Complexity Analysis of the Interaction between Government Carbon Quota Mechanism and Manufacturers’ Emission Reduction Strategies under Carbon Cap-and-Trade Mechanism

Abstract

Based on different carbon quota trading mechanisms, the price and emission reduction strategies of oligopoly manufacturers in the low-carbon market and the government carbon quota mechanism are considered. A dynamic game evolution model of the two oligopoly manufacturers with competitive relations is established. The stability of the equilibrium point of the game model, the price adjustment speed of the decision variable, the impact of carbon emission reduction investment, and the government carbon quota on the system are discussed. Through nonlinear dynamics research, it is found that the advantage of the grandfathering method is that it is conducive to maintaining market stability when the government’s carbon quota decision changes; the advantage of the benchmarking method is that when manufacturers formulate price adjustment strategies, the benchmarking method carbon quota mechanism has a stronger stability range for the market, the manufacturer’s profit price adjustment speed is positively correlated, and the government carbon quota decision and emission reduction investment are also positively correlated. Decision makers need to choose appropriate carbon quota mechanisms and manufacturers’ emission reduction strategies according to actual market changes to maintain supply chain stability.

1. Introduction

As global industrialization advances, the issue of global warming has become increasingly pronounced. Statistical data indicate that by 2023, the global average temperature will have risen by 1.42 °C compared to pre-industrial levels (average from 1850 to 1900) [1]. Addressing climate change is essential for achieving sustainable development worldwide. One widely accepted method to combat climate change involves reducing greenhouse gas emissions, specifically through lowering carbon emissions [2]. As the largest developing country, China is actively engaged in carbon emission reduction efforts. Internationally, China has participated in agreements such as the Paris Agreement, committing to sustainable development. In September 2020, China announced its goal to reach peak carbon emissions by 2030 and achieve carbon neutrality by 2060 [3]. So far, China has become the world’s largest carbon trading market. There are currently two common initial free quota methods: the grandfathering method and the benchmarking method [4]. The grandfathering method allocates free quotas based on a company’s historical emissions, while the benchmarking method bases free quotas on the emission levels of industry peers [5]. These initial free quota methods effectively incentivize enterprises to participate in emission reduction. Until now, the cumulative transaction volume of carbon emission quotas in the national carbon market has reached 230 million tons, with a transaction volume of CNY 10.48 billion. However, due to the unique characteristics of each method, specific market conditions and expected goals still need to be considered when using them.

With the gradual enhancement of environmental awareness, in addition to the government’s implementation of emission reduction policies, consumers are also increasingly inclined towards green consumption. Manufacturers, in order to obtain more benefits and government support, will, to some extent, cater to market demand and adopt emission reduction measures. For example, Apple has developed a green supply chain management system encompassing green procurement, production, packaging, and recycling. Gree Electric Appliances focuses on energy-saving and environmental protection in its green supply chain management, while BYD, an electric vehicle manufacturer, actively promotes green supply chain practices. In practice, market demand uncertainty, input cost fluctuations, and varying enterprise sizes necessitate optimal decision-making by supply chain participants to maximize benefits [6]. Rational decision-makers must, therefore, balance technological and cost inputs, competitive intensity, and pricing decisions [7]. However, due to information asymmetry and transmission delays, enterprises often have incomplete information about the market, introducing further uncertainties in decision-making [8]. Within the duopoly supply chain, manufacturers of varying strengths make different investment and pricing decisions for carbon emission reduction, and members at the same level exhibit competitive behaviors. Decisions in the duopoly supply chain must also consider consumer acceptance and market impacts, which exacerbate conflicts of interest among network members. Thus, to coordinate competitive relationships and maintain market stability, it is essential to analyze market equilibrium conditions, including pricing and profit optimization.

This paper investigates the impact of carbon trading prices, oligopolistic competition, and carbon quota constraints on market stability, emission reduction rates, pricing, and profit within a duopoly low-carbon supply chain. It aims to address the following questions:

• What is the relationship between decision adjustments by manufacturers and market equilibrium under limited rationality?
• How do manufacturers’ carbon emission reduction decisions, product pricing, and profits evolve under the carbon quota method?
• What is the relationship between government-imposed carbon quotas and manufacturers’ emission reduction decisions?

To explore these issues, this study considers the complexities of the low-carbon supply chain’s carbon trading mechanism and the interactions of emission reduction technology investments. It constructs the demand and profit functions of a duopoly supply chain and employs the Bertrand game to determine optimal product pricing and profit. Finally, the study establishes a system dynamic equation and uses numerical simulations to investigate the effects of oligopolistic competition, carbon trading prices, and carbon quota constraints on product pricing and profits.

The remainder of the article is organized as follows: Section 2 reviews the previous literature, analyzing the advantages and disadvantages before determining the innovation of this paper. Section 3 addresses the research questions by proposing hypotheses and establishing a dynamic game model for the supply chain. Optimal equilibrium decisions of supply chain members and the stability conditions of the system are obtained through reverse solution. Different parameter changes yield optimal solutions. Section 4 discusses the model’s establishment and system stability conditions under various operating modes, concluding that specific parameter considerations lead to a stable operating state for the supply chain. In Section 5, supply chain profits are compared, and a hybrid control strategy is applied to stabilize the dynamic system. Finally, Section 6 provides the article’s conclusion and suggestions.

2. Literature Review

The relevant literature encompasses three main areas: research on low-carbon supply chains and their emission reduction decision-making processes, studies on competitive markets with duopolies, and the application of chaos theory to management science issues.

2.1. Research on Low Carbon Supply Chains and Their Decision-Making on Emission Reduction

Extensive research has been conducted on low-carbon supply chains. Ma and Wang [9] examined competition within the clothing supply chain under a low-carbon economy and determined that centralized decision-making benefits supply chain members when the low-carbon investment coefficient is moderate, while decentralized decision-making is advantageous when the investment is either large or small. Fu et al. [10] identified key factors affecting low-carbon practices in the green construction supply chain. Yu et al. [11] found that appropriate carbon taxes incentivize manufacturers in both vertical and horizontal dual-chain systems to reduce emissions. Huang et al. [12] demonstrated that government intervention can enhance technological innovation and decrease the costs associated with low-carbon inputs in urban and rural markets. Lin et al. [13] investigated the impact of retailers’ social preferences on pricing and emission reduction decisions, concluding that higher social preferences among retailers favor carbon reduction and stabilize the supply chain. In carbon abatement investment decision-making, Jauhari et al. [14] explored a hybrid production context, finding that controlling costs and emissions while transitioning to greener production facilities can effectively reduce total supply chain emissions and increase product prices. Liu et al. [15] showed that knowledge sharing and carbon tax rates significantly affect carbon emission benefits and the selection of optimal emission reduction technologies. Kang et al. [16] highlighted the free-rider problem in emission reduction investment, which can deter proactive emission reduction efforts. Handayani et al. [17] concluded that the production level, carbon emission level, and emission threshold can have a significant influence on the generation of total carbon emissions.

