Research on Green Supply Chain Financing Decisions Based on Inter-Chain Competition and Implicit Equity Consideration

Abstract

This paper addresses green supply chain financing decisions based on inter-chain competition and implicit equity considerations. First, considering the different situations of green supply chain members when facing green investment costs, it separately establishes green supply chain revenue Cournot competition models under the three models of no-financing, manufacturer financing, and retailer financing, as well as two implicit equity holding models. Second, it calculates the optimal order quantity and product greenness of the green supply chain under different scenarios. Finally, mathematical derivation and numerical simulation explore the effects of consumers’ green preference, capital opportunity costs, and other factors on product greenness, optimal order quantity, and supply chain members’ revenue. The results of this paper show that regardless of the financing model or who owns the implicit equity, there are optimal order quantities and optimal product greenness in the green supply chain. Furthermore, under inter-chain competition and implicit equity considerations, the manufacturer in the green supply chain always has financing motivations, but it prefers to be financed by the well-funded retailer. However, the retailer will consider financing only when consumer preference for green products in the green supply chain is below a specific threshold. Moreover, compared to the manufacturer financing model, the product greenness under the retailer financing model is high. It performs better in terms of environmental protection, which contributes to improving the entire supply chain and environmental performance.

1. Introduction

With the deepening of global economic integration, balancing economic growth with environmental protection has become an urgent challenge (Chen et al., 2023, [1]). To address resource and environmental constraints and promote industrial green transformation, China is actively constructing green supply chains centered on green products, leveraging technological innovation and industrial upgrading to drive green and low-carbon development [2]. The State Council has repeatedly emphasized the importance of establishing a sound economic system for green, low-carbon, and circular development, regarding it as a critical pathway toward sustainable development. Meanwhile, growing consumer environmental awareness has created broad market prospects for green supply chains.

However, the high costs associated with green technologies pose significant barriers to corporate green transformation. Many manufacturing enterprises struggle to undertake green innovation due to financial constraints, leading to pronounced financing challenges. Existing studies often assume that manufacturers independently bear green investments [3], largely overlooking the financing difficulties faced by capital-constrained manufacturers. In practice, well-funded retailers such as Walmart and Costco may directly invest in manufacturers’ green production initiatives, thereby sharing decision-making power over greenness. This reality underscores the importance of selecting appropriate financing subjects within green supply chains—a topic that has not been systematically explored in the literature.

In the context of economic globalization, the competition among enterprises has evolved from a simple competition of products and services to a comprehensive competition among supply chains [4]. This change in the competitive landscape will have a far-reaching impact on the development and construction of a green supply chain. The competition between supply chains not only promotes the enterprises’ investment in the green technology field but also prompts them to adopt more active and innovative strategies in green supply chain management [5]. Furthermore, frequent transactions and stable demand relationships among supply chain members contribute to potential value between them, known as implicit equity [6]. As an informal equity relationship among supply chain members, it is not specifically manifested in financial statements but has a non-negligible role in the financing and production decisions of green supply chains. Nevertheless, few studies have investigated how inter-chain competition and implicit equity jointly shape the selection of financing modes in green supply chains.

Against this background, the options between the retailer financing model and the manufacturer financing model become a crucial decision point. In the decision-making process, many retailers and manufacturers conduct an in-depth assessment of their willingness to bear the costs of green investment based on their interests. For example, Walmart has proposed the Billion Metric Tons Emission Reduction Project, which aims to reduce greenhouse gas emissions by one billion metric tons by 2030, and through the Sustainable Supply Chain Financing Initiative, it supports manufacturers in implementing green production projects. Apple has also launched a USD 4.7 billion green bond program, which raises funds through a bond issue specifically to support sustainability projects in its supply chain. Therefore, there is a need to determine which supply chain member (retailer or manufacturer) shows stronger motivation to adopt green technologies and which financing model can more efficiently contribute to the continuous progress and development of green supply chains. Especially crucial is the impact of intense competition among supply chains and implicit equity on financing decisions for green supply chains. However, few studies have systematically analyzed which financing models are more effective at stimulating green technological progress and enhancing overall supply chain sustainability when competing scenarios interact with implicit equity. In this regard, this paper argues that the following issues need to be clarified.

In the green supply chain, what is the mechanism of influence between parameters such as product order quantity, product greenness, consumer green preference, and financial opportunity cost when considering inter-chain competition and implicit equity?

Considering inter-chain competition and implicit equity, what are the equilibrium financing models chosen by decision-makers (manufacturers and retailers) in two supply chains under different financing modes?

This paper consists of seven parts. Section 1 is the research background. Section 2 is the literature review and problem formulation. Section 3 is the model assumptions and definition of variable symbols. Section 4 constructs a green supply chain decision-making model based on inter-chain competition and implicit equity considerations. Section 5 is the analysis of green supply chain financing decision-making based on inter-chain competition and implicit equity considerations. Section 6 is the numerical simulation study, and Section 7 is the research conclusion.

2. Literature Review and Research Issues

2.1. Decision Coordination and Financing Decisions in the Green Supply Chain

(i)

Coordination of decision-making in the green supply chain

Encouraging manufacturers to produce more environmentally friendly products through improved environmental performance is a crucial step in supporting sustainable development. Addressing coordination issues in decision-making within green supply chains is particularly important to facilitating this process. Taleizadeh et al. [7] studied a two-echelon supply chain and found that cooperative models yield higher profits than non-cooperative ones, with wholesale price contracts achieving the highest supply chain profits and coordinating the entire chain. Mohsin et al. [8] developed a differential game model for green supply chains and demonstrated that centralized decision-making enhances the level of green technology and total channel profit, while a dynamic wholesale price mechanism coordinates the behavior of manufacturers and retailers. Yu et al. [9] proposed a joint contract to coordinate supply chain operations, which not only reduces prices in both direct and traditional channels but also improves product greenness and overall demand. Han et al. [10] established a dual-channel green supply chain model under the “dual-carbon” goal and concluded that a two-part pricing contract serves as an effective coordination mechanism to achieve a win–win outcome among members. Zhang and Qu [11] emphasized that an efficient coordination mechanism can enhance supply chain performance, thereby generating greater benefits for all members.

(ii)

Financing decisions in the green supply chain

Green manufacturers often face a dilemma of capital shortages due to the substantial green investment costs incurred when producing green products, but little research has examined financing decision-making issues in green supply chains under funding constraints. Yang and Duan [12] found that when manufacturers in a green supply chain lack funds, retailers should adopt the financing strategy of prepayment and motivate manufacturers to make green technological innovations to improve product greenness and expand market competitiveness. Dressler and Mugerman [13] looked at how investors and institutions respond to environmental, social, and governance (ESG) factors: institutional investors’ behavior regarding environmental proposals and corporate governance can inform the broader significance of green financing decisions. Additionally, Li et al. [14] found that even the transparency of supply chains (e.g., how much firms disclose about their supply chain practices) can influence institutional investor preferences. Reza-Gharehbag et al. [15] showed that government intervention policies have a positive impact on the development of new green products in risk-averse and capital-constrained supply chains. Ji et al. [16] found that bank green credit financing plays an essential role in the low-carbon transition for capital-constrained suppliers, which can stimulate the enthusiasm of suppliers for carbon emission reduction, and that the hybrid credit model helps to alleviate financing pressure. Wu et al. [17] studied the impact of blockchain technology on manufacturers’ financing strategies and found that when the investment efficiency of blockchain technology is high, the commercial credit financing strategy is the optimal choice.

2.2. Competitive Relationships Between Supply Chains and Their Effects

(i)

Study of competitive relationships among supply chains

McGuire and Staelin [18] (1983) first explored the competitive relationship between supply chains. In recent years, the competitive relationship between supply chains has received more attention from scholars. Fang and Shou [19] constructed a Cournot competition model with supply uncertainty between two supply chains and found that, in the process of supply chain competition, there are many potential risks in addition to the uncertainty of demand and supply. Jamali and Rasti-Barzoki [20] explored the pricing of non-green and green products, as well as the greenness decision problem, in the context of two dual-channel supply chains competing with each other and found that products in scenarios with centralized decision-making are greener. Feng and Liu [21] comprehensively compared the game models of non-cooperative and cooperative advertising from the perspective of a symmetric competitive supply chain and found that the price of goods is jointly affected by competition and the long-term utility of advertising. Du and Zhang [22] studied the financing strategies of competitive supply chains under financial constraints and concluded that, except for the no-financing model, all other financing models diminish the increase in the expected profits of the members of the competitive supply chain when the intensity of inter-chain competition is higher.

(ii)

Study on the impact of competition among supply chains

In the process of supply chain competition, companies must analyze their competitors’ strategies and continuously improve their management approaches to enhance their competitiveness and market position. Liu and Ji [23] studied an inter-chain competition model involving manufacturers and exclusive retailers, analyzing the impact of such competition and consumers’ green preferences on product selection and revenue. Sarkar and Bhadouriya [24] examined a two-tier supply chain with a single retailer and multiple manufacturers, and found that increased competition among manufacturers leads to higher product green quality. Shang et al. [25] demonstrated that policy facilitation is crucial for shifting firm-level competition to inter-supply chain competition, particularly when price competition is moderate or green-level competition is weak, and that collaboration among manufacturers can effectively increase retailers’ profits. Yang et al. [26] showed that manufacturers with higher technological capabilities exhibit stronger incentives to invest in environmentally friendly technologies. Wang et al. [27] found that inter-chain competition not only enhances product sustainability but also improves overall supply chain performance. Tang et al. [28] concluded that under decentralized channel structures, firms can increase total profit, expand market share, and strengthen overall competitiveness through extended warranty service strategies.

2.3. The Meaning of Implicit Equity in Supply Chains and Its Implications

Regarding the study of implicit equity, Yan and Lin [6] emphasize that in reality transactions, a stable and long-term trading relationship can effectively reduce transaction costs, promote the synergistic effects of the assets of both parties, and achieve better economic benefits, thus highlighting the significant value of such a relationship. It follows that this stable and long-term trading relationship has value, and the value of quantifying the long-term and stable cooperative relationship between the trading parties is known as implicit equity [29,30,31]. Liu and Cruz [32] showed that when suppliers sell parts or raw materials to manufacturers, they should consider not only the existing earnings, but also the value of implicit equity. Wang et al. [33] used the capital asset pricing model with the net present value method to quantitatively analyze hidden equity, and the study showed that when the incomplete information game gradually reaches the equilibrium state, the suppliers who consider hidden equity can obtain bank credit loans at a lower cost. Hou et al. [34] studied a supply chain finance system with e-commerce interventions in the context of implicit equity. They found that e-commerce platforms that consider implicit equity earn higher economic returns when raising funds, when quantifying implicit equity using a discounted dividend model.

2.4. Presentation of the Problem

In summary, the existing literature has mainly studied the green supply chain from the perspectives of decision coordination and financing decisions, exploring the competitive relationship between supply chains and their impact, while few scholars have studied the issue of implicit equity between members within the supply chain. Therefore, existing studies need to be expanded further in the following aspects. Firstly, the current literature primarily focuses on decision coordination and financing decisions within a single green supply chain, with limited studies considering financing issues in the green supply chain under inter-supply chain competition. Secondly, while the existing literature primarily explores the competitive relationships and their impacts on supply chains, few studies have delved into the unique competitive dynamics and potential impacts between the green supply chain and the traditional supply chain. Thirdly, the existing literature has not clarified the impact mechanism of variables such as consumers’ green preferences and the opportunity cost of capital on the optimal order quantity and product greenness under inter-chain competition and implicit equity, or on the choice of equilibrium financing models for members of the two supply chains.

