in Subsidy Strategies

Authors: Yingwei Ju
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Trade-in is one of the most widely used recycling methods in practice. A key problem in the trade-in process is the dynamic pricing strategy under different trade-in subsidy strategies. Consumers are divided into myopic and strategic consumers. Considering three situations, namely, (1) no trade-in subsidy strategy, (2) the firm’s trade-in subsidy strategy, and (3) the government’s trade-in subsidy strategy, the dynamic game equilibrium between firms and consumers is analyzed. The optimal dynamic pricing strategy is obtained by constructing a two-stage dynamic game model. The results show that (1) the number of consumers who purchase next-generation new products is not impacted by the trade-in subsidy strategy. However, the number of consumers who engage in trade-ins is impacted by the trade-in subsidy strategy. (2) The number of consumers who engage in trade-ins with the trade-in subsidy strategy is larger than that without the trade-in subsidy strategy. (3) Compared with the government’s trade-in subsidy strategy, the firm’s trade-in subsidy strategy can more effectively motivate consumers to engage in trade-in activity.


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## Body

## 1. Introduction

### 1.1. Background

Trade-in program is one of the most widely used recycling methods in practice. For example, mobile phone, air conditioners, and computers can be recycled through trade-in programs (Altug and Aydinliyim [1]). An important problem faced by firms during trade-ins is the product’s pricing problem. To retain old consumers and encourage consumers to buy products many times, firms often offer a discount price for first-generation products (Klemperer [2]), which we call the trade-in rebate. Due to the product life cycle and product upgrades during trade-ins, firms are often faced with the dynamic pricing problem (Cachon and Feldman [3]; Correa et al. [4]). A dynamic pricing strategy means that the enterprise determines the selling price for each period. Many firms have used the dynamic pricing strategy during their sales process. For instance, many producers, such as Huawei and Samsung, make pricing decisions for each generation’s products dynamically. Under the dynamic pricing strategy, in the first period, customers purchase first-generation products and firms determine the selling price of these first-generation products. In the second period, customers choose to purchase second-generation products using trade-ins, and enterprises decide the selling price of new-generation products.In general, when enterprises determine to adopt the dynamic pricing strategy, they should know the effect of customers’ strategic choice behavior. The strategic customer can estimate the price of future products and strategically choose when to buy. Customers’ strategic behavior mainly expresses two aspects, that is, on the one hand, customers can get the retail price of next-generation products in many ways. On the other hand, when firms offer trade-ins, customers will select whether and when to engage in trade-ins.In view of the important influence of trade-ins in improving demand for new products and the circular economy, many governments have pointed out a series of trade-in policies to improve customers to engage in trade-ins. For example, the United States offers a subsidy of $3500–$4500 to customers. The Chinese government subsidizes 4000–6000 yuan for customers who engage in vehicle trade-in activities. In addition to the government providing trade-in subsidies, some enterprises also provide trade-in subsidies to encourage repeat purchases and sales of new products. For example, Huawei offers a subsidy of 300 yuan for customers. Apple offers a maximum subsidy of 1000 yuan for customers. The dynamic pricing strategy for successive-generation products considering consumer choice behavior and different trade-in subsidy strategies should be worthy of research. In this paper, the following questions are addressed: (a) What are the optimal pricing strategies and trade-in strategies for firms under different trade-in subsidy strategies. (b) How would the different trade-in subsidy strategies influence consumers’ choice behavior and the firm’s optimal dynamic pricing strategy.First, the two-stage game model was built based on myopic customers, strategic customers, and different trade-in subsidy strategies. It is assumed that the enterprise sells two-generation products. First, the enterprise sells the first-generation products to myopic customers and strategic customers. In the second period, the enterprise sells the second-generation products and offers trade-ins for myopic and strategic consumers under different trade-in subsidy strategies. Second, myopic and strategic customers’ choice behavior and pricing strategy within each period are analyzed. Third, the impact of different trade-in subsidy strategies on two kinds of customers’ choice behavior and the firm’s optimal dynamic pricing strategy is analyzed.The main results are summarized as follows: From the perspective of consumers, (1) in the first period, some consumers would like to purchase the second-generation new products, some consumers prefer to purchase the first-generation new products at a discount price, and others would like to engage in trade-in activity. (2) In the second period, many customers who engage in the trade-in activity are impacted by the trade-in subsidy strategy. The number of consumers who engage in trade-ins with the trade-in subsidy strategy is larger than that without it. From the view of the enterprise, (1) the enterprise’s profit in the dynamic pricing strategy is decreasing when the discount factor and the innovation incremental value are at a higher level, (2) the firm can obtain much more profit when two kinds of consumers exist in the market, and (3) with the increase in trade-in subsidies, the trade-in rebate decreases.


### 1.2. Contribution Statement

First, this study discusses the optimal dynamic pricing strategy under different trade-in subsidies which includes no trade-in subsidy strategy, the firm’s trade-in subsidy strategy, and the government’s trade-in subsidy strategy. Second, this study is the first to characterize the impact of myopic consumers and strategic consumers’ choice behavior on the dynamic pricing strategy under different trade-in subsidy strategies.The remainder of this study is organized as follows: Section2 points out the literature review. Section 3 provides the model and the notation. Section 4 analyzes the enterprise’s dynamic pricing strategy during two periods. In Section 5, the results under three decision-making models are compared, and Section 6 concludes the paper.


## 1.1. Background

Trade-in program is one of the most widely used recycling methods in practice. For example, mobile phone, air conditioners, and computers can be recycled through trade-in programs (Altug and Aydinliyim [1]). An important problem faced by firms during trade-ins is the product’s pricing problem. To retain old consumers and encourage consumers to buy products many times, firms often offer a discount price for first-generation products (Klemperer [2]), which we call the trade-in rebate. Due to the product life cycle and product upgrades during trade-ins, firms are often faced with the dynamic pricing problem (Cachon and Feldman [3]; Correa et al. [4]). A dynamic pricing strategy means that the enterprise determines the selling price for each period. Many firms have used the dynamic pricing strategy during their sales process. For instance, many producers, such as Huawei and Samsung, make pricing decisions for each generation’s products dynamically. Under the dynamic pricing strategy, in the first period, customers purchase first-generation products and firms determine the selling price of these first-generation products. In the second period, customers choose to purchase second-generation products using trade-ins, and enterprises decide the selling price of new-generation products.In general, when enterprises determine to adopt the dynamic pricing strategy, they should know the effect of customers’ strategic choice behavior. The strategic customer can estimate the price of future products and strategically choose when to buy. Customers’ strategic behavior mainly expresses two aspects, that is, on the one hand, customers can get the retail price of next-generation products in many ways. On the other hand, when firms offer trade-ins, customers will select whether and when to engage in trade-ins.In view of the important influence of trade-ins in improving demand for new products and the circular economy, many governments have pointed out a series of trade-in policies to improve customers to engage in trade-ins. For example, the United States offers a subsidy of $3500–$4500 to customers. The Chinese government subsidizes 4000–6000 yuan for customers who engage in vehicle trade-in activities. In addition to the government providing trade-in subsidies, some enterprises also provide trade-in subsidies to encourage repeat purchases and sales of new products. For example, Huawei offers a subsidy of 300 yuan for customers. Apple offers a maximum subsidy of 1000 yuan for customers. The dynamic pricing strategy for successive-generation products considering consumer choice behavior and different trade-in subsidy strategies should be worthy of research. In this paper, the following questions are addressed: (a) What are the optimal pricing strategies and trade-in strategies for firms under different trade-in subsidy strategies. (b) How would the different trade-in subsidy strategies influence consumers’ choice behavior and the firm’s optimal dynamic pricing strategy.First, the two-stage game model was built based on myopic customers, strategic customers, and different trade-in subsidy strategies. It is assumed that the enterprise sells two-generation products. First, the enterprise sells the first-generation products to myopic customers and strategic customers. In the second period, the enterprise sells the second-generation products and offers trade-ins for myopic and strategic consumers under different trade-in subsidy strategies. Second, myopic and strategic customers’ choice behavior and pricing strategy within each period are analyzed. Third, the impact of different trade-in subsidy strategies on two kinds of customers’ choice behavior and the firm’s optimal dynamic pricing strategy is analyzed.The main results are summarized as follows: From the perspective of consumers, (1) in the first period, some consumers would like to purchase the second-generation new products, some consumers prefer to purchase the first-generation new products at a discount price, and others would like to engage in trade-in activity. (2) In the second period, many customers who engage in the trade-in activity are impacted by the trade-in subsidy strategy. The number of consumers who engage in trade-ins with the trade-in subsidy strategy is larger than that without it. From the view of the enterprise, (1) the enterprise’s profit in the dynamic pricing strategy is decreasing when the discount factor and the innovation incremental value are at a higher level, (2) the firm can obtain much more profit when two kinds of consumers exist in the market, and (3) with the increase in trade-in subsidies, the trade-in rebate decreases.


## 1.2. Contribution Statement

First, this study discusses the optimal dynamic pricing strategy under different trade-in subsidies which includes no trade-in subsidy strategy, the firm’s trade-in subsidy strategy, and the government’s trade-in subsidy strategy. Second, this study is the first to characterize the impact of myopic consumers and strategic consumers’ choice behavior on the dynamic pricing strategy under different trade-in subsidy strategies.The remainder of this study is organized as follows: Section2 points out the literature review. Section 3 provides the model and the notation. Section 4 analyzes the enterprise’s dynamic pricing strategy during two periods. In Section 5, the results under three decision-making models are compared, and Section 6 concludes the paper.


