Coordination Analysis of the Recycling and Remanufacturing Closed-Loop Supply Chain Considering Consumers’ Low Carbon Preference and Government Subsidy

Abstract

Guided by the goals of carbon peaking and carbon neutrality, in order to coordinate the recycling and remanufacturing closed-loop supply chain (CLSC), considering the strengthened low-carbon awareness of consumers and the high cost of carbon emission reduction(CER), realizing that the demand of remanufactured product is affected by both market price and manufacturer’s CER effort and that the manufacturer is responsible for CER, this study focuses on the hybrid recycling between the manufacturer and retailer. Based on Stackelberg game theory, it constructs the profit models for various interested parties under centralized and decentralized decision-making models to study consumer low carbon preference coefficient and government CER subsidy coefficient, in order to work out the optimal pricing strategy and the level of manufacturer’s CER effort under two decision models, and designs a cost-and-benefit-sharing contract to coordinate the supply chain. The results show that: (1) total recycling amount and total profit under CLSC are negatively correlated with recycling channel competition coefficient; (2) government CER subsidy and consumers’ low carbon preference help enhance both total profit under CLSC and the zeal of manufacturers for CER; and (3) the parameters of contracts in some circumstances contribute to alleviating the competition between manufacturer and retailer cycling channels and increasing the total recycling amount under CLSC. Meanwhile, the profit for various interested parties and total profit under CLSC, and the level of a manufacturer’s CER effort, can be simultaneously improved. Consequently, the Pareto improvement and sustainable development for the CLSC can be achieved.

1. Introduction

In recent years, with the emissions of greenhouse gases and the shortening of the life cycles of electronic products and household appliances, global warming and the shortage of resources have already attracted attention from all countries in the world. Reduced carbon emissions and recycling of resources have become key measures to solve the current problems of continuous environmental degradation and climate warming, thereby promoting sustainable development of the global economy [1]. In order to reduce carbon emissions, international organizations have issued a series of conventions and policies to limit greenhouse gas emissions, such as the Kyoto Protocol and the Paris Agreement [2], which have promoted countries to formulate quantitative emissions reduction plans, and governments have also taken corresponding emissions reduction measures [3]. As a large emitter of carbon dioxide and other greenhouse gases, China started the pilot work of carbon emission trading in seven regions, including Beijing and Shenzhen, in 2011, and proposed the goals of “carbon peaking” by 2030 and “carbon neutralization” by 2060. This requires the government to control the carbon emissions produced by enterprises in the production process, and implement strong policy measures to effectively promote the progress of CER [4].

The investment of enterprises in CER will occupy the capital of production activities and lead to a decrease in enterprises’ enthusiasm in CER. Therefore, the Chinese government has introduced a variety of preferential policies to reduce the cost for enterprises of CER, such as tax incentives, subsidies, preferential government procurement, etc., among which government subsidies are recognized as effective policies [5]. Furthermore, to encourage resource recycling, the China Municipal Government has successively issued the Green Procurement Guide for Enterprises, the Guiding Opinions on Accelerating the Establishment and Perfection of Green and Low Carbon Circular Development Economic System, and so on, actively guiding enterprises to establish green supply chains, realizing green, low-carbon, and circular development, and encouraging more and more manufacturers to integrate recycling and remanufacturing into their production [6]. Compared with manufacturing new products, remanufacturing products can not only save resources and reduce costs, but also reduce carbon emissions [7]. This requires recycling enough waste products to meet the needs of remanufacturing production. Therefore, in addition to the traditional independent recycling of manufacturers, there are multiple recycling models, such as manufacturers entrusting retailers or third-party recyclers to carry out recycling, and hybrid recycling between two parties (including Manufacturer-Retailer, Manufacturer-Third Party recycler, etc.). However, competition between various recycling channels is unavoidable. Simultaneously with the development of e-commerce, many manufacturers have opened online stores on e-commerce platforms, thus forming sales competition with retailers. In the case of competition between sales channels and recycling channels, manufacturers and retailers are both partners and competitors. Conflicts of interest and competition in mixed channels will reduce the overall benefits of the CLSC. As a result, it is of great practical significance to study how to make pricing and emissions reduction decisions of the recycling and remanufacturing CLSC to improve the total recycling amount and total profit of the supply chain and reduce carbon emissions.

This paper mainly discusses the following questions:

• When there are conflicts between sales channels and recovery channels, how do the manufacturer and the retailer set wholesale prices, retail prices and recycling prices in the CLSC, and in addition, what is the difference between the total recycling amount and total profit of CLSC, and the level of CER efforts of the manufacturer, under decentralized and centralized decision-making models?
• What is the correlation between consumers’ low carbon preference coefficient and government CER subsidy coefficient, and the level of manufacturer CER? What is the impact on a manufacturer’s and retailer’s pricing decisions?
• Can the “CER cost sharing + revenue sharing” contract improve the profits of various interested parties and total profit of the supply chain, as well as the level of the manufacturer’s CER? What is the best way to set contract parameters?

2. Literature Review

This article involves three aspects of research – CLSC under the background of the goals of carbon peaking and carbon neutrality, CLSC considering consumers’ low carbon preference, and CLSC considering government subsidy.

2.1. CLSC under the Background of the Goals of Carbon Peaking and Carbon Neutrality

With the state and government strongly advocating energy conservation and emissions reduction, theoretical research on CER is constantly being enriched. The research field related to this paper is CLSC management under the background of the goals of carbon peaking and carbon neutrality, which typically includes the research contents of manufacturer’s production, pricing, inventory management, emissions reduction decisions, recycling channel selection, and CLSC coordination.

Guided by the goals of carbon peaking and carbon neutrality, in response to the problem of production and pricing decisions in the CLSC, the pricing research of the remanufacturing CLSC is divided into two categories: homogenization and heterogeneity pricing methods of remanufactured and new products. The market sale prices of new and remanufactured products are the same in the homogenization pricing strategy, whereas in the heterogeneous pricing strategy, new and remanufactured products have different or dynamic prices [8]. When the quality of remanufactured products is inferior to that of new products, their prices are usually lower. However, differentiated pricing strategy tells customers in some way that lower-priced products are remanufactured and of poor quality, and thus customers will tend to choose new products. This pricing strategy may have a negative impact on the development of remanufactured products [9]. When remanufactured products are priced the same as new products, Savaskan et al. [10] argued that the supply chain benefits from a homogeneous pricing strategy. Furthermore, the carbon emissions of remanufactured products are lower than those of new products, meeting the needs of consumers with low carbon preferences. Ganet al. [11] investigated the pricing decision of a two-level CLSC under the carbon trading policy, taking into account consumers’ different willingness to pay for new and remanufactured products. Zheng et al. [12] established a game model of duopoly manufacturers to implement waste recycling and low-carbon advertising investment strategies under the carbon tax mechanism, and discussed manufacturers’ different low carbon strategy choices and recycling price decisions based on the state’s restrictions on manufacturers’ carbon emissions; Qi et al. [13] studied the optimal pricing decision of a supply chain consisting of one supplier and two competitors. Yang et al. [14] considered the single-rate carbon tax policy and the excessive progressive carbon tax policy for the CLSC network composed of multiple manufacturing and remanufacturing factories and multiple demand markets, and compared and analyzed the influence of the two carbon tax policies on new product production, pricing, carbon emissions, and profits of the members of the CLSC network. Liu et al. [15] used the minimum economic cost, lowest carbon emission, and maximum social benefit as optimization objectives, aiming at the sustainable CLSC network optimization, and considering the influence of flexible supply strategy and facility transformation on network optimization.

In light of the CER decision-making problem of the recycling and remanufacturing CLSC, Zhang et al. [16] studied a CLSC decision-making model in which retailers are responsible for recycling under the carbon quota policy, and compared and analyzed the optimal decision with or without the carbon quota constraint. On this basis, the two-part pricing contract and payment transfer mechanism were combined to realize system coordination in three scenarios: no carbon emissions reduction technology, carbon emissions reduction technology consideration, and CER technology subsidy. Li et al. [17] discussed the impact of manufacturer and retailer competition intensity on the level of CER and recovery. Shi [18] investigated the impact of low carbon products and waste low carbon products on expected utility, and realized the coordination of remanufacturing CLSC through positive channel revenue sharing and reverse channel risk sharing contracts; Li et al. [19] constructed non-cooperation, one-way cooperation, two-way cooperation and cooperation in the context of CER, to explore the optimal cooperation mode. In addition, some scholars also consider the quality and cost in the remanufacturing process. Yang et al. [20] studied the influence of recycling quality and carbon transaction price on product pricing, optimal emissions reduction, and the upper limit of investment cost for emissions reduction, and built a Stackelberg game model composed of manufacturers and retailers to study the influence of recycling quality and carbon transaction price on product pricing, optimal emissions reduction, and the upper limit of investment cost for emissions reduction; Zhang et al. [21] explored the impact of abatement cost factors and remanufacturing abatement factors on the profits and total carbon emissions of CLSC, taking into account the price difference between new and remanufactured products.