Against this backdrop, this study adopts the Cournot competition theory, green supply chain management theory, and relational contract theory as its core framework to construct an analytical model from the perspective of inter-chain competition and implicit equity cross-ownership. Specifically, Cournot competition theory provides the foundation for analyzing the competitive behavior of green and traditional supply chains in output markets. By extending this theory to differentiated green product markets, the study examines how inter-chain competition influences firms’ optimal production and product greenness decisions, taking into account consumer green preferences, thereby addressing the shortcomings of the existing literature, which often focuses on single-chain analysis while neglecting inter-chain strategic interactions. Green supply chain management theory provides the basis for identifying key variables, such as product greenness and consumer green preferences, as well as their underlying mechanisms. It not only incorporates environmental dimensions into operational and financing decisions but also reveals, through model simulations, the mechanisms by which green preferences and the opportunity cost of capital influence supply chain equilibrium, thereby expanding the explanatory boundaries of this theory in competitive dual-chain environments. Relationship contract theory provides a theoretical foundation for understanding implicit equity arrangements within supply chains, treating implicit equity as a governance mechanism based on long-term cooperation and informal commitments. It analyzes its role in alleviating financing constraints and coordinating member interests, addressing the existing research’s neglect of intra-chain governance structures and financing model selection. This study extends the application of Cournot competition from symmetric firms to heterogeneous supply chain competition scenarios involving green and non-green supply chains, introduces relational contracts into the context of green supply chain financing, and clarifies the interactive effects of multiple exogenous variables under competitive and governance mechanisms. It integrates and expands the explanatory scope of existing theories, deepening the theoretical understanding of green supply chain financing and coordination in competitive environments.

This paper takes two secondary supply chains composed of a retailer and a manufacturer producing different products as the research object and studies the decision-making problem of green supply chain financing based on inter-chain competition and implicit equity considerations. First, based on the analysis of the green supply chain financing model, a revenue model of each member of the two supply chains based on inter-chain competition and implicit equity consideration is constructed. Second, the optimal order quantity and the optimal product greenness under each financing model are derived inversely through the determination of the revenue function of each member of the supply chain. Finally, through mathematical derivation and numerical simulation analysis, the effects of consumer green preference, capital opportunity cost, and other variables on the optimal order quantity, product greenness, and supply chain revenue are explored, and the optimal financing model choices of each supply chain decision-making subject under different financing models are comparatively studied.

The main contributions of this paper are as follows. (i) Combining the practical background of green supply chain financing, this paper explores the financing decision-making of supply chain members based on inter-chain competition and implicit equity, enriching relevant research in fields such as green supply chain financing. (ii) Based on the Cournot competition model, it constructs supply chain revenue models based on inter-chain competition and implicit equity considerations, and the optimal order quantity and the optimal product greenness are calculated, which provide decision-making references for supply chain members to choose financing models. (iii) Through mathematical derivation and simulation analysis, the influence mechanisms of consumer green preference, capital opportunity cost, and other variables on the optimal order quantity, the optimal product greenness, and the benefit of supply chain decision-makers are investigated. (iv) Through comparative studies, it is found that, in supply chains based on inter-chain competition and implicit equity considerations, the decision makers of both supply chains will choose their respective equilibrium financing models.

3. Model Assumptions and Definition of Variable Symbols

3.1. Analysis of Green Supply Chain Financing Model Based on Inter-Chain Competition

This paper investigates whether members of the green supply chain are willing to finance their green investment costs in the context of inter-chain competition and implicit equity. The study constructs a model of competition between two secondary supply chains consisting of a retailer and a manufacturer in Cournot, where the first chain is the traditional supply chain (focusing on the production of conventional products) and the second is the green supply chain (with the option to produce green products). Firstly, the manufacturer and the retailer in each supply chain engage in an internal game to determine their production and distribution strategies. Subsequently, the two supply chains compete in the market and continuously adjust their strategies to gain a competitive advantage [35]. Furthermore, the competition between the two supply chains operates under a Nash equilibrium, meaning that members of the supply chains make their respective decisions simultaneously. Figure 1 depicts the structural form of competition between supply chains.

In green supply chain management, the manufacturer often faces the problem of insufficient capital. Compared to the production process of traditional products, the manufacture of green products typically incurs high green investment costs, which makes it difficult for the manufacturer with limited capital to undertake the research, development, and production of green products on their own. Therefore, the manufacturer usually needs to seek external financing, such as loans from banks or other financial institutions. Alternatively, some well-funded retailers may choose to bear the cost of green investments themselves to determine the product’s greenness. In this paper, referring to the method of Zhang et al. [36], it is assumed that the green investment cost is $C\left(e\right)=k{e}^2/2$, where C(e) is satisfied when $C^{\prime }(e)>0$ and $C^{″}(e)>0$; as the product greenness increases, the green investment cost has an accelerating upward trend, while the amount of financing required is $L=k{e}^2/2$.

Furthermore, this paper refers to the methods of Shi et al. [37] and Wu et al. [38] regarding the inverse demand function, where the inverse demand functions of the two supply chains are $p_1=a_1-q_1-b{q}_2-{\theta }_{1}e$ and $p_2=a_2-q_2-b{q}_1+{\theta }_{2}e$, respectively. The selling price (p_i) of the retailer in its supply chain is affected by three factors, including the order quantity of its own supply chain (q_i), the order quantity of the competitor’s supply chain (q_3-i), and the product greenness (e). We assume that consumers are environmentally conscious, which can be regarded as a significant motivation for investing in green production. Due to consumers’ green preferences, the green level of products in the green supply chain serves as an important factor stimulating demand for such products. Consequently, a higher green grade for a product leads to a larger order quantity. Meanwhile, for the traditional supply chain, the order quantity of non-green products decreases owing to the growing personal and environmental awareness among consumers.

3.2. Determination of Implicit Equity in Green Supply Chains Based on Inter-Chain Competition

Implicit equity develops among supply chain members because of their frequent transactions and relatively stable demand relationships. For ease of calculation, it is necessary to quantify the implicit equity, referring to the method of Wang et al. [33], using NC_j to denote the cash flow of supply chain member j from the other party of the supply chain in stage t, and the amount of cash flow in each stage is equal; thus, the following equation can be derived.

$$E_j={\displaystyle \frac{N{C}_j}{(1+\alpha )}}+{\displaystyle \frac{N{C}_j}{{(1+\alpha )}^2}}+\cdots +{\displaystyle \frac{N{C}_j}{{(1+\alpha )}^t}}\cdots$$

(1)

where $j\in \left\{m,r\right\}$, E_j denotes the value of the implicit equity owned by j, and α denotes the discount rate, which reflects the risk of obtaining cash flows. Equation (1) then simplifies to $E_j={\displaystyle \frac{N{C}_j}{{\alpha }_j}},j\in \left\{m,r\right\}$. If the growth rate of supply chain members and the retention rate of the transaction relationship are also considered, the expression for implicit equity is as follows

$$E_j={\phi }_j(1+g_j){\displaystyle \frac{N{C}_j}{(1+{\alpha }_j)}}+{\phi }_j^2{(1+g_j)}^2{\displaystyle \frac{N{C}_j}{{(1+\alpha )}^2}}+\cdots +{\phi }_j^t{(1+g_j)}^t{\displaystyle \frac{N{C}_j}{{(1+\alpha )}^t}}\cdots j\in \left\{m,r\right\}$$

(2)

where ${\phi }_j$ denotes the probability that j is retained in the next stage and $0<{\phi }_j<1$. $g_j$ represents the growth rate of j and $0\le g_j<\alpha$. Thus, Equation (2) simplifies to $E_j={\displaystyle \frac{N{C}_j}{1+\alpha /\phi (1+g_j)-1}},j\in \left\{m,r\right\}$. Furthermore, in this paper, for the simplicity of the calculation, let ${\displaystyle \frac{1}{1+\alpha /\phi (1+g_j)-1}}={\mu }_j$.

Thus, the manufacturer’s net cash flow would be equal to wholesale less the cost of raw materials, or $N{C}_m=(w_i-c)q_i$, and the retailer’s net cash flow would be equal to total retail less the cost of wholesale, or $N{C}_r=(p_i-w_i)q_i$. Thus, combining the above formulas, the value of the implicit equity owned by the manufacturer would be $E_m={\mu }_m(w_i-c)q_i$, and the value of the implicit equity owned by the retailer would be $E_r={\mu }_r(p_i-w_i)q_i$.

3.3. Assumptions (Summary of Main Assumptions in Appendix A)

(i)

The retailers and manufacturers are risk-neutral and perfectly rational, so their decision choices will depend entirely on maximizing their returns.

(ii)

The information available to members in the supply chains and between chains is perfectly symmetric.

(iii)

For the orders of the retailers, the manufacturers produce no product surplus, meaning that the market demand is equal to the number of orders.

(iv)

Considers two scenarios for implicit equity, with the first scenario being that the manufacturer owns the value of the implicit equity and the second scenario being that the retailer owns the value of the implicit equity.

3.4. Variable Symbols

i denotes the ith supply chain, where i = 1 denotes the traditional supply chain, and i = 2 denotes the green supply chain.

j denotes the supply chain member, where j = r denotes the retailer, and j = m denotes the manufacturer.

a denotes the market basic demand of the ith supply chain.

b denotes the substitution coefficient of two products, 0 ≤ b ≤ 1.

θ_i denotes the green preference of consumers in the ith supply chain, θ₁ < θ₂.

c denotes the unit cost of producing traditional products, excluding the green investment cost.

w_i denotes the wholesale price of the unit product from the manufacturer to the retailer in the ith supply chain, where w₂ ≥ w₁.

p_i denotes the selling price per unit of product in the ith supply chain, p_i > w_i > c.

k denotes the green investment cost coefficient, k > 0.

I_j denotes the opportunity cost of financing for supply chain member j, the loan interest rate.

u_j denotes the probability of financing success for supply chain member j.

e denotes the product greenness.

π_ji denotes the revenue of member j in the ith supply chain.

q_i denotes the order quantity of the retailer in the ith supply chain.

q_i,g denotes the order quantity of retailers in supply chain i when the green supply chain is successfully financed.

q_i,t denotes the order quantity of retailers in supply chain i when the green supply chain is not successfully financed.

E_j denotes the implicit equity value owned by j.

4. A Decision Model for Green Supply Chain Financing Based on Inter-Chain Competition and Implicit Equity Considerations

To deeply study the order quantity and product greenness of the green supply chain under different financing models of inter-chain competition and implicit equity consideration, three financing models are considered in this section, including the non-financing model (NF), the manufacturer financing model (MF), and the retailer financing model (RF). Additionally, two ways of owning implicit equity value, i.e., the manufacturer or retailer owning implicit equity value, are also considered to construct the supply chain decision model. Based on the principle of maximizing the returns of supply chain members, the existence of the optimal order quantity and the optimal level of product greenness of the supply chain is mathematically deduced to reveal the profound effects of different financing models and implicit equity holdings on the operation of the green supply chain.