## 2. Literature Review

Many scholars have studied pricing decision issues in trade-ins. Some papers discussed the static pricing strategy problem in trade-ins. Ray et al. [5] discussed the optimal static pricing and the trade-in strategy. Huang et al. [6] considered the static pricing strategy in an automobile supply chain. Correa et al. [4] pointed out that the static pricing policy could reduce the waiting behavior of strategic customers and increase the enterprise’s profit. Han et al. [7] discussed the situations when enterprises should provide a trade-old for remanufactured (TOR) program. Chen et al. [8] analyzed three different choices of the enterprise. Ma et al. [9] analyzed one enterprise’s static pricing strategy and obtained the thresholds that determined whether the enterprise provides “trade-old for new” and “trade-old for remanufactured” programs. Miao et al. [10] analyzed three static pricing decision models with trade-ins. Cao et al. [11] analyzed the static pricing strategy in trade-in programs in B2C platforms. Cao et al. [12] discussed the optimal static pricing strategy and the optimal trade-in strategy of an enterprise subject to carbon tax policies. Hu et al. [13] discussed the enterprise’s static pricing strategy considering the trade-in duration. Sheu and Choi [14] used one multimethodological approach to explore trade-in-upgrade-related pricing decisions.Some scholars have discussed the dynamic pricing strategy problem in trade-ins. Zhu et al. [15] discussed the effect of a competitive environment on dynamic pricing strategy during the trade-in program. Xiao et al. [16] considered a firm that offered both trade-in options and discussed the dynamic pricing strategy during two periods. Liu et al. [17] compared the difference between static pricing strategies and dynamic pricing strategies. In the previous sections, some researchers analyzed the influence of customers’ strategic behavior on both static pricing strategy and dynamic pricing strategy. Huang et al. [6], Zhu et al. [15], Chen and Hsu [8], Xiao and Zhou [16], and Sheu and Choi [14] considered the effect of customers’ strategic behavior on pricing strategies. Moreover, they discussed the advantages and usage situations of different trade-in strategies. Some other papers that considered trade-in rebate decisions are as follows: Ray et al. [5] discussed optimal pricing and trade-in rebate decisions. Considering the strategic consumer and trade-in rebate, Liu et al. [17] discussed the difference between the static pricing strategy and the dynamic pricing strategy. Some papers explore the trade-in subsidy strategy offered by enterprises or governments. Levinthal and Purohit [18] pointed out that the enterprise uses trade-in rebates to improve product quality. Fudenberg and Tirole [19] discussed the optimal pricing and trade-in rebate decisions for a firm. Kim et al. [20] built one analytical model to discuss customers’ choices during trade-in program. Sana [21] investigated a reduction-inventory model where regular preventive maintenance starts at the end of production for smooth functioning in the next cycle.As is known to us, only a few scholars discuss the effect of the trade-in subsidy on the pricing strategy. For example, Huang et al. [6] considered one automobile supply chain and analyzed the impact of subsidy strategy on stimulating consumers’ trade-in transactions. Cao et al. [12] explored the optimal static pricing strategy and third-party collection authorization strategies for the manufacturer. Meng et al. [22] investigated the optimal government subsidy and its impact on the operation of the CLSC. He et al. [23] discussed the pricing decisions for the firm and government’s subsidy policy.This study contributes to the literature in following aspects: Some studies discuss the effect of myopic customers or strategic customers on the static pricing strategy or the dynamic pricing strategy in trade-ins. Others consider the optimal trade-in rebate decision problem during trade-ins. However, they do not analyze the impact of different trade-in subsidy on consumers’ choice and on firms’ pricing strategy. To fill the gap in the literature, this study discusses the optimal dynamic pricing strategy and the optimal trade-in rebate strategy. Moreover, this study considers the effect of myopic customers and strategic customers’ choice on the pricing strategy under the situations of different trade-in subsidy strategies and dual rollover strategies. Then, the effect of different trade-in subsidy strategies on consumers’ choice and the enterprise’s pricing strategy is compared.


## 3. The Model

Considering the two-stage game model between firms and customers, firms sell products by means of a dual rollover strategy; in period 1, the enterprise sells the first-generation new productsX1 at the retail price p1. The manufacturing cost is c1 (i.e., c1≤p1). The customers’ valuation of first-generation new products X1 is v1 (i.e., p1≤v1), which follows a [0,1] uniform distribution. We assume that it has D (i.e., =1) potential consumers in the market. Based on the consumer’s purchase behavior, α0≤α≤1 myopic consumers and 1−α strategic consumers were considered. Strategic customers analyze the opportunity to purchase products in period 2. However, myopic customers will not think to purchase products; when the retail price of the first-generation products is no larger than its valuation, myopic consumers will make purchase decisions. Assuming that the utility of the consumer decreases with the usage time, the discount factor δ∈0,1 is introduced.In period 2, the enterprise sells the second-generation new productsX2 at price p2, and the enterprise sells the first-generation products X1 in the market at discount price pd0≤pd≤p1,0≤pd≤p2. To retain the old consumers, we motivate consumers to buy repeatedly and stabilize market share, and the firm provides the trade-in service. The customers who have purchased the products X1 returns the old products X1 to the enterprise and get the certain trade-in rebate pt0≤pt≤p2 when the customers purchase X2, and the manufacturing cost is c2 (i.e., c2≤p2). The customers’ valuation of the second-generation new products is v2 (i.e., p2≤v2), where v2=θ+1v1, Each old product has a salvage value s0≤s≤θ. Figure 1 illustrates the time sequence of events.Figure 1
Sequence of events under a dynamic pricing strategy.In Figure1, at the beginning of the period 1, based on the number of consumers waiting for the second period, the firm decides the unit retail price p1. After that, two kinds of customers will determine whether to purchase them. At the beginning of the period 2, based on the unit retail price p1 and the expectation of the market demand, the firm determines the unit retail price p2, the discount price pd, and trade-in rebate pt.We consider the following decision-making models (see Figure2).(a)
Model (N), neither the firm nor the government provides the trade-in subsidy strategy (Figure2(a)).(b)
Model (R), the firm provides the trade-in subsidy strategy (Figure2(b)). Any customers who purchase the next generation new products and return his or her old generation products can obtain a specific trade-in subsidy sr from the firm.(c)
InModel (G), the government provides the trade-in subsidy strategy (Figure2(c)). Any customers who purchase the next generation new products and return his or her old generation products can obtain a specific trade-in subsidy sg from the government.Figure 2
Decision-making models. (a) Model N. (b) Model R. (c) Model G.
(a)(b)(c)Moreover, for easier analysis, the notations are summarized in Table1.Table 1
Model parameters.
ParametersDefinitionsqi2TNumber of customers who engage in trade-in program in period 2, where,i∈m,s, m means myopic consumers, s means strategic consumersqi2NTNumber of customers who do not engage in trade-ins in period 2qi2BNumber of customers who want to buy the second-generation new products in period 2qi2NBNumber of customers who do not want to buy the second-generation new products in period 2qi2BdNumber of customers who want to buy first-generation new products in period 2 at a discount priceqi1BNumber of customers who want to buy first-generation new products in period 1p1Retail pricep2Retail pricepdDiscount priceptFirm offers a trade-in rebatesrThe trade-in subsidy provided by the firmsgThe trade-in subsidy provided by the governmentδDiscount factorθInnovation incremental valuec1The manufacturing cost of the first-generation new productsc2The manufacturing cost of the next generation new productsv1Customers’ valuation of the first-generation new productsv2Customers’ valuation of the next generation new productsαProportion of the myopic consumerπ2Firm’s profit in the second periodπtFirm’s total profit in the two periods


## 4. Decision-Making Models

Because the government or the firm can provide different trade-in subsidy strategies for customers, the impact of different trade-in subsidy strategies on customers’ choice and enterprises’ dynamic pricing strategies will be discussed. We characterize the outcomes of each decision-making model as shown in Figure2.


### 4.1. Benchmark: Without Trade-in Subsidies (Model N)

#### 4.1.1. Consumers’ Strategic Choice Behavior

First, myopic consumers’ choice behavior is analyzed. When the retail price of the first-generation products is no higher than its valuation, myopic consumers will make purchase decisions. When myopic customers with a valuation higher thanτ1m purchase X1 in period 1, myopic consumers with a valuation lower than τ1m choose to wait. In period 2, myopic customers include X1 -holders and non- X1 -holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt or to keep using X1 (the utility is v1). The non- X1 -holders decide whether to buy X1 at the discount price (the utility is v1−pd) or buy X2 at the regular price (the utility is 1+θv1−p2. Comparing the myopic consumers’ surplus, their purchasing decision can be obtained in the second period. The X1 -holders with a valuation higher than τ2m choose to trade-in X1 for X2. For the non-X1-holders, it exists two different selections. In case I(II), the customers would like to buy X2X1, the non- X1 -holders with a valuation between τ3m and τ1m will purchase X2X1, and the non- X1 -holders with a valuation between τ4m and τ3m will purchase X1X2, while those with a valuation lower than τ4m will buy nothing.Next, strategic customer choice behavior is analyzed. Strategic customers can analyze the choice to purchase products in period 2, and the customers can compare the utility gotten from the first period with the delayed period to utilise and choose the best purchase chance. When strategic consumers with the valuation larger thanτ1s purchase X1 in period 1, strategic customers with a valuation lower than τ1s choose to wait. In period 2, strategic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 with the discount price (the utility is v1−pd) or to buy X2 with the regular price (the utility is 1+θv1−p2).


#### 4.1.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed,π2N is the enterprise’s profit in period 2, and the optimization problem is as follows:(1)maxp2,pt,pdπ2N=p2−c2−ptqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(2)s.t.0≤qm2B≤α−qm1B,(3)0≤qm2Bd≤α−qm1B,(4)qm2Bd+qm2B≤α−qm1B,(5)qm2T≤qm1B,(6)0≤pt≤p2,(7)0≤pd≤p1,(8)0≤qs2B≤1−α−qs1B,(9)0≤qs2Bd≤1−α−qs1B,(10)qs2Bd+qs2B≤1−α−qs1B,(11)qs2T≤qs1B.Constraints (2)–(4) guarantee that myopic consumers who will not hold X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (5) makes sure that the X1-holders could trade-in X1 for X2. Constraints (8)–(10) guarantee that strategic consumers will not hold X1, buy X2 at p2, or purchase X1 with pd in the second period. Constraint (11) makes sure that the X1-holders will trade-in X1 for X2.Next, based on customer’s demand under different choice conditions, the firm should decide the retail pricep1 in the first period to maximize the total profit πtN of two periods.(12)maxπtN=p1−c1qm1B+qs1B+π2N∗,(13)s.t.0≤qm1B≤α,(14)0≤qs1B≤1−α,(15)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (13)–(15) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution, when the firm uses the dynamic pricing strategy, can be obtained in Proposition 1 (See Appendix for the proof of Proposition 1).Proposition 1.
When the enterprise uses the dynamic pricing strategy, the equilibrium results are derived as follows:(i)
In case I, the equilibrium prices and trade-in rebate are as follows:(16)p1∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,p2∗=c22+1+θ2+c14θ+11+θ43θ−14δθ−θ−1−41+θ2,pt∗=1−θ8+c19θ+11+θ43θ−14δθ−θ−5−41+θ2,pd∗=18+c14θ+143θ−14δθ−θ−1−41+θ2+c12.(ii)
In case I, the equilibrium quantities are as follows:(17)qm2B∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qs2B∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qm1B∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs2Bd∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qm2Bd∗=1+δ+2c12θ+12θ+1+c11+θ2,qs2T∗=1−α−c22θ,qm2T∗=α−c22θ,qs1B∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2.(iii)
In case II, the equilibrium prices and trade-in rebate are as follows:(18)p1∗=θ+c12θ+12θ+13+δ,p2∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pd∗=θ+2c12θ+12θ+13+δ,pt∗=θ1+θ+2c12θ+11+θ2θ+13+δ−2c11+θ4−θ4.(iv)
In case II, the equilibrium quantities are as follows:(19)qm1B∗=αθ+1−θ2θ+1,qm2T∗=α−2c24θ−14,qs1B∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qs2Bd∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2Bd∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qm2B∗=c22θ−1+θ2θ−c221+θ,qs2B∗=c22θ−1+θ2θ−c221+θ,qs2T∗=1−α−2c24θ−14.Lemma 1.
Under the situation of without trade-in subsidy strategy, (i) the retail price is larger than the discount price, and (ii) the customers who have bought the first-generation’s old products is larger than the customers who engage in trade-in program.
We can get the following conclusions:①
In the case without the trade-in subsidy strategy, in period 2, the retail price of the second-generation new products is larger than the discount price.②
Comparing with the choice behavior of different types of consumers, it was found thatqm1B∗>qm2T∗;qs1B∗>qs2T∗. This means that customers who have purchased old products in period 1 do not necessarily engage in trade-ins in period 2. Some customers can choose to buy the second-generation new products; in case I, strategic customers select to buy the new-generation products directly.