These studies confirm that enterprises in different industries can reduce carbon emissions and increase profits through reasonable carbon reduction decisions, providing important insights for low-carbon supply chain management. However, when companies face different carbon reduction policies formulated by the government, manufacturers’ carbon reduction decisions need to be adjusted according to the different characteristics of the policies. Unfortunately, there has been relatively little consideration given to the impact of carbon policies on corporate decision-making and the adjustments that companies should make to address these effects. Therefore, the research has strong attributes. Additionally, carbon taxes have a significant impact on emission reduction benefits and technology choices, while the free-rider problem in emission reduction investment hinders active emission reduction efforts. Moreover, production levels and emission thresholds have a crucial impact on total emissions.

2.2. Research on the Competitive Market with Duopoly Enterprises

Research on competitive markets with duopoly enterprises explores the unique dynamics and strategic interactions in markets dominated by two major firms. This structure significantly deviates from monopolistic and perfectly competitive markets, with each firm’s actions directly impacting the other. Classic models, such as Cournot’s quantity competition and Bertrand’s price competition, provide theoretical foundations, while modern research incorporates factors like product differentiation, capacity constraints, and regulatory impacts. Understanding these dynamics is vital for policymakers to foster competition, prevent anti-competitive practices, and promote innovation. Thus, the study of duopolistic competition remains a critical and evolving field in economic research. Scholars have also focused on the market competition of duopoly enterprises. Jena et al. [18] studied cooperation and competition issues in a closed-loop supply chain. Bera [19] used the Cournot duopoly game method to study the sales quantity strategy of differentiated intelligent SSC. Santanu Sinha et al. [20] showed that duopoly competition can make consumers better off or worse off depending on the degree of product differentiation and the type of product, while coordination enhances overall supply-chain profitability. Ding et al. [21] constructed a competitive duopoly model with implicit collusion, revealing that such collusion leads to higher equilibrium prices than the Bertrand equilibrium. Huang et al. [22] studied price competition and cooperation in a two-tier supply chain, finding that profitability for duopoly retailers depends on their cooperation mode. Yan et al. [23] introduced competitive intensity factors in bi-oligopolistic markets, analyzing the effects on decentralized supply chain decision-making. Ding et al. [24] developed a Stackelberg game model with two competing retailers, showing benefits to both consumers and manufacturers from fierce retailer competition. Zhao et al. [25] created a duopoly Gounod game model under a hybrid carbon policy framework to provide theoretical insights for low-carbon supply chain decisions. Jin et al. [26] used the Gounod oligopoly model to determine Nash equilibrium in electricity supply and constructed a bank performance guarantee model based on profit functions in electricity sales.

These studies on duopoly markets examine the dynamics and strategic interactions between two dominant firms. Classic models like Cournot and Bertrand provide theoretical foundations, while modern research considers product differentiation, capacity constraints, and regulatory influences. Key findings highlight the impact of cooperation and competition in closed-loop supply chains, sales strategies for differentiated smart supply chains, and implicit collusion. However, research gaps exist in dynamic competition, consumer behavior, and emerging technologies. Addressing these gaps will enhance the theoretical and practical understanding of duopoly market competition.

2.3. Research on Chaos Theory in Management Issues

Initially, supply chain management focused on linear and deterministic models to optimize operations. However, as businesses encounter increasing complexity and unpredictability, researchers are beginning to recognize the limitations of these traditional approaches [27].

Research on chaos theory in management explores how small changes can lead to vastly different outcomes in complex systems. It helps understand market volatility, leadership dynamics, and decision-making, emphasizing flexibility, adaptability, and decentralized decision-making [28]. By recognizing chaotic elements, leaders can better navigate uncertainty, predict market behavior, and foster innovation, sustaining competitive advantage in turbulent environments. Chaos theory provides a framework for understanding the nonlinear dynamic and emergent behavior inherent in supply chain systems. Studies have shown chaotic behavior in all aspects of supply chain operations, including demand forecasting, inventory management, production scheduling, and distribution logistics [29]. These findings highlight the need for new approaches that can manage and control the chaos inherent in supply chains. Li et al. [30] applied chaos theory to study order decision-making and complexity in dual-channel supply chains. Sheng et al. [31] proposed a new research paradigm for supply chain resilience based on complex system thinking. Zhang et al. [32] investigated carbon emission reduction in fresh food supply chains using chaos theory. Ma and Wang [33] and Huang et al. [34] applied chaos theory to competition issues in clothing and shipping supply chains, respectively.

These studies highlight chaotic behavior in demand forecasting, operations management, and production scheduling, emphasizing the need for innovative approaches. Chaos theory provides a framework for understanding nonlinear dynamics in supply chains. Research applications include order decisions in dual-channel supply chains, supply chain resilience, carbon reduction, and competition in apparel and shipping. Despite its importance in managing supply chain complexity, there is limited focus on corporate carbon reduction decisions and government carbon policy implementation. This article uses chaos theory to address these issues.

2.4. Contribution Statements

The above literature provides us with insights into the importance and complexity of supply chain management in the context of duopoly competition. At the same time, it emphasizes the need for more seamless integration between carbon emission reduction investments in low-carbon supply chains and behavioral factors related to enterprises to improve carbon target achievement and operational efficiency within the supply chain.

The existing literature on low-carbon supply chain carbon reduction investment mainly focuses on behavioral factors related to enterprises, often neglecting the combination with government-led carbon policies. To address this gap, this study introduces government carbon quota constraints into the decision-making process of low-carbon supply chain networks, emphasizing the impact of these constraints on the behavior of supply chain members. Moreover, in the decision-making of duopoly enterprises, only static games are generally discussed, and dynamic games and their development trends, as well as corresponding countermeasures for these problems, are rarely considered. Our research uses chaos theory to consider the dynamic development between government carbon quotas and manufacturer emission reduction strategies, aiming to strengthen communication and interaction between the government and enterprises, assist various subjects in making more appropriate decisions at various stages, and promote the early realization of social carbon goals. Table 1 presents the differences between our studies and the previous literature.

