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

. 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 diferent trade-in subsidy strategies. Consumers are divided into myopic and strategic consumers. Considering three situations, namely, (1) no trade-in subsidy strategy, (2) the frm’s trade-in subsidy strategy, and (3) the government’s trade-in subsidy strategy, the dynamic game equilibrium between frms and consumers is analyzed. Te optimal dynamic pricing strategy is obtained by constructing a two-stage dynamic game model. Te 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) Te 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 frm’s trade-in subsidy strategy can more efectively motivate consumers to engage in trade-in activity.


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 frms during trade-ins is the product's pricing problem. To retain old consumers and encourage consumers to buy products many times, frms often ofer a discount price for frst-generation products (Klemperer [2]), which we call the trade-in rebate. Due to the product life cycle and product upgrades during trade-ins, frms 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 frms 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 frst period, customers purchase frst-generation products and frms determine the selling price of these frst-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 efect of customers' strategic choice behavior. Te 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 frms ofer trade-ins, customers will select whether and when to engage in trade-ins.
In view of the important infuence 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 ofers a subsidy of $3500-$4500 to customers. Te 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 ofers a subsidy of 300 yuan for customers. Apple ofers a maximum subsidy of 1000 yuan for customers. Te dynamic pricing strategy for successive-generation products considering consumer choice behavior and diferent 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 frms under different trade-in subsidy strategies. (b) How would the diferent trade-in subsidy strategies infuence consumers' choice behavior and the frm's optimal dynamic pricing strategy.
First, the two-stage game model was built based on myopic customers, strategic customers, and diferent tradein subsidy strategies. It is assumed that the enterprise sells two-generation products. First, the enterprise sells the frstgeneration products to myopic customers and strategic customers. In the second period, the enterprise sells the second-generation products and ofers trade-ins for myopic and strategic consumers under diferent trade-in subsidy strategies. Second, myopic and strategic customers' choice behavior and pricing strategy within each period are analyzed. Tird, the impact of diferent trade-in subsidy strategies on two kinds of customers' choice behavior and the frm's optimal dynamic pricing strategy is analyzed.
Te main results are summarized as follows: From the perspective of consumers, (1) in the frst period, some consumers would like to purchase the second-generation new products, some consumers prefer to purchase the frst-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. Te 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 proft in the dynamic pricing strategy is decreasing when the discount factor and the innovation incremental value are at a higher level, (2) the frm can obtain much more proft when two kinds of consumers exist in the market, and (3) with the increase in trade-in subsidies, the trade-in rebate decreases.

Contribution Statement.
First, this study discusses the optimal dynamic pricing strategy under diferent trade-in subsidies which includes no trade-in subsidy strategy, the frm's trade-in subsidy strategy, and the government's tradein subsidy strategy. Second, this study is the frst to characterize the impact of myopic consumers and strategic consumers' choice behavior on the dynamic pricing strategy under diferent trade-in subsidy strategies.
Te remainder of this study is organized as follows: Section 2 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.

Literature Review
Many scholars have studied pricing decision issues in tradeins. 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 proft. Han et al. [7] discussed the situations when enterprises should provide a trade-old for remanufactured (TOR) program. Chen et al. [8] analyzed three diferent 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 efect of a competitive environment on dynamic pricing strategy during the trade-in program. Xiao et al. [16] considered a frm that ofered both trade-in options and discussed the dynamic pricing strategy during two periods. Liu et al. [17] compared the diference between static pricing strategies and dynamic pricing strategies. In the previous sections, some researchers analyzed the infuence 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 efect of customers' strategic behavior on pricing strategies. Moreover, they discussed the advantages and usage situations of diferent 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 diference between the static pricing strategy and the dynamic pricing strategy. Some papers explore the trade-in subsidy strategy ofered 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 frm. 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 efect 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. 2 Mathematical Problems in Engineering [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 frm and government's subsidy policy.
Tis study contributes to the literature in following aspects: Some studies discuss the efect 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 diferent tradein subsidy on consumers' choice and on frms' pricing strategy. To fll the gap in the literature, this study discusses the optimal dynamic pricing strategy and the optimal tradein rebate strategy. Moreover, this study considers the efect of myopic customers and strategic customers' choice on the pricing strategy under the situations of diferent trade-in subsidy strategies and dual rollover strategies. Ten, the efect of diferent trade-in subsidy strategies on consumers' choice and the enterprise's pricing strategy is compared.