2.2. CLSC Considering Consumers’ Low Carbon Preference

As people’s awareness of environmental protection grows, more and more consumers will consider low carbon attributes when purchasing products and are willing to pay more for them. Many countries and regions have begun to implement “carbon label” policies to guide and encourage consumers to purchase low carbon products. Consumers’ low carbon preference must be considered in the operation of the entire recycling and remanufacturing CLSC.

Wang et al. [22] investigated the impact of consumers’ low carbon awareness on CER strategies, comparing and analyzing the total profit and emissions reduction effect under decentralized and centralized decision-making. Ji et al. [23] studied the impact of carbon quota allocation methods on the CLSC by incorporating consumers’ low carbon preference into the model. The findings indicate that the benchmark method can more effectively promote the production and sale of low carbon products. Sun et al. [24] analyzed the impact of the proportion of low carbon preference consumers, the acceptance of ordinary consumers of reproduction and carbon trading prices on CER and pricing decisions, and show that the improvement of these three variables can promote manufacturers to improve their CER rate. Xing et al. [25] built a competitive CLSC consisting of one green product manufacturer, one ordinary product manufacturer, and one retailer to explore the impact of consumers’ green preference, green investment coefficient, and manufacturers’ competition intensity on supply chain product selection and node enterprise profits. Coskun et al. [26] proposed a goal planning model that focuses on three consumer groups (green, inconsistent, and red consumers) to investigate the impact of consumers’ green expectations on the supply chain network; Wu et al. [27] studied the influence of recycling quality and carbon trading price on product pricing, optimal emissions reduction, and investment cost ceiling of emissions reduction. Yang et al. [28] investigated the impact of total carbon control and trading, as well as consumer preference, on manufacturers’ channel choice; Zhang et al. [29] built a decision model of two-stage differential pricing for a single monopoly manufacturer, and compared and analyzed the effects of different subsidy policies and consumers’ low carbon preference on manufacturers’ emissions reduction decisions. In addition, since the carbon emission of remanufactured products is lower than that of new productions, low carbon preference consumers are willing to buy remanufactured productions. Cao et al. [30] constructed a two-stage CLSC model to analyze the influence of consumers’ attention to remanufacturing on optimal decision-making.

2.3. CLSC Considering Government Subsidy

Government policies frequently play a crucial role in the growth of an entire industry [31]. The industry of recycling and remanufacturing in China is still in its infancy. Through subsidies and other incentives, the government can effectively increase the enthusiasm of businesses to engage in recycling and remanufacturing of waste products and CER, thereby promoting the sustainable development of the recycling and remanufacturing industry. With government participation, numerous academics are currently discussing the issues of CLSC recovery and emissions reduction decisions.

Jena et al. [32] investigated the impact of government incentives on the efficiency of recovery and overall performance of CLSC. According to studies by Guo et al. [33], government subsidies can increase supply chain members’ profits and reduce the start-up conditions for manufacturers in order to reduce emissions. Zhang et al. [34] discussed the dynamic game between the government and the members of the dual-channel CLSC with or without government reward and punishment. Wang et al. [35] investigated the impact of government reward and punishment, profit-sharing, and cost-sharing coefficients on the equilibrium decision of CLSC; Li et al. [36] focused on the impact of carbon subsidies on the decision-making mode of a remanufacturing CLSC, and revealed when and how the government implemented carbon subsidies to encourage enterprises to reduce carbon emissions. However, the majority of the preceding studies assume that market demand is only related to product retail prices, without taking into account the impact of the carbon reduction ranking of enterprises on consumer purchasing behavior. Yang et al. [37] studied the impact of government subsidy on CER decisions of CLSC members, and found that government subsidy can effectively increase the profits of CLSC and reduce carbon emissions; meanwhile, Shang et al. [38] discussed the influence of government subsidy policies on optimal pricing decisions considering consumption preferences. Heydari et al. [39] investigated how a manufacturer and a retailer increase customers’ willingness to return old products by offering discounts or direct charges, and how the government improves supply chain coordination by offering various incentives (tax exemption and subsidies) to CLSC members. Shu et al. [40] examined how the “old-for-new” model affects consumers’ willingness to buy remanufactured products in the context of a carbon tax and government subsidies, as well as the optimal pricing and production decisions of manufacturers and remanufacturers. When the government subsidizes remanufactured products, Xu et al. [41] discussed the impact of consumers’ low carbon preference on the remanufacturing activities and the supply chain decisions; He et al. [42] examined the incentive effect of government subsidy policy and discovered that as consumer preferences improved and potential benefits to manufacturers increased, the incentive effect of subsidy policy was significantly enhanced; in addition, Mi et al. [43] discussed the best government subsidy policy under the coexistence of trade-old-for-new and trade-old-for-remanufactured programs.

To sum up, most of the existing research on the CLSC assumes that the manufacturer independently recycles, or the manufacturer entrusts the retailer to recycle, or the manufacturer entrusts the third-party recycler to recycle, or a form of hybrid recycling, focusing on the pricing decision of the CLSC, without considering the influence of competition among recycling channels on the decision-making of each member and total recycling amount of CLSC. Additionally, the current literature mainly discusses the impact of consumers’ low carbon preference on carbon quota allocation method, CER rate, CER cost and consumers’ acceptance of remanufactured products. Few literatures have included consumers’ low carbon preference into the influencing factors of market demand, and the existing literature on government subsidy mainly focuses on the impact of different types of government subsidy policies, carbon taxes and subsidy for recycling waste products on supply chain pricing decisions, while the literature on the impact of government CER subsidy on manufacturers’ CER efforts is less populated. However, there is even less literature that comprehensively considers the impact of consumers’ low carbon preference and government CER subsidy on the total profit and total recycling amount of CLSC and the level of manufacturer’s CER in the scenario of Manufacturer-Retailer (MR) hybrid recycling. In view of this, on the basis of the above literature studies, the Stackelberg game model of the recycling and remanufacturing CLSC consisting of a single manufacturer and a single retailer was constructed in this paper under the condition of competition between sales channels and recycling channels. Total recycling amount and total profit of the supply chain under decentralized and centralized decision-making as well as the level of the manufacturer’s CER were compared and analyzed, and the contract of “CER cost sharing + revenue sharing” was designed to coordinate the CLSC. Finally, the impact of recycling channel competition coefficient, consumers’ low carbon preference coefficient, and government CER subsidy coefficient on interested parties’ pricing decision-making, the supply chain’s total recycling amount, and total profit is analyzed by numerical examples, and the profit of each main body and the total profit of the supply chain before and after the coordination is compared. This paper provides a theoretical basis for the optimal pricing and emissions reduction decision-making of node enterprises in the recycling and remanufacturing CLSC.

3. Model Description and Assumptions

3.1. Description of the Problem and Symbolic Meaning

The paper considers a CLSC system with a single dominant manufacturer W and a single retailer R. Waste products can be recycled by both manufacturer and retailer. Manufacturers can choose to use raw materials or waste products for product production in the forward supply chain production link; in the sales process, manufacturers wholesale their products to retailers at wholesale price w through traditional channels, and then retailers sell their products to consumers at price $p_r$; on the other hand, they directly sell their products to end consumers at price $p_m$ through direct sales channels. In the CLSC, both manufacturers and retailers participate in recycling, and manufacturers and retailers compete in sales and recycling. The government has offered subsidies to encourage companies to reduce carbon emissions. Figure 1 depicts the corresponding CLSC structure diagram, where the solid line represents forward logistics and the dotted line represents reverse logistics. In addition,

Table 1. Major notations in the model.

Parameters	Definitions
$a$	Capacity of the market
$C_m$	Unit cost of manufacturing new products with brand-new parts
$C_r$	Unit cost of remanufacturing the recycled waste products to produce new products
$w$	Unit wholesale price of the manufacturer
$h(0<h<1)$	Manufacturer’s online market proportion coefficient
$l_1$	Sensitivity coefficient of demand price
$l_2$	Cross price elasticity coefficient
$p_{mi}(i=1,2,3)$	Direct selling price of products sold in manufacturer’s direct selling channel
$p_{ri}(i=1,2,3)$	Retail price of products sold in retailer’s retail channel
${\epsilon }_i(i=1,2,3)$	The level of manufacturer’s CER effort
$e_0$	Unit initial carbon emissions of new products
$\gamma$	Consumer’s low carbon preference coefficient (referring to consumers’ willingness to buy remanufactured products)
$k$	Influence coefficient of the level of manufacturer’s CER effort on CER cost
$s(0<s<1)$	Government CER subsidy coefficient
$g$	Transfer payments made by the manufacturer to the retailer
$p_{1i}(i=1,2,3)$	Unit recycling price paid by the manufacturer to consumers
$p_{2i}(i=1,2,3)$	Unit recycling price paid by the retailer to consumers
${\beta }_1$	Sensitivity coefficient of consumers to recycling price
${\beta }_2$	Recycling channel competition coefficient
$D_{mi}(i=1,2,3)$	Demand of direct selling channel
$D_{ri}(i=1,2,3)$	Demand of retail channel
$D_i(i=1,2,3)$	Total market demand
$Q_{mi}(i=1,2,3)$	Manufacturer’s recycling amount
$Q_{ri}(i=1,2,3)$	Retailer’s recycling amount
$Q_i(i=1,2,3)$	Total recycling amount under CLSC
${\Pi }_{mi}(i=1,2,3)$	Manufacturer’s profit
${\Pi }_{ri}(i=1,2,3)$	Retailer’s profit
${\Pi }_i(i=1,2,3)$	Total profit under CLSC

Note: $i=1,2,3$ denotes decentralized, centralized and coordinated decision-making respectively.

shows other related symbols and variables in the model.