### 4.2. The Firm Provides Trade-in Subsidies (Model R)

To retain old consumers, we motivate consumers to buy products frequently, reduce the encroachment effect from the secondary market, and promote product upgrading; the firm provides trade-in subsidiessr to customers who engage in trade-ins.


#### 4.2.1. Consumers’ Strategic Choice Behavior

First, when the retail price is no higher than its valuation, myopic consumers will make purchase decisions. When myopic customers with a valuation larger thanτ1m buy X1 in period 1, myopic customers with a valuation lower than τ1m choose to wait. In period 2, myopic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sr) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or buy X2 at a regular price (the utility is 1+θv1−p2). Comparing the myopic customers’ utility, the X1-holders with the valuation larger than τ2m choose to trade-in X1 for X2, and the others will keep using X1. For the non-X1-holders, in case I(II), the customers would like to buy X2X1, the non-X1-holders with a valuation between τ3m and τ1m will purchase X2X1, and the non-X1-holders with the valuation between τ4m and τ3m buys X1X2.Next, strategic customer choice behavior is analyzed. When strategic customers with the valuation larger thanτ1s buy X1 in the first period, strategic consumers with a valuation lower than τ1s choose to wait. In the second period, strategic customers include X1-holders and the non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sr) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or to buy X2 at the regular price (the utility is 1+θv1−p2).


#### 4.2.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed,π2R is the enterprise’s profit in the second period, and the optimization problem is that(20)maxp2,pt,pdπ2R=p2−c2−pt−srqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(21)s.t.0≤qm2B≤α−qm1B,(22)0≤qm2Bd≤α−qm1B,(23)qm2Bd+qm2B≤α−qm1B,(24)qm2T≤qm1B,(25)0≤pt≤p2,(26)0≤pd≤p1,(27)0≤qs2B≤1−α−qs1B,(28)0≤qs2Bd≤1−α−qs1B,(29)qs2Bd+qs2B≤1−α−qs1B,(30)qs2T≤qs1B.Constraints (21)–(23) guarantee that myopic consumers will not hold X1 purchase X2 at p2 or purchase X1 with pd in period 2. Constraint (24) makes sure that the X1-holders trade-in X1 for X2. Constraints (27)–(29) guarantee that strategic consumers will not hold X1 purchase X2 at p2 or purchase X1 with pd in period 2. Constraint (30) makes sure that the X1-holders trade-in X1 for X2.Next, based on the consumer’s demand under different choice conditions, the firm should decide the retail pricep1 in the first period to maximize the total profit πtR of the two periods. π2R∗ is the firm’s optimal profit in the second period.(31)maxp1πtR=p1−c1qm1B+qs1B+π2R∗,(32)s.t.0≤qm1B≤α,(33)0≤qs1B≤1−α,(34)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (32)–(34) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution when the firm uses the dynamic pricing strategy can be obtained in Proposition 2 (See Appendix for the proof of Proposition 2).Proposition 2.
If the firm provides a trade-in subsidy, under a dynamic pricing strategy, it has one perfect equilibrium.(i)
In case I, the equilibrium solutions are as follows:(35)p1∗∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,pd∗∗=18+c14θ+143θ−14δθ−θ−1−41+θ2+c12,pt∗∗=1−θ8+c19θ+11+θ43θ−14δθ−θ−5−41+θ2−sr,p2∗∗=c22+θ+12+c1θ+14θ+143θ−14δθ−θ−1−41+θ2.(ii)
In case II, the equilibrium quantities are as follows:(36)qm2B∗∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qm2Bd∗∗=1+δ+2c12θ+12θ+1+c11+θ2,qs2B∗∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qs2Bd∗∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qs1B∗∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2,qm1B∗∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs2T∗∗=34−α−c22θ+srθ,qm2T∗∗=α−c22θ−14+srθ.(iii)
In case I, the equilibrium solutions are as follows:(37)p2∗∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pt∗∗=θθ+1+2c12θ+1θ+12θ+13+δ−2c1θ+14−θ4−sr2,p1∗∗=θ+c12θ+12θ+13+δ,pd∗∗=θ+2c12θ+12θ+13+δ.(iv)
In case II, the equilibrium quantities are as follows:(38)qs1B∗∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qs2B∗∗=c22θ−1+θ2θ−c221+θ,qm2T∗∗=α−2c24θ−14+sr2θ,qm2Bd∗∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qs2T∗∗=1−α−2c24θ−14+sr2θ,qm1B∗∗=αθ+1−θ2θ+1,qs2Bd∗∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2B∗∗=c22θ−1+θ2θ−c221+θ.Lemma 2.
When the enterprise uses the trade-in subsidy strategy, (i) the retail price is larger than the discount price and (ii) number of customers who have purchased the first-generation old products is larger than number of customers who engage in trade-ins in period 2.
Based on Lemma2, the following conclusions are obtained as follows:①
With the enterprise’s trade-in subsidy strategy, under two cases, the retail price of the second-generation new products is larger than the discount price. Some myopic consumers and strategic customers prefer to purchase first-generation new products with discount prices, while others prefer to buy second-generation new products.②
Comparing with the choice behavior of different types of consumers, it was found thatqm1B∗∗>qm2T∗∗;qs1B∗∗>qs2T∗∗. This means that customers who have purchased old products in period 1 do not necessarily engage in trade-ins in period 2. Some customers will choose to purchase the second-generation new products, and other consumers prefer to purchase the first-generation new products at a discounted price.


### 4.3. The Government Provides Trade-in Subsidies (Model G)

To stimulate the demand for new products and play an important role in the circular economy, the government provides trade-in subsidiessg to consumers who engage in trade-ins.


#### 4.3.1. Consumers’ Strategic Choice Behavior

While analyzing myopic consumers’ choice behavior, the retail price of the first-generation products is no higher than its valuation, and myopic consumers make purchase decisions. When myopic consumers with the valuation larger thanτ1m buys X1 in the first period, myopic consumers with the valuation smaller than τ1m select to wait. In period 2, myopic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sg) or to keep using X1 (the utility is v1). The non-X1-holders determine whether to purchase X1 at a discount price (the utility is v1−pd) or to purchase X2 at a regular price (the utility is 1+θv1−p2). Comparing the myopic consumers’ utility, their purchasing decision in period 2 can be obtained. The X1-holders with the valuation larger than τ2m choose to trade-in X1 for X2, and the others will keep using X1. The non-X1-holders have two different selections. In case I(II), the customers would like to buy X2X1, the non-X1-holders between τ3m and τ1m will buy X2X1, and the non-X1-holders between τ4m and τ3m would like to buy X1X2.Next, strategic customer choice behavior is analyzed. When strategic customers with the valuation larger thanτ1s buy X1 in period 1, strategic customers with the valuation lower than τ1s choose to wait. Then, in the second period, strategic customers include X1-holders and the non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sg) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or to buy X2 at a regular price (the utility is 1+θv1−p2).


#### 4.3.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed as follows:(39)maxp2,pt,pdπ2G=p2−c2−ptqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(40)s.t.0≤qm2B≤α−qm1B,(41)0≤qm2Bd≤α−qm1B,(42)qm2Bd+qm2B≤α−qm1B,(43)qm2T≤qm1B,(44)0≤pt≤p2,(45)0≤pd≤p1,(46)0≤qs2B≤1−α−qs1B,(47)0≤qs2Bd≤1−α−qs1B,(48)qs2Bd+qs2B≤1−α−qs1B,(49)qs2T≤qs1B.Constraints (40)–(42) guarantee that myopic consumers will not hold X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (43) makes sure that the X1-holders trade-in X1 for X2. Constraints (46)–(48) guarantee that strategic consumers will not hold-X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (49) makes sure that the X1-holders trade-in X1 for X2.Next, based on consumer’s demand under different choice conditions, the firm should decide the retail pricep1 to maximize the total profit πtG of two periods. π2G∗ is the enterprise’s optimal profit in period 2.(50)maxp1πtG=p1−c1qm1B+qs1B+π2G∗,(51)s.t.0≤qm1B≤α,(52)0≤qs1B≤1−α,(53)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (51)–(53) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution when the firm uses the dynamic pricing strategy can be obtained in Proposition 3 (See Appendix for the proof of Proposition 3).Proposition 3.
If the government provides a trade-in subsidy, under a dynamic pricing strategy, it shows perfect equilibrium.(i)
In case I, the equilibrium solutions are as follows:(54)p2∗∗∗=c22+θ+12+c14θ+1θ+143θ−14δθ−θ−1−41+θ2,pt∗∗∗=1−θ8+c19θ+1θ+143θ−14δθ−θ−5−41+θ2−sg,p1∗∗∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,pd∗∗∗=c12+18+c14θ+143θ−14δθ−θ−1−41+θ2.(ii)
In case I, the equilibrium quantities are as follows:(55)qm2B∗∗∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qm2Bd∗∗∗=1+δ+2c12θ+12θ+1+c11+θ2,qm1B∗∗∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs1B∗∗∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2,qs2B∗∗∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qs2Bd∗∗∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qs2T∗∗∗=1−α−2c24θ−14+sg2θ,qm2T∗∗∗=α−2c24θ−14+sg2θ.(iii)
In case II, the equilibrium solutions are as follows:(56)p2∗∗∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pt∗∗∗=θθ+1+2c12θ+1θ+12θ+13+δ−2c1θ+14−θ4−sg2,p1∗∗∗=θ+c12θ+12θ+13+δ,pd∗∗∗=θ+2c12θ+12θ+13+δ.(iv)
In case II, the equilibrium quantities are as follows:(57)qs1B∗∗∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qm2B∗∗∗=c22θ−1+θ2θ−c221+θ,qs2Bd∗∗∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2Bd∗∗∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qs2B∗∗∗=c22θ−1+θ2θ−c221+θ,qm2T∗∗∗=α−2c24θ−14+sg2θ,qs2T∗∗∗=1−α−2c24θ−14+sg2θ,qm1B∗∗∗=αθ+1−θ2θ+1.Lemma 3.
When the government provides trade-in subsidy strategy, (i) the retail price of the second-generation new products is larger than the discount price, (ii) the number of consumers who have bought the first-generation old products is larger than the customers who engage in trade-ins in period 2, and (iii) the trade-in rebate will decrease with an increase in the government’s trade-in subsidy.
Based on Lemma3, the following conclusions are obtained:①
Under two cases, in the second period, the retail price of the second-generation new products is larger than the discount price of the first-generation new products. Some myopic customers and strategic customers prefer to purchase first-generation new products with discount prices.②
Comparing the choice behavior of different types of customers, it was found thatqm1B∗∗∗>qm2T∗∗∗;qs1B∗∗∗>qs2T∗∗∗. This means that customers who have purchased the first-generation old products in the first period do not necessarily engage in trade-ins. Some consumers will select to purchase the second-generation new products, and other consumers prefer to purchase the first-generation new products with the discount price.
From Propositions1 to 3, the equilibrium results in two cases under Model N, Model R, and Model G can be obtained. The following Lemma 4 is obtained:Lemma 4.
For different models, (i) in the first period, no myopic customers purchase the first-generation product whenc1≥c1∗; (ii) in the first period, no strategic customers purchase the first-generation product when c1≥c2∗, and (iii) c1∗>c2∗.In the first period, Lemma4 points out the thresholds under which customers do not buy the first-generation products. The first threshold implies that the manufacturing cost of the first-generation new product should be less than c1∗; this suggests that the sales quantity from myopic customers is positive. Similarly, when c1≥c2∗, it ensures that the sales quantity from the strategic consumer is positive. Of course, the thresholds c1∗ and c2∗ mean the consumers’ trade-off between the valuation of the first-generation new product v1 and the retail price p1 should pay for the first-generation new product.Lemma 5.
For different models, (i) in the second period, no myopic customers purchase the second-generation new product whenc2≥c1∗∗; (ii) in the second period, no strategic customers purchase the second-generation new product when c2≥c2∗∗; and (iii) c1∗∗=c2∗∗.In the second period, Lemma5 suggests the thresholds under which customers do not buy the second-generation products. The first threshold implies that the manufacturing cost of the second-generation product is less than c1∗∗, which suggests that when c2≥c1∗∗, the sales quantity from the myopic consumer is positive. When c2≥c2∗∗, it means that the sales quantity from the strategic consumer is positive. However, c1∗∗=c2∗∗ implies that sales quantity from the strategic consumer is equivalent to that from the myopic consumer. Therefore, the purchase behavior of consumers in two selling periods was considered. The second selling period is the last period, and there, it is not meaningful for the strategic consumer to choose to wait. Thus, some strategic consumers choose to purchase the second-generation product to meet needs. The number of strategic customers who purchase the second-generation product is equivalent to the number of myopic customers who purchase the second-generation product.