The Model
Considering the two-stage game model between frms and customers, frms sell products by means of a dual rollover strategy; in period 1, the enterprise sells the frst-generation new products X 1 at the retail price p 1 . Te manufacturing cost is c 1 (i.e., c 1 ≤ p 1 ). Te customers' valuation of frstgeneration new products X 1 is v 1 (i.e., p 1 ≤ v 1 ), 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, α(0 ≤ 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 frstgeneration 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 products X 2 at price p 2 , and the enterprise sells the frstgeneration products X 1 in the market at discount price To retain the old consumers, we motivate consumers to buy repeatedly and stabilize market share, and the frm provides the trade-in service. Te customers who have purchased the products X 1 returns the old products X 1 to the enterprise and get the certain trade-in rebate p t (0 ≤ p t ≤ p 2 ) when the customers purchase X 2 , and the manufacturing cost is c 2 (i.e., c 2 ≤ p 2 ). Te customers' valuation of the second-generation new products is v 2 (i.e., p 2 ≤ v 2 ), where v 2 � (θ + 1)v 1 , Each old product has a salvage value s(0 ≤ s ≤ θ). Figure 1 illustrates the time sequence of events.
In Figure 1, at the beginning of the period 1, based on the number of consumers waiting for the second period, the frm decides the unit retail price p 1 . 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 p 1 and the expectation of the market demand, the frm determines the unit retail price p 2 , the discount price p d , and trade-in rebate p t .
We consider the following decision-making models (see Figure 2).
(a) Model (N), neither the frm nor the government provides the trade-in subsidy strategy (Figure 2(a)). (b) Model (R), the frm provides the trade-in subsidy strategy ( Figure 2(b)). Any customers who purchase the next generation new products and return his or her old generation products can obtain a specifc trade-in subsidy s r from the frm. (c) InModel (G), the government provides the trade-in subsidy strategy (Figure 2(c)). Any customers who purchase the next generation new products and return his or her old generation products can obtain a specifc trade-in subsidy s g from the government.
Moreover, for easier analysis, the notations are summarized in Table 1.

Decision-Making Models
Because the government or the frm can provide diferent trade-in subsidy strategies for customers, the impact of diferent 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 Figure 2.

Consumers' Strategic Choice
Behavior. First, myopic consumers' choice behavior is analyzed. When the retail price of the frst-generation products is no higher than its valuation, myopic consumers will make purchase decisions. When myopic customers with a valuation higher than τ m 1 purchase X 1 in period 1, myopic consumers with a valuation lower than τ m 1 choose to wait. In period 2, myopic customers include X 1 -holders and non-X 1 -holders. Te X 1 -holders decide whether to trade-in X 1 for X 2 (the utility is (1 + θ)v 1 − p 2 + p t ) or to keep using X 1 (the utility is v 1 ). Te non-X 1 -holders decide whether to buy X 1 at the discount price (the utility is v 1 − p d ) or buy X 2 at the regular price (the utility is (1 + θ)v 1 − p 2 ). Comparing the myopic consumers' surplus, their purchasing decision can be obtained in the second period. Te X 1 -holders with a valuation higher than τ m 2 choose to trade-in X 1 for X 2 . For the non-X 1 -holders, it exists two diferent selections. In case I(II), the customers would like to buy X 2 (X 1 ), the non-X 1 -holders with a valuation between τ m 3 and τ m 1 will purchase X 2 (X 1 ), and the non-X 1 -holders with a valuation between τ m 4 and τ m 3 will purchase X 1 (X 2 ), while those with a valuation lower than τ m 4 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 frst period with the delayed period to utilise and choose the best purchase chance. When strategic    Retail price Firm ofers a trade-in rebate s r Te trade-in subsidy provided by the frm s g Te trade-in subsidy provided by the government δ Discount factor θ Innovation incremental value c 1 Te manufacturing cost of the frst-generation new products c 2 Te manufacturing cost of the next generation new products v 1 Customers' valuation of the frst-generation new products v 2 Customers' valuation of the next generation new products α Proportion of the myopic consumer π 2 Firm's proft in the second period π t Firm's total proft in the two periods 4 Mathematical Problems in Engineering consumers with the valuation larger than τ s 1 purchase X 1 in period 1, strategic customers with a valuation lower than τ s 1 choose to wait. In period 2, strategic customers include X 1 -holders and non-X 1 -holders. Te X 1 -holders decide whether to trade-in X 1 for X 2 (the utility is (1 + θ)v 1 − p 2 + p t ) or to keep using X 1 (the utility is v 1 ). Te non-X 1 -holders decide whether to buy X 1 with the discount price (the utility is v 1 − p d ) or to buy X 2 with the regular price (the utility is (1 + θ)v 1 − p 2 ).