3.2. Assumptions

The assumptions of this model are as follows.

Hypothesis 1:

Without losing generality, the manufacturer’s production cost needs to be met  $C_m>C_r$, that is, the production cost $\Delta =C_m-C_r>0$ saved by recycling products can ensure the enthusiasm of the manufacturer to participate in recycling and remanufacturing production. Moreover, in order to encourage manufacturers to reduce carbon emissions, the government gives manufacturer CER subsidy coefficient$s$ per unit product to reduce manufacturers’ emissions reduction costs;

Hypothesis 2:

Referring to the market demand models of Gurnani et al. [44] and Li et al. [45], the price competition between traditional retail channels and direct selling channels in the market, as well as the influence of a manufacturer’s CER efforts on market demand, direct selling channels, retail channels, and market aggregate demand function are expressed as Equations (1)–(3), respectively:

$$D_{mi}=ah-l_1{p}_{mi}+l_2{p}_{ri}+\gamma {\epsilon }_i{e}_0$$

(1)

$$D_{ri}=a\left(1-h\right)-l_1{p}_{ri}+l_2{p}_{mi}+\gamma {\epsilon }_i{e}_0$$

(2)

$$D_i=a-\left(l_1-l_2\right)\left(p_{ri}+p_{mi}\right)+2\gamma {\epsilon }_i{e}_0$$

(3)

Hypothesis 3:

Consumers’ price sensitivity and cross-price elasticity coefficient are greater than consumers’ low carbon preference coefficient, and the influence of channel price on their own demand is greater than that of cross-price, namely: $l_1>l_2>\gamma >0$;

Hypothesis 4:

Considering that the cost of CER is affected by the level of the manufacturer’s CER, and referring to the research of Ha et al. [46] and Zhou et al. [47], the function of the manufacturer’s CER cost is shown in Equation (4) below:

$$C\left({\epsilon }_i\right)=k{\left({\epsilon }_i{e}_0\right)}^2/2$$

(4)

Hypothesis 5:

Manufacturer and retailer collaborate on recycling projects. The manufacturer, on the one hand, recycles waste products directly from consumers at recycling price$p_1$; on the other hand, the manufacturer entrusts the retailer to recycle waste products from consumers at recycling price$p_2$, and then recycle waste products from the retailer with transfer payment. Due to the inevitable competition between manufacturer and retailer in the recycling market, the functions of manufacturer, retailer, and the total recycling amount are expressed in Equations (5)–(7), respectively, referring to the recycling function models of Li et al. [48].

$$Q_{mi}={\beta }_1{p}_{1i}-{\beta }_2{p}_{2i}$$

(5)

$$Q_{ri}=(1-n){\beta }_0+{\beta }_1{p}_{2i}-{\beta }_2{p}_{1i}$$

(6)

$$Q_i=\left({\beta }_1-{\beta }_2\right)\left(p_{1i}+p_{2i}\right)$$

(7)

where ${\beta }_1>{\beta }_2$, it shows that the amount of recycling increases with the increase in its own recycling price, and decreases with the decrease in its competitors’ recycling price. This hypothesis has been proved in the research of literature [48]:

Hypothesis 6:

To ensure that all members in the CLSC are profitable, the following parameter relationships are assumed: $p_m>w>C_m>C_r>0; p_r>w>C_m>C_r>0$.

4. Model Decisions

4.1. Decentralized Decision-Making Model (Model D)

In the recycling and remanufacturing CLSC, both manufacturer and retailer make corresponding pricing decisions with the goal of maximizing their own benefits when making decentralized decisions. The profit functions of manufacturer and retailer are shown in Formulae (8) and (9):

$${\Pi }_{m1}=D_m\left(p_m-C_m\right)+D_r\left(w-C_m\right)+\Delta \left(Q_m+Q_r\right)-p_1{Q}_m-g{Q}_r-\left(1-s\right)k{\left(\epsilon e_0\right)}^2/2$$

(8)

$${\Pi }_{R1}=\left(p_r-w\right)D_r+\left(g-p_2\right)Q_r$$

(9)

The manufacturer and the retailer form a Stackelberg game in the CLSC system, with the manufacturer as the leader. The decision sequence is that the manufacturer first determines wholesale price $w$, direct selling price $p_m$, recycling price $p_1$, transfer payment $g$, and the level of CER effort $\epsilon$. The retailer then determines the optimal retail price $p_r$ and recycling price $p_2$ by maximizing its own profits. When making decisions, the manufacturer’s interests must be balanced against the retailer’s strategy, so the backward induction method is used, and the solution process is as follows.

For the retailer, the best retail price $p_r$ and recycling price $p_2$ are determined by maximizing their own profits. By taking partial derivatives of $p_r$ and $p_2$ in Equation (9), we can get:

$$\frac{\partial {\Pi }_{r1}}{\partial p_r}=a+{\mathrm{wl}}_1+l_2{p}_m{-2l}_1{p}_r-ah+\gamma \epsilon e_0$$

$$\frac{\partial {\Pi }_{r1}}{\partial p_2}={\beta }_1{(g-p}_2)+(n-1){\beta }_0-{\beta }_1{p}_2+{\beta }_2{p}_1$$

Thus, the Hessian matrix of the retailer’s profit function is:

$$H=(\begin{array}{cc}-2l_1& 0\\ 0& -2{\beta }_1\end{array})$$

According to the hypothesis of the model, that is, $l_1>0;{\beta }_1>0$, we get $4l_1{\beta }_1>0$. Thus, the Hessian matrix is negative definite, and the profit function has a unique maximum value. Therefore, joining $\frac{\partial {\Pi }_{r1}}{\partial \Pi p_r}=0$ with $\frac{\partial {\Pi }_{r1}}{\partial \Pi p_2}=0$, we get:

$$p_2=\frac{g{\beta }_1-{\beta }_0+n{\beta }_0+p_1{\beta }_2}{2{\beta }_1}$$

$$p_r=\frac{a+w{l}_1+l_2{p}_m-ah+\gamma \epsilon e_0}{2l_1}$$

Substitute the obtained results into Equation (8), and establish the following first-order conditions:

$$\frac{\partial {\Pi }_{m1}}{\partial g}=\frac{{\beta }_0\left(n-1\right)+\Delta \left({\beta }_1-{\beta }_2\right)}{2}+p_1{\beta }_2-g{\beta }_1$$

$$\frac{\partial {\Pi }_{m1}}{\partial \epsilon }=\frac{\gamma e_0\left(p_m-C_m\right)\left(2l_1+l_2\right)}{2l_1}-\frac{\gamma e_0\left(C_m-w\right)}{2}+k\epsilon {e_0}^2(s-1)$$

$$\frac{\partial {\Pi }_{m1}}{\partial w}=\frac{2l_2{p}_m+\gamma \epsilon e_0+ah+C_m(l_1-l_2)-2w{l}_1-a}{2}-a(h-1)$$

$$\frac{\partial {\Pi }_{m1}}{\partial p_m}=\frac{\begin{array}{l}{l}_2\gamma \epsilon e_0+C_m(2l_1^2-l_2^2-l_1{l}_2)+a{l}_2(1-h)\\ +2l_1{l}_{2}w+2p_m(l_2^2-l_1^2)\end{array}}{2l_1}-l_1{p}_m+ah+\gamma \epsilon e_0$$

$$\frac{\partial {\Pi }_{m1}}{\partial p_1}=\frac{2p_1({\beta }_2^2-{\beta }_1^2)-\Delta ({\beta }_2^2-{\beta }_1{\beta }_2-2{\beta }_1^2)+(n-1){\beta }_0{\beta }_2}{2}+g{\beta }_2-n{\beta }_0-p_1\beta$$