## 4.1. Benchmark: Without Trade-in Subsidies (Model N)

### 4.1.1. Consumers’ Strategic Choice Behavior

First, myopic consumers’ choice behavior is analyzed. When the retail price of the first-generation products is no higher than its valuation, myopic consumers will make purchase decisions. When myopic customers with a valuation higher thanτ1m purchase X1 in period 1, myopic consumers with a valuation lower than τ1m choose to wait. In period 2, myopic customers include X1 -holders and non- X1 -holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt or to keep using X1 (the utility is v1). The non- X1 -holders decide whether to buy X1 at the discount price (the utility is v1−pd) or buy X2 at the regular price (the utility is 1+θv1−p2. Comparing the myopic consumers’ surplus, their purchasing decision can be obtained in the second period. The X1 -holders with a valuation higher than τ2m choose to trade-in X1 for X2. For the non-X1-holders, it exists two different selections. In case I(II), the customers would like to buy X2X1, the non- X1 -holders with a valuation between τ3m and τ1m will purchase X2X1, and the non- X1 -holders with a valuation between τ4m and τ3m will purchase X1X2, while those with a valuation lower than τ4m will buy nothing.Next, strategic customer choice behavior is analyzed. Strategic customers can analyze the choice to purchase products in period 2, and the customers can compare the utility gotten from the first period with the delayed period to utilise and choose the best purchase chance. When strategic consumers with the valuation larger thanτ1s purchase X1 in period 1, strategic customers with a valuation lower than τ1s choose to wait. In period 2, strategic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 with the discount price (the utility is v1−pd) or to buy X2 with the regular price (the utility is 1+θv1−p2).


### 4.1.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed,π2N is the enterprise’s profit in period 2, and the optimization problem is as follows:(1)maxp2,pt,pdπ2N=p2−c2−ptqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(2)s.t.0≤qm2B≤α−qm1B,(3)0≤qm2Bd≤α−qm1B,(4)qm2Bd+qm2B≤α−qm1B,(5)qm2T≤qm1B,(6)0≤pt≤p2,(7)0≤pd≤p1,(8)0≤qs2B≤1−α−qs1B,(9)0≤qs2Bd≤1−α−qs1B,(10)qs2Bd+qs2B≤1−α−qs1B,(11)qs2T≤qs1B.Constraints (2)–(4) guarantee that myopic consumers who will not hold X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (5) makes sure that the X1-holders could trade-in X1 for X2. Constraints (8)–(10) guarantee that strategic consumers will not hold X1, buy X2 at p2, or purchase X1 with pd in the second period. Constraint (11) makes sure that the X1-holders will trade-in X1 for X2.Next, based on customer’s demand under different choice conditions, the firm should decide the retail pricep1 in the first period to maximize the total profit πtN of two periods.(12)maxπtN=p1−c1qm1B+qs1B+π2N∗,(13)s.t.0≤qm1B≤α,(14)0≤qs1B≤1−α,(15)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (13)–(15) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution, when the firm uses the dynamic pricing strategy, can be obtained in Proposition 1 (See Appendix for the proof of Proposition 1).Proposition 1.
When the enterprise uses the dynamic pricing strategy, the equilibrium results are derived as follows:(i)
In case I, the equilibrium prices and trade-in rebate are as follows:(16)p1∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,p2∗=c22+1+θ2+c14θ+11+θ43θ−14δθ−θ−1−41+θ2,pt∗=1−θ8+c19θ+11+θ43θ−14δθ−θ−5−41+θ2,pd∗=18+c14θ+143θ−14δθ−θ−1−41+θ2+c12.(ii)
In case I, the equilibrium quantities are as follows:(17)qm2B∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qs2B∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qm1B∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs2Bd∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qm2Bd∗=1+δ+2c12θ+12θ+1+c11+θ2,qs2T∗=1−α−c22θ,qm2T∗=α−c22θ,qs1B∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2.(iii)
In case II, the equilibrium prices and trade-in rebate are as follows:(18)p1∗=θ+c12θ+12θ+13+δ,p2∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pd∗=θ+2c12θ+12θ+13+δ,pt∗=θ1+θ+2c12θ+11+θ2θ+13+δ−2c11+θ4−θ4.(iv)
In case II, the equilibrium quantities are as follows:(19)qm1B∗=αθ+1−θ2θ+1,qm2T∗=α−2c24θ−14,qs1B∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qs2Bd∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2Bd∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qm2B∗=c22θ−1+θ2θ−c221+θ,qs2B∗=c22θ−1+θ2θ−c221+θ,qs2T∗=1−α−2c24θ−14.Lemma 1.
Under the situation of without trade-in subsidy strategy, (i) the retail price is larger than the discount price, and (ii) the customers who have bought the first-generation’s old products is larger than the customers who engage in trade-in program.
We can get the following conclusions:①
In the case without the trade-in subsidy strategy, in period 2, the retail price of the second-generation new products is larger than the discount price.②
Comparing with the choice behavior of different types of consumers, it was found thatqm1B∗>qm2T∗;qs1B∗>qs2T∗. This means that customers who have purchased old products in period 1 do not necessarily engage in trade-ins in period 2. Some customers can choose to buy the second-generation new products; in case I, strategic customers select to buy the new-generation products directly.


## 4.1.1. Consumers’ Strategic Choice Behavior

First, myopic consumers’ choice behavior is analyzed. When the retail price of the first-generation products is no higher than its valuation, myopic consumers will make purchase decisions. When myopic customers with a valuation higher thanτ1m purchase X1 in period 1, myopic consumers with a valuation lower than τ1m choose to wait. In period 2, myopic customers include X1 -holders and non- X1 -holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt or to keep using X1 (the utility is v1). The non- X1 -holders decide whether to buy X1 at the discount price (the utility is v1−pd) or buy X2 at the regular price (the utility is 1+θv1−p2. Comparing the myopic consumers’ surplus, their purchasing decision can be obtained in the second period. The X1 -holders with a valuation higher than τ2m choose to trade-in X1 for X2. For the non-X1-holders, it exists two different selections. In case I(II), the customers would like to buy X2X1, the non- X1 -holders with a valuation between τ3m and τ1m will purchase X2X1, and the non- X1 -holders with a valuation between τ4m and τ3m will purchase X1X2, while those with a valuation lower than τ4m will buy nothing.Next, strategic customer choice behavior is analyzed. Strategic customers can analyze the choice to purchase products in period 2, and the customers can compare the utility gotten from the first period with the delayed period to utilise and choose the best purchase chance. When strategic consumers with the valuation larger thanτ1s purchase X1 in period 1, strategic customers with a valuation lower than τ1s choose to wait. In period 2, strategic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 with the discount price (the utility is v1−pd) or to buy X2 with the regular price (the utility is 1+θv1−p2).


## 4.1.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed,π2N is the enterprise’s profit in period 2, and the optimization problem is as follows:(1)maxp2,pt,pdπ2N=p2−c2−ptqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(2)s.t.0≤qm2B≤α−qm1B,(3)0≤qm2Bd≤α−qm1B,(4)qm2Bd+qm2B≤α−qm1B,(5)qm2T≤qm1B,(6)0≤pt≤p2,(7)0≤pd≤p1,(8)0≤qs2B≤1−α−qs1B,(9)0≤qs2Bd≤1−α−qs1B,(10)qs2Bd+qs2B≤1−α−qs1B,(11)qs2T≤qs1B.Constraints (2)–(4) guarantee that myopic consumers who will not hold X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (5) makes sure that the X1-holders could trade-in X1 for X2. Constraints (8)–(10) guarantee that strategic consumers will not hold X1, buy X2 at p2, or purchase X1 with pd in the second period. Constraint (11) makes sure that the X1-holders will trade-in X1 for X2.Next, based on customer’s demand under different choice conditions, the firm should decide the retail pricep1 in the first period to maximize the total profit πtN of two periods.(12)maxπtN=p1−c1qm1B+qs1B+π2N∗,(13)s.t.0≤qm1B≤α,(14)0≤qs1B≤1−α,(15)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (13)–(15) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution, when the firm uses the dynamic pricing strategy, can be obtained in Proposition 1 (See Appendix for the proof of Proposition 1).Proposition 1.
When the enterprise uses the dynamic pricing strategy, the equilibrium results are derived as follows:(i)
In case I, the equilibrium prices and trade-in rebate are as follows:(16)p1∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,p2∗=c22+1+θ2+c14θ+11+θ43θ−14δθ−θ−1−41+θ2,pt∗=1−θ8+c19θ+11+θ43θ−14δθ−θ−5−41+θ2,pd∗=18+c14θ+143θ−14δθ−θ−1−41+θ2+c12.(ii)
In case I, the equilibrium quantities are as follows:(17)qm2B∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qs2B∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qm1B∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs2Bd∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qm2Bd∗=1+δ+2c12θ+12θ+1+c11+θ2,qs2T∗=1−α−c22θ,qm2T∗=α−c22θ,qs1B∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2.(iii)
In case II, the equilibrium prices and trade-in rebate are as follows:(18)p1∗=θ+c12θ+12θ+13+δ,p2∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pd∗=θ+2c12θ+12θ+13+δ,pt∗=θ1+θ+2c12θ+11+θ2θ+13+δ−2c11+θ4−θ4.(iv)
In case II, the equilibrium quantities are as follows:(19)qm1B∗=αθ+1−θ2θ+1,qm2T∗=α−2c24θ−14,qs1B∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qs2Bd∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2Bd∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qm2B∗=c22θ−1+θ2θ−c221+θ,qs2B∗=c22θ−1+θ2θ−c221+θ,qs2T∗=1−α−2c24θ−14.Lemma 1.
Under the situation of without trade-in subsidy strategy, (i) the retail price is larger than the discount price, and (ii) the customers who have bought the first-generation’s old products is larger than the customers who engage in trade-in program.
We can get the following conclusions:①
In the case without the trade-in subsidy strategy, in period 2, the retail price of the second-generation new products is larger than the discount price.②
Comparing with the choice behavior of different types of consumers, it was found thatqm1B∗>qm2T∗;qs1B∗>qs2T∗. This means that customers who have purchased old products in period 1 do not necessarily engage in trade-ins in period 2. Some customers can choose to buy the second-generation new products; in case I, strategic customers select to buy the new-generation products directly.