4.1. Green Supply Chain Financing Decision Model Based on Inter-Chain Competition and Implicit Equity Considerations Under the No-Financing Model

This section discusses the case of the green supply chain without financing, where two competing supply chains have opted to produce the conventional product. In this decentralized decision-making environment, both the retailer and the manufacturer aim to maximize their respective returns. Moreover, the calculation reveals that the optimal order quantity remains the same regardless of who holds the implicit equity value. Therefore, this section does not discuss them separately. Based on this analysis, the retailers of the two supply chains simultaneously determine their order quantity, $q_i(i=1,2)$, and the retail price, $p_i(i=1,2)$. Therefore, under the no-financing model, and considering implicit equity, we can derive the revenue functions for both the manufacturer and the retailer.

$${\pi }_{mi}^{NF}=(w_i-c)q_i+E_m=(1+{\mu }_m)(w_i-c)q_i$$

(3)

$${\pi }_{ri}^{NF}=(p_i-w_i)q_i+E_r=(1+{\mu }_r)(a_i-q_i-b{q}_{3-i}-w_i)q_i$$

(4)

Proposition 1.

In the absence of a financing model and considering implicit equity, the existence of optimal order quantities for the two supply chains can be obtained, which are $q_1^{{NF}^{*}}$and $q_2^{{NF}^{*}}$.

Proof. 

We take the first-order derivatives of Equation (4) with respect to q₁ and q₂. The first-order derivatives are, respectively,

$${{\pi }^{\prime }}_{r1}^{NF}={\displaystyle \frac{\partial {\pi }_{r1}^{NF}}{\partial q_1}}=(1+{\mu }_r)(a_1-2q_1-b{q}_2-w_1)$$

(5)

$${{\pi }^{\prime }}_{r2}^{NF}={\displaystyle \frac{\partial {\pi }_{r2}^{NF}}{\partial q_2}}=(1+{\mu }_r)(a_2-2q_2-b{q}_1-w_2)$$

(6)

Hence, it takes the second-order derivatives of Equations (5) and (6) concerning q₁ and q₂, respectively, to obtain ${{\pi }^{″}}_{r2}^{NF}={{\pi }^{″}}_{r2}^{NF}=-2$. The second-order derivatives are all less than 0, which indicates that Equation (4) concerning $q_i(i=1,2)$ is a convex function, and the function has a maximum revenue and an optimal order quantity. Setting Equations (5) and (6) to equal 0 and solving them simultaneously, we obtain the optimal order quantities.

$$q_1^{NF*}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}$$

(7)

$$q_2^{NF*}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

(8)

In summary, this proves the existence of the optimal order quantity in the case of a no-financing model and considering implicit equity. □

4.2. Green Supply Chain Financing Decision Model Based on Inter-Chain Competition and Implicit Equity Considerations Under the Manufacturer Financing Model

In the context of green supply chain systems, the manufacturer undertakes the crucial task of producing green products. However, financial constraints often hinder its ability to complete production. Consequently, this section will analyze whether the manufacturer in the green supply chain opts to seek financing to overcome these challenges. Specifically, the manufacturer may need to apply for a loan from an external financial institution, such as a bank, with the loan amount denoted as $L=k{e}^2/2$. The financing process inherently involves a certain degree of risk; thus, the financial institution will evaluate the manufacturer’s loan application and may agree to provide the loan with a probability denoted as $u_m\in \left(0,1\right)$. Additionally, the amount $I_m\in \left(0,1\right)$ represents the opportunity cost of capital, which corresponds to the interest rate associated with the loan.

First, in the green supply chain, the financially constrained manufacturer is required to establish their level of green efforts and submit loan applications to financial institutions before the commencement of production. Second, if the manufacturer receives a loan, the retailers within both supply chains will concurrently determine their order quantity, $q_{i,g}\left(i=1,2\right)$. Subsequently, the manufacturer will complete the production of the green product and facilitate its delivery to the retailer. Upon fulfilling the delivery, the manufacturer will be obligated to repay the financial institution the principal amount of the loan with the appropriate interest. Finally, if the loan application is denied, the manufacturer within the green supply chain will continue to produce the traditional product, while the retailers in both supply chains will simultaneously determine their respective order quantities, $q_{i,t}\left(i=1,2\right)$, and proceed with the production and sales processes of the entire supply chain sequentially. Therefore, we can obtain the revenue functions for each member (manufacturer and retailer, hereafter) within the two supply chains (green supply chain and traditional supply chain, hereafter).

$${\pi }_{m2}^{MF}=u_m\left[(w_2-c)q_{2,g}-(1+I_m){\displaystyle \frac{k{e}^2}{2}}\right]+(1-u_m)(w_2-c)q_{2,t}+E_m$$

(9)

$${\pi }_{r2}^{MF}=u_m(p_{2,g}-w_2)q_{2,g}+(1-u_m)(p_{2,t}-w_2)q_{2,t}+E_r$$

(10)

$${\pi }_{m1}^{MF}=u_m(w_1-c)q_{1,g}+(1-u_m)(w_1-c)q_{1,t}+E_m$$

(11)

$${\pi }_{r1}^{MF}=u_m(p_{1,g}-w_1)q_{1,g}+(1-u_m)(p_{1,t}-w_1)q_{1,t}+E_r$$

(12)

4.2.1. Green Supply Chain Financing Decision Model Based on Inter-Chain Competition Consideration Under the Implicit Equity Value Owned by the Manufacturer

Under the condition that the manufacturer owns the value of implicit equity, by quantifying the implicit equity, we can determine that the manufacturer who owns the value of implicit equity is $E_m={\mu }_m\left[u_m(w_i-c)q_{i,g}+(1-u_m)(w_i-c)q_{i,t}\right]$, and the retailer is $E_r=0$. By substituting these two equations into Equations (9)–(12), we obtain the two supply chain members’ revenue functions under this condition.

$${\pi }_{m2}^{MF}=(1+{\mu }_m)\left[u_m(w_2-c)q_{2,g}+(1-u_m)(w_2-c)q_{2,t}\right]-u_m(1+I_m){\displaystyle \frac{k{e}^2}{2}}$$

(13)

$${\pi }_{r2}^{MF}=u_m(p_{2,g}-w_2)q_{2,g}+(1-u_m)(p_{2,t}-w_2)q_{2,t}$$

(14)

$${\pi }_{m1}^{MF}=(1+{\mu }_m)\left[u_m(w_1-c)q_{1,g}+(1-u_m)(w_1-c)q_{1,t}\right]$$

(15)

$${\pi }_{r1}^{MF}=u_m(p_{1,g}-w_1)q_{1,g}+(1-u_m)(p_{1,t}-w_1)q_{1,t}$$

(16)

Proposition 2.

Under the manufacturer financing model and when the manufacturer holds an implicit equity stake in the retailer, when the manufacturer of the green supply chain obtains financing, the optimal order quantity,  $q_{i,g}^{MF*}$, and the optimal product greenness, $e^{MF*}$, exist in both supply chains.

Proof. 

When the manufacturer obtains financing, the order quantity of the green supply chain retailer is q_2,g, and the corresponding retail price is $p_{2,g}=a_2-q_{2,g}-b{q}_{1,g}+{\theta }_{2}e$. When the manufacturer does not obtain financing, the order quantity of the green supply chain retailer is q_2,t, and the corresponding retail price is $p_{2,t}=a_2-q_{2,t}-b{q}_{1,t}$. Hence, substituting p_2,g and p_2,t into Equation (14) yields

$${\pi }_{r2}^{MF}=u_m(a_2-q_{2,g}-b{q}_{1,g}+{\theta }_{2}e-w_2)q_{2,g}+(1-u_m)(a_2-q_{2,t}-b{q}_{1,t}-w_2)q_{2,t}$$

(17)

The first-order derivatives of Equation (17) with respect to q_2,g and q_2,t are, respectively

$${{\pi }^{\prime }}_{r2}^{MF}={\displaystyle \frac{\partial {\pi }_{r2}^{MF}}{\partial q_{2,g}}}=u_m(a_2-2q_{2,g}-b{q}_{1,g}+{\theta }_{2}e-w_2)$$

(18)

$${{\pi }^{\prime }}_{r2}^{MF}={\displaystyle \frac{\partial {\pi }_{r2}^{MF}}{\partial q_{2,t}}}=(1-u_m)(a_2-2q_{2,t}-b{q}_{1,t}-w_2)$$

(19)

We continue to find the second-order derivatives of Equations (18) and (19) concerning q_2,g and q_2,t, respectively, and this yields ${{\pi }^{″}}_{r2}^{MF}={\displaystyle \frac{{\partial }^2{\pi }_{r2}^{MF}}{\partial q_{2,g}^2}}=-2u_m$ and ${{\pi }^{″}}_{r2}^{MF}={\displaystyle \frac{{\partial }^2{\pi }_{r2}^{MF}}{\partial q_{2,t}^2}}=-2(1-u_m)$. Since $u_m\in \left(0,1\right)$, the second-order derivatives are all less than 0, which indicates that Equation (17) is a convex function with respect to q_2,g and q_2,t. Therefore, Equation (17) has a maximum revenue value and the corresponding optimal ordering quantity.

Similarly, when the manufacturer obtains financing, the order quantity of the traditional supply chain retailer is q_1,g, and the corresponding retail price of the product is $p_{1,g}=a_1-q_{1,g}-b{q}_{2,g}-{\theta }_{1}e$. When the manufacturer does not obtain financing, the order quantity of the traditional supply chain retailer is q_1,t, and the corresponding retail price of the product is $p_{1,t}=a_1-q_{1,t}-b{q}_{2,t}$. Hence, substituting p_1,g and p_1,t into Equation (16) yields

$${\pi }_{r1}^{MF}=u_m(a_1-q_{1,g}-b{q}_{2,g}-{\theta }_{1}e-w_1)q_{1,g}+(1-u_m)(a_1-q_{1,t}-b{q}_{2,t}-w_1)q_{1,t}$$

(20)

The first-order derivatives of Equation (20) with respect to q_1,g and q_1,t are, respectively,

$${{\pi }^{\prime }}_{r1}^{MF}={\displaystyle \frac{\partial {\pi }_{r1}^{MF}}{\partial q_{1,g}}}=u_m(a_1-2q_{1,g}-b{q}_{2,g}-{\theta }_{1}e-w_1)$$

(21)

$${{\pi }^{\prime }}_{r1}^{MF}={\displaystyle \frac{\partial {\pi }_{r1}^{MF}}{\partial q_{1,t}}}=(1-u_m)(a_1-2q_{1,t}-b{q}_{2,t}-w_1)$$

(22)

We continue to find the second-order derivatives of Equations (21) and (22) concerning q_1,g and q_1,t, and the second-order derivatives are all less than 0, which indicates that Equation (20) is a convex function with respect to q_1,g and q_1,t. Therefore, Equation (20) has a maximum revenue value and the corresponding optimal ordering quantity.