Thus, the Hessian matrix of the manufacturer’s profit function is:

$$H=\left(\begin{array}{ccccc}-{\beta }_1& 0& 0& 0& {\beta }_2\\ 0& (s-1)k{e_0}^2& \gamma e_0/2& \gamma e_0(2l_1+l_2)/2l_1& 0\\ 0& \gamma e_0/2& -2l_1& l_2& 0\\ 0& \gamma e_0(2l_1+l_2)/2l_1& l_2& (l_2^2-2l_1^2)/l_1& 0\\ {\beta }_2& 0& 0& 0& {\beta }_2^2-{\beta }_1^2-{\beta }_1\end{array}\right)$$

According to the hypothesis of the model, that is, $0<s<{1;l}_1>l_2>0;{\beta }_1>{\beta }_2>0$; $k>0$, we get:

$$H_1=-{\beta }_1<0;$$

$$H_2=H_1\left(s-1\right)k{e}_0^2>0;$$

$$H_3=-2l_1{H}_2<0;$$

$$H_4=\left[\left(l_2^2-2l_1^2\right)/l_1\right]H_3>0;$$

$$H_5=\left({\beta }_2^2-{\beta }_1^2-{\beta }_1\right)H_4<0;$$

Thus, the Hessian matrix is negative definite, and the profit function has a unique maximum value. Therefore, associating the first-order condition $\frac{\partial {\Pi }_{m1}}{\partial g}=0;\frac{\partial {\Pi }_{m1}}{\partial \epsilon }=0;$ $\frac{\partial {\Pi }_{m1}}{\partial w}=0;\frac{\partial {\Pi }_{m1}}{\partial p_m}=0;\frac{\partial {\Pi }_{m1}}{\partial p_1}=0$, the optimal decision of the manufacturer can be obtained by solving as follows:

$$p_{11}^{*}=g_1^{*}=\frac{\Delta }{2}$$

$$p_{m1}^{*}=\frac{\begin{array}{c}4k{l}_1\left(1-s\right)\left(C_m{l}_2^2-ah{l}_1+ah{l}_2-a{l}_2-C_m{l}_1^2\right)+\\ {\gamma }^2\left(2C_m{l}_2^2+6C_m{l}_1^2+8C_m{l}_1{l}_2-a{l}_1+2ah{l}_2\right)\end{array}}{2\left(l_1+l_2\right)\left[{\gamma }^2\left(3l_1+l_2\right)+4k{l}_1\left(1-s\right)\left(l_2-l_1\right)\right]}$$

$${\epsilon }_1^{*}=\frac{\gamma \left(3C_m{l}_1^2-C_m{l}_2^2-2C_m{l}_1{l}_2-ah{l}_1+ah{l}_2-a{l}_1-a{l}_2\right)}{e_0\left[{\gamma }^2\left(3l_1+l_2\right)+4k{l}_1\left(1-s\right)\left(l_2-l_1\right)\right]}$$

$$w_1^{*}=\frac{\begin{array}{c}4k{l}_1\left(1-s\right)\left(C_m{l}_2^2+ah{l}_1-ah{l}_2-a{l}_1-C_m{l}_1^2\right)+\\ {\gamma }^2\left(2C_m{l}_2^2+2a{l}_1+a{l}_2+6C_m{l}_1^2+8C_m{l}_1{l}_2-4ah{l}_1-2ah{l}_2\right)\end{array}}{2\left(l_1+l_2\right)\left[{\gamma }^2\left(3l_1+l_2\right)+4k{l}_1\left(1-s\right)\left(l_2-l_1\right)\right]}$$

The asterisk indicates that it is the optimal solution of the decision variable, and the asterisks of the decision variable later in paper all mean this.

Substituting $w_1^{*},p_{m1}^{*},p_{11}^{*},g_1^{*},{\epsilon }_1^{*}$ into the retailer’s optimal solution, so the retailer’s optimal decision can be obtained as follows:

$$p_{21}^{*}=\frac{\Delta \left({\beta }_2+{\beta }_1\right)}{4{\beta }_1}$$

$$p_{r1}^{*}=\frac{\begin{array}{c}2k{l}_1\left(1-s\right)\left(C_m{l}_2^2-C_m{l}_1{l}_2-2ah{l}_2+3ah{l}_1-3a{l}_1-C_m{l}_1^2\right)+2k{l}_2^2\left(1-s\right)(a\\ +C_m{l}_2-ah)+{\gamma }^2\left(2C_m{l}_2^2+3a{l}_1+2a{l}_2+6C_m{l}_1^2+8C_m{l}_1{l}_2-6ah{l}_1-4ah{l}_2\right)\end{array}}{2\left(l_1+l_2\right)\left[{\gamma }^2\left(3l_1+l_2\right)+4k{l}_1\left(1-s\right)\left(l_2-l_1\right)\right]}$$

4.2. Centralized Decision-Making Model (Model C)

Under centralized decision-making, the manufacturer and the retailer are regarded as a whole, and they negotiate to determine the recycling price and selling price with the common goal of maximizing total profit of CLSC. They are unified decision-making bodies, regardless of the leader and follower. The total profit of the entire supply chain is as follows:

$${\Pi }_2=D_m\left(p_m-C_m\right)+D_r\left(p_r-C_m\right)+\Delta \left(Q_m+Q_r\right)-p_1{Q}_m-p_2{Q}_r-\left(1-s\right)k{\left(\epsilon e_0\right)}^2/2$$

(10)

Taking the second partial derivative with respect to $\epsilon ,p_m,p_r,p_2$ according to Equation (10), the Hessian matrix of profit function is:

$$H=\left(\begin{array}{cccc}-k{e}_0(1-s)& \gamma & \gamma & 0\\ 0& {\gamma }^2-2k{l}^1(1-s)& 2k{l}_2(1-s)& 0\\ 0& 0& 4(l_1+l_2)[{\gamma }^2-k(1-s)(l_1-l_2)& 0\\ 0& 0& 0& -2{\beta }_1\end{array}\right)$$

According to the hypothesis of the model, that is, $0<s<{1;l}_1>l_2>0;{\beta }_1>{\beta }_2>0$; $k>0;0<\gamma <1$, so when ${\gamma }^2-k(1-s)(l_1-l_2)<0,$ we get:

$$H_1=-k{e}_0\left(1-s\right)<0;$$

$$H_2=H_1\left[2k{l}_1\left(1-s\right)-{\gamma }^2\right]>0;$$

$$H_3=4\left(l_1+l_2\right)\left[{\gamma }^2-k\left(1-s\right)\left(l_1-l_2\right)\right]H_2<0;$$

$$H_4=-2{\beta }_1{H}_3>0;$$

Thus, the Hessian matrix is negative definite, and the profit function has a unique maximum value. Thus, associating the first-order condition $\frac{\partial {\Pi }_2}{\partial \epsilon }=0$, we can yield:

$${\epsilon }_2=\frac{\gamma (p_m+p_r-2C_m)}{\left(1-s\right){ke}_0}$$

Substituting this into Equation (10), then taking the partial derivative of $p_m$ and associating the first order condition $\frac{\partial {\Pi }_2}{\partial p_m}=0$, we get:

$$p_{m2}=\frac{k\left(1-s\right)\left[ah+2l_2{p}_r+C_m\left(l_1-l_2\right)\right]+{\gamma }^2\left(p_r-2C_m\right)}{2k{l}_1\left(1-s\right)-{\gamma }^2}$$

Substituting this into Equation (10), then taking the partial derivative of $p_r$ and associating the first order condition $\frac{\partial {\Pi }_2}{\partial p_r}=0$, we get:

$$p_{r2}^{*}=\frac{2k\left(1-s\right)\left(C_m{l}_2^2-a{l}_1+ah{l}_1-ah{l}_2-C_m{l}_1^2\right)+{\gamma }^2\left(a+4C_m{l}_1+4C_m{l}_2-2ah\right)}{4\left(l_1+l_2\right)\left[{\gamma }^2+k\left(1-s\right)\left(l_2-l_1\right)\right]}$$

Taking the partial derivative of $p_2$ in Equation (10) and associating the first-order condition $\frac{\partial {\Pi }_2}{\partial p_2}=0$, we get:

$$p_{22}=\frac{\Delta \left({\beta }_1-{\beta }_2\right)+2p_1{\beta }_2}{2{\beta }_1}$$

Substituting this into Equation (10), then taking the partial derivative of $p_1$ and associating the first order condition $\frac{\partial {\Pi }_2}{\partial p_1}=0$, we get:

$$p_{12}=\frac{\Delta }{2}$$

Based on the above solution, the optimal pricing strategy for the manufacturer and the retailer under centralized decision-making and the optimal level of CER can be obtained as follows:

$$p_{12}^{*}=p_{22}^{*}=\frac{\Delta }{2}$$

$${\epsilon }_2^{*}=\frac{\gamma \left(2C_m{l}_1-2C_m{l}_2-a\right)}{e_0\left[{\gamma }^2+k\left(1-s\right)\left(l_2-l_1\right)\right]}$$

$$p_{r2}^{*}=\frac{2k\left(1-s\right)\left(C_m{l}_2^2-a{l}_1+ah{l}_1-ah{l}_2-C_m{l}_1^2\right)+{\gamma }^2\left(a+4C_m{l}_1+4C_m{l}_2-2ah\right)}{4\left(l_1+l_2\right)\left[{\gamma }^2+k\left(1-s\right)\left(l_2-l_1\right)\right]}$$

$$p_{m2}^{*}=\frac{2k\left(1-s\right)\left(C_m{l}_2^2-a{l}_2+ah{l}_2-ah{l}_1-C_m{l}_1^2\right)+{\gamma }^2\left(4C_m{l}_1+4C_m{l}_2+2ah-a\right)}{4\left(l_1+l_2\right)\left[{\gamma }^2+k\left(1-s\right)\left(l_2-l_1\right)\right]}$$

4.3. Analysis of Equilibrium Results of Different Decision-Making Models

4.3.1. The Influence of Each Parameter on the Equilibrium Result

Proposition 1:

The recycling price of the manufacturer and the retailer are negatively correlated with the production costs of remanufactured products in both decision-making models.