## 4.2. The Firm Provides Trade-in Subsidies (Model R)

To retain old consumers, we motivate consumers to buy products frequently, reduce the encroachment effect from the secondary market, and promote product upgrading; the firm provides trade-in subsidiessr to customers who engage in trade-ins.


### 4.2.1. Consumers’ Strategic Choice Behavior

First, when the retail price is no higher than its valuation, myopic consumers will make purchase decisions. When myopic customers with a valuation larger thanτ1m buy X1 in period 1, myopic customers with a valuation lower than τ1m choose to wait. In period 2, myopic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sr) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or buy X2 at a regular price (the utility is 1+θv1−p2). Comparing the myopic customers’ utility, the X1-holders with the valuation larger than τ2m choose to trade-in X1 for X2, and the others will keep using X1. For the non-X1-holders, in case I(II), the customers would like to buy X2X1, the non-X1-holders with a valuation between τ3m and τ1m will purchase X2X1, and the non-X1-holders with the valuation between τ4m and τ3m buys X1X2.Next, strategic customer choice behavior is analyzed. When strategic customers with the valuation larger thanτ1s buy X1 in the first period, strategic consumers with a valuation lower than τ1s choose to wait. In the second period, strategic customers include X1-holders and the non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sr) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or to buy X2 at the regular price (the utility is 1+θv1−p2).


### 4.2.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed,π2R is the enterprise’s profit in the second period, and the optimization problem is that(20)maxp2,pt,pdπ2R=p2−c2−pt−srqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(21)s.t.0≤qm2B≤α−qm1B,(22)0≤qm2Bd≤α−qm1B,(23)qm2Bd+qm2B≤α−qm1B,(24)qm2T≤qm1B,(25)0≤pt≤p2,(26)0≤pd≤p1,(27)0≤qs2B≤1−α−qs1B,(28)0≤qs2Bd≤1−α−qs1B,(29)qs2Bd+qs2B≤1−α−qs1B,(30)qs2T≤qs1B.Constraints (21)–(23) guarantee that myopic consumers will not hold X1 purchase X2 at p2 or purchase X1 with pd in period 2. Constraint (24) makes sure that the X1-holders trade-in X1 for X2. Constraints (27)–(29) guarantee that strategic consumers will not hold X1 purchase X2 at p2 or purchase X1 with pd in period 2. Constraint (30) makes sure that the X1-holders trade-in X1 for X2.Next, based on the consumer’s demand under different choice conditions, the firm should decide the retail pricep1 in the first period to maximize the total profit πtR of the two periods. π2R∗ is the firm’s optimal profit in the second period.(31)maxp1πtR=p1−c1qm1B+qs1B+π2R∗,(32)s.t.0≤qm1B≤α,(33)0≤qs1B≤1−α,(34)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (32)–(34) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution when the firm uses the dynamic pricing strategy can be obtained in Proposition 2 (See Appendix for the proof of Proposition 2).Proposition 2.
If the firm provides a trade-in subsidy, under a dynamic pricing strategy, it has one perfect equilibrium.(i)
In case I, the equilibrium solutions are as follows:(35)p1∗∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,pd∗∗=18+c14θ+143θ−14δθ−θ−1−41+θ2+c12,pt∗∗=1−θ8+c19θ+11+θ43θ−14δθ−θ−5−41+θ2−sr,p2∗∗=c22+θ+12+c1θ+14θ+143θ−14δθ−θ−1−41+θ2.(ii)
In case II, the equilibrium quantities are as follows:(36)qm2B∗∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qm2Bd∗∗=1+δ+2c12θ+12θ+1+c11+θ2,qs2B∗∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qs2Bd∗∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qs1B∗∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2,qm1B∗∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs2T∗∗=34−α−c22θ+srθ,qm2T∗∗=α−c22θ−14+srθ.(iii)
In case I, the equilibrium solutions are as follows:(37)p2∗∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pt∗∗=θθ+1+2c12θ+1θ+12θ+13+δ−2c1θ+14−θ4−sr2,p1∗∗=θ+c12θ+12θ+13+δ,pd∗∗=θ+2c12θ+12θ+13+δ.(iv)
In case II, the equilibrium quantities are as follows:(38)qs1B∗∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qs2B∗∗=c22θ−1+θ2θ−c221+θ,qm2T∗∗=α−2c24θ−14+sr2θ,qm2Bd∗∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qs2T∗∗=1−α−2c24θ−14+sr2θ,qm1B∗∗=αθ+1−θ2θ+1,qs2Bd∗∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2B∗∗=c22θ−1+θ2θ−c221+θ.Lemma 2.
When the enterprise uses the trade-in subsidy strategy, (i) the retail price is larger than the discount price and (ii) number of customers who have purchased the first-generation old products is larger than number of customers who engage in trade-ins in period 2.
Based on Lemma2, the following conclusions are obtained as follows:①
With the enterprise’s trade-in subsidy strategy, under two cases, the retail price of the second-generation new products is larger than the discount price. Some myopic consumers and strategic customers prefer to purchase first-generation new products with discount prices, while others prefer to buy second-generation new products.②
Comparing with the choice behavior of different types of consumers, it was found thatqm1B∗∗>qm2T∗∗;qs1B∗∗>qs2T∗∗. This means that customers who have purchased old products in period 1 do not necessarily engage in trade-ins in period 2. Some customers will choose to purchase the second-generation new products, and other consumers prefer to purchase the first-generation new products at a discounted price.


## 4.2.1. Consumers’ Strategic Choice Behavior

First, when the retail price is no higher than its valuation, myopic consumers will make purchase decisions. When myopic customers with a valuation larger thanτ1m buy X1 in period 1, myopic customers with a valuation lower than τ1m choose to wait. In period 2, myopic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sr) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or buy X2 at a regular price (the utility is 1+θv1−p2). Comparing the myopic customers’ utility, the X1-holders with the valuation larger than τ2m choose to trade-in X1 for X2, and the others will keep using X1. For the non-X1-holders, in case I(II), the customers would like to buy X2X1, the non-X1-holders with a valuation between τ3m and τ1m will purchase X2X1, and the non-X1-holders with the valuation between τ4m and τ3m buys X1X2.Next, strategic customer choice behavior is analyzed. When strategic customers with the valuation larger thanτ1s buy X1 in the first period, strategic consumers with a valuation lower than τ1s choose to wait. In the second period, strategic customers include X1-holders and the non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sr) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or to buy X2 at the regular price (the utility is 1+θv1−p2).


## 4.2.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed,π2R is the enterprise’s profit in the second period, and the optimization problem is that(20)maxp2,pt,pdπ2R=p2−c2−pt−srqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(21)s.t.0≤qm2B≤α−qm1B,(22)0≤qm2Bd≤α−qm1B,(23)qm2Bd+qm2B≤α−qm1B,(24)qm2T≤qm1B,(25)0≤pt≤p2,(26)0≤pd≤p1,(27)0≤qs2B≤1−α−qs1B,(28)0≤qs2Bd≤1−α−qs1B,(29)qs2Bd+qs2B≤1−α−qs1B,(30)qs2T≤qs1B.Constraints (21)–(23) guarantee that myopic consumers will not hold X1 purchase X2 at p2 or purchase X1 with pd in period 2. Constraint (24) makes sure that the X1-holders trade-in X1 for X2. Constraints (27)–(29) guarantee that strategic consumers will not hold X1 purchase X2 at p2 or purchase X1 with pd in period 2. Constraint (30) makes sure that the X1-holders trade-in X1 for X2.Next, based on the consumer’s demand under different choice conditions, the firm should decide the retail pricep1 in the first period to maximize the total profit πtR of the two periods. π2R∗ is the firm’s optimal profit in the second period.(31)maxp1πtR=p1−c1qm1B+qs1B+π2R∗,(32)s.t.0≤qm1B≤α,(33)0≤qs1B≤1−α,(34)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (32)–(34) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution when the firm uses the dynamic pricing strategy can be obtained in Proposition 2 (See Appendix for the proof of Proposition 2).Proposition 2.
If the firm provides a trade-in subsidy, under a dynamic pricing strategy, it has one perfect equilibrium.(i)
In case I, the equilibrium solutions are as follows:(35)p1∗∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,pd∗∗=18+c14θ+143θ−14δθ−θ−1−41+θ2+c12,pt∗∗=1−θ8+c19θ+11+θ43θ−14δθ−θ−5−41+θ2−sr,p2∗∗=c22+θ+12+c1θ+14θ+143θ−14δθ−θ−1−41+θ2.(ii)
In case II, the equilibrium quantities are as follows:(36)qm2B∗∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qm2Bd∗∗=1+δ+2c12θ+12θ+1+c11+θ2,qs2B∗∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qs2Bd∗∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qs1B∗∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2,qm1B∗∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs2T∗∗=34−α−c22θ+srθ,qm2T∗∗=α−c22θ−14+srθ.(iii)
In case I, the equilibrium solutions are as follows:(37)p2∗∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pt∗∗=θθ+1+2c12θ+1θ+12θ+13+δ−2c1θ+14−θ4−sr2,p1∗∗=θ+c12θ+12θ+13+δ,pd∗∗=θ+2c12θ+12θ+13+δ.(iv)
In case II, the equilibrium quantities are as follows:(38)qs1B∗∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qs2B∗∗=c22θ−1+θ2θ−c221+θ,qm2T∗∗=α−2c24θ−14+sr2θ,qm2Bd∗∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qs2T∗∗=1−α−2c24θ−14+sr2θ,qm1B∗∗=αθ+1−θ2θ+1,qs2Bd∗∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2B∗∗=c22θ−1+θ2θ−c221+θ.Lemma 2.
When the enterprise uses the trade-in subsidy strategy, (i) the retail price is larger than the discount price and (ii) number of customers who have purchased the first-generation old products is larger than number of customers who engage in trade-ins in period 2.
Based on Lemma2, the following conclusions are obtained as follows:①
With the enterprise’s trade-in subsidy strategy, under two cases, the retail price of the second-generation new products is larger than the discount price. Some myopic consumers and strategic customers prefer to purchase first-generation new products with discount prices, while others prefer to buy second-generation new products.②
Comparing with the choice behavior of different types of consumers, it was found thatqm1B∗∗>qm2T∗∗;qs1B∗∗>qs2T∗∗. This means that customers who have purchased old products in period 1 do not necessarily engage in trade-ins in period 2. Some customers will choose to purchase the second-generation new products, and other consumers prefer to purchase the first-generation new products at a discounted price.