The article introduces the contributions and topics of previous research, as shown in Table 1 below:

Table 1. Classifying based on article features.

Author	Double Oligarchy	Green Supply Chain	Carbon Quota	Static Game	Dynamic Game	Complex Dynamics
Fu et al. [10]		√	√	√
Huang et al. [12]		√	√		√
Handayani et al. [17]		√	√	√
Ma and Wang [9]		√	√		√	√
Jena S K et al. [18]	√			√
Subhamoy Bera [19]	√	√		√
Huang et al. [22]	√				√
Santanu Sinha [20]	√		√	√
Li et al. [30]						√
Brianzoni et al. [27]		√			√	√
Our work	√	√	√	√	√	√

Table 1. Classifying based on article features.

Author	Double Oligarchy	Green Supply Chain	Carbon Quota	Static Game	Dynamic Game	Complex Dynamics
Fu et al. [10]		√	√	√
Huang et al. [12]		√	√		√
Handayani et al. [17]		√	√	√
Ma and Wang [9]		√	√		√	√
Jena S K et al. [18]	√			√
Subhamoy Bera [19]	√	√		√
Huang et al. [22]	√				√
Santanu Sinha [20]	√		√	√
Li et al. [30]						√
Brianzoni et al. [27]		√			√	√
Our work	√	√	√	√	√	√

The contributions of this paper can be summarized as follows:

• Compared with Table 1, the key distinction of this paper lies in its proposal of a new paradigm that integrates supply chain management with carbon reduction strategies. It incorporates environmental sustainability into supply chain activities and streamlines processes to minimize environmental impact. The introduction of nonlinear dynamics is utilized to formulate dynamic equations, while chaos theory is employed to comprehend the nonlinearity of these equations. Chaos theory is then applied to analyze the intricate and unpredictable behaviors within supply chain operations, ultimately enhancing the stability and performance of enterprise operations.
• This study examines the evolving relationship between government carbon quotas and manufacturers’ emission reduction strategies. It aims to enhance communication and collaboration between the government and businesses, support entities in making informed decisions at different stages, bridge the divide between corporate carbon reduction choices and government policy implementation, and accelerate the achievement of societal carbon targets.
• In supply chain research, the complexity of decision-making in duopoly supply chains and market competition often leads to the failure of many traditional contracts. The existing literature ignores the interdependence between supply chain members, examines the interaction and strategy of two dominant enterprises in the market, and uses uncertain demand to make decisions to improve competitive positioning.
• In previous studies on corporate decision-making, only static games were discussed, and dynamic games and their development trends, as well as corresponding countermeasures to these problems, were rarely considered. The integration of government-led carbon policies was neglected, leading to an underestimation of the importance of the government in supply chain decision-making.

This study comprehensively examines factors that affect the stability of the supply chain system, such as the duopoly market, market competition, and government policies, with a special focus on the pricing decision-making behavior of supply chain manufacturers. Combined with government-led carbon policies, this study analyzes the dynamic interaction between the regulatory framework and the power of enterprises themselves. Through comprehensive long-term dynamic game analysis, this study aims to provide theoretical guidance for supply chain members to respond to and adopt effective strategies in a complex market environment, protect their own interests, and ensure that the supply chain operates smoothly while complying with carbon emission reduction requirements.

3. Model Description and Construction

3.1. Model Description

Assume that there are two competing secondary supply chains in the market, including a manufacturer (M) and a retailer (S), and the two supply chains make decisions under the constraints of carbon policy. The decision-makers of these two supply chains are both limited rational participants, limited by incomplete market information. They operate a kind of alternative homogeneous product and play a game between supply chains on price and carbon reduction decisions to maximize their own profits. Therefore, the Bertrand model game, which is a classic economic model that describes price competition, is selected. The model assumes that there are two or more manufacturers in the market, and they compete by setting product prices rather than setting production quantities. Among them, the manufacturer is a leader with sufficient resources. The green level, price, and marketing cost of the product are all determined by it. The retailer is only responsible for sales, and customers in the market have a strong desire to consume green and low-carbon products. The specific theoretical framework of the competition model is shown in Figure 1, and the meaning of the symbols involved is shown in

Table 2. Symbols and instructions.

Symbol	Illustrate
Superscript
$F$	The grandfathering method
$B$	The benchmarking method
Decision variables
$p_1$	Unit product price of supply chain 1
$p_2$	Unit product price of supply chain 2
${\tau }_1$	Manufacturer 1s’ $\left(M_1\right)$ carbon reduction levels
${\tau }_2$	Manufacturer 2s’ $\left(M_2\right)$ carbon reduction levels
$A$	The carbon emissions cap of the grandfathering method
$B$	The carbon emission cap of the benchmarking method
Model parameters
$a_1$	Potential market demand for supply chain 1
$a_2$	Potential market demand for supply chain 2
$\beta$	The degree of price competition between supply chains
$v_1$	Manufacturer 1s’ $\left(M_1\right)$ carbon reduction levels are attractive to consumers
$v_2$	Manufacturer 2s’ $\left(M_2\right)$ carbon reduction levels are attractive to consumers
${\mu }_1$	Consumer sensitivity to carbon allowances for product of $M_1$
${\mu }_2$	Consumer sensitivity to carbon allowances for product of $M_2$
$e$	Original emissions per unit product
$p_e$	Carbon trading price
$g_1$	Product price adjustment parameters for $M_1$
$g_2$	Product price adjustment parameters for $M_2$
$g_3$	Carbon reduction cost adjustment parameters for $M_1$
$g_4$	Carbon reduction cost adjustment parameters for $M_2$
$g_5$	Government adjusts market carbon quota parameters
Dependent Variable
$D_1$	Retailer’s Market Demand of Supply Chain 1
$D_2$	Retailer’s Market Demand of Supply Chain 2
${\pi }_{SC}^F$	Overall supply chain profit in the grandfathering method
${\pi }_1^F$	Manufacturer 1s’ $\left(M_1\right)$ profits in the grandfathering method
${\pi }_2^F$	Manufacturer 1s’ $(M_2)$ profits in the grandfathering method
${\pi }_{SC}^B$	Overall profit of the supply chain in the benchmarking method
${\pi }_1^B$	Manufacturer 1s’ $\left(M_1\right)$ profit in the benchmarking method
${\pi }_2^B$	Manufacturer 1s’ $(M_2)$ profit in the benchmarking method

.