Proof: 

It is demonstrated that the optimal decision of the two decision models is as follows:

$$\frac{\partial p_{11}^{*}}{\partial C_r}=\frac{\partial p_{12}^{*}}{\partial C_r}=\frac{\partial p_{22}^{*}}{\partial C_r}=-\frac{1}{2}<0;\frac{\partial p_{21}^{*}}{\partial C_r}=-\frac{{\beta }_1+{\beta }_2}{4{\beta }_1}<0$$

Proposition 1 shows that when the production cost of remanufactured products decreases, the manufacturer’s enthusiasm for producing remanufactured products increases, prompting the manufacturer to increase the recycling price to increase total recycling amount under CLSC.

Proposition 2:

The manufacturer’s direct selling price and CER level, as well as the retailer’s retail price, are positively correlated with consumers’ low carbon preference coefficient in both decision-making models.

Proof: 

It is demonstrated that the optimal decision of the two decision models is as follows:

$$\begin{array}{c}\frac{\partial {\epsilon }_1^{*}}{\partial \gamma }=\frac{{\epsilon }_1^{*}}{\gamma }+\frac{2\gamma \left(3l_1+l_2\right){\epsilon }_1^{*}}{4k{l}_1\left(1-s\right)\left(l_1-l_2\right)-{\gamma }^2\left(3l_1+l_2\right)}>0;\\ \frac{\partial {\epsilon }_2^{*}}{\partial \gamma }=\frac{{\epsilon }_2^{*}}{\gamma }+\frac{2\gamma {\epsilon }_2^{*}}{k\left(1-s\right)\left(l_1-l_2\right)-{\gamma }^2}>0\end{array}$$

$$\frac{\partial p_{m1}^{*}}{\partial \gamma }=\frac{4\gamma \left(l_1+l_2\right)(3l_1+l_2)p_{m1}^{*}-2\gamma \left(2C_m{l}_2^2+8C_m{l}_1{l}_2+2ah{l}_2-a{l}_2+6C_m{l}_1^2\right)}{\left(l_1+l_2\right)\left[4k{l}_1\left(1-s\right)\left(l_1-l_2\right)-{\gamma }^2\left(3l_1+l_2\right)\right]}>0;$$

$$\frac{\partial p_{m1}^{*}}{\partial \gamma }=\frac{8\gamma \left(l_1+l_2\right)p_{m2}^{*}-2\gamma \left(a+4C_m{l}_1+4C_m{l}_1-2ah\right)}{4\left(l_1+l_2\right)\left[k\left(1-s\right)\left(l_1-l_2\right)-{\gamma }^2\right]}>0;$$

By the same token: $\frac{\partial p_{r1}^{*}}{\partial \gamma }>0;\frac{\partial p_{r2}^{*}}{\partial \gamma }>0$

Proposition 2 indicates that when consumers’ low carbon preference coefficient increases, the market demand for remanufactured product expands, prompting the manufacturer to produce more remanufactured products. At the same time, the manufacturer increases its CER investment, forcing the manufacturer to compensate for the cost of CER by raising the selling price, and the retailer raises its retail price accordingly.

Proposition 3:

In both decision-making models, selling prices of the manufacturer and the retailer and the level of the manufacturer’s CER effort are positively related to government CER subsidy coefficient.

Proof: 

It is demonstrated that the optimal decision of the two decision models is as follows:

$$\begin{gathered} \frac{\partial {\epsilon }_1^{*}}{\partial s}=\frac{4k{l}_1\left(l_1-l_2\right){\epsilon }_1^{*}}{4k{l}_1\left(1-s\right)\left(l_1-l_2\right)-{\gamma }^2\left(3l_1+l_2\right)}>0; \\ \frac{\partial {\epsilon }_2^{*}}{\partial s}=\frac{k\left(l_1-l_2\right){\epsilon }_2^{*}}{k\left(1-s\right)\left(l_1-l_2\right)-{\gamma }^2}>0; \end{gathered}$$

$$\frac{\partial p_{m1}^{*}}{\partial s}=\frac{2k{l}_1\left(C_m{l}_2^2-ah{l}_1+ah{l}_2-a{l}_2-C_m{l}_1^2\right)-4k{l}_1\left(l_2^2-l_1^2\right)p_{m1}^{*}}{\left(l_1+l_2\right)\left[4k{l}_1\left(1-s\right)\left(l_1-l_2\right)-{\gamma }^2\left(3l_1+l_2\right)\right]}>0;$$

$$\frac{\partial p_{m2}^{*}}{\partial s}=\frac{2k\left(C_m{l}_2^2-ah{l}_1+ah{l}_2-a{l}_2-C_m{l}_1^2\right)-4k\left(l_2^2-l_1^2\right)p_{m2}^{*}}{4\left(l_1+l_2\right)\left[k\left(1-s\right)-{\gamma }^2\right]}>0;$$

By the same token:$\frac{\partial p_{r1}^{*}}{\partial s}>0;\frac{\partial p_{r2}^{*}}{\partial s}>0$

Proposition 3 shows that when the government CER subsidy coefficient increases, the cost of the manufacturer’s CER is reduced, thus improving the zeal of the manufacturer for CER, making the level of manufacturer’s CER effort rise, reducing the carbon emissions of the whole CLSC, generating good social benefits, and improving the corporate image. Therefore, consumers are more willing to buy remanufactured products produced by the manufacturer, thus expanding market demand and increasing selling price.

Proposition 4:

Under the two decision-making models, the total recycling amount under CLSC is negatively related to the recycling channel competition coefficient.

Proof: 

It is demonstrated that the optimal decision of the two decision models is as follows:

$$\begin{gathered} \frac{\partial p_{11}^{*}}{\partial {\beta }_2}=\frac{\partial p_{12}^{*}}{\partial {\beta }_2}=\frac{\partial p_{22}^{*}}{\partial {\beta }_2}=0;\frac{\partial p_{21}^{*}}{\partial {\beta }_2}=\frac{\Delta }{4{\beta }_1}>0; \\ \frac{\partial Q_1}{\partial {\beta }_2}=\frac{-\Delta \left(2{\beta }_1+{\beta }_2\right)}{2{\beta }_1}<0;\frac{\partial Q_2}{\partial {\beta }_2}=-\Delta <0 \end{gathered}$$

The proposition shows that the greater the recycling channel competition coefficient, the less total recycling amount there is under CLSC. This is because when the competition between the manufacturer and retailer in recycling is relatively fierce, as the leader of CLSC, the manufacturer’s pricing decisions are almost unaffected by the recycling channel competition coefficient. By contrast, the retailer can only attract consumers by increasing the recycling price to improve competitiveness in the recycling market, leading to higher recycling costs and reducing the zeal of the retailer for recycling, thus reducing the total recycling amount under CLSC.

4.3.2. Comparative Analysis of Equilibrium Results of Different Decision-Making Models

Proposition 5:

The level of the manufacturer’s CER effort under centralized decision-making is higher than that under decentralized decision-making, that is, ${\epsilon }_2^{*}>{\epsilon }_1^{*}$.

Proof: 

According to the equilibrium results of the above model, we can get:

$${\epsilon }_1^{*}=\frac{\gamma \left[\left(p_{m1}-C_m\right)+l_1\left(w-C_m\right)\right]}{2k{l}_1{e}_0\left(1-s\right)}$$

$${\epsilon }_2^{*}=\frac{\gamma \left(p_{m2}+p_{r2}-2C_m\right)}{k{e}_0\left(1-s\right)}$$

According to Hypothesis 6, all members are profitable, so:

$$p_{m2}-C_m>0;p_{r2}-C_m>0;p_{m1}-C_m>w-C_m>0$$

Then:

$$\frac{{\epsilon }_2^{*}}{{\epsilon }_1^{*}}=\frac{2l_1\left(p_{m2}+p_{r2}-2C_m\right)}{\left(p_{m1}-C_m\right)+l_1\left(w-C_m\right)}>1$$

Proposition 5 demonstrates that in the decentralized decision-making mode, the manufacturer only pursues the maximization of its own interests, and reduces CER costs by reducing CER effort, making the level of the manufacturer’s CER effort under the decentralized decision-making lower than that of the centralized decision-making, resulting in a double marginal effect. As a result, measures should be taken to encourage manufacturer to improve CER effort in order to reduce carbon emissions throughout the supply chain.