## 4.3. The Government Provides Trade-in Subsidies (Model G)

To stimulate the demand for new products and play an important role in the circular economy, the government provides trade-in subsidiessg to consumers who engage in trade-ins.


### 4.3.1. Consumers’ Strategic Choice Behavior

While analyzing myopic consumers’ choice behavior, the retail price of the first-generation products is no higher than its valuation, and myopic consumers make purchase decisions. When myopic consumers with the valuation larger thanτ1m buys X1 in the first period, myopic consumers with the valuation smaller than τ1m select to wait. In period 2, myopic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sg) or to keep using X1 (the utility is v1). The non-X1-holders determine whether to purchase X1 at a discount price (the utility is v1−pd) or to purchase X2 at a regular price (the utility is 1+θv1−p2). Comparing the myopic consumers’ utility, their purchasing decision in period 2 can be obtained. The X1-holders with the valuation larger than τ2m choose to trade-in X1 for X2, and the others will keep using X1. The non-X1-holders have two different selections. In case I(II), the customers would like to buy X2X1, the non-X1-holders between τ3m and τ1m will buy X2X1, and the non-X1-holders between τ4m and τ3m would like to buy X1X2.Next, strategic customer choice behavior is analyzed. When strategic customers with the valuation larger thanτ1s buy X1 in period 1, strategic customers with the valuation lower than τ1s choose to wait. Then, in the second period, strategic customers include X1-holders and the non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sg) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or to buy X2 at a regular price (the utility is 1+θv1−p2).


### 4.3.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed as follows:(39)maxp2,pt,pdπ2G=p2−c2−ptqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(40)s.t.0≤qm2B≤α−qm1B,(41)0≤qm2Bd≤α−qm1B,(42)qm2Bd+qm2B≤α−qm1B,(43)qm2T≤qm1B,(44)0≤pt≤p2,(45)0≤pd≤p1,(46)0≤qs2B≤1−α−qs1B,(47)0≤qs2Bd≤1−α−qs1B,(48)qs2Bd+qs2B≤1−α−qs1B,(49)qs2T≤qs1B.Constraints (40)–(42) guarantee that myopic consumers will not hold X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (43) makes sure that the X1-holders trade-in X1 for X2. Constraints (46)–(48) guarantee that strategic consumers will not hold-X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (49) makes sure that the X1-holders trade-in X1 for X2.Next, based on consumer’s demand under different choice conditions, the firm should decide the retail pricep1 to maximize the total profit πtG of two periods. π2G∗ is the enterprise’s optimal profit in period 2.(50)maxp1πtG=p1−c1qm1B+qs1B+π2G∗,(51)s.t.0≤qm1B≤α,(52)0≤qs1B≤1−α,(53)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (51)–(53) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution when the firm uses the dynamic pricing strategy can be obtained in Proposition 3 (See Appendix for the proof of Proposition 3).Proposition 3.
If the government provides a trade-in subsidy, under a dynamic pricing strategy, it shows perfect equilibrium.(i)
In case I, the equilibrium solutions are as follows:(54)p2∗∗∗=c22+θ+12+c14θ+1θ+143θ−14δθ−θ−1−41+θ2,pt∗∗∗=1−θ8+c19θ+1θ+143θ−14δθ−θ−5−41+θ2−sg,p1∗∗∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,pd∗∗∗=c12+18+c14θ+143θ−14δθ−θ−1−41+θ2.(ii)
In case I, the equilibrium quantities are as follows:(55)qm2B∗∗∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qm2Bd∗∗∗=1+δ+2c12θ+12θ+1+c11+θ2,qm1B∗∗∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs1B∗∗∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2,qs2B∗∗∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qs2Bd∗∗∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qs2T∗∗∗=1−α−2c24θ−14+sg2θ,qm2T∗∗∗=α−2c24θ−14+sg2θ.(iii)
In case II, the equilibrium solutions are as follows:(56)p2∗∗∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pt∗∗∗=θθ+1+2c12θ+1θ+12θ+13+δ−2c1θ+14−θ4−sg2,p1∗∗∗=θ+c12θ+12θ+13+δ,pd∗∗∗=θ+2c12θ+12θ+13+δ.(iv)
In case II, the equilibrium quantities are as follows:(57)qs1B∗∗∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qm2B∗∗∗=c22θ−1+θ2θ−c221+θ,qs2Bd∗∗∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2Bd∗∗∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qs2B∗∗∗=c22θ−1+θ2θ−c221+θ,qm2T∗∗∗=α−2c24θ−14+sg2θ,qs2T∗∗∗=1−α−2c24θ−14+sg2θ,qm1B∗∗∗=αθ+1−θ2θ+1.Lemma 3.
When the government provides trade-in subsidy strategy, (i) the retail price of the second-generation new products is larger than the discount price, (ii) the number of consumers who have bought the first-generation old products is larger than the customers who engage in trade-ins in period 2, and (iii) the trade-in rebate will decrease with an increase in the government’s trade-in subsidy.
Based on Lemma3, the following conclusions are obtained:①
Under two cases, in the second period, the retail price of the second-generation new products is larger than the discount price of the first-generation new products. Some myopic customers and strategic customers prefer to purchase first-generation new products with discount prices.②
Comparing the choice behavior of different types of customers, it was found thatqm1B∗∗∗>qm2T∗∗∗;qs1B∗∗∗>qs2T∗∗∗. This means that customers who have purchased the first-generation old products in the first period do not necessarily engage in trade-ins. Some consumers will select to purchase the second-generation new products, and other consumers prefer to purchase the first-generation new products with the discount price.
From Propositions1 to 3, the equilibrium results in two cases under Model N, Model R, and Model G can be obtained. The following Lemma 4 is obtained:Lemma 4.
For different models, (i) in the first period, no myopic customers purchase the first-generation product whenc1≥c1∗; (ii) in the first period, no strategic customers purchase the first-generation product when c1≥c2∗, and (iii) c1∗>c2∗.In the first period, Lemma4 points out the thresholds under which customers do not buy the first-generation products. The first threshold implies that the manufacturing cost of the first-generation new product should be less than c1∗; this suggests that the sales quantity from myopic customers is positive. Similarly, when c1≥c2∗, it ensures that the sales quantity from the strategic consumer is positive. Of course, the thresholds c1∗ and c2∗ mean the consumers’ trade-off between the valuation of the first-generation new product v1 and the retail price p1 should pay for the first-generation new product.Lemma 5.
For different models, (i) in the second period, no myopic customers purchase the second-generation new product whenc2≥c1∗∗; (ii) in the second period, no strategic customers purchase the second-generation new product when c2≥c2∗∗; and (iii) c1∗∗=c2∗∗.In the second period, Lemma5 suggests the thresholds under which customers do not buy the second-generation products. The first threshold implies that the manufacturing cost of the second-generation product is less than c1∗∗, which suggests that when c2≥c1∗∗, the sales quantity from the myopic consumer is positive. When c2≥c2∗∗, it means that the sales quantity from the strategic consumer is positive. However, c1∗∗=c2∗∗ implies that sales quantity from the strategic consumer is equivalent to that from the myopic consumer. Therefore, the purchase behavior of consumers in two selling periods was considered. The second selling period is the last period, and there, it is not meaningful for the strategic consumer to choose to wait. Thus, some strategic consumers choose to purchase the second-generation product to meet needs. The number of strategic customers who purchase the second-generation product is equivalent to the number of myopic customers who purchase the second-generation product.


## 4.3.1. Consumers’ Strategic Choice Behavior

While analyzing myopic consumers’ choice behavior, the retail price of the first-generation products is no higher than its valuation, and myopic consumers make purchase decisions. When myopic consumers with the valuation larger thanτ1m buys X1 in the first period, myopic consumers with the valuation smaller than τ1m select to wait. In period 2, myopic customers include X1-holders and non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sg) or to keep using X1 (the utility is v1). The non-X1-holders determine whether to purchase X1 at a discount price (the utility is v1−pd) or to purchase X2 at a regular price (the utility is 1+θv1−p2). Comparing the myopic consumers’ utility, their purchasing decision in period 2 can be obtained. The X1-holders with the valuation larger than τ2m choose to trade-in X1 for X2, and the others will keep using X1. The non-X1-holders have two different selections. In case I(II), the customers would like to buy X2X1, the non-X1-holders between τ3m and τ1m will buy X2X1, and the non-X1-holders between τ4m and τ3m would like to buy X1X2.Next, strategic customer choice behavior is analyzed. When strategic customers with the valuation larger thanτ1s buy X1 in period 1, strategic customers with the valuation lower than τ1s choose to wait. Then, in the second period, strategic customers include X1-holders and the non-X1-holders. The X1-holders decide whether to trade-in X1 for X2 (the utility is 1+θv1−p2+pt+sg) or to keep using X1 (the utility is v1). The non-X1-holders decide whether to buy X1 at a discount price (the utility is v1−pd) or to buy X2 at a regular price (the utility is 1+θv1−p2).