Firm's Dynamic Pricing Decisions.
Based on the consumer's market demand under diferent choice conditions, the optimal dynamic pricing decision model in period 2 is constructed, π N 2 is the enterprise's proft in period 2, and the optimization problem is as follows: Constraints (2)-(4) guarantee that myopic consumers who will not hold X 1 buy X 2 at p 2 or purchase X 1 with p d in the second period. Constraint (5) makes sure that the X 1 -holders could trade-in X 1 for X 2 . Constraints (8)- (10) guarantee that strategic consumers will not hold X 1 , buy X 2 at p 2 , or purchase X 1 with p d in the second period. Constraint (11) makes sure that the X 1 -holders will trade-in X 1 for X 2 .
Next, based on customer's demand under diferent choice conditions, the frm should decide the retail price p 1 in the frst period to maximize the total proft π N t of two periods.
where the frst item expresses the proft gained by the frm from selling the products X 1 , and the second item expresses the enterprise's optimal proft in period 2. Constraints (13)-(15) make sure that the frst-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution, when the frm 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: (ii) In case I, the equilibrium quantities are as follows: (iii) In case II, the equilibrium prices and trade-in rebate are as follows: Mathematical Problems in Engineering (iv) In case II, the equilibrium quantities are as follows:

Te Firm Provides Trade-in Subsidies (Model R).
To retain old consumers, we motivate consumers to buy products frequently, reduce the encroachment efect from the secondary market, and promote product upgrading; the frm provides trade-in subsidies s r to customers who engage in trade-ins.

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 than τ m 1 buy X 1 in period 1, myopic customers with a valuation lower than τ m 1 choose to wait. In period 2, myopic customers include X 1 -holders and non-X 1 -holders. Te X 1 -holders decide whether to trade-in X 1 for X 2 (the utility is (1 + θ)v 1 − p 2 + p t + s r ) or to keep using X 1 (the utility is v 1 ). Te non-X 1 -holders decide whether to buy X 1 at a discount price (the utility is v 1 − p d ) or buy X 2 at a regular price (the utility is (1 + θ)v 1 − p 2 ). Comparing the myopic customers' utility, the X 1 -holders with the valuation larger than τ m 2 choose to trade-in X 1 for X 2 , and the others will keep using X 1 . For the non-X 1 -holders, in case I(II), the customers would like to buy X 2 (X 1 ), the non-X 1 -holders with a valuation between τ m 3 and τ m 1 will purchase X 2 (X 1 ), and the non-X 1 -holders with the valuation between τ m 4 and τ m 3 buys X 1 (X 2 ). Next, strategic customer choice behavior is analyzed. When strategic customers with the valuation larger than τ s 1 buy X 1 in the frst period, strategic consumers with a valuation lower than τ s 1 choose to wait. In the second period, strategic customers include X 1 -holders and the non-X 1 -holders. Te X 1 -holders decide whether to trade-in X 1 for X 2 (the utility is (1 + θ)v 1 − p 2 + p t + s r ) or to keep using X 1 (the utility is v 1 ). Te non-X 1 -holders decide whether to buy X 1 at a discount price (the utility is v 1 − p d ) or to buy X 2 at the regular price (the utility is (1 + θ)v 1 − p 2 ).

Firm's Dynamic Pricing Decisions.
Based on the consumer's market demand under diferent choice conditions, the optimal dynamic pricing decision model in period 2 is constructed, π R 2 is the enterprise's proft in the second period, and the optimization problem is that

Mathematical Problems in Engineering
Constraints (21)-(23) guarantee that myopic consumers will not hold X 1 purchase X 2 at p 2 or purchase X 1 with p d in period 2. Constraint (24) makes sure that the X 1 -holders trade-in X 1 for X 2 . Constraints (27)-(29) guarantee that strategic consumers will not hold X 1 purchase X 2 at p 2 or purchase X 1 with p d in period 2. Constraint (30) makes sure that the X 1 -holders trade-in X 1 for X 2 .
Next, based on the consumer's demand under diferent choice conditions, the frm should decide the retail price p 1 in the frst period to maximize the total proft π R t of the two periods. π R * 2 is the frm's optimal proft in the second period. max where the frst item expresses the proft gained by the frm from selling the products X 1 , and the second item expresses the enterprise's optimal proft in period 2. Constraints (32)-(34) make sure that the frst-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution when the frm uses the dynamic pricing strategy can be obtained in Proposition 2 (See Appendix for the proof of Proposition 2).