This makes Equations (18), (19), (21) and (22) equal to 0, solving Equations (18) and (21) and Equations (19) and (22), respectively, for the optimal order quantity:

$$q_{2,g}^{MF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}}$$

(23)

$$q_{1,g}^{MF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_1+b{\theta }_2)}{4-b^2}}$$

(24)

$$q_{2,t}^{MF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

(25)

$$q_{1,t}^{MF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}$$

(26)

We substitute Equations (23) and (25) in Equation (13).

$$\begin{array}{cc}{\pi }_{m2}^{MF}& =(1+{\mu }_m)u_m\left[(w_2-c){\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)+e(2{\theta }_2+b{\theta }_1)}{4-b^2}}\right.\\ & \left.+(1-u_m)(w_2-c){\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}\right]-u_m(1+I_m){\displaystyle \frac{k{e}^2}{2}}\end{array}$$

We take the first-order derivative of the above equation with respect to e and make the first-order derivative equal to 0 to obtain the solution.

$$e^{MF}={\displaystyle \frac{(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)(w_2-c)}{(4-b^2)k(1+I_m)}}$$

(27)

We substitute Equation (27) into Equations (23) and (24) and summarize the optimal order quantity and the optimal product greenness of the two supply chains.

$$q_{2,g}^{MF\ast }={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{(1+{\mu }_m){(2{\theta }_2+b{\theta }_1)}^2(w_2-c)}{{(4-b^2)}^{2}k(1+I_m)}}$$

(28)

$$q_{1,g}^{MF\ast }={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)(2{\theta }_1+b{\theta }_2)(w_2-c)}{{(4-b^2)}^{2}k(1+I_m)}}$$

(29)

$$e^{MF\ast }={\displaystyle \frac{(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)(w_2-c)}{(4-b^2)k(1+I_m)}}$$

(30)

$$q_{2,t}^{MF\ast }={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

$$q_{1,t}^{MF\ast }={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}$$

In summary, it is demonstrated that under the manufacturer financing model and when the manufacturer holds the implicit equity of the retailer, the optimal order quantity, q^MF*, and the optimal product greenness, e^MF*, of the two supply chains are obtained. □

4.2.2. Green Supply Chain Financing Decision Model Based on Inter-Chain Competition Considering the Implicit Equity Value Owned by the Retailer

Under the condition that the retailer owns the value of implicit equity, by quantifying the implicit equity, we can determine that the retailer who owns the value of implicit equity is $E_r={\mu }_r\left[u_m(p_{i,g}-w_2)q_{i,g}+(1-u_m)(p_{i,t}-w_2)q_{i,t}\right]$, and the manufacturer is $E_m=0$. By substituting these two equations into Equations (9)–(12), we obtain the two supply chain members’ revenue functions under this condition.

$${\pi }_{m2}^{MF}=u_m\left[(w_2-c)q_{2,g}-(1+I_m){\displaystyle \frac{k{e}^2}{2}}\right]+(1-u_m)(w_2-c)q_{2,t}$$

(31)

$${\pi }_{r2}^{MF}=(1+{\mu }_r)\left[u_m(p_{2,g}-w_2)q_{2,g}+(1-u_m)(p_{2,t}-w_2)q_{2,t}\right]$$

(32)

$${\pi }_{m1}^{MF}=u_m(w_1-c)q_{1,g}+(1-u_m)(w_1-c)q_{1,t}$$

(33)

$${\pi }_{r1}^{MF}=(1+{\mu }_r)\left[u_m(p_{1,g}-w_1)q_{1,g}+(1-u_m)(p_{1,t}-w_1)q_{1,t}\right]$$

(34)

Proposition 3.

Under the manufacturer financing model and when the retailer holds an implicit equity stake in the manufacturer, when the manufacturer of the green supply chain obtains financing, an optimal order quantity,  $q_{i,g}^{MF*}$, exists, and optimal product greenness, $e^{MF*}$, exists in both supply chains.

Proof. 

When the manufacturer obtains financing, the order quantity of the green supply chain retailer is q_2,g, and the corresponding retail price is $p_{2,g}=a_2-q_{2,g}-b{q}_{1,g}+{\theta }_{2}e$. When the manufacturer does not obtain financing, the order quantity of the green supply chain retailer is q_2,t, and the corresponding retail price is $p_{2,t}=a_2-q_{2,t}-b{q}_{1,t}$. Hence, substituting p_2,g and p_2,t into Equation (32) yields

$${\pi }_{r2}^{MF}=(1+{\mu }_r)\left[u_m(a_2-q_{2,g}-b{q}_{1,g}+{\theta }_{2}e-w_2)q_{2,g}+(1-u_m)(a_2-q_{2,t}-b{q}_{1,t}-w_2)q_{2,t}\right]$$

(35)

The first-order derivatives of Equation (35) with respect to q_2,g and q_2,t are, respectively,

$${{\pi }^{\prime }}_{r2}^{MF}={\displaystyle \frac{\partial {\pi }_{r2}^{MF}}{\partial q_{2,g}}}=(1+{\mu }_r)u_m(a_2-2q_{2,g}-b{q}_{1,g}+{\theta }_{2}e-w_2)$$

(36)

$${{\pi }^{\prime }}_{r2}^{MF}={\displaystyle \frac{\partial {\pi }_{r2}^{MF}}{\partial q_{2,t}}}=(1+{\mu }_r)(1-u_m)(a_2-2q_{2,t}-b{q}_{1,t}-w_2)$$

(37)

We continue to find the second-order derivatives of Equations (36) and (37) concerning q_2,g and q_2,t, respectively, and we obtain ${{\pi }^{″}}_{r2}^{MF}={\displaystyle \frac{{\partial }^2{\pi }_{r2}^{MF}}{\partial q_{2,g}^2}}=-2(1+{\mu }_r)u_m$ and ${{\pi }^{″}}_{r2}^{MF}={\displaystyle \frac{{\partial }^2{\pi }_{r2}^{MF}}{\partial q_{2,t}^2}}=-2(1+{\mu }_r)(1-u_m)$. Since $u_m\in \left(0,1\right)$ and ${\mu }_r>0$, the second-order derivatives are all less than 0, which indicates that Equation (35) is a convex function with respect to q_2,g and q_2,t. Therefore, Equation (35) has a maximum revenue value and the corresponding optimal ordering quantity.

Similarly, when the manufacturer obtains financing, the order quantity of the traditional supply chain retailer is q_1,g, and the corresponding retail price of the product is $p_{1,g}=a_1-q_{1,g}-b{q}_{2,g}-{\theta }_{1}e$. When the manufacturer does not obtain financing, the order quantity of the traditional supply chain retailer is q_1,t, and the corresponding retail price of the product is $p_{1,t}=a_1-q_{1,t}-b{q}_{2,t}$. Hence, substituting p_1,g and p_1,t into Equation (34) yields

$${\pi }_{r1}^{MF}=(1+{\mu }_r)\left[u_m(a_1-q_{1,g}-b{q}_{2,g}-{\theta }_{1}e-w_1)q_{1,g}+(1-u_m)(a_1-q_{1,t}-b{q}_{2,t}-w_1)q_{1,t}\right]$$

(38)

The first-order derivatives of Equation (38) with respect to q_1,g and q_1,t are, respectively,

$${{\pi }^{\prime }}_{r1}^{MF}={\displaystyle \frac{\partial {\pi }_{r1}^{MF}}{\partial q_{1,g}}}=(1+{\mu }_r)u_m(a_1-2q_{1,g}-b{q}_{2,g}-{\theta }_{1}e-w_1)$$

(39)

$${{\pi }^{\prime }}_{r1}^{MF}={\displaystyle \frac{\partial {\pi }_{r1}^{MF}}{\partial q_{1,t}}}=(1+{\mu }_r)(1-u_m)(a_1-2q_{1,t}-b{q}_{2,t}-w_1)$$

(40)

We continue to find the second-order derivatives of Equations (39) and (40) concerning q_1,g and q_1,t. The second-order derivatives are all less than 0, which indicates that Equation (38) is a convex function with respect to q_1,g and q_1,t. Therefore, Equation (38) has a maximum revenue value and the corresponding optimal ordering quantity.

This makes Equations (36), (37), (39), and (40) equal to 0, solving Equations (36) and (39) and Equations (37) and (40), respectively, for the optimal order quantity:

$$q_{2,g}^{MF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}}$$

(41)

$$q_{1,g}^{MF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_1+b{\theta }_2)}{4-b^2}}$$

(42)

$$q_{2,t}^{MF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

(43)

$$q_{1,t}^{MF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}$$

(44)

We substitute Equations (41) and (43) into Equation (31).

$${\pi }_{m2}^{MF}=u_m\left[(w_2-c){\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)+e(2{\theta }_2+b{\theta }_1)}{4-b^2}}-(1+I_m){\displaystyle \frac{k{e}^2}{2}}\right]+(1-u_m)(w_2-c){\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

We take the first-order derivative of the above equation with respect to e and make the first-order derivative equal to 0 to obtain the solution:

$$e^{MF}={\displaystyle \frac{(2{\theta }_2+b{\theta }_1)(w_2-c)}{(4-b^2)k(1+I_m)}}$$

(45)

We substitute Equation (45) into Equations (41) and (42) and summarize the optimal order quantity and the optimal product greenness of the two supply chains.

$$q_{2,g}^{MF\ast }={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{{(2{\theta }_2+b{\theta }_1)}^2(w_2-c)}{{(4-b^2)}^{2}k(1+I_m)}}$$

$$q_{1,g}^{MF\ast }={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{(2{\theta }_2+b{\theta }_1)(2{\theta }_1+b{\theta }_2)(w_2-c)}{{(4-b^2)}^{2}k(1+I_m)}}$$

$$e^{MF\ast }={\displaystyle \frac{(2{\theta }_2+b{\theta }_1)(w_2-c)}{(4-b^2)k(1+I_m)}}$$

$$q_{2,t}^{MF\ast }={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

$$q_{1,t}^{MF\ast }={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}$$

In summary, it is demonstrated that under the manufacturer financing model and when the retailer holds the implicit equity of the manufacturer, the optimal order quantity, q^MF*, and the optimal product greenness, e^MF*, of the two supply chains are obtained. □

4.3. Green Supply Chain Financing Decision Model Based on Inter-Chain Competition and Implicit Equity Considerations Under Retailer Financing Model

With the growing consumer demand for green products, the retailer within the green supply chain must face the challenge of fulfilling this demand. However, the manufacturer in the green supply chain often struggles to produce green products effectively due to a lack of financial resources. This limitation not only restricts the retailer’s ability to sell green products but may also diminish its competitiveness in the marketplace. Therefore, this section will focus on the issue of whether the well-financed retailer in the green supply chain chooses to bear the cost of green investments when faced with insufficient production capacity for green products from the manufacturer. For example, prominent retailers such as Walmart and Costco, aiming to meet consumer demand and enhance their brand reputation, not only participate directly in determining product greenness, but also bear all the capital manufacturers require to produce green products [37].

This paper assumes that a retailer with sufficient financial resources will evaluate the manufacturer’s production capacity, credit rating, and other relevant factors to determine whether to provide financing, with the financing amount denoted as $L=k{e}^2/2$. Moreover, in the context of the green supply chain, the retailer agrees to finance the manufacturer with an exogenous probability, $u_r\in \left(0,1\right)$, which is determined before the sales start and influenced by various factors. Additionally, consider $I_r\in \left(0,1\right)$ the opportunity cost of financing, representing the interest on the potential returns from the diverted funds. Furthermore, if the well-funded retailer chooses to finance the manufacturer independently, effectively bearing the entire cost of the green investment, the retailer will gain the final say on product greenness and, accordingly, will require that the manufacturer produce the green product according to its specific standards.