(1)

Mixed carbon policies include carbon cap policies and carbon trading policies. Carbon cap policies include the grandfathering method and benchmarking method. The carbon emission cap in the grandfathering method is $A$, and the carbon emission cap in the benchmarking method is $B$, both of which are initial carbon quotas set by the government for the market free of charge. Under the constraints of carbon trading policy, carbon trading price $p_e$ is an exogenous variable. If the carbon emissions of manufacturers during the production process exceed the carbon emission cap set by the government, they need to purchase carbon emission rights in the carbon trading market to ensure the normal production activities of the manufacturers. After the production activities are completed, if the carbon emissions are lower than the carbon emission cap set by the government, the manufacturers can sell the remaining carbon emission rights through the carbon trading market to make a profit.

(2)

Under the constraints of the carbon cap-and-trade mechanism, in order to reduce carbon emissions, supply chain manufacturers under the duopoly competition invest in carbon reduction technologies and produce and operate the same products. The quality of the products produced is the same, and there is no difference; the marginal production cost is 0. Assume that each manufacturer acts simultaneously in setting product prices, that each manufacturer knows the other’s price-setting strategy, and that any one manufacturer can meet all market demand. Therefore, manufacturers in the supply chain mainly consider the investment cost of technology emission reduction, the sales revenue of carbon emission reduction quotas, and the product sales revenue.

(3)

The carbon emission reduction cost that manufacturers need to pay in the low-carbon production process is ${\displaystyle \frac{1}{2}}\phi {\tau }_i^2$, $\phi$ represents the carbon emission reduction cost coefficient, and its value is $0<\phi <1$, ${\tau }_i$ represent the manufacturer $i$’s carbon emission reduction rate.

(4)

Considering that the products of the two supply chains are homogeneous and substitutable, there is a market competition relationship, which is specifically manifested as price competition. The parameter $\beta \in \left(0,1\right)$ represents the market competition intensity between products. When $\beta$ tends to 1, it means that the market competition is more intense. Since consumers have green product consumption preferences, let $v_{i,i=1,2}$ be the degree of consumer sensitivity to the manufacturer’s carbon reduction technology inputs, and let ${\mu }_{i,i=1,2}$ be the degree of consumer sensitivity to carbon limits.

3.2. Model Construction

Based on the above assumptions, under the background of carbon cap-and-trade mechanism constraints, the manufacturer’s profit is composed of sales revenue, carbon trading revenue, and carbon reduction technology input costs. Therefore, the demand function and profit function of the grandfathering method and benchmarking method manufacturers can be expressed as follows.

Based on the above assumptions, the demand functions for the grandfathering method and benchmarking method manufacturers are as follows, respectively, shown as (1) and (2):

$$\{\begin{array}{c}{D}_1=a_1-p_1+\beta p_2+v_1{\tau }_1+{\mu }_{1}A\\ D_2=a_2-p_2+\beta p_1+v_2{\tau }_2+{\mu }_{2}A\end{array}$$

(1)

$$\{\begin{array}{c}{D}_1=a_1-p_1+\beta p_2+v_1{\tau }_1+{\mu }_{1}B\\ D_2=a_2-p_2+\beta p_1+v_2{\tau }_2+{\mu }_{2}B\end{array}$$

(2)

where, $p_{i,i=1,2}$ represents the retailer’s product pricing; $a_{i,i=1,2}$ refers to the initial market influence of a product, $a_1+a_2$, which is less than or equal to the total market size. This set of demand functions is an extension of the linear demand model, taking into account the impact of various factors on demand, such as commodity prices, substitute prices, emission reductions and limits, etc. Its theoretical basis is the consumer behavior theory and demand elasticity theory in microeconomics [35].

3.2.1. The Grandfathering Method

The profit function is shown in Formula (3):

$$\{\begin{array}{c}{\pi }_1^F=p_1{D}_1-\left(e\left(1-{\tau }_1\right)D_1-A\right)p_e-{\displaystyle \frac{1}{2}}\phi {\tau }_1^2-{\displaystyle \frac{1}{2}}\rho A^2\\ {\pi }_2^F=p_2{D}_2-(e\left(1-{\tau }_2\right)D_2-A)p_e-{\displaystyle \frac{1}{2}}\phi {\tau }_2^2-{\displaystyle \frac{1}{2}}\rho A^2\\ {\pi }_{SC}^F=p_1{D}_1+p_2{D}_2-e\left(\left(1-{\tau }_1\right)D_1+\left(1-{\tau }_2\right)D_2\right)-2A{p}_e-{\displaystyle \frac{1}{2}}\phi ({\tau }_1^2+{\tau }_2^2)-\rho A^2\end{array}$$

(3)

This profit function is inspired by previous literature [9,13,21]. By taking partial derivatives of manufacturer 1 and manufacturer 2 and the sum of their profits with respect to $p_1$, $p_2$, ${\tau }_1$, ${\tau }_2$ and $A$, where ${\displaystyle \frac{1}{2}}\rho A^2$ represents the carbon cap, we can effectively reflect the increasing cost as the carbon cap increases. This is because manufacturers must emit within this limit, and if they exceed the limit, they need to purchase additional emission permits.

We can obtain the marginal profit of the manufacturer as shown in Formula (4):

$$\{\begin{array}{c}{\displaystyle \frac{\partial {\pi }_1^F}{\partial p_1}}=a_1-2p_1+\beta p_2+{\mu }_{1}A+{\tau }_1{v}_1+e{p}_e\left(1-{\tau }_1\right)\\ {\displaystyle \frac{\partial {\pi }_2^F}{\partial p_2}}=a_2-2p_2+\beta p_1+{\mu }_{2}A+{\tau }_2{v}_2+e{p}_e\left(1-{\tau }_2\right)\\ {\displaystyle \frac{\partial {\pi }_1^F}{\partial {\tau }_1}}=v_1{p}_1-\phi {\tau }_1+e{p}_e\left(a_1-p_1+\beta p_2+{\mu }_{1}A-\left(1-2{\tau }_1\right)v_1\right)\\ {\displaystyle \frac{\partial {\pi }_2^F}{\partial {\tau }_2}}=v_2{p}_2-\phi {\tau }_2+e{p}_e\left(a_2-p_2+\beta p_1+{\mu }_{2}A-\left(1-2{\tau }_2\right)v_2\right)\\ {\displaystyle \frac{\partial {\pi }_{SC}^F}{\partial A}}={\mu }_1{p}_1+{\mu }_2{p}_2-2A\rho -2p_e-e{p}_e\left({\mu }_1\left(1-{\tau }_1\right)+{\mu }_2\left(1-{\tau }_2\right)\right)\end{array}$$

(4)

Bishci and Naimzada [36], Zhuang Jixiang et al. [37], and others studied the limited rationality model of oligopoly competition. In the real economy, the market may have incomplete information, and enterprises are not completely rational when making production decisions. A more reasonable assumption is that enterprises adjust production or prices according to their marginal profit situation. When profits increase, enterprises will increase production or prices accordingly. Dixit [38] called this production adjustment method the short-sighted behavior of enterprises.