Proposition 6:

The total market demand under centralized decision-making is higher than that under decentralized decision-making, that is, $D_2>D_1$.

Proof: 

Because of:

$$\frac{p_{m1}^{*}+p_{r1}^{*}}{p_{m2}^{*}+p_{r2}^{*}}=\frac{2k{l}_1\left(1-s\right)\left[3a\left(1-h\right)+4\left(l_1+l_2\right)p_{m1}+C_m\left(l_1-l_2\right)\right]+3{\gamma }^2\left[\left(p_{m1}-C_m\right)+l_1\left(w-C_m\right)\right]}{k\left(1-s\right)\left[a\left(1-h\right)+2\left(l_1+l_2\right)p_{m2}+C_m\left(l_1-l_2\right)\right]-4{\gamma }^2{C}_m}$$

We can get:

$$\frac{p_{m1}^{*}+p_{r1}^{*}}{p_{m2}^{*}+p_{r2}^{*}}>1$$

So:

$$D_1-D_2=\left(l_1-l_2\right)\left(p_{m2}^{*}+p_{r2}^{*}-p_{m1}^{*}-p_{r1}^{*}\right)+2\gamma e_0\left({\epsilon }_1^{*}-{\epsilon }_2^{*}\right)<0$$

This proposition demonstrates that in the decentralized decision-making model, the competition between direct selling channel and retail channel compels the retailer to raise retail price to ensure its own profits, which reduces consumers’ purchase intention and leads to a decrease in market demand. By comparison, under centralized decision-making, the manufacturer and the retailer jointly negotiate sales price based on the market situation, which weakens the impact of the competition between direct selling channels and retail channels and makes the total market demand under centralized decision-making higher than that under decentralized decision-making.

Proposition 7:

The retailer’s recycling price under centralized decision-making is higher than that under decentralized decision-making, in addition, the manufacturer is the leader of CLSC and can set price independently, so that the manufacturer’s recycling price and transfer payment are almost not affected by the decision choice, resulting in a higher total recycling amount under centralized decision than under decentralized decision-making, that is:

$$g_1^{*}=p_{11}^{*}=p_{12}^{*}=p_{22}^{*}>p_{21}^{*};Q_2>Q_1$$

Proof: 

According to the equilibrium results of the above model, we can get:

$$g_1^{*}=p_{11}^{*}=p_{12}^{*}=\frac{\Delta }{2}$$

$$p_{22}^{*}-p_{21}^{*}=\frac{\Delta \left({\beta }_1-{\beta }_2\right)}{4{\beta }_1}>0$$

$$Q_2-Q_1=\left({\beta }_1-{\beta }_2\right)\left(p_{12}^{*}+p_{22}^{*}-p_{11}^{*}-p_{21}^{*}\right)>0$$

Proposition 7 demonstrates that, under decentralized decision-making, the retailer reduces the recycling price in order to reduce recycling cost, which reduces consumers’ willingness to participate in recycling. At the same time, in order to obtain enough waste products for recycling and remanufacturing, the manufacturer takes the initiative to increase recycling price, which intensifies the competition between the manufacturer and retailer in the recycling market, resulting in less total recycling volume of CLSC than under the centralized decision-making model.

5. Coordination of the Recycling and Remanufacturing CLSC

In order to improve the CER level of CLSC, the manufacturer can negotiate with the retailer to jointly bear CER cost. In addition, the manufacturer can transfer a certain percentage of revenue to the retailer (including sales revenue of the direct marketing channel and the production cost saved by recycling and remanufacturing) to stabilize the cooperative relationship with the retailer. Assuming the proportion of CER costs shared by the retailer in the contract of “CER cost sharing + revenue sharing” is $x(0<x<1)$, and then the proportion of CER costs borne by the manufacturer is $1-x$, the manufacturer will share a proportion of $\theta (0<\theta <1)$ of the benefits with the retailer. The profit of the manufacturer and the retailer under this contract are:

$${\Pi }_{m3}=D_r\left(w-C_m\right)+\left(1-\theta \right)\left[D_m\left(p_m-C_m\right)+\Delta \left(Q_m+Q_r\right)-p_1{Q}_m-g{Q}_r\right]-\left(1-x-s\right)k{\left(\epsilon e_0\right)}^2/2$$

(11)

$${\Pi }_{R3}=\left(p_r-w\right)D_r+\left(g-p_2\right)Q_r+\theta \left[D_m\left(p_m-C_m\right)+\Delta \left(Q_m+Q_r\right)-p_1{Q}_m-g{Q}_r\right]-xk{\left(\epsilon e_0\right)}^2/2$$

(12)

At this time, $w$ and $g$ are the adjustment variables, and the backward induction method is adopted. The solution process is as follows:

For retailers, the best retail price $p_r$ and recycling price $p_2$ are determined by maximizing their own profits. Taking the second partial derivatives of $p_r$ and $p_2$ in Equation (12), the Hessian matrix of the retailer’s profit function is:

$$H=(\begin{array}{cc}-2l_1& 0\\ 0& -2{\beta }_1\end{array})$$

According to the hypothesis of the model, that is, $l_1>0;{\beta }_1>0$, then we get $4l_1{\beta }_1>0$, so the Hessian matrix is negative definite, and the profit function has a unique maximum value. Therefore, associating $\frac{\partial {\Pi }_{r3}}{\partial \Pi p_r}=0$ and $\frac{\partial {\Pi }_{r3}}{\partial \Pi p_2}=0$, we get:

$$p_r=\frac{a+w{l}_1+l_2{p}_m+\theta l_2\left(p_m-C_m\right)-ah+\gamma \epsilon e_0}{2l_1}$$

$$p_2=\frac{g{\beta }_1\left(1-\theta \right)+p_1{\beta }_2\left(1+\theta \right)+\theta \Delta \left({\beta }_1-{\beta }_2\right)}{2{\beta }_1}$$

Substituting the obtained results into Equation (11), and taking the second partial derivatives of $p_m,p_1,\epsilon$ in Equation (11), so the Hessian matrix of manufacturer’s profit function is:

$$H=\left(\begin{array}{ccc}(1-\theta )(x{l}_2^2-2l_1{l}_2-l_2^2)/(l_1+l_2)& (1-\theta )(2l_1\gamma e_0+3l_2\gamma e_0)/2(l_1+l_2)& 0\\ A& x{e}_0(l_{1}k{e}_{0}s+l_{2}k{e}_{0}s-l_{1}k{e}_0-l_{2}k{e}_0)/(l_1+l_2)& 0\\ 0& 0& -[2{\beta }_1^2-(1+\theta ){\beta }_2^2]\end{array}\right)$$

Including: $A=x{e}_0(2l_1\gamma +3l_2\gamma -2l_1\gamma x-3l_2\gamma x)/2(l_1+l_2)$;

According to the hypothesis of the model, that is, $l_1>l_2>0;0<x<1;0<\theta <1;{\beta }_1>{\beta }_2$, we get:

$$\begin{array}{l}{H}_1=(1-\theta )(x{l}_2^2-2l_1{l}_2-l_2^2)/(l_1+l_2)<0;\\ H_2=[x{e}_0(l_{1}k{e}_{0}s+l_{2}k{e}_{0}s-l_{1}k{e}_0-l_{2}k{e}_0)/(l_1+l_2)]H_1>0;\\ H_3=-[2{\beta }_1^2-(1+\theta ){\beta }_2^2]H_2<0;\end{array}$$

Thus, the Hessian matrix is negative definite, and the profit function has a unique maximum value. Therefore, associating $\frac{\partial {\Pi }_{m3}}{\partial \Pi p_m}=0;\frac{\partial {\Pi }_{m3}}{\partial \Pi \epsilon }=0$ and $\frac{\partial {\Pi }_{m3}}{\partial \Pi p_1}=0$, we get the optimal decision of the manufacturer as follows:

$$p_{13}^{*}=\frac{2g{\beta }_1{\beta }_2\left(1-\theta \right)-f_3-2\Delta {\beta }_2\left({\beta }_2-{\beta }_1\right)\theta }{2\left[(2{\beta }_1^2-(1+\theta ){\beta }_2^2)\right]}$$

$${\epsilon }_3^{*}=\frac{f_{15}+w{f}_{16}+\theta f_{17}-\theta w{f}_{18}}{e_0\left[f_9+\theta f_8+4l_1{l}_2^{2}kx\left(1+\theta \right)-8kx{l}_1^3\right]}$$

$$p_{m3}^{*}=\frac{f_4+w{f}_5+x{f}_6+\theta f_7+4\theta xk{C}_m{l}_1{l}_2^2-4kwx{l}_1^2{l}_2}{f_9+\theta f_8+4l_1{l}_2^{2}kx\left(1+\theta \right)-8kx{l}_1^3}$$