Based on the above analysis, first, the retailer in the green supply chain determines the desired level of product greenness and evaluates the qualification of the manufacturer before production begins. Second, if the manufacturer’s qualification evaluation results are satisfactory, the retailer will choose to bear the green investment cost independently. At this point, the retailers in both supply chains will simultaneously decide on their respective order quantity, $q_{i,g}(i=1,2)$. The manufacturers will then produce green products based on these orders and deliver them to the retailers, who will lose a certain amount of opportunity cost during the sales process. Finally, if the manufacturer’s qualification results are unsatisfactory, the retailer in the green supply chain will no longer bear the green investment cost, and the manufacturer will maintain the production of traditional products. At this point, the retailers in both supply chains will make decisions on their respective order quantities, $q_{i,t}\left(i=1,2\right)$, and finalize the production and sales process of the entire supply chain. Therefore, we can obtain the revenue functions for each member of the two supply chains.

$${\pi }_{m2}^{RF}=u_r(w_2-c)q_{2,g}+(1-u_r)(w_2-c)q_{2,t}+E_m$$

(46)

$${\pi }_{r2}^{RF}=u_r\left[(p_{2,g}-w_2)q_{2,g}-(1+I_r){\displaystyle \frac{k{e}^2}{2}}\right]+(1-u_r)(p_{2,t}-w_2)q_{2,t}+E_r$$

(47)

$${\pi }_{m1}^{RF}=u_r(w_1-c)q_{1,g}+(1-u_r)(w_1-c)q_{1,t}+E_m$$

(48)

$${\pi }_{r1}^{RF}=u_r(p_{1,g}-w_1)q_{1,g}+(1-u_r)(p_{1,t}-w_1)q_{1,t}+E_r$$

(49)

4.3.1. Green Supply Chain Financing Decision Model Based on Inter-Chain Competition Consideration Under the Implicit Equity Value Owned by the Manufacturer

Under the condition that the manufacturer owns the value of implicit equity, by quantifying the implicit equity, we can determine that the manufacturer who owns the value of implicit equity is $E_m={\mu }_m\left[u_r(w_i-c)q_{i,g}+(1-u_r)(w_i-c)q_{i,t}\right]$, and the retailer is $E_r=0$. By substituting these two equations into Equations (46)–(49), we obtain the two supply chain members’ revenue functions under this condition.

$${\pi }_{m2}^{RF}=(1+{\mu }_m)\left[u_r(w_2-c)q_{2,g}+(1-u_r)(w_2-c)q_{2,t}\right]$$

(50)

$${\pi }_{r2}^{RF}=u_r\left[(p_{2,g}-w_2)q_{2,g}-(1+I_r){\displaystyle \frac{k{e}^2}{2}}\right]+(1-u_r)(p_{2,t}-w_2)q_{2,t}$$

(51)

$${\pi }_{m1}^{RF}=(1+{\mu }_m)\left[u_r(w_1-c)q_{1,g}+(1-u_r)(w_1-c)q_{1,t}\right]$$

(52)

$${\pi }_{r1}^{RF}=u_r(p_{1,g}-w_1)q_{1,g}+(1-u_r)(p_{1,t}-w_1)q_{1,t}$$

(53)

Proposition 4.

Under the retailer financing model and when the manufacturer holds an implicit equity stake in the retailer, when the retailer in the green supply chain chooses to bear the green investment cost, the optimal order quantity, $q_{i,g}^{RF*}$, and the optimal product greenness, $e^{RF*}$, exist in both supply chains.

Proof. 

When the retailer chooses to bear the cost of green investment, the order quantity of the green supply chain retailer is q_2,g, and the corresponding retail price is $p_{2,g}=a_2-q_{2,g}-b{q}_{1,g}+{\theta }_{2}e$. When the retailer chooses not to bear the cost of green investment, the order quantity of the green supply chain retailer is q_2,t, and the corresponding retail price is $p_{2,t}=a_2-q_{2,t}-b{q}_{1,t}$. Hence, substituting p_2,g and p_2,t into Equation (51) yields

$${\pi }_{r2}^{RF}=u_r\left[(a_2-q_{2,g}-b{q}_{1,g}+{\theta }_{2}e-w_2)q_{2,g}-(1+I_r){\displaystyle \frac{k{e}^2}{2}}\right]+(1-u_r)(a_2-q_{2,t}-b{q}_{1,t}-w_2)q_{2,t}$$

(54)

The first-order derivatives of Equation (54) with respect to q_2,g and q_2,t are, respectively,

$${{\pi }^{\prime }}_{r2}^{RF}={\displaystyle \frac{\partial {\pi }_{r2}^{RF}}{\partial q_{2,g}}}=u_r(a_2-2q_{2,g}-b{q}_{1,g}+{\theta }_{2}e-w_2)$$

(55)

$${{\pi }^{\prime }}_{r2}^{RF}={\displaystyle \frac{\partial {\pi }_{r2}^{RF}}{\partial q_{2,t}}}=(1-u_r)(a_2-2q_{2,t}-b{q}_{1,t}-w_2)$$

(56)

We continue to find the second-order derivatives of Equations (55) and (56) concerning q_2,g and q_2,t, respectively, and we obtain ${{\pi }^{″}}_{r2}^{RF}={\displaystyle \frac{{\partial }^2{\pi }_{r2}^{RF}}{\partial q_{2,g}^2}}=-2u_r$ and ${{\pi }^{″}}_{r2}^{RF}={\displaystyle \frac{{\partial }^2{\pi }_{r2}^{RF}}{\partial q_{2,t}^2}}=-2(1-u_r)$. Since $u_r\in \left(0,1\right)$, the second-order derivatives are all less than 0, which indicates that Equation (54) is a convex function with respect to q_2,g and q_2,t. Therefore, Equation (54) has a maximum revenue value and the corresponding optimal ordering quantity.

Similarly, when the retailer bears the cost of green investments, the order quantity of the traditional supply chain retailer is q_1,g, and the corresponding retail price of the product is $p_{1,g}=a_1-q_{1,g}-b{q}_{2,g}-{\theta }_{1}e$. When the retailer chooses not to bear the cost of green investments, the order quantity of the traditional supply chain retailer is q_1,t, and the corresponding retail price of the product is $p_{1,t}=a_1-q_{1,t}-b{q}_{2,t}$. Hence, substituting p_1,g and p_1,t into Equation (53) yields

$${\pi }_{r1}^{RF}=u_r(a_1-q_{1,g}-b{q}_{2,g}-{\theta }_{1}e-w_1)q_{1,g}+(1-u_r)(a_1-q_{1,t}-b{q}_{2,t}-w_1)q_{1,t}$$

(57)

The first-order derivatives of Equation (57) with respect to q_1,g and q_1,t are, respectively,

$${{\pi }^{\prime }}_{r1}^{RF}={\displaystyle \frac{\partial {\pi }_{r1}^{RF}}{\partial q_{1,g}}}=u_r(a_1-2q_{1,g}-b{q}_{2,g}-{\theta }_{1}e-w_1)$$

(58)

$${{\pi }^{\prime }}_{r1}^{RF}={\displaystyle \frac{\partial {\pi }_{r1}^{RF}}{\partial q_{1,t}}}=(1-u_r)(a_1-2q_{1,t}-b{q}_{2,t}-w_1)$$

(59)

We continue to find the second-order derivatives of Equations (58) and (59) concerning q_1,g and q_1,t. The second-order derivatives are all less than 0, which indicates that Equation (57) is a convex function with respect to q_1,g and q_1,t. Therefore, Equation (57) has a maximum revenue value and the corresponding optimal ordering quantity.

This makes Equations (55), (56), (58), and (59) equal to 0, solving Equations (55) and (58) and Equations (56) and (59), respectively, for the optimal order quantity:

$$q_{2,g}^{RF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}}$$

(60)

$$q_{1,g}^{RF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_1+b{\theta }_2)}{4-b^2}}$$

(61)

$$q_{2,t}^{RF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

(62)

$$q_{1,t}^{RF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}$$

(63)

We substitute Equations (60) and (63) into Equation (54):

$${\pi }_{r2}^{RF}=u_r\left[(a_2-(1+b){\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}-(1-b){\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}}+{\theta }_{2}e-w_2)\right.\times$$

$$\begin{gathered} \left.\left.\left({\displaystyle \frac{(2a_2-w_2)-b(a_1-w_1)}{4-b^2}}\right.+{\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}}\right)-(1+I_r){\displaystyle \frac{k{e}^2}{2}}\right]+(1-{\mu }_r)\times (a_2- \\ {\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}-{\displaystyle \frac{2b(a_1-w_1)-b(a_2-w_2)}{4-b^2}}-w_2){\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}} \end{gathered}$$

We take the first-order derivative of the above equation with respect to e and make the first-order derivative equal to 0 to obtain the solution:

$$e^{RF}={\displaystyle \frac{2(2{\theta }_2-b^2{\theta }_2-b{\theta }_1)\left[2(a_2-w_2)-b(a_1-w_1)\right]}{k(1+I_r)b^4+4b^3{\theta }_1{\theta }_2+(6{{\theta }_1}^2+8{{\theta }_2}^2-8k-8k{I}_r)b^2+8b{\theta }_1{\theta }_2-8{{\theta }_2}^2+16k(I_r+1)}}$$

(64)

We substitute Equation (64) into Equations (60) and (61) and summarize the optimal order quantity and the optimal product greenness of the two supply chains.

$$\begin{gathered} q_{2,g}^{RF\ast }={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{2B(2{\theta }_2+b{\theta }_1)(2a_2-2w_2-b{a}_1+b{w}_1)}{A(4-b^2)}} \\ {q}_{1,g}^{RF\ast }={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{2B(2{\theta }_1+b{\theta }_2)(2a_2-2w_2-b{a}_1+b{w}_1)}{A(4-b^2)}} \end{gathered}$$

(65)

$$e^{RF\ast }={\displaystyle \frac{2B(2a_2-2w_2-b{a}_1+b{w}_1)}{A}}$$

(66)

$$q_{2,t}^{RF\ast }={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

$$q_{1,t}^{RF\ast }={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}$$

$$A=k(1+I_r)b^4+4b^3{\theta }_1{\theta }_2+(6{{\theta }_1}^2+8{{\theta }_2}^2-8k-8k{I}_r)b^2+8b{\theta }_1{\theta }_2-8{{\theta }_2}^2+16k(I_r+1)$$

$$B=2{\theta }_2-b^2{\theta }_2-b{\theta }_1$$

In summary, it is demonstrated that under the retailer financing model and when the manufacturer holds the implicit equity of the retailer, the optimal order quantity, q^RF*, and the optimal product greenness, e^RF*, of the two supply chains are obtained. □

4.3.2. Green Supply Chain Financing Decision Model Based on Inter-Chain Competition Consideration Under the Implicit Equity Value Owned by the Retailer

Under the condition that the retailer owns the value of implicit equity, by quantifying the implicit equity, we can determine that the retailer who owns the value of implicit equity is $E_r={\mu }_r\left[u_r(p_{i,g}-w_2)q_{i,g}+(1-u_r)(p_{i,t}-w_2)q_{i,t}\right]$, and the manufacturer is $E_m=0$. By substituting these two equations into Equations (46)–(49), we obtain the two supply chain members’ revenue functions under this condition.

$${\pi }_{m2}^{RF}=u_r(w_2-c)q_{2,g}+(1-u_r)(w_2-c)q_{2,t}$$

(67)

$${\pi }_{r2}^{RF}=(1+{\mu }_r)\left[u_r(p_{2,g}-w_2)q_{2,g}+(1-u_r)(p_{2,t}-w_2)q_{2,t}\right]-u_r(1+I_r){\displaystyle \frac{k{e}^2}{2}}$$

(68)

$${\pi }_{m1}^{RF}=u_r(w_1-c)q_{1,g}+(1-u_r)(w_1-c)q_{1,t}$$

(69)

$${\pi }_{r1}^{RF}=(1+{\mu }_r)\left[u_r(p_{1,g}-w_1)q_{1,g}+(1-u_r)(p_{1,t}-w_1)q_{1,t}\right]$$

(70)

Proposition 5.