Thus, the dynamic game model of five discrete prices and emission reduction strategies of the duopoly manufacturers in the grandfathering method is established as shown in system (5):

$$\{\begin{array}{c}{p}_1^F\left(t+1\right)=p_1^F\left(t\right)+g_1{p}_1^F\left(t\right){\displaystyle \frac{\partial {\pi }_1^F}{\partial p_1}}\\ p_2^F\left(t+1\right)=p_2^F\left(t\right)+g_2{p}_2^F\left(t\right){\displaystyle \frac{\partial {\pi }_2^F}{\partial p_2}}\\ {\tau }_1^F\left(t+1\right)={\tau }_1^F\left(t\right)+g_3{\tau }_1^F\left(t\right){\displaystyle \frac{\partial {\pi }_1^F}{\partial {\tau }_1}}\\ {\tau }_2^F\left(t+1\right)={\tau }_2^F\left(t\right)+g_4{\tau }_2^F\left(t\right){\displaystyle \frac{\partial {\pi }_2^F}{\partial {\tau }_2}}\\ A\left(t+1\right)=A\left(t\right)+g_{5}A\left(t\right){\displaystyle \frac{\partial {\pi }_{SC}^F}{\partial A}}\end{array}$$

(5)

where $g_i>0\left(i=1,2,3,4,5\right)$ represents the decision adjustment speed of the duopoly manufacturer.

3.2.2. The Benchmarking Method

The profit in the benchmarking method is shown in Formula (6):

$$\{\begin{array}{c}{\pi }_1^B=p_1{D}_1-\left(e\left(1-{\tau }_1\right)-B\right)D_1{p}_e-{\displaystyle \frac{1}{2}}\phi {\tau }_1^2-{\displaystyle \frac{1}{2}}\rho B^2\\ {\pi }_2^B=p_2{D}_2-\left(e\left(1-{\tau }_2\right)-B\right)D_2{p}_e-{\displaystyle \frac{1}{2}}\phi {\tau }_2^2-{\displaystyle \frac{1}{2}}\rho B^2\\ {\pi }_{SC}^B=p_1{D}_1+p_2{D}_2-\left(\left(e\left(1-{\tau }_1\right)-B\right)D_1+\left(e\left(1-{\tau }_2\right)-B\right)D_2\right)p_e-{\displaystyle \frac{1}{2}}\phi ({\tau }_1^2+{\tau }_2^2)-\rho B^2\end{array}$$

(6)

Similar to (3), taking partial derivatives of the profits of manufacturer 1 and manufacturer 2 and the sum of their profits with respect to $p_1$, $p_2$, ${\tau }_1$, ${\tau }_2$ and $B$, where ${\displaystyle \frac{1}{2}}\rho B^2$ represents the carbon cap, can effectively reflect the increasing cost as the carbon cap increases. This is because manufacturers must emit within this limit, and if they exceed the limit, they need to purchase additional emission permits.

We can obtain the marginal profit of the manufacturer as shown in Formula (7):

$$\{\begin{array}{c}{\displaystyle \frac{\partial {\pi }_1^B}{\partial p_1}}=a_1-2p_1+\beta p_2+{\mu }_{1}B+{\tau }_1{v}_1+p_e\left(e\right(1-{\tau }_1)-B)\\ {\displaystyle \frac{\partial {\pi }_2^B}{\partial p_2}}=a_2-2p_2+\beta p_1+{\mu }_{2}b+{\tau }_2{v}_2+p_e\left(e\right(1-{\tau }_2)-B)\\ {\displaystyle \frac{\partial {\pi }_1^B}{\partial {\tau }_1}}=v_1{p}_1-\phi {\tau }_1+e{p}_e\left(a_1-p_1+\beta p_2+{\mu }_{1}B+{\tau }_1{v}_1)-p_e\left(\right(1-{\tau }_1)-B)v_1\right)\\ {\displaystyle \frac{\partial {\pi }_2^B}{\partial {\tau }_2}}=v_2{p}_2-\phi {\tau }_2+e{p}_e\left(a_2-p_2+\beta p_1+{\mu }_{2}B+{\tau }_2{v}_2)-p_e\left(\right(1-{\tau }_2)-B)v_2\right)\\ {\displaystyle \frac{\partial {\pi }_{SC}^B}{\partial B}}={\mu }_1{p}_1+{\mu }_2{p}_2-2B\rho -p_e\left({\mu }_1\left(e\left(1-{\tau }_1\right)-B\right)+{\mu }_2\left(e\left(1-{\tau }_2\right)-B\right)\right)\end{array}$$

(7)

The duopoly manufacturers can only obtain partial market information when making decisions and are in a state of limited rationality. Therefore, the manufacturer’s next decision will be adjusted according to the marginal utility of this period. When the marginal utility of this period is greater than zero, the manufacturer will increase the speed of decision adjustment in the next period; when the marginal utility of this period is less than zero, the manufacturer will increase the speed of decision adjustment in the next period.

Thus, the dynamic game model of five discrete prices and emission reduction strategies of the duopoly manufacturers in the grandfathering method is established as shown in system (8):

$$\{\begin{array}{c}{p}_1^B\left(t+1\right)=p_1^B\left(t\right)+g_1{p}_1^B\left(t\right){\displaystyle \frac{\partial {\pi }_1^B}{\partial p_1}}\\ p_2^B\left(t+1\right)=p_2^B\left(t\right)+g_2{p}_2^B\left(t\right){\displaystyle \frac{\partial {\pi }_2^B}{\partial p_2}}\\ {\tau }_1^B\left(t+1\right)={\tau }_1^B\left(t\right)+g_3{\tau }_1^B\left(t\right){\displaystyle \frac{\partial {\pi }_1^B}{\partial {\tau }_1}}\\ {\tau }_2^B\left(t+1\right)={\tau }_2^B\left(t\right)+g_4{\tau }_2^B\left(t\right){\displaystyle \frac{\partial {\pi }_2^B}{\partial {\tau }_2}}\\ B\left(t+1\right)=B\left(t\right)+g_{5}B\left(t\right){\displaystyle \frac{\partial {\pi }_{SC}^B}{\partial B}}\end{array}$$

(8)

Similar to (5), $g_i>0\left(i=1,2,3,4,5\right)$ represents the decision adjustment speed of the duopoly manufacturer.