Proposition .
If the frm 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: (ii) In case II, the equilibrium quantities are as follows:

Mathematical Problems in Engineering
(iii) In case I, the equilibrium solutions are as follows: (iv) In case II, the equilibrium quantities are as follows:

Lemma . 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 frstgeneration old products is larger than number of customers who engage in trade-ins in period 2.
Based on Lemma 2, the following conclusions are obtained as follows:

Te 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 subsidies s g to consumers who engage in trade-ins.

Consumers' Strategic Choice
Behavior. While analyzing myopic consumers' choice behavior, the retail price of the frst-generation products is no higher than its valuation, and myopic consumers make purchase decisions. When myopic consumers with the valuation larger than τ m 1 buys X 1 in the frst period, myopic consumers with the valuation smaller than τ m 1 select to wait. In period 2, myopic customers include X 1 -holders and non-X 1 -holders. Te X 1 -holders decide whether to trade-in X 1 for X 2 (the utility is (1 + θ)v 1 − p 2 + p t + s g ) or to keep using X 1 (the utility is v 1 ). Te non-X 1 -holders determine whether to purchase X 1 at a discount price (the utility is v 1 − p d ) or to purchase X 2 at a regular price (the utility is (1 + θ)v 1 − p 2 ). Comparing the myopic consumers' utility, their purchasing decision in period 2 can be obtained. Te X 1 -holders with the valuation larger than τ m 2 choose to trade-in X 1 for X 2 , and the others will keep using X 1 . Te non-X 1 -holders have two diferent selections. In case I(II), the customers would like to buy X 2 (X 1 ), the non-X 1 -holders between τ m 3 and τ m 1 will buy X 2 (X 1 ), and the non-X 1 -holders between τ m 4 and τ m 3 would like to buy X 1 (X 2 ).
Next, strategic customer choice behavior is analyzed. When strategic customers with the valuation larger than τ s 1 buy X 1 in period 1, strategic customers with the valuation lower than τ s 1 choose to wait. Ten, in the second period, strategic customers include X 1 -holders and the non-X 1 -holders. Te X 1 -holders decide whether to trade-in X 1 for X 2 (the utility is (1 + θ)v 1 − p 2 + p t + s g ) or to keep using X 1 (the utility is v 1 ). Te non-X 1 -holders decide whether to buy X 1 at a discount price (the utility is v 1 − p d ) or to buy X 2 at a regular price (the utility is (1 + θ)v 1 − p 2 ).

Firm's Dynamic Pricing Decisions.
Based on the consumer's market demand under diferent choice conditions, the optimal dynamic pricing decision model in period 2 is constructed as follows: Constraints (40)-(42) guarantee that myopic consumers will not hold X 1 buy X 2 at p 2 or purchase X 1 with p d in the second period. Constraint (43) makes sure that the X 1 -holders trade-in X 1 for X 2 . Constraints (46)-(48) guarantee that strategic consumers will not hold-X 1 buy X 2 at p 2 or purchase X 1 with p d in the second period. Constraint (49) makes sure that the X 1 -holders trade-in X 1 for X 2 .
Next, based on consumer's demand under diferent choice conditions, the frm should decide the retail price p 1 to maximize the total proft π G t of two periods. π G * 2 is the enterprise's optimal proft in period 2.
where the frst item expresses the proft gained by the frm from selling the products X 1 , and the second item expresses the enterprise's optimal proft in period 2. Constraints (51)-(53) make sure that the frst-period quantity and price are non-negative. Based on the previous analysis, the equilibrium solution when the frm 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: (ii) In case I, the equilibrium quantities are as follows: Mathematical Problems in Engineering (iii) In case II, the equilibrium solutions are as follows: (iv) In case II, the equilibrium quantities are as follows:

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 frst-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 Lemma 3, 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 frst-generation new products. Some myopic customers and strategic customers prefer to purchase frst-generation new products with discount prices. In the frst period, Lemma 4 points out the thresholds under which customers do not buy the frst-generation products. Te frst threshold implies that the manufacturing cost of the frst-generation new product should be less than c (1) * ; this suggests that the sales quantity from myopic customers is positive. Similarly, when c 1 ≥ c (2) * , it ensures that the sales quantity from the strategic consumer is positive. Of course, the thresholds c (1) * and c (2) * mean the consumers' trade-of between the valuation of the frstgeneration new product (v 1 ) and the retail price (p 1 ) should pay for the frst-generation new product. In the second period, Lemma 5 suggests the thresholds under which customers do not buy the second-generation products. Te frst threshold implies that the manufacturing cost of the second-generation product is less than c (1) * * , which suggests that when c 2 ≥ c (1) * * , the sales quantity from the myopic consumer is positive. When c 2 ≥ c (2) * * , it means that the sales quantity from the strategic consumer is positive. However, c (1) * * � c (2) * * implies that sales quantity from the strategic consumer is equivalent to that from the myopic consumer. Terefore, the purchase behavior of consumers in two selling periods was considered. Te second selling period is the last period, and there, it is not meaningful for the strategic consumer to choose to wait. Tus, some strategic consumers choose to purchase the second-generation product to meet needs. Te number of strategic customers who purchase the second-generation product is equivalent to the number of myopic customers who purchase the second-generation product.
First, the previous result shows that in period 1, in two cases, the retail price of the frst-generation products is equivalent in three decision-making models. Te sales quantities to two kinds of consumers are equivalent in three decision-making models. Neither the frm nor the government provides the trade-in subsidy in the frst period.
Second, in the second period, in two cases, the retail price of the second-generation products is equivalent in three decision-making models. Te discount price of the frst-generation products is also equivalent in three decisionmaking models. Tis also leads to the result that the sales quantities of the second-generation products are equivalent.
Tird, in the second period, since providing trade-ins obtains the residual value, the frm has an incentive to provide trade-ins. Regardless of whether the frm 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. Tis means that trade-in subsidies can efectively 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.
Ten, the impact of the innovation incremental value θ and the discount factor δ on the equilibrium results is analyzed, and some management insights are obtained.
Proposition 5 shows that the retail price of the secondgeneration new products and the trade-in rebate increases with increasing innovation incremental value. Te 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 Proposition 6 shows that in case I, consumers prefer to buy the second-generation new products X 2 , the pricing of the frst-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 proft.

Te Diferent Trade-in Subsidy Strategies.
In this subsection, the impact of trade-in subsidy strategy on the consumer's choice behavior and the frm's dynamic pricing strategy is analyzed. Te equilibrium solution under three decision-making models is very complex, and the numerical example is used in this subsection. Te relevant parameters are as follows: From Figures 3 and 4, it can be seen that in the second period, with the increase in the frm's or the government's trade-in subsidy, the customers who engage in trade-in activity increases. Tis can refect that the frm's or the government's trade-in subsidy strategy can efectively motivate consumers to engage in trade-in activity. Moreover, compared with the government's trade-in subsidy, the frm's trade-in subsidy can more efectively motivate consumers to participate in the trade-in activity.

Conclusions
Trade-ins are one of the most widely used recycling methods in practice and can efectively 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 diferent trade-in subsidy strategies and trade-in rebates. Tus, 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 frm's trade-in subsidy strategy (Model R), and the government's trade-in subsidy strategy (Model G); the dynamic game equilibrium between frms and consumers is analyzed. Te optimal dynamic pricing strategy is obtained by constructing a two-stage dynamic game model. Te efect 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 frst period, in two cases, the retail price of the frst-generation products is equivalent in three decision-making models. Te sales quantities of two kinds of consumers are equivalent in three decision-making models. Neither the frm nor the government provides the trade-in subsidy in the frst period; thus, there are no changes in the retail price of the frst-generation products and the sales quantities to customers. (ii) In the second period, in two cases, the retail price of the secondgeneration products is equivalent in three decision-making models. Te discount price of the frst-generation products is also equivalent in three decision-making models. Tis also leads to the result that the sales quantities of the secondgeneration 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) Te customers who participate in trade-ins with the trade-in subsidy strategy are larger than that without the trade-in subsidy strategy.
For frm proft, (i) the enterprise's proft in the dynamic pricing strategy decreases when the discount factor and the innovation incremental value are at a higher level. (ii) Te enterprise's proft increases in the dynamic pricing strategy when the discount factor is at a low level. (iii) Te frm uses the dynamic pricing strategy, and the frm can obtain much proft when two diferent 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. Te frm 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 efect of diferent trade-in subsidy strategies on the enterprise's dynamic pricing strategy. However, we do not discuss the impact of diferent trade-in subsidy strategies on the frm's static pricing strategy. Moreover, we do not compare the diference between the dynamic pricing and the static pricing under diferent trade-in subsidy strategies. Some interesting questions can be analyzed in the future. First, the impact of diferent trade-in subsidy strategies on the frm's static pricing strategy could be explored. Second, when it has asymmetric information, how customers make the selection is of interest. How should frms make decisions accordingly is discussed. Tird, we can discuss the diference between two pricing strategies under diferent trade-in subsidy strategies.
Moreover, the authors can obtain the equilibrium prices and trade-in rebate.
Te proof process in Case II is the same as in Case I. □

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te author declares that he has no conficts of interest.