Under the retailer financing model and when the retailer holds an implicit equity stake in the manufacturer, when the retailer in the green supply chain chooses to bear the green investment cost, the optimal order quantity, $q_{i,g}^{RF*}$, and the optimal product greenness,$e^{RF*}$, exist in both supply chains.

Proof. 

When the retailer chooses to bear the cost of green investment, the order quantity of the green supply chain retailer is q_2,g, and the corresponding retail price is $p_{2,g}=a_2-q_{2,g}-b{q}_{1,g}+{\theta }_{2}e$. When the retailer chooses not to bear the cost of green investment, the order quantity of the green supply chain retailer is q_2,t, and the corresponding retail price is $p_{2,t}=a_2-q_{2,t}-b{q}_{1,t}$. Hence, substituting p_2,g and p_2,t into Equation (68) yields

$${\pi }_{r2}^{RF}=(1+{\mu }_r)\left[u_r(a_2-q_{2,g}-b{q}_{1,g}+{\theta }_{2}e-w_2)q_{2,g}+(1-u_r)(a_2-q_{2,t}-b{q}_{1,t}-w_2)q_{2,t}\right]-u_r(1+I_r){\displaystyle \frac{k{e}^2}{2}}$$

(71)

The first-order derivatives of Equation (71) with respect to q_2,g and q_2,t are, respectively,

$${{\pi }^{\prime }}_{r2}^{RF}={\displaystyle \frac{\partial {\pi }_{r2}^{RF}}{\partial q_{2,g}}}=(1+{\mu }_r)u_r(a_2-2q_{2,g}-b{q}_{1,g}+{\theta }_{2}e-w_2)$$

(72)

$${{\pi }^{\prime }}_{r2}^{RF}={\displaystyle \frac{\partial {\pi }_{r2}^{RF}}{\partial q_{2,t}}}=(1+{\mu }_r)(1-u_r)(a_2-2q_{2,t}-b{q}_{1,t}-w_2)$$

(73)

We continue to find the second-order derivatives of Equations (72) and (73) concerning q_2,g and q_2,t, respectively, and we obtain ${{\pi }^{″}}_{r2}^{RF}={\displaystyle \frac{{\partial }^2{\pi }_{r2}^{RF}}{\partial q_{2,g}^2}}=-2u_r(1+{\mu }_r)$ and ${{\pi }^{″}}_{r2}^{RF}={\displaystyle \frac{{\partial }^2{\pi }_{r2}^{RF}}{\partial q_{2,t}^2}}=-2(1-u_r)(1+{\mu }_r)$. Since $u_r\in \left(0,1\right),{\mu }_r>0$, the second-order derivatives are all less than 0, which indicates that Equation (71) is a convex function with respect to q_2,g and q_2,t. Therefore, Equation (71) has a maximum revenue value and the corresponding optimal ordering quantity.

Similarly, when the retailer bears the cost of green investments, the order quantity of the traditional supply chain retailer is q_1,g, and the corresponding retail price of the product is $p_{1,g}=a_1-q_{1,g}-b{q}_{2,g}-{\theta }_{1}e$. When the retailer chooses not to bear the cost of green investments, the order quantity of the traditional supply chain retailer is q_1,t, and the corresponding retail price of the product is $p_{1,t}=a_1-q_{1,t}-b{q}_{2,t}$. Hence, substituting p_1,g and p_1,t into Equation (70) yields

$${\pi }_{r1}^{RF}=(1+{\mu }_r)\left[u_r(a_1-q_{1,g}-b{q}_{2,g}-{\theta }_{1}e-w_1)q_{1,g}+(1-u_r)(a_1-q_{1,t}-b{q}_{2,t}-w_1)q_{1,t}\right]$$

(74)

The first-order derivatives of Equation (74) with respect to q_1,g and q_1,t are, respectively,

$${{\pi }^{\prime }}_{r1}^{RF}={\displaystyle \frac{\partial {\pi }_{r1}^{RF}}{\partial q_{1,g}}}=(1+{\mu }_r)u_r(a_1-2q_{1,g}-b{q}_{2,g}-{\theta }_{1}e-w_1)$$

(75)

$${{\pi }^{\prime }}_{r1}^{RF}={\displaystyle \frac{\partial {\pi }_{r1}^{RF}}{\partial q_{1,t}}}=(1+{\mu }_r)(1-u_r)(a_1-2q_{1,t}-b{q}_{2,t}-w_1)$$

(76)

We continue to find the second-order derivatives of Equations (75) and (76) concerning q_1,g and q_1,t. The second-order derivatives are all less than 0, which indicates that Equation (74) is a convex function with respect to q_1,g and q_1,t. Therefore, Equation (74) has a maximum revenue value and the corresponding optimal ordering quantity.

This makes Equations (72), (73), (75) and (76) equal to 0, solving Equations (72) and (75) and Equations (73) and (76), respectively, for the optimal order quantity:

$$q_{2,g}^{RF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}}$$

(77)

$$q_{1,g}^{RF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_1+b{\theta }_2)}{4-b^2}}$$

(78)

$$q_{2,t}^{RF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}$$

(79)

$$q_{1,t}^{RF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}$$

(80)

We substitute Equations (77) and (80) into Equation (71):

$$\begin{gathered} {\pi }_{r2}^{RF}=(1+{\mu }_r)\left[u_r(a_2-(1+b){\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}-(1-b){\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}}+{\theta }_{2}e-w_2)\right.\times \\ ({\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}})+(1-u_r)(a_2-{\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}- \\ \left.{\displaystyle \frac{2b(a_1-w_1)-b(a_2-w_2)}{4-b^2}}-w_2){\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}\right]-u_r(1+I_r){\displaystyle \frac{k{e}^2}{2}} \end{gathered}$$

We take the first-order derivative of the above equation with respect to e and make the first-order derivative equal to 0 to obtain the solution:

$$e^{RF}={\displaystyle \frac{2(1+{\mu }_r)(2{\theta }_2-b^2{\theta }_2-b{\theta }_1)\left[2(a_2-w_2)-b(a_1-w_1)\right]}{k(1+I_r)b^4+4b^3{\theta }_1{\theta }_2+(6{{\theta }_1}^2+8{{\theta }_2}^2-8k-8k{I}_r)b^2+8b{\theta }_1{\theta }_2-8{{\theta }_2}^2+16k(I_r+1)}}$$

(81)

We substitute Equation (81) into Equations (77) and (78) and summarize the optimal order quantity and the optimal product greenness of the two supply chains:

$$q_{2,g}^{RF\ast }={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{2B(2{\theta }_2+b{\theta }_1)(2a_2-2w_2-b{a}_1+b{w}_1)}{A(4-b^2)}}$$

(82)

$$q_{1,g}^{RF\ast }={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{2B(2{\theta }_1+b{\theta }_2)(2a_2-2w_2-b{a}_1+b{w}_1)}{A(4-b^2)}}$$

(83)

$$\begin{gathered} e^{RF\ast }={\displaystyle \frac{2B(2a_2-2w_2-b{a}_1+b{w}_1)}{A}} \\ {q}_{2,t}^{RF\ast }={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}} \\ {q}_{1,t}^{RF\ast }={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}} \\ A=k(1+I_r)b^4+4b^3{\theta }_1{\theta }_2+(6{{\theta }_1}^2+8{{\theta }_2}^2-8k-8k{I}_r)b^2+8b{\theta }_1{\theta }_2-8{{\theta }_2}^2+16k(I_r+1) \\ B=(1+{\mu }_r)(2{\theta }_2-b^2{\theta }_2-b{\theta }_1) \end{gathered}$$

(84)

In summary, it is demonstrated that under the retailer financing model and when the retailer holds the implicit equity of the manufacturer, the optimal order quantity, q^RF*, and the optimal product greenness, e^RF*, of the two supply chains are obtained. □

5. Analysis of Green Supply Chain Financing Decisions Based on Inter-Chain Competition and Implicit Equity Consideration

In Section 4, we constructed revenue functions under different financing models and implicit equity holding. Subsequently, we determined the optimal order quantity and level of product greenness under each model. To further reveal the relationship between key variables and green supply chain financing decisions, we will analyze the dynamic relationships between optimal order quantity, product greenness, supply chain members’ returns, and critical variables. This section systematically examines how these core variables fluctuate regarding the parameters involved, providing a more comprehensive and in-depth exploration of financing decisions within green supply chains.

5.1. Analysis of the Impact of Optimal Order Quantity on Green Supply Chain Financing Decisions

When solving for the optimal order quantity, q*, in Section 4, all the variables are treated as constants. However, as the competition among supply chains becomes more intense, these variables will change, leading to changes in, for example, green investment costs and retail prices, which will indirectly affect the optimal order quantity. Therefore, to better understand the dynamic relationship between these variables, the following section discusses the relationship between the optimal order quantity, q*, and the key variables.

Theorem 1.

Under the non-financing model and manufacturer financing model, and considering implicit equity, the optimal order quantity of the green supply chain increases with the increase in the underlying market demand of the green supply chain and the wholesale price of the traditional supply chain. Also, it decreases with the decrease in the wholesale price of the green supply chain and the underlying market demand of the traditional supply chain.

Proof. 