4. Analysis of Balance State and Stability Analysis

4.1. Equilibrium Point

In the Bertrand model, two firms set product prices simultaneously to compete for market share. It is assumed that each firm knows the other’s price-setting strategy and that any firm can meet all market demand. Each firm maximizes profit by taking the derivative of price and setting it to zero, which means that marginal profit is zero, and finds the best pricing strategy. Equate Equations (4) and (7) to zero. The result of this process is the Nash equilibrium point, at which no firm can increase profit by changing price, and the price eventually drops to the marginal cost level, which represents the stable state. In system (5) and system (8), $p_i\left(t+1\right)=p_i\left(t\right)$, ${\tau }_i\left(t+1\right)={\tau }_i\left(t\right)$, $A\left(t+1\right)=A\left(t\right)$, $B\left(t+1\right)=B\left(t\right), \left(i=1,2,3\right)$, respectively, and we obtain 32 equilibrium points for each system $E_j\left(j=1,2,\cdots ,32\right)$, where the equilibrium points $E_1~E_{31}$ are boundary points. As mentioned in the Bertrand equilibrium analysis, when p1 < p2, in this scenario, there is no zero carbon reduction in the market, which does not align with the definition of a low-carbon supply chain. Moreover, the government’s carbon limit is considered, eliminating the possibility of a zero market situation. Consequently, the boundary points in the results lack practical significance. In the grandfathering method $E_{32}^F=\left(p_{1 }^{F^{*}},p_{2 }^{F^{*}},{\tau }_1^{F^{*}},{\tau }_2^{F^{*}},A^{*}\right)$ and the benchmarking method $E_{32}^B=\left(p_{1 }^{B^{*}},p_{2 }^{B^{*}},{\tau }_{1 }^{B^{*}},{\tau }_{2 }^{B^{*}},B^{*}\right)$, they are all equilibrium points; see the Appendix A and Appendix B for the proof process.

4.2. Qualitative Analysis of Equilibrium Stability

In order to vividly demonstrate the stability characteristics of the system, the system’s stability region is determined through numerical simulation, considering the constraints of the Nash equilibrium points $E_{32}^F$ and $E_{32}^B$, known as the Jury criterion. The system parameter values are obtained according to previous research as follows: $a_1=15$, $a_2=10$, $\beta =0.2$, $e=0.3$, $p_e=0.3$, $\phi =0.6$, $\rho =0.7$, ${\mu }_1=0.3$, ${\mu }_2=0.3$, $v_1=0.2$, $v_2=0.2$. According to the system stability conditions, the three-dimensional stability domains of the grandfathering method and the benchmarking method are simulated, respectively, as shown in Figure 2.

Figure 2 shows the three-dimensional stability domain of the system under the grandfathering method and the benchmarking method. When the decision adjustment speeds of manufacturer 1 and manufacturer 2 are both within the stability domain, after a finite number of decision games in the retail industry, the system will reach a stable state at the equilibrium point.

It can be seen from Figure 2a,b that as $g_3$ increases, the stability domain of the system increases; this indicates that the greater the fluctuation of manufacturer 1’s emission reduction decision, the larger the stable range of manufacturer 1 and manufacturer 2’s price decisions, and the fluctuation of emission reduction decisions is beneficial to manufacturer price decisions and efficient market operation; under the same conditions, the stability domain of the grandfathering method is larger than that of the benchmarking method. It can be concluded that as it increases, the grandfathering method is more beneficial to the operation of mall files and the profitability of the $g_3$ supply chain members than the benchmarking method.

5. Numerical Simulation

5.1. Impact of Dynamic Adjustment Parameters on System Stability

Through the analysis of the system stability domain, it can be seen that the adjustment speed of decision-making has an important impact on the stability of the system. The following takes $g_1$ and $g_5$ as examples to analyze the impact of the adjustment speed of decision-making on the stability of the system.

5.1.1. Impact of Product Price Adjustment Parameters for $M_1$ Changes in the System

Figure 3 shows the bifurcation diagram of the system (5) and (8) with $g_1$ (product price adjustment parameters for $M_1$) fixed other decision variables’ adjustment parameters, $g_2=0.03$, $g_3=0.042$, $g_4=0.042$, $g_5=0.04$, respectively, in the grandfathering method and the benchmarking method, in which the y-axis represents the impact of changes in $g_1$ on the decision variables ($p_1,p_2,{\tau }_1,{\tau }_2,A,B$) in the two systems. In the grandfathering method, when $g_1=0.0647$, the system bifurcates and begins to enter the double-period bifurcation state. As $g_1$ increases, the system enters the chaotic state from the double-period state. In the benchmarking method, when $g_1=0.0684$, the system enters the double-period bifurcation state and then enters the chaotic state from the double-period state. Separately at steady state, ${\tau }_1=4.5$ and ${\tau }_2=4.3$ in the grandfathering method, and ${\tau }_1=14.3$ and ${\tau }_2=14.1$ in the benchmarking method.

From the above, we can see that the price adjustment speed of $M_1$ increases, the market is prone to instability and chaos, and the grandfathering carbon quota mechanism falls into chaos earlier than the benchmarking carbon quota mechanism. In the market stability stage, the equilibrium solutions of the product prices of the two oligopoly manufacturers are the same in the grandfathering method and the benchmarking method, but the equilibrium solutions of emission reduction input and carbon quota are different. The equilibrium solution of emission reduction input and carbon quota in the benchmarking method is larger. This shows that the higher the carbon quota set by the government, the more it can encourage manufacturers to reduce emissions.