Substituting $p_{m3}^{*},p_{13}^{*},{\epsilon }_3^{*}$ into the retailer’s optimal solution, the retailer’s optimal decision can be obtained as follows:

$$p_{23}^{*}=\frac{4g\left(1-\theta \right){\beta }_1^3-f_1-\theta f_2}{4{\beta }_1\left[2{\beta }_1^2-\left(1+\theta \right){\beta }_2^2\right]}$$

$$p_{r3}^{*}=\frac{f_{10}+\theta f_{11}+x{f}_{12}+\theta x{f}_{13}+\theta w{f}_{14}-4k{l}_1^{3}wx}{f_9+\theta f_8+4l_1{l}_2^{2}kx\left(1+\theta \right)-8kx{l}_1^3}$$

Including:

$f_1=\Delta {\beta }_2^3+\Delta {\beta }_1{\beta }_2^2-2\Delta {\beta }_1^2{\beta }_2;$

$f_2=\Delta {\beta }_2^3-4\Delta {\beta }_1^3+\Delta {\beta }_1{\beta }_2^2+2\Delta {\beta }_1^2{\beta }_2;$

$f_3=\Delta {\beta }_2^2-2\Delta {\beta }_1^2+\Delta {\beta }_1{\beta }_2;$

$f_4=k{l}_1(1-s)(2C_m{l}_1^2-C_m{l}_2^2-C_m{l}_1{l}_2+2ah{l}_1-ah{l}_2+a{l}_2)-{\gamma }^2{C}_m(l_2^2+6l_1^2+5l_1{l}_2);$

$f_5={\gamma }^2{l}_1(2l_1+l_2)+4k{l}_1^2{l}_2(1-s);$

$f_6=2k{l}_1(C_m{l}_2^2-2C_m{l}_1^2+C_m{l}_1{l}_2-2ah{l}_1+ah{l}_2-a{l}_2);$

$f_7={\gamma }^2{C}_m(4l_1^2+l_1{l}_2+l_2^2)-4C_{m}k{l}_1{l}_2^2(1-s);$

$f_8={\gamma }^2(2l_1+l_2{)}^2-4k{l}_1{l}_2^2(1-s);$

$f_9=4k{l}_1(1-s)(2l_1^2-l_2^2)-{\gamma }^2(2l_1+l_2{)}^2;$

$\begin{array}{l}{f}_{10}=k(1-s)(4a{l}_1^2-C_m{l}_2^3-a{l}_2^2-C_m{l}_1{l}_2^2+2C_m{l}_1^2{l}_2-4ah{l}_1^2+ah{l}_2^2+2ah{l}_1{l}_2)+{\gamma }^2\\ (4ah{l}_1+2ah{l}_2-2a{l}_1-a{l}_2-C_m{l}_2^2-4C_m{l}_1^2-5C_m{l}_1{l}_2);\end{array}$

$\begin{array}{l}{f}_{11}=k{l}_2(1-s)(2ah{l}_1+ah{l}_2-a{l}_2-C_m{l}_2^2-C_m{l}_1{l}_2-2C_m{l}_1^2)+{\gamma }^2(2a{l}_1+a{l}_2-4ah{l}_1-2ah{l}_2+\\ 2C_m{l}_2^2+2C_m{l}_1^2+3C_m{l}_1{l}_2);\end{array}$

$f_{12}=k(C_m{l}_2^3-4a{l}_1^2-a{l}_2^2+C_m{l}_1{l}_2^2-2C_m{l}_1^2{l}_2+4ah{l}_1^2-ah{l}_2^2-2ah{l}_1{l}_2);$

$f_{13}=k{l}_2(C_m{l}_2^2+a{l}_2-ah{l}_2+C_m{l}_1{l}_2+2C_m{l}_1^2-2ah{l}_1);$

$f_{14}={\gamma }^2(2l_1^2-l_2^2+l_1{l}_2);$

$f_{15}=\gamma (C_m{l}_2^3-8C_m{l}_1^3-ah{l}_2^2+a{l}_2^2+2a{l}_1{l}_2+3C_m{l}_1{l}_2^2-4C_m{l}_1^2{l}_2+4ah{l}_1^2-ah{l}_2^2);$

$f_{16}=4l_1^2\gamma (l_1+l_2);$

$f_{17}=\gamma (4C_m{l}_1^3-C_m{l}_2^3-a{l}_2^2-2a{l}^{1}l{}_2+C_m{l}_1{l}_2^2+4C_m{l}_1^2{l}_2-4ah{l}_1^2+ah{l}_2^2);$

$f_{18}=4\gamma l_1{l}_2(l_1+l_2)$

At this time, when the parameters in the contract of “CER cost sharing + revenue sharing” meet the following condition, the contract can coordinate CLSC:

$$\begin{array}{c}\widehat{\theta }=\frac{4\Delta {\beta }_1^4-2\Delta {\beta }_1^2{\beta }_2^2+2f_3{\beta }_1^2-f_1{\beta }_2-4\Delta {\beta }_1^3{\beta }_2+2\Delta {\beta }_1{\beta }_2^3}{4\Delta {\beta }_1^3{\beta }_2+f_2{\beta }_2+2\Delta {\beta }_1^2{\beta }_2^2+2\Delta {\beta }_1{\beta }_2^3}\\ \widehat{w}=\frac{e_0{\epsilon }_2^{*}\left[f_9+\theta f_8+4l_1{l}_2^{2}kx\left(1+\theta \right)-8kx{l}_1^3\right]-f_{15}-\theta f_{17}}{f_{16}-\theta f_{18}}\\ \widehat{g}=\frac{4\Delta {\beta }_1^3-2\Delta {\beta }_1{\beta }_2^2+\theta \left(f_2-2\Delta {\beta }_1{\beta }_2^2\right)+f_1}{4{\beta }_1^3\left(1-\theta \right)}\end{array}$$

Proof: 

Associating $p_{r3}^{*}=p_{r2}^{*},p_{13}^{*}=p_{23}^{*}=p_{12}^{*},{\epsilon }_3^{*}={\epsilon }_2^{*}$ yields $\widehat{\theta },\widehat{w}$ and $\widehat{g}$, where there exist $0\le \widehat{x}\le 1$ such that the coordinated $p_{r3}^{*},p_{13}^{*},p_{23}^{*},{\epsilon }_3^{*}$ is equal to the optimal value under centralized decision-making, thus achieving coordination of the supply chain.

6. Numerical Examples

In order to further verify the correctness of the above conclusions and the feasibility of contract coordination, numerical simulation is carried out on the equilibrium results under different decisions. By referring to parameter settings of the Zhang et al. [49] and Chen et al. [50] on the premise of combining the model assumption of the paper, the values of key parameters are set as follows:

$$a=200;{\beta }_1=3;C_m=45;C^r=15;k=3;l_1=4;l_2=3;e_0=3;h=0.4$$

6.1. Comparative Analysis of Centralized or Decentralized Decision-Making

The comparison of optimal decision and profit results of the recycling and remanufacturing CLSC under different models as shown in

Table 2. Comparison results of optimal decision and profit under different models.

Model	w	pm	pr	p1	p2	g	ɛ	Q	Π
Model D	65.608	73.197	80.938	15	10	15	3.440	50	2393.786
Model C	/	73.339	76.197	15	15	/	7.560	60	2721.332

.

It can be seen from Table 2 that under the decentralized decision-making model, all participants in the CLSC aim to maximize their own interests. In order to reduce the cost of CER and safeguard their own interests, the manufacturer chooses to reduce the level of CER effort, while the retailer chooses to increase retail price and reduce recycling price to obtain more profits, which reduces the enthusiasm of consumers to participate in recycling and remanufacturing. As a result, the total market demand and total recycling amount of the CLSC decrease, making the total profit of decentralized decision-making lower than that of centralized decision-making. Therefore, certain strategies need to be taken to improve total recycling amount and total profit under CLSC and the level of manufacturers’ CER efforts, so as to improve the overall performance of the recycling and remanufacturing CLSC.

6.2. Sensitivity Analysis of System Parameters

6.2.1. Analysis of the Influence of Recycling Channel Competition Coefficient

Assuming all other parameters remain constant, based on the constraints of this paper, the total recycling amount and total profit of CLSC under centralized and decentralized decision-making models are compared and analyzed.

It can be seen from Figure 2a,b that the greater the recycling channel competition coefficient is, the lower the total recycling amount and total profit under CLSC are, which is consistent with the relevant content of proposition 4. This shows that when the competition between manufacturer and retailer in the recycling market is relatively fierce, retailers pay higher costs for recycling waste products, which reduces the enthusiasm of the retailer to carry out recycling work, thus reducing total recycling volume of the supply chain and even leading to an insufficient recycling amount to meet the requirements of remanufacturing. However, the centralized decision-making mode weakens the competition between the manufacturer and the retailer cycling channel, making the total recycling amount and total profit of the supply chain higher than for the decentralized decision-making.