Firstly, through Section 3.1 and Section 3.2, it can be seen that the optimal order quantity of the green supply chain in the non-financing model is Equation (8), and, in the manufacturer’s model, it is Equation (28). Taking their first-order derivatives with respect to a₁, w₁, a₂, and w₂, we obtain

$$\begin{gathered} {q^{\prime }}_2^{NF*}={q^{\prime }}_{2,g}^{MF*}={\displaystyle \frac{\partial q_2^{NF*}/\partial q_{2,g}^{MF*}}{\partial a_1}}={\displaystyle \frac{-b}{4-b^2}} \\ {q^{\prime }}_2^{NF*}={q^{\prime }}_{2,g}^{MF*}={\displaystyle \frac{\partial q_2^{NF*}/\partial q_{2,g}^{MF*}}{\partial w_1}}={\displaystyle \frac{b}{4-b^2}} \\ {q^{\prime }}_2^{NF*}={q^{\prime }}_{2,g}^{MF*}={\displaystyle \frac{\partial q_2^{NF*}/\partial q_{2,g}^{MF*}}{\partial a_2}}={\displaystyle \frac{2}{4-b^2}} \\ {q^{\prime }}_2^{NF*}={\displaystyle \frac{\partial q_2^{NF*}}{\partial w_2}}={\displaystyle \frac{-2}{4-b^2}} \\ {q^{\prime }}_{2,g}^{MF*}={\displaystyle \frac{\partial q_{2,g}^{MF*}}{\partial w_2}}={\displaystyle \frac{-2}{4-b^2}}+{\displaystyle \frac{(1+{\mu }_m){(2{\theta }_2+b{\theta }_1)}^2}{{(4-b^2)}^{2}k(1+I_m)}} \end{gathered}$$

Because 0 ≤ b ≤ 1, therefore, ${q^{\prime }}_2^{NF*}={q^{\prime }}_{2,g}^{MF*}={\displaystyle \frac{\partial q_2^{NF*}/\partial q_{2,g}^{MF*}}{\partial a_1}}<0$, ${q^{\prime }}_2^{NF*}={q^{\prime }}_{2,g}^{MF*}={\displaystyle \frac{\partial q_2^{NF*}/\partial q_{2,g}^{MF*}}{\partial w_2}}<0$, ${q^{\prime }}_2^{NF\ast }={q^{\prime }}_{2,g}^{MF\ast }={\displaystyle \frac{\partial q_2^{NF*}/\partial q_{2,g}^{MF*}}{\partial w_1}}>0$, ${q^{\prime }}_2^{NF*}={\displaystyle \frac{\partial q_2^{NF*}}{\partial a_2}}>0$, ${q^{\prime }}_{2,g}^{MF*}={\displaystyle \frac{\partial q_{2,g}^{MF*}}{\partial a_2}}>0$. The first-order derivative greater than 0 indicates that the optimal order quantity varies positively with that variable (a₂ and w₁), and the first-order derivative less than 0 indicates that the optimal order quantity varies inversely with that variable (a₁ and w₂). Furthermore, the calculations also reveal similar conclusions in other cases. □

In summary, Theorem 1 shows that there is an inverse relationship between the optimal order quantity of the green supply chain and the wholesale price under the no-financing and manufacturer financing models. Specifically, without considering other external factors, the higher the wholesale price, the lower the order quantity. Furthermore, due to the existing inter-supply chain competition, when the traditional supply chain raises the wholesale price of its products, the market competitiveness of the green supply chain’s products will be enhanced, which will lead to an increase in the number of orders. Therefore, boosting the base demand of the market and capturing the potential market share have become the core strategies to enhance their competitiveness. To achieve these objectives, decision-makers within the supply chain must formulate a comprehensive set of strategies, such as trying to reduce production costs and wholesale prices, thereby ensuring product quality while increasing order quantities, which will further enhance the overall profitability and market competitiveness of the supply chain.

Theorem 2.

Under the manufacturer financing model and retailer financing model and considering implicit equity, when green supply chain financing is successful, the optimal order quantity of the green supply chain is positively related to consumers’ green preferences and inversely related to the manufacturer’s opportunity cost of capital and the green investment cost coefficient.

Proof. 

Considering the proof in Section 3.2 above, the optimal order quantity of the green supply chain when the manufacturer obtains financing is Equation (28), which is obtained by taking the first-order derivatives with respect to ${\theta }_2,{\theta }_1,k,I_m$ to obtain.

$$\begin{gathered} {\displaystyle \frac{\partial q_{2,g}^{MF\ast }}{\partial {\theta }_2}}={\displaystyle \frac{4(1+{\mu }_m)(w_2-c)(2{\theta }_2+b{\theta }_1)}{{(4-b^2)}^{2}k(1+I_m)}} \\ {\displaystyle \frac{\partial q_{2,g}^{MF\ast }}{\partial {\theta }_1}}={\displaystyle \frac{2b(1+{\mu }_m)(w_2-c)(2{\theta }_2+b{\theta }_1)}{{(4-b^2)}^{2}k(1+I_m)}} \\ {\displaystyle \frac{\partial q_{2,g}^{MF\ast }}{\partial k}}=-{\displaystyle \frac{(1+{\mu }_m){(2{\theta }_2+b{\theta }_1)}^2(w_2-c)}{{(4-b^2)}^2{k}^2(1+I_m)}} \\ {\displaystyle \frac{\partial q_{2,g}^{MF\ast }}{\partial I_m}}=-{\displaystyle \frac{(1+{\mu }_m){(2{\theta }_2+b{\theta }_1)}^2(w_2-c)}{{(4-b^2)}^{2}k{(1+I_m)}^2}} \end{gathered}$$

Because $w_2>c>0,{\mu }_m>0,{\theta }_2>0,{\theta }_1>0,0\le b\le 1,k>0,I_m>0$, therefore, ${\displaystyle \frac{\partial q_{2,g}^{MF\ast }}{\partial {\theta }_2}}>0$, ${\displaystyle \frac{\partial q_{2,g}^{MF\ast }}{\partial {\theta }_1}}>0$, ${\displaystyle \frac{\partial q_{2,g}^{MF\ast }}{\partial k}}<0$, ${\displaystyle \frac{\partial q_{2,g}^{MF\ast }}{\partial I_m}}<0$. Hence, the optimal order quantity is positively related to consumers’ green preferences and inversely related to the manufacturer’s opportunity cost of capital and the coefficient of green investment cost. The above process only demonstrates the relationship between the optimal order quantity of the green supply chain and these variables when manufacturer financing is successful. Similarly, the same relationship can be found between the optimal order quantity and these variables in the retailer financing model. □

Theorem 2 shows that, regardless of the retailer financing model or the manufacturer financing model, when green supply chain financing is successful, the enhancement of consumers’ preferences for green products leads them to be more willing to pay higher prices for these products, which drives the increase in the order quantity of green products. However, the rise in the opportunity cost of capital for the manufacturer and the increase in the cost coefficient of green investment may inhibit the order quantity of green products. To address this issue, the manufacturer in the green supply chain may resort to lowering the greenness of their products to control costs. At the same time, the manufacturer may set higher wholesale prices for the retailer to obtain greater profit margins. Hence, green supply chain members should try to increase consumers’ green preference for green products and reduce the capital opportunity cost and green investment cost coefficient to obtain more orders and ultimately increase revenue.

Theorem 3.

Under the manufacturer financing model and retailer financing model, and considering implicit equity, when green supply chain financing succeeds, the optimal order quantity of both supply chains is positively related to the product greenness.

Proof. 

Considering the proof in Section 3.2 and Section 3.3 above, it can be concluded that the optimal order quantities of the two supply chains when the green supply chain obtains financing under either the manufacturer financing model or the retailer financing model can be found in Equations (23), (24), (41), (42), (60), (61), (77) and (78), which are collapsed and can be obtained as follows.

$$q_{2,g}^{MF}=q_{2,g}^{RF}={\displaystyle \frac{2(a_2-w_2)-b(a_1-w_1)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_2+b{\theta }_1)}{4-b^2}}$$

(85)

$$q_{1,g}^{MF}=q_{1,g}^{RF}={\displaystyle \frac{2(a_1-w_1)-b(a_2-w_2)}{4-b^2}}+{\displaystyle \frac{e(2{\theta }_1+b{\theta }_2)}{4-b^2}}$$

(86)

The first-order derivatives of Equations (85) and (86) concerning e are

$$\begin{gathered} {q^{\prime }}_{2,g}^{MF}={q^{\prime }}_{2,g}^{RF}={\displaystyle \frac{\partial q_{2,g}^{MF}/\partial q_{2,g}^{RF}}{\partial e}}={\displaystyle \frac{2{\theta }_2+b{\theta }_1}{4-b^2}} \\ {q^{\prime }}_{1,g}^{MF}={q^{\prime }}_{1,g}^{RF}={\displaystyle \frac{\partial q_{1,g}^{MF}/\partial q_{1,g}^{RF}}{\partial e}}={\displaystyle \frac{2{\theta }_1+b{\theta }_2}{4-b^2}} \end{gathered}$$

Because ${\theta }_1>0,{\theta }_2>0,0\le b\le 1$, therefore, ${q^{\prime }}_{2,g}^{MF}={q^{\prime }}_{2,g}^{RF}={\displaystyle \frac{\partial q_{2,g}^{MF}/\partial q_{2,g}^{RF}}{\partial e}}>0$, ${q^{\prime }}_{1,g}^{MF}={q^{\prime }}_{1,g}^{RF}={\displaystyle \frac{\partial q_{1,g}^{MF}/\partial q_{1,g}^{RF}}{\partial e}}>0$. Hence, both in the manufacturer financing model and the retailer financing model, the optimal order quantity of the two supply chains is positively related to the product greenness. □

Theorem 3 shows that, under the retailer financing model and the manufacturer financing model, when green supply chain financing is successful, the increase in product greenness leads to a rise in the order quantity of the green supply chain. This is due to the growing environmental awareness of consumers, which makes them more inclined to purchase goods with higher greenness, thus driving up the sales volume of green products. Moreover, although there is a competitive relationship between the green supply chain and the traditional supply chain in the market, improvements in product greenness can also lead to an increase in the sales of traditional products. This is because, as product greenness continues to improve, the price of green products will continue to rise. Meanwhile, the prices of traditional products experience a relative decline, giving traditional products a more price-competitive advantage. Thus, the lower the green preference of consumers, the higher their preference for lower-priced traditional products, which leads to an increase in the number of orders for traditional products. However, this increase does not lead to an increase in total revenue for the traditional supply chain retailer but rather a decrease in overall revenue due to a more significant drop in prices. In contrast, the green supply chain can achieve higher returns while enhancing product greenness. Therefore, improving product greenness not only helps the green supply chain capture potential demand markets and enhance supply chain competitiveness but also contributes to better environmental performance and shapes a positive environmental image for the company.

5.2. Analysis of the Impact of Product Greenness on Green Supply Chain Financing Decisions

In the previous section, when studying the optimal product greenness, e*, of the green supply chain, the significant variables, such as consumer’s green preference, θ_i, and capital opportunity cost, I_j, are considered constants, yet different θ_i and I_j values will lead to changes in the manufacturer’s green production costs, as well as the retail price of the product, and also lead to consequent changes in the optimal product greenness of the green supply chain. Therefore, the relationship between product greenness and core variables will be explored next.

Theorem 4.

Under the manufacturer financing model and considering implicit equity, when green supply chain financing is successful, product greenness is inversely related to production costs, green investment cost coefficients, and the opportunity cost of the manufacturer’s capital, and positively related to consumers’ green preferences and the wholesale price of the green supply chain.

Proof. 