In discrete dynamic systems, the chaotic attractor is an indivisible bounded point set composed of an infinite number of unstable points, which is an important feature of the system that dissipates power. Figure 4 shows the chaotic attractor of the system for other decision variable adjustment parameters: $g_1=0.03$, $g_2=0.03$, and $g_5=0.04$. Initial value sensitivity is another important feature of chaos. In Figure 5, when keeping the price $p_1$ and $p_2$ unchanged, the initial value of $A$ or $B$ (carbon cap) changes by 0.001; after this game, the price trajectory fluctuates violently, and the trend is unpredictable. That is, a slight change in the initial value of the chaotic system will cause a dramatic change and evolution of the system. When the market is in a chaotic state, market competitors will not be able to predict the changing trend of the market, which is not conducive to market participants making long-term decisions.

5.1.2. Impact of Carbon Limit Decision Variable Adjustment Parameters Changes on the System

Figure 5 shows the bifurcation diagram of the system with $g_5$ (carbon limit decision variable adjustment parameters) fixed other decision variables adjustment parameters, $g_1=0.03$, $g_2=0.03$, $g_3=0.042$, $g_4=0.042$, respectively, in the grandfathering method and the benchmarking method, in which the y-axis represents the impact of changes in $g_5$ on the decision variables ($p_1,p_2,{\tau }_1,{\tau }_2,A,B$) in the two systems. In the grandfathering method, when $g_5=0.2534$, the system bifurcates and begins to enter the double-period bifurcation state. As $g_5$ increases, the system enters the chaotic state from the double-period state. In the benchmarking method, when $g_5=0.1198$, the system enters the double-period bifurcation state and then enters the chaotic state from the double-period state. Separately at steady state, ${\tau }_1=5$ and ${\tau }_2=4.6$ in the grandfathering method and ${\tau }_1=15$ and ${\tau }_2=14.9$ in the benchmarking method.

As can be seen from the above, compared with Figure 3, as the government’s carbon quota adjustment speed increases, the benchmarking carbon quota mechanism falls into chaos earlier than the grandfathering carbon quota mechanism. This shows that the government should take a prudent attitude when making decisions on benchmarking carbon quotas compared to the grandfathering carbon quota mechanism; otherwise, it will disrupt the manufacturer’s decision and cause the market to fall into chaos. The impact of the carbon quota adjustment speed on the manufacturer’s emission reduction technology is greater than the impact on product prices. Therefore, the effect on $g_3$, $g_4$ (decision variable adjustment parameters) with the change in $g_5$ is further explored in Figure 6. As can be seen from Figure 6, the stabilization range of $g_3$, $g_4$ is gradually expanding with the increase in $g_5$.

Conclusion 1: When the product price adjustment range is too large, the system is prone to instability and chaos. The grandfathering carbon quota mechanism falls into chaos earlier than the benchmarking carbon quota mechanism, and the benchmarking market regulation ability is stronger. When the government carbon limit adjustment range is too large, the benchmarking carbon quota mechanism falls into chaos earlier than the grandfathering carbon quota mechanism, and the grandfathering market regulation ability is stronger.

5.2. Effect of Adjustment Parameter Changes on Profits

Supply chain members mostly use profit as a business goal to measure corporate performance. However, corporate profit revenue is often inseparable from the market demand and product prices of its products. Therefore, this paper mainly analyzes the impact of changes in demand and price adjustment decision parameters on retailers’ profits.

Figure 7a,b show the bifurcation of the change in demand for manufacturer 1 and manufacturer 2 as $g_1$ increases in the grandfathered and baseline methods, respectively. In the grandfathering method, when $g_1=0.0647$, the demand of manufacturer 1 and manufacturer 2 enters a multiplicative period bifurcation and enters a chaotic state as $g_1$ increases. In the benchmarking method, when $g_1=0.0647$, the demand of manufacturer 1 and manufacturer 2 enters a multiplicative period bifurcation and enters a chaotic state as $g_1$ increases. As shown in Figure 7, in the two carbon quota mechanisms, the market demand of manufacturer 1 is greater than that of manufacturer 2 in the stable stage of market demand; in the stable stage, the demand of manufacturers in the benchmarking carbon quota mechanism is significantly higher than that in the grandfathering method. This shows that the size of the market demand share is unrelated to the carbon quota mechanism, and the size of the market demand is related to the carbon quota mechanism.

Figure 8, respectively, shows the impact on average profits of price adjustment speeds $g_1$ and $g_2$ (decision variable adjustment parameters) for the two oligopolistic manufacturers under the grandfathering method and benchmarking method, where the z-axis represents profit $\pi$. The top half of both Figure 8a,b shows the profit of manufacturer 1, and the bottom half shows the profit of manufacturer 2; compared with (a), the change in profit under the benchmarking method in (b) is smoother and less volatile. From Figure 8c,d, it can be seen more clearly that the sum of oligopoly manufacturers’ profits in the grandfathered approach is more volatile as $g_1$ and $g_2$ (decision variable adjustment parameters) increase, and the sum of oligopoly manufacturers’ profits in the benchmarked approach is less volatile as $g_1$ and $g_2$ (decision variable adjustment parameters) increase, which suggests that the benchmarked approach may be more stable in dealing with oligopoly markets. The profit trend obtained by manufacturers is consistent with the trend of market demand changes in Figure 7, and manufacturers can obtain greater profits under the benchmarking method.

Conclusion 2: When the system is in a stable state, the duopoly manufacturers can obtain higher profits in the benchmarking carbon quota mechanism; when the system is in a chaotic state, the profit fluctuations of the duopoly manufacturers in the grandfathering carbon quota mechanism are greater. At this time, in order to stabilize the system, manufacturers should control the price adjustment speed within a reasonable range.

5.3. Impact of Parameter Changes on the System

In reality, in addition to the price adjustment strategies that manufacturers can implement in the low-carbon market, changes in factors such as $\beta$ (the degree of competition with other oligopolistic manufacturers), $\nu$ (the impact of their own $v$ carbon reduction investment levels on consumers), and $\mu$ (the impact of the government’s carbon quota mechanism on manufacturers) will also have an impact on the system. This section focuses on two factors, $\beta$ and $\mu$.

5.3.1. Impact of $\beta$ Changes on the System

Figure 9a,b, respectively, show the bifurcation of the system with increasing $\beta$ for manufacturer 1 and manufacturer 2 in the grandfathering and benchmarking method. The y-axis represents the impact of changes in $\beta$ on the decision variables ($p_1,p_2,{\tau }_1,{\tau }_2,A,B$) in the two systems. In the grandfathering method, when $\beta =1.016$, the demand of manufacturer 1 and manufacturer 2 enters a two-fold periodic bifurcation and enters a chaotic state as $\beta$ increases; similarly, in the benchmarking method, when $\beta =1.02$, the demand of manufacturer 1 and manufacturer 2 enters a times-two periodic bifurcation and enters a chaotic state as $\beta$ increases. As shown in Figure 9, the competition among oligopoly manufacturers is maintained within a certain range, and the market is in a stable state; excessive competition causes the two manufacturers to fall into the same chaos. Oligopoly competition among manufacturers will not change due to different carbon quota mechanisms.