## 4.3.2. Firm’s Dynamic Pricing Decisions

Based on the consumer’s market demand under different choice conditions, the optimal dynamic pricing decision model in period 2 is constructed as follows:(39)maxp2,pt,pdπ2G=p2−c2−ptqm2T+qs2T+p2−c2qm2B+qs2B+pd−c1qm2Bd+qs2Bd,(40)s.t.0≤qm2B≤α−qm1B,(41)0≤qm2Bd≤α−qm1B,(42)qm2Bd+qm2B≤α−qm1B,(43)qm2T≤qm1B,(44)0≤pt≤p2,(45)0≤pd≤p1,(46)0≤qs2B≤1−α−qs1B,(47)0≤qs2Bd≤1−α−qs1B,(48)qs2Bd+qs2B≤1−α−qs1B,(49)qs2T≤qs1B.Constraints (40)–(42) guarantee that myopic consumers will not hold X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (43) makes sure that the X1-holders trade-in X1 for X2. Constraints (46)–(48) guarantee that strategic consumers will not hold-X1 buy X2 at p2 or purchase X1 with pd in the second period. Constraint (49) makes sure that the X1-holders trade-in X1 for X2.Next, based on consumer’s demand under different choice conditions, the firm should decide the retail pricep1 to maximize the total profit πtG of two periods. π2G∗ is the enterprise’s optimal profit in period 2.(50)maxp1πtG=p1−c1qm1B+qs1B+π2G∗,(51)s.t.0≤qm1B≤α,(52)0≤qs1B≤1−α,(53)0≤p1≤1,where the first item expresses the profit gained by the firm from selling the products X1, and the second item expresses the enterprise’s optimal profit in period 2. Constraints (51)–(53) make sure that the first-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution when the firm uses the dynamic pricing strategy can be obtained in Proposition 3 (See Appendix for the proof of Proposition 3).Proposition 3.
If the government provides a trade-in subsidy, under a dynamic pricing strategy, it shows perfect equilibrium.(i)
In case I, the equilibrium solutions are as follows:(54)p2∗∗∗=c22+θ+12+c14θ+1θ+143θ−14δθ−θ−1−41+θ2,pt∗∗∗=1−θ8+c19θ+1θ+143θ−14δθ−θ−5−41+θ2−sg,p1∗∗∗=3θ−12δθ−θ−5−1+θ22θ+1+c14,pd∗∗∗=c12+18+c14θ+143θ−14δθ−θ−1−41+θ2.(ii)
In case I, the equilibrium quantities are as follows:(55)qm2B∗∗∗=θ+129θ+1+8θ+323θ−14δθ−θ−5−41+θ2−c22θ−18,qm2Bd∗∗∗=1+δ+2c12θ+12θ+1+c11+θ2,qm1B∗∗∗=α−2θ+12p1−c13θ−16δθ−θ−1−1+θ2,qs1B∗∗∗=1−α−8θ2p1−c13θ−16δθ−θ−5−1+θ2,qs2B∗∗∗=θ9θ+1+7θ−143θ−14δθ−θ−5−1+θ2−c22θ−18,qs2Bd∗∗∗=θ1−δ+c1θ+12θ+13+δ+c12+θθ,qs2T∗∗∗=1−α−2c24θ−14+sg2θ,qm2T∗∗∗=α−2c24θ−14+sg2θ.(iii)
In case II, the equilibrium solutions are as follows:(56)p2∗∗∗=2c24+θ1+θ+2c12θ+11+θ2θ+13+δ−c11+θ4,pt∗∗∗=θθ+1+2c12θ+1θ+12θ+13+δ−2c1θ+14−θ4−sg2,p1∗∗∗=θ+c12θ+12θ+13+δ,pd∗∗∗=θ+2c12θ+12θ+13+δ.(iv)
In case II, the equilibrium quantities are as follows:(57)qs1B∗∗∗=1−α2θ+13+δ−θ1−δ−2c12θ+11−δ2θ+13+δ,qm2B∗∗∗=c22θ−1+θ2θ−c221+θ,qs2Bd∗∗∗=θ1−δ+2c12θ+11−δ2θ+13+δ+c11+θ4θ,qm2Bd∗∗∗=θ2+δ+2c12θ+12+δ2θ+13+δ−2c1+c11+θ4θ,qs2B∗∗∗=c22θ−1+θ2θ−c221+θ,qm2T∗∗∗=α−2c24θ−14+sg2θ,qs2T∗∗∗=1−α−2c24θ−14+sg2θ,qm1B∗∗∗=αθ+1−θ2θ+1.Lemma 3.
When the government provides trade-in subsidy strategy, (i) the retail price of the second-generation new products is larger than the discount price, (ii) the number of consumers who have bought the first-generation old products is larger than the customers who engage in trade-ins in period 2, and (iii) the trade-in rebate will decrease with an increase in the government’s trade-in subsidy.
Based on Lemma3, the following conclusions are obtained:①
Under two cases, in the second period, the retail price of the second-generation new products is larger than the discount price of the first-generation new products. Some myopic customers and strategic customers prefer to purchase first-generation new products with discount prices.②
Comparing the choice behavior of different types of customers, it was found thatqm1B∗∗∗>qm2T∗∗∗;qs1B∗∗∗>qs2T∗∗∗. This means that customers who have purchased the first-generation old products in the first period do not necessarily engage in trade-ins. Some consumers will select to purchase the second-generation new products, and other consumers prefer to purchase the first-generation new products with the discount price.
From Propositions1 to 3, the equilibrium results in two cases under Model N, Model R, and Model G can be obtained. The following Lemma 4 is obtained:Lemma 4.
For different models, (i) in the first period, no myopic customers purchase the first-generation product whenc1≥c1∗; (ii) in the first period, no strategic customers purchase the first-generation product when c1≥c2∗, and (iii) c1∗>c2∗.In the first period, Lemma4 points out the thresholds under which customers do not buy the first-generation products. The first threshold implies that the manufacturing cost of the first-generation new product should be less than c1∗; this suggests that the sales quantity from myopic customers is positive. Similarly, when c1≥c2∗, it ensures that the sales quantity from the strategic consumer is positive. Of course, the thresholds c1∗ and c2∗ mean the consumers’ trade-off between the valuation of the first-generation new product v1 and the retail price p1 should pay for the first-generation new product.Lemma 5.
For different models, (i) in the second period, no myopic customers purchase the second-generation new product whenc2≥c1∗∗; (ii) in the second period, no strategic customers purchase the second-generation new product when c2≥c2∗∗; and (iii) c1∗∗=c2∗∗.In the second period, Lemma5 suggests the thresholds under which customers do not buy the second-generation products. The first threshold implies that the manufacturing cost of the second-generation product is less than c1∗∗, which suggests that when c2≥c1∗∗, the sales quantity from the myopic consumer is positive. When c2≥c2∗∗, it means that the sales quantity from the strategic consumer is positive. However, c1∗∗=c2∗∗ implies that sales quantity from the strategic consumer is equivalent to that from the myopic consumer. Therefore, the purchase behavior of consumers in two selling periods was considered. The second selling period is the last period, and there, it is not meaningful for the strategic consumer to choose to wait. Thus, some strategic consumers choose to purchase the second-generation product to meet needs. The number of strategic customers who purchase the second-generation product is equivalent to the number of myopic customers who purchase the second-generation product.


## 5. Analysis of Results

### 5.1. Optimal Dynamic Pricing Decisions and Demand

We can get the following propositions by comparing the equilibrium results:Proposition 4.
Comparing the equilibrium results in previous models, the following is obtained:(i)
In case I, in the first period,p1∗=p1∗∗=p1∗∗∗, qm1B∗=qm1B∗∗=qm1B∗∗∗, qs1B∗=qs1B∗∗=qs1B∗∗∗; in the second period, p2∗=p2∗∗=p2∗∗∗, pd∗=pd∗∗=pd∗∗∗, qm2B∗=qm2B∗∗=qm2B∗∗∗, qs2B∗=qs2B∗∗=qs2B∗∗∗, qm2Bd∗=qm2Bd∗∗=qm2Bd∗∗∗, qs2Bd∗=qs2Bd∗∗=qs2Bd∗∗∗;if sr>sg>θ/2, pt∗∗<pt∗∗∗<pt∗, qm2T∗<qm2T∗∗∗<qm2T∗∗, qs2T∗<qs2T∗∗∗<qs2T∗∗.(ii)
In case II, in the first period,p1∗=p1∗∗=p1∗∗∗qm1B∗=qm1B∗∗=qm1B∗∗∗, qs1B∗=qs1B∗∗=qs1B∗∗∗; in the second period, p2∗=p2∗∗=p2∗∗∗, pd∗=pd∗∗=pd∗∗∗, qm2B∗=qm2B∗∗=qm2B∗∗∗, qs2B∗=qs2B∗∗=qs2B∗∗∗, qm2Bd∗=qm2Bd∗∗=qm2Bd∗∗∗, qs2Bd∗=qs2Bd∗∗=qs2Bd∗∗∗;if sr>sg, pt∗∗<pt∗∗∗<pt∗, qm2T∗<qm2T∗∗∗<qm2T∗∗, qs2T∗<qs2T∗∗∗<qs2T∗∗.First, the previous result shows that in period 1, in two cases, the retail price of the first-generation products is equivalent in three decision-making models. The sales quantities to two kinds of consumers are equivalent in three decision-making models. Neither the firm nor the government provides the trade-in subsidy in the first period.Second, in the second period, in two cases, the retail price of the second-generation products is equivalent in three decision-making models. The discount price of the first-generation products is also equivalent in three decision-making models. This also leads to the result that the sales quantities of the second-generation products are equivalent.Third, in the second period, since providing trade-ins obtains the residual value, the firm has an incentive to provide trade-ins. Regardless of whether the firm or the government provides trade-in subsidies, the customers who engage in trade-ins under Model R and Model G is higher than that under Model N. This means that trade-in subsidies can effectively motivate customers to engage in trade-ins. In addition, a very interesting phenomenon was found, in which the trade-in rebate under Model N is higher than that under Model R and Model G.Then, the impact of the innovation incremental valueθ and the discount factor δ on the equilibrium results is analyzed, and some management insights are obtained.Proposition 5.
.(i)
In case I, under three decision-making models, there is∂p2∗/∂θ>0,∂p2∗∗/∂θ>0,∂p2∗∗∗/∂θ>0, ∂pd∗/∂θ>0,∂pd∗∗/∂θ>0,∂pd∗∗∗/∂θ>0, ∂pt∗/∂θ>0,∂pt∗∗/∂θ>0,∂pt∗∗∗/∂θ>0.(ii)
In case II, under three decision-making models, there is∂p2∗/∂θ>0,∂p2∗∗/∂θ>0,∂p2∗∗∗/∂θ>0, ∂pd∗/∂θ>0,∂pd∗∗/∂θ>0,∂pd∗∗∗/∂θ>0, ∂pt∗/∂θ>0,∂pt∗∗/∂θ>0,∂pt∗∗∗/∂θ>0.Proposition5 shows that the retail price of the second-generation new products and the trade-in rebate increases with increasing innovation incremental value. The larger the innovation incremental value, the larger the value of the second-generation new products, and the higher the retail price of the second-generation new products. At the same time, the surplus value of new products is very high; when consumers engage in trade-ins, the consumer will obtain a relatively high discount price.Proposition 6.
(i)
In case I, under three decision-making models, there is∂p1∗/∂δ>0,∂p1∗∗/∂δ>0,∂p1∗∗∗/∂δ>0, ∂p2∗/∂δ>0,∂p2∗∗/∂δ>0,∂p2∗∗∗/∂δ>0, ∂pd∗/∂δ>0,∂pd∗∗/∂δ>0,∂pd∗∗∗/∂δ>0.(ii)
In case II, under three decision-making models, there is∂p1∗/∂δ>0,∂p1∗∗/∂δ>0,∂p1∗∗∗/∂δ>0, ∂p2∗/∂δ>0,∂p2∗∗/∂δ>0,∂p2∗∗∗/∂δ>0, ∂pd∗/∂δ>0,∂pd∗∗/∂δ>0,∂pd∗∗∗/∂δ>0.Proposition6 shows that in case I, consumers prefer to buy the second-generation new products X2, the pricing of the first-generation new products increases with the increase in the discount factor. Because the consumer prefers to purchase the second-generation new products, the enterprise should reduce the retail price of the second-generation new products to meet the needs of consumers and obtain much more profit.