6.2.2. Analysis of the Influence of Consumers’ Low Carbon Preference Coefficient

Assuming all other parameters remain constant, based on the constraints of this paper, the comparative analysis is made on the manufacturer’s direct selling price, retailer’s retail price and total profit of CLSC under centralized and decentralized decision-making at time $\gamma \in [0,1]$.

As can be seen from Figure 3a–c, with the increase in consumers’ low carbon preference coefficient, the manufacturer’s direct selling price and retailer’s retail price rise, making total profit of CLSC augment accordingly, which is consistent with the relevant content of proposition 2. This indicates that with the increase in consumers’ low carbon preference coefficient, consumers’ requirements for the low carbon attributes of products improve—that is, they are more willing to buy remanufactured products and pay a higher price, prompting the manufacturer to produce more remanufactured products. At the same time, the manufacturer increases its investment in CER, which reduces the carbon emissions of the entire supply chain, increases the social benefits of the manufacturer, and further drives consumers’ demand for manufacturer’s products, thus increasing sales volume, sales revenue and total profit of CLSC. In addition, it can also be seen from Figure 3c that the continuous growth of the retailer’s retail price under decentralized decision-making reduces consumers’ willingness to buy, leading to a decline in total profit of the supply chain under decentralized decision-making. At the same time, by comparing the equilibrium results of the two decision-making models, it can be seen that the manufacturer’s direct selling price under decentralized decision-making is less affected by the choice of decision-making mode, and the retailer’s retail price under decentralized decision-making is higher than that under centralized decision-making. This indicates that decentralized decision-making has a bilateral effect, which leads to the decline of the overall benefits of the recycling and remanufacturing CLSC.

6.2.3. Analysis of the Influence of Government CER Subsidy Coefficient

Assuming all other parameters remain constant, based on the constraints in this paper, we compare selling prices of the manufacturer and the retailer and total profit of CLSC under centralized and decentralized decision-making at time $s\in [0,0.8]$.

It can be seen from Figure 4a–c that the selling prices of the manufacturer and the retailer and the total profit of CLSC show an upward trend with the increase in government CER subsidy coefficient, which is consistent with the relevant content of proposition 3. This indicates that with the increase in government CER subsidy coefficient, the manufacturer’s CER cost is greatly reduced, and the manufacturer is encouraged to actively reduce carbon emissions, so that the carbon emissions of the entire supply chain are reduced, generating good social benefits, improving the corporate image, expanding market demand, and thus increasing total profit under CLSC. Therefore, government CER subsidy policy is conducive to improving the manufacturer’s zeal for CER and total profit of the whole CLSC.

In addition, it can also be seen from Figure 4c that total profit growth of the supply chain under decentralized decision-making is relatively small, and it starts to decline gradually when the government CER subsidy coefficient gets bigger. This is because under the decentralized decision-making process, the retailers’ sales price increases greatly, which reduces consumers’ purchase intentions and leads to a decline in market demand, thus reducing the total profit of the supply chain.

6.2.4. Analysis of the Common Influence of ³ and S on CLSC

1. Impact Analysis on the Level of Manufacturer’s CER Effort

Assuming all other parameters remain constant, based on the constraints in this paper, a comparison is made between the level of the manufacturer’s CER effort under centralized and decentralized decision-making at time $\gamma \in [0,1]$ and $s\in [0,0.6]$.

As shown in Figure 5, the level of the manufacturer’s CER effort is improving as the government CER subsidy coefficient increases, and the rise in consumers’ low carbon preference coefficient prompts the manufacturer to increase CER investment, thus improving the level of CER. Thus, it can be seen that the optimal level of manufacturer’s CER effort is increasing under the combined influence of government CER subsidy policy and consumers’ low carbon preference, and the optimal CER level under centralized decision-making is always higher than that under decentralized decision-making.

2. Impact Analysis on Total Market Demand

Assuming all other parameters remain constant, based on the constraints in this paper, total market demand under centralized and decentralized decision-making is compared and analyzed at time $\gamma \in [0,1]$ and $s\in [0,0.6]$.

It can be seen from Figure 6 that under the joint influence of government CER subsidy coefficient and the consumers’ low carbon preference coefficient, with the increase in both, total market demand shows an upward trend, and total market demand under centralized decision-making is higher than that under decentralized decision-making. This shows that government CER subsidy reduces the CER cost of manufacturers and incentivizes them to produce more remanufactured products; at the same time, with the increase in low carbon preference coefficient, consumers’ willingness to buy remanufactured products increases, which makes total demand for recycling and remanufacturing CLSC rise.

6.3. Analysis of the Results of ContractCoordination

According to the equilibrium results of the contract parameters under the coordination decision, both the contract parameters and the adjustment variables are related to the recycling channel competition coefficient. From this, it can be concluded that under the contract coordination of “CER cost sharing + revenue sharing”, the profit of the manufacturer and the retailer change with ${\beta }_2$.

It can be seen from Figure 7a,b that the contract of “CER cost sharing + revenue sharing” can increase the respective profit of the manufacturer and the retailer, thus increasing the total profit of the supply chain. In addition, it can be seen from Figure 7b that the trend of curve change after coordination is slower than that before coordination, which indicates that the contract contributes to alleviating the competition between manufacturer and retailer cycling channel, improving the profit for various interested parties along the supply chain and the total profit of the whole supply chain, and thus realizing Pareto improvement for the CLSC.

7. Conclusions

This study analyzes the decision-making on pricing, recycling, and CER in the recycling and remanufacturing CLSC, in the context of carbon peaking and carbon neutrality. Considering the fact that demand is affected by price and manufacturer’s level of CER effort, the study constructs a recycling and remanufacturing CLSC game model based on the situation that the manufacturer bears the CER alone and both the manufacturer and the retailer participate in recycling. Then, under centralized and decentralized decision-making models, the study works out the optimal pricing strategy and level of CER by manufacturer, and discusses the impact of recycling channel competition coefficient, consumers’ low carbon preference coefficient, and government CER subsidy coefficient on interested parties’ pricing decision-making, as well as the supply chain’s total recycling amount and profit. For double marginal effects under the decentralized decision-making model, the study proposes to coordinate with the cost-and-benefit-sharing contract and compares the profit of various interested parties and that of the supply chain before and after the coordination. The study shows that:

(1)

In the recycling and remanufacturing CLSC, the total supply chain recycling amount, total profit, and manufacturer’s CER effort level under the centralized decision-making model are higher than those under the decentralized decision-making model. The competition between recycling channels will reduce the total recycling amount, and the total profit of the supply chain is negatively related to the recycling channel competition coefficient.

(2)

The price set by the manufacturer and retailer under the centralized decision-making model is lower than that under the decentralized model, which encourages consumers’ purchasing intention and correspondingly expands market demand. In addition, the manufacturer’s CER effort level and his direct selling price, the retailer’s retail price, and the total profit of the supply chain are all positively related to consumers’ low carbon preference coefficient and government CER subsidy coefficient. Under the decentralized decision-making model, however, the retailer sets a higher price and the continuous increase in retail price will eventually damage the overall profit of the supply chain.

(3)

The cost-and-benefit-sharing contract can coordinate CLSC. The parameters of the contract in some circumstances can mitigate the competition between recycling channels and hence increase the total recycling amount of the supply chain, the profit of each involved party and that of the entire supply chain, as well as the CER effort level of the manufacturer, which consequently leads to the Pareto improvement of CLSC.

This paper not only enriches the theoretical research of the future sustainable development of the recycling and remanufacturing CLSC, but also provides some recommendations for the decision-making of each main body of CLSC. These are as follows. (1) Government departments should actively promote the significance of low carbon, provide CER subsidies to manufacturers with high CER inputs, encourage consumers to purchase remanufactured products and increase market demand, thereby reducing carbon emissions throughout CLSC. (2) As the leader of the CLSC, manufacturers should focus on the sustainable development of the entire CLSC, attach importance to the economic and environmental benefits brought by CER and remanufacturing, grasp the development trend in conjunction with policies, actively innovate green production technologies and remanufacturing technologies, and fairly balance the interests between themselves and the environment. (3) As retailers are closer to the consumer market, they should give manufacturers timely feedback on market demand information to prompt manufacturers to increase their investment in CER, innovate manufacturing technologies, and produce remanufactured products, which is of great practical importance for energy conservation, emissions reduction, environmental protection, and resource consumption reduction.

The limitation of this study is that it mainly concerns CER input by manufacturers in a CLSC, but does not consider other parties’ CER efforts. Future research will consider the pricing and CER decision-making model under multiplayer participation, which helps to analyze the pricing and CER in CLSC in a more comprehensive and systematic way.