Figure 10a,b show the effect of $\beta$ (the coefficient of oligopolistic competition) on $\pi$ (profits) under the grandfathering and benchmarking approaches, respectively. The stabilized, bifurcated, and chaotic states of manufacturer 1 and manufacturer 2 profits in Figure 10 are consistent with the description in Figure 9. As shown in Figure 10, when the competition among oligopoly manufacturers is maintained within a certain range, the profits of manufacturer 1 and manufacturer 2 are in a stable rising state; when the competition is excessive, the two manufacturers fall into the same chaos. Within the stable range of the system, the rate of increase in the profits of manufacturer 1 and manufacturer 2 under the benchmarking carbon quota mechanism is greater than the rate of increase in the profits of the two manufacturers under the grandfathering method. It can be seen that in oligopoly competition within the stable range of the system, the benchmarking carbon quota mechanism is more conducive to enterprises obtaining higher profits.

5.3.2. Impact of ${\mu }_1$, ${\mu }_2$ Changes on the System

Figure 11a,b show the bifurcation diagrams of the system with ${\mu }_1$ increasing manufacturer 1 and manufacturer 2 in the grandfathering method and the benchmarking method, respectively, in which the y-axis represents the impact of changes in ${\mu }_1$ on the decision variables ($p_1,p_2,{\tau }_1,{\tau }_2,A,B$) in the two systems. In the grandfather method, the system enters the chaotic state later when the first bifurcation occurs at ${\mu }_1=0.405$; in the benchmark method, the system enters the chaotic state earlier when the first bifurcation point is at ${\mu }_1=0.3602$.

It can be seen that the grandfathering method government carbon quota mechanism has a higher risk aversion level for manufacturers; the system under the benchmarking method is more sensitive to ${\mu }_1$ changes, and even small fluctuations may cause the system to quickly enter an unpredictable and difficult-to-control chaotic state, increasing the difficulty and complexity of decision-making.

Figure 12a,b also corroborate this view. Unlike Figure 11, the degree of oscillation of the government carbon quota amount is significantly larger than the span of the chaotic region of ${\mu }_1$ as ${\mu }_2$ increases to enter the chaotic stage in Figure 12, which also proves that manufacturer 2 is more sensitive to the change in the government carbon quota amount.

Conclusion 3: As the competition among oligopolistic manufacturers increases, the system goes through a doubling period to a chaotic state. In oligopolistic competition, the benchmarking method has a stronger market regulation capability and brings equal profits to manufacturers than the grandfathering method. As the influence of government carbon quotas on manufacturers increases, the system becomes more sensitive to the benchmarking method, which increases the difficulty and complexity of decision-making.

6. Conclusions

This paper presents a decision-making game model for duopoly manufacturers in a competitive relationship. It examines the effects of decision variables, price adjustments, and carbon reduction investments by low-carbon manufacturers in the green consumer market under the government’s carbon quota mechanism. The study particularly focuses on the influence of grandfathering and benchmarking carbon quota systems on market stability and manufacturer behavior.

The key findings are as follows:

(1) System instability and chaos: Large adjustments in product prices lead to instability and chaos. Grandfathering often falls into chaos earlier than benchmarking, indicating that the market regulation ability under benchmarking is stronger. On the contrary, excessive adjustments in government carbon quotas cause benchmarking to fall into chaos earlier than grandfathering, indicating that grandfathering has stronger regulatory ability. (2) Profitability under stability and chaos: When the system is stable, the profits of duopoly manufacturers under the benchmarking carbon quota mechanism are higher. In a chaotic state, the profit fluctuations under the grandfathering system are greater. To stabilize the system, manufacturers should moderate the speed of price adjustments. (3) The impact of competition and government quotas: As oligopoly competition intensifies, the system transitions from the doubling period to the chaos period. Compared with grandfather clauses, benchmarking provides manufacturers with stronger market supervision and higher profits. The increase in the impact of government carbon quotas makes the system more sensitive under the benchmark and complicates decision-making.

Suggestions for improvement:

(1) Strengthening price adjustment strategies: Manufacturers should avoid excessive price adjustments to prevent system instability. Detailed market analysis and incremental price adjustments are recommended to maintain stability. Implementing predictive analysis and market simulation can help predict the impact of price changes and optimize adjustments. (2) Carbon quota mechanism selection: Manufacturers should comprehensively evaluate the stability and market supervision capabilities of carbon quota mechanisms. For short-term market stability, benchmarking may be preferable, while grandfather clauses may provide long-term flexibility. A hybrid approach can also be considered to balance short-term stability and long-term adaptability. (3) Competitiveness and adaptability: In the context of fierce market competition and strict government carbon policies, manufacturers must enhance competitiveness and adaptability. It is essential to develop agile business strategies, flexible production plans, and dynamic carbon reduction plans. Investing in advanced technologies and continuous improvement practices can further enhance resilience to market and policy fluctuations. (4) Risk management and policy sensitivity: Due to the system’s high sensitivity to benchmarking methods, manufacturers should conduct comprehensive policy impact analysis, multi-scenario evaluation, and simulations to mitigate decision risks. Engaging in active dialogue with policymakers and participating in industry forums can offer valuable insights into regulatory trends, allowing manufacturers to foresee and adapt to policy changes. (5) Holistic management approach: Manufacturers should adopt a holistic approach to supply chain management, combining environmental sustainability with operational efficiency. This includes leveraging digital tools for real-time monitoring and decision support, facilitating collaboration with supply chain partners, and aligning corporate strategy with broader sustainability goals. Continuous learning and adaptation are key to thriving in a dynamically evolving market environment.

Conclusions and recommendations are shown Figure 13, and the connections found therein are such that by focusing on these key connections, manufacturers can better align their strategies with the conclusions drawn from the research.

There are still shortcomings in this study, such as adding a recycling link to the supply chain and considering factors such as the investment in recycling products to construct higher-order dynamic equations. This will make the research on carbon reduction in green supply chains more complete and provide more effective solutions and methods for practical production and management.