### 5.2. The Different Trade-in Subsidy Strategies

In this subsection, the impact of trade-in subsidy strategy on the consumer’s choice behavior and the firm’s dynamic pricing strategy is analyzed. The equilibrium solution under three decision-making models is very complex, and the numerical example is used in this subsection. The relevant parameters are as follows:c1=2, c2=3, tm=6, t∈0,6, k=1, D=1, θ=0.5, α=0.5, δ=0.6, sg∈1,6, sr∈1,6.From Figures3 and 4, it can be seen that in the second period, with the increase in the firm’s or the government’s trade-in subsidy, the customers who engage in trade-in activity increases. This can reflect that the firm’s or the government’s trade-in subsidy strategy can effectively motivate consumers to engage in trade-in activity. Moreover, compared with the government’s trade-in subsidy, the firm’s trade-in subsidy can more effectively motivate consumers to participate in the trade-in activity.Figure 3
(a) The impact of a firm’s trade-in subsidy strategy on consumers’ choice behavior in case I. (b) The impact of the government’s trade-in subsidy strategy on consumers’ choice behavior in case I.
(a)(b)Figure 4
(a) The impact of a firm’s trade-in subsidy strategy on consumers’ choice behavior in case II. (b) The impact of the government’s trade-in subsidy strategy on consumers’ choice behavior in case II.
(a)(b)


## 5.1. Optimal Dynamic Pricing Decisions and Demand

We can get the following propositions by comparing the equilibrium results:Proposition 4.
Comparing the equilibrium results in previous models, the following is obtained:(i)
In case I, in the first period,p1∗=p1∗∗=p1∗∗∗, qm1B∗=qm1B∗∗=qm1B∗∗∗, qs1B∗=qs1B∗∗=qs1B∗∗∗; in the second period, p2∗=p2∗∗=p2∗∗∗, pd∗=pd∗∗=pd∗∗∗, qm2B∗=qm2B∗∗=qm2B∗∗∗, qs2B∗=qs2B∗∗=qs2B∗∗∗, qm2Bd∗=qm2Bd∗∗=qm2Bd∗∗∗, qs2Bd∗=qs2Bd∗∗=qs2Bd∗∗∗;if sr>sg>θ/2, pt∗∗<pt∗∗∗<pt∗, qm2T∗<qm2T∗∗∗<qm2T∗∗, qs2T∗<qs2T∗∗∗<qs2T∗∗.(ii)
In case II, in the first period,p1∗=p1∗∗=p1∗∗∗qm1B∗=qm1B∗∗=qm1B∗∗∗, qs1B∗=qs1B∗∗=qs1B∗∗∗; in the second period, p2∗=p2∗∗=p2∗∗∗, pd∗=pd∗∗=pd∗∗∗, qm2B∗=qm2B∗∗=qm2B∗∗∗, qs2B∗=qs2B∗∗=qs2B∗∗∗, qm2Bd∗=qm2Bd∗∗=qm2Bd∗∗∗, qs2Bd∗=qs2Bd∗∗=qs2Bd∗∗∗;if sr>sg, pt∗∗<pt∗∗∗<pt∗, qm2T∗<qm2T∗∗∗<qm2T∗∗, qs2T∗<qs2T∗∗∗<qs2T∗∗.First, the previous result shows that in period 1, in two cases, the retail price of the first-generation products is equivalent in three decision-making models. The sales quantities to two kinds of consumers are equivalent in three decision-making models. Neither the firm nor the government provides the trade-in subsidy in the first period.Second, in the second period, in two cases, the retail price of the second-generation products is equivalent in three decision-making models. The discount price of the first-generation products is also equivalent in three decision-making models. This also leads to the result that the sales quantities of the second-generation products are equivalent.Third, in the second period, since providing trade-ins obtains the residual value, the firm has an incentive to provide trade-ins. Regardless of whether the firm or the government provides trade-in subsidies, the customers who engage in trade-ins under Model R and Model G is higher than that under Model N. This means that trade-in subsidies can effectively motivate customers to engage in trade-ins. In addition, a very interesting phenomenon was found, in which the trade-in rebate under Model N is higher than that under Model R and Model G.Then, the impact of the innovation incremental valueθ and the discount factor δ on the equilibrium results is analyzed, and some management insights are obtained.Proposition 5.
.(i)
In case I, under three decision-making models, there is∂p2∗/∂θ>0,∂p2∗∗/∂θ>0,∂p2∗∗∗/∂θ>0, ∂pd∗/∂θ>0,∂pd∗∗/∂θ>0,∂pd∗∗∗/∂θ>0, ∂pt∗/∂θ>0,∂pt∗∗/∂θ>0,∂pt∗∗∗/∂θ>0.(ii)
In case II, under three decision-making models, there is∂p2∗/∂θ>0,∂p2∗∗/∂θ>0,∂p2∗∗∗/∂θ>0, ∂pd∗/∂θ>0,∂pd∗∗/∂θ>0,∂pd∗∗∗/∂θ>0, ∂pt∗/∂θ>0,∂pt∗∗/∂θ>0,∂pt∗∗∗/∂θ>0.Proposition5 shows that the retail price of the second-generation new products and the trade-in rebate increases with increasing innovation incremental value. The larger the innovation incremental value, the larger the value of the second-generation new products, and the higher the retail price of the second-generation new products. At the same time, the surplus value of new products is very high; when consumers engage in trade-ins, the consumer will obtain a relatively high discount price.Proposition 6.
(i)
In case I, under three decision-making models, there is∂p1∗/∂δ>0,∂p1∗∗/∂δ>0,∂p1∗∗∗/∂δ>0, ∂p2∗/∂δ>0,∂p2∗∗/∂δ>0,∂p2∗∗∗/∂δ>0, ∂pd∗/∂δ>0,∂pd∗∗/∂δ>0,∂pd∗∗∗/∂δ>0.(ii)
In case II, under three decision-making models, there is∂p1∗/∂δ>0,∂p1∗∗/∂δ>0,∂p1∗∗∗/∂δ>0, ∂p2∗/∂δ>0,∂p2∗∗/∂δ>0,∂p2∗∗∗/∂δ>0, ∂pd∗/∂δ>0,∂pd∗∗/∂δ>0,∂pd∗∗∗/∂δ>0.Proposition6 shows that in case I, consumers prefer to buy the second-generation new products X2, the pricing of the first-generation new products increases with the increase in the discount factor. Because the consumer prefers to purchase the second-generation new products, the enterprise should reduce the retail price of the second-generation new products to meet the needs of consumers and obtain much more profit.


## 5.2. The Different Trade-in Subsidy Strategies

In this subsection, the impact of trade-in subsidy strategy on the consumer’s choice behavior and the firm’s dynamic pricing strategy is analyzed. The equilibrium solution under three decision-making models is very complex, and the numerical example is used in this subsection. The relevant parameters are as follows:c1=2, c2=3, tm=6, t∈0,6, k=1, D=1, θ=0.5, α=0.5, δ=0.6, sg∈1,6, sr∈1,6.From Figures3 and 4, it can be seen that in the second period, with the increase in the firm’s or the government’s trade-in subsidy, the customers who engage in trade-in activity increases. This can reflect that the firm’s or the government’s trade-in subsidy strategy can effectively motivate consumers to engage in trade-in activity. Moreover, compared with the government’s trade-in subsidy, the firm’s trade-in subsidy can more effectively motivate consumers to participate in the trade-in activity.Figure 3
(a) The impact of a firm’s trade-in subsidy strategy on consumers’ choice behavior in case I. (b) The impact of the government’s trade-in subsidy strategy on consumers’ choice behavior in case I.
(a)(b)Figure 4
(a) The impact of a firm’s trade-in subsidy strategy on consumers’ choice behavior in case II. (b) The impact of the government’s trade-in subsidy strategy on consumers’ choice behavior in case II.
(a)(b)


## 6. Conclusions

Trade-ins are one of the most widely used recycling methods in practice and can effectively motivate consumers to return their old products. One of the key problems in the trade-in process is the pricing strategy in the presence of different trade-in subsidy strategies and trade-in rebates. Thus, from the angle of game theory, in the market, consumers are divided into myopic consumers and strategic consumers, considering three situations of no trade-in subsidy strategy (Model N), the firm’s trade-in subsidy strategy (Model R), and the government’s trade-in subsidy strategy (Model G); the dynamic game equilibrium between firms and consumers is analyzed. The optimal dynamic pricing strategy is obtained by constructing a two-stage dynamic game model. The effect of the trade-in subsidy strategy on myopic customers’ and strategic customers’ choice behavior and on the enterprise’s dynamic pricing strategy is analyzed. Finally, the following important conclusions are obtained:For pricing and quantity, (i) in the first period, in two cases, the retail price of the first-generation products is equivalent in three decision-making models. The sales quantities of two kinds of consumers are equivalent in three decision-making models. Neither the firm nor the government provides the trade-in subsidy in the first period; thus, there are no changes in the retail price of the first-generation products and the sales quantities to customers. (ii) In the second period, in two cases, the retail price of the second-generation products is equivalent in three decision-making models. The discount price of the first-generation products is also equivalent in three decision-making models. This also leads to the result that the sales quantities of the second-generation products are equivalent.For consumer choice behavior, (i) on the one hand, in two cases, under three decision-making models, the customers who purchase two generations products are not impacted by the trade-in subsidy strategy. On the other hand, the customers who participate in trade-ins are impacted by the trade-in subsidy strategy. (ii) The customers who participate in trade-ins with the trade-in subsidy strategy are larger than that without the trade-in subsidy strategy.For firm profit, (i) the enterprise’s profit in the dynamic pricing strategy decreases when the discount factor and the innovation incremental value are at a higher level. (ii) The enterprise’s profit increases in the dynamic pricing strategy when the discount factor is at a low level. (iii) The firm uses the dynamic pricing strategy, and the firm can obtain much profit when two different kinds of customers exist in the market.Managerial Insights.(1)
Customers should carefully consider whether to participate in trade-ins based on the relationship between the residual value of the old product and the trade-in rebate.(2)
To encourage consumers to repeat purchases and retain old consumers, the enterprise provides a trade-in subsidy to customers. The firm should determine an appropriate amount of trade-in subsidy when they implement the dynamic pricing strategy.(3)
To improve the recycling and reuse of old products, the government can provide trade-in subsidies to consumers.In this study, we only discuss the effect of different trade-in subsidy strategies on the enterprise’s dynamic pricing strategy. However, we do not discuss the impact of different trade-in subsidy strategies on the firm’s static pricing strategy. Moreover, we do not compare the difference between the dynamic pricing and the static pricing under different trade-in subsidy strategies. Some interesting questions can be analyzed in the future. First, the impact of different trade-in subsidy strategies on the firm’s static pricing strategy could be explored. Second, when it has asymmetric information, how customers make the selection is of interest. How should firms make decisions accordingly is discussed. Third, we can discuss the difference between two pricing strategies under different trade-in subsidy strategies.



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