Under the manufacturer financing model and the two implicit equity holding methods, when manufacturer financing is successful, the above model part of the solution process can determine the product greenness under the two implicit equity holding methods using Equations (27) and (45) and the first-order derivatives of Equation (27) concerning $c,k,I_m,{\theta }_2,{\theta }_1,w_2$:

$$\begin{gathered} {e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial c}}={\displaystyle \frac{-(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)}{(4-b^2)k(1+I_m)}} \\ {e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial k}}={\displaystyle \frac{-(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)(w_2-c)(4-b^2)(1+I_m)}{{\left[(4-b^2)k(1+I_m)\right]}^2}} \\ {e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial I_m}}={\displaystyle \frac{-(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)(w_2-c)(4-b^2)k}{{\left[(4-b^2)k(1+I_m)\right]}^2}} \\ {e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial {\theta }_2}}={\displaystyle \frac{2(1+{\mu }_m)(w_2-c)}{(4-b^2)k(1+I_m)}} \\ {e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial {\theta }_1}}={\displaystyle \frac{b(1+{\mu }_m)(w_2-c)}{(4-b^2)k(1+I_m)}} \\ {e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial w_2}}={\displaystyle \frac{(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)}{(4-b^2)k(1+I_m)}} \end{gathered}$$

Because ${\mu }_m>0,w_2>c,{\theta }_2>0,{\theta }_1>0,0\le b\le 1,k>0,I_m>0$, therefore, ${e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial {\theta }_2}}>0$, ${e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial {\theta }_1}}>0$, ${e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial w_2}}>0$, ${e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial c}}<0$, ${e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial k}}<0$,${e^{\prime }}^{MF}={\displaystyle \frac{\partial e^{MF}}{\partial I_m}}<0$. The first-order derivative greater than 0 indicates that the product greenness is positively related to the variable (${\theta }_2,{\theta }_1,w_2$), and the first-order derivative less than 0 indicates that the product greenness is negatively related to the variable ($c,k,I_m$). Similarly, the derivatives of Equation (45) concerning these variables also lead to the same conclusion. □

Theorem 4 shows that regardless of the type of implicit equity holding, when the manufacturer in the green supply chain faces financing choices and the market presents a combination of higher wholesale prices and relatively lower production costs, it can achieve significant economic returns. This scenario also provides significant incentives for the manufacturer to increase its green capital investment, thereby enhancing product greenness. Furthermore, as consumers’ preference for green products increases, they are willing to pay higher prices for greener products. This market preference not only boosts the competitiveness of green products but also pushes the manufacturer to improve the product’s greenness further. However, it is also important to note that rising capital opportunity costs and green investment cost coefficients for the manufacturer may lead to higher production costs, which could force the manufacturer to reduce product greenness to control costs. To address this issue, cooperation between the manufacturer and the retailer is crucial, not only to drive the manufacturer to continuously enhance product greenness, but also to enhance overall environmental performance.

Theorem 5.

Under the retailer financing model and considering implicit equity, when green supply chain financing is successful, the product greenness is inversely proportional to the wholesale price of the green supply chain, positively proportional to the wholesale price of the traditional supply chain, and inversely proportional to the coefficient of the green investment cost. At the same time, when condition A is satisfied, it is also inversely proportional to the opportunity cost of the retailer’s capital.

Proof. 

Under the retailer financing model, either the manufacturer holds the implicit equity of the retailer or the retailer holds the implicit equity of the manufacturer; the optimal product greenness is found in the proof process in Section 4.3, above. These processes are Equations (64) and (81) for the two different implicit equity holdings, respectively. Taking the first-order derivatives of Equation (81) concerning w₁, w₂, k, and I_r, yields

$$\begin{gathered} {e^{\prime }}^{RF}={\displaystyle \frac{\partial e^{RF}}{\partial w_1}}={\displaystyle \frac{b\times B}{A}} \\ {e^{\prime }}^{RF}={\displaystyle \frac{\partial e^{RF}}{\partial w_2}}={\displaystyle \frac{-2\times B}{A}} \\ {e^{\prime }}^{RF}={\displaystyle \frac{\partial e^{RF}}{\partial k}}={\displaystyle \frac{-2B\times D\times \left[{(b^2-4)}^2(1+I_r)\right]}{A^2}} \\ {e^{\prime }}^{RF}={\displaystyle \frac{\partial e^{RF}}{\partial I_r}}={\displaystyle \frac{-2B\times D\times \left[b^{4}k-8b^2(1+k)+16k\right]}{A^2}} \\ A=k(1+I_r)b^4+4b^3{\theta }_1{\theta }_2+(6{{\theta }_1}^2+8{{\theta }_2}^2-8k-8k{I}_r)b^2+8b{\theta }_1{\theta }_2-8{{\theta }_2}^2+16k(I_r+1) \\ B=(1+{\mu }_r)(2{\theta }_2-b^2{\theta }_2-b{\theta }_1) \\ D=(2a_2-2w_2-b{a}_1+b{w}_1) \end{gathered}$$

Because $A>0,B>0,b>0,D>0,{(b^2-4)}^2\times (1+I_r)>0$, therefore, ${e^{\prime }}^{RF}={\displaystyle \frac{\partial e^{RF}}{\partial w_1}}>0$, ${e^{\prime }}^{RF}={\displaystyle \frac{\partial e^{RF}}{\partial w_2}}<0$, ${e^{\prime }}^{RF}={\displaystyle \frac{\partial e^{RF}}{\partial k}}<0$. Moreover, when $b^{4}k-8b^2(1+k)+16k>0$, ${e^{\prime }}^{RF}={\displaystyle \frac{\partial e^{RF}}{\partial I_r}}<0$. This proves the relationship between product greenness and the variation in the relevant variables. Similarly, the process of establishing a relationship between Equation (64) and the variation in the relevant variables is the same as that of Equation (81). □

Theorem 5 shows that under the retailer financing model, regardless of the implicit equity holding, the retailer tends to reduce the product greenness when the wholesale price of products in the green supply chain is high. Conversely, it is more likely to increase product greenness when the wholesale price of products in the traditional supply chain is low. This is in contrast to the conclusion of Theorem 4, where higher wholesale prices mean higher revenues for the manufacturer, so this tends to produce greener products to meet market demand. However, for the retailer downstream in the supply chain, higher wholesale prices directly increase its costs, making it more likely to reduce product greenness to maintain profitability in the face of cost pressures. Moreover, since the selling price of a product is linked to the order quantity in both supply chains, the retailer must consider the impact of competing supply chains when making decisions. Specifically, when the wholesale price of the traditional supply chain rises, the green supply chain gains a price advantage. This motivates the retailer in the green supply chain to increase product greenness to attract more consumers. Moreover, under certain conditions, a higher coefficient of the opportunity cost of capital and green investment costs for the retailer can increase financing costs, which may lead to the retailer choosing to lower product greenness to reduce costs, aligning with the conclusion of Theorem 4. Therefore, during the financing process, the retailer not only needs to pay attention to the improvement of product greenness to enhance competitiveness, but also should strengthen the examination and study of competitors’ market information, which provides more accurate market development strategies and can help to achieve the optimization of both economic benefits and environmental performance.

5.3. Analysis of the Impact of Supply Chain Members’ Returns on Green Supply Chain Financing Decisions

In the previous section, the optimal order quantity and the optimal product greenness under various modes were all derived in the model construction process. These variables are next substituted into the revenue model, and the relationship between the revenue of each member of the supply chain and the core variables is investigated.

Theorem 6.

When green supply chain financing is successful, traditional supply chain manufacturers’ returns are inversely related to the opportunity cost of capital, retailers’ returns are positively related to the opportunity cost of capital, and green supply chain manufacturers’ and retailers’ returns are inversely related to the opportunity cost of capital.

Proof. 

The optimal order quantity and the green level of the product of the green supply chain under inter-chain competition with implicit equity considerations are proved in Propositions 2 and 3, partially by substituting them into the gain where they —are not substituted as a whole since A and B are independent of I_m—and substituting Equations (28)–(30) into Equations (13) and (15) to obtain

$$\begin{gathered} {\pi }_{m2}^{MF}=(1+{\mu }_m)\left[u_m(w_2-c)(q_{2,t}+{\displaystyle \frac{(1+{\mu }_m){(2{\theta }_2+b{\theta }_1)}^2(w_2-c)}{{(4-b^2)}^{2}k(1+I_m)}})+(1-u_m)(w_2-c)q_{2,t}\right]-{\displaystyle \frac{u_m(1+I_m)k}{2}}{\left[{\displaystyle \frac{(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)(w_2-c)}{(4-b^2)k(1+I_m)}}\right]}^2 \\ {\pi }_{m1}^{MF}=(1+{\mu }_m)\left[u_m(w_1-c)(q_{1,t}+{\displaystyle \frac{(1+{\mu }_m)(2{\theta }_2+b{\theta }_1)(2{\theta }_1+b{\theta }_2)(w_2-c)}{{(4-b^2)}^{2}k(1+I_m)}})+(1-u_m)(w_1-c)q_{1,t}\right] \end{gathered}$$

The first-order derivatives of the above equation with respect to I_m are obtained.

$$\begin{gathered} {\displaystyle \frac{\partial {\pi }_{m2}^{MF}}{\partial I_m}}=-{\displaystyle \frac{u_m{(2{\theta }_2+b{\theta }_1)}^2{(w_2-c)}^2{(1+{\mu }_m)}^2}{2{(4-b^2)}^{2}k{(1+I_m)}^2}} \\ {\displaystyle \frac{\partial {\pi }_{m1}^{MF}}{\partial I_m}}=-{\displaystyle \frac{u_m(2{\theta }_2+b{\theta }_1)(2{\theta }_1+b{\theta }_2)(w_2-c)(w_1-c){(1+{\mu }_m)}^2}{{(4-b^2)}^{2}k{(1+I_m)}^2}} \end{gathered}$$

Because $u_m>0,w_2\ge w_1>c,{\theta }_2>0,{\theta }_1>0,k>0,I_m>0$, therefore, ${\displaystyle \frac{\partial {\pi }_{m2}^{MF}}{\partial I_m}}<0$, ${\displaystyle \frac{\partial {\pi }_{m1}^{MF}}{\partial I_m}}<0$. The above demonstrates that the manufacturer’s returns of the two supply chains are inversely related to the opportunity cost of capital. Similarly, the relationship between the retailer’s return and the opportunity cost of capital is ${\displaystyle \frac{\partial {\pi }_{r2}^{MF}}{\partial I_r}}<0$ and ${\displaystyle \frac{\partial {\pi }_{r1}^{MF}}{\partial I_r}}>0$. Thus, Theorem 6 is proved. □

In summary, Theorem 6 shows that in the green supply chain, when the opportunity cost of the manufacturer’s capital increases, this leads to an increase in financing cost, which in turn results in a decrease in revenue. Therefore, the manufacturer may choose to lower the environmental standards of its products to control costs and maintain its revenue levels. However, raising the environmental standards of products requires the concerted efforts of many parties, and the manufacturer may raise wholesale prices to increase its revenue and invest in environmentally friendly technologies. Governments and social institutions should promote a shift in demand toward green products by increasing consumer awareness of environmental protection through publicity. At the same time, governments can provide incentives for green product production and implement strict regulations to reduce cost pressures on manufacturers while ensuring enforcement of environmental standards. Furthermore, strengthening cooperation between research institutions and enterprises, along with increasing funding for research and development of environmental technologies, is crucial for promoting technological innovation. These measures will help to reduce pollution, improve the ecological environment, and promote sustainable economic development, ultimately achieving a win–win situation for both the environment and the economy.

6. Simulation Studies

To verify the reasonableness of the model’s construction and to analyze whether the manufacturer or the retailer has the motivation to finance the production of green products, as well as which party is more inclined to pay the green investment cost, in this paper, we use simulation analysis to more clearly compare the optimal order quantity, product greenness, and supply chain members’ revenue under different financing modes and different implicit equity holdings with changes in each parameter, referring to the parameter settings of Wang et al. [33], Zhang et al. [36], Xiao and Luo [39], and Heydari et al. [40]. In this paper, it is assumed that the parameter settings are as follows: c = USD 10/unit, w₁ = USD 40/unit, w₂ = USD 45/unit, a₁ = a₂ = 300, b = 0.3, u_m = 0.5, u_r = 0.7, θ₁ = 1, θ₂ = 1.2, I_r = I_m = 0.1, k = 20, and μ_m = μ_r = 0.5. The simulation analysis is carried out based on the above parameter settings using the MATLAB 2021a software.

6.1. Analysis of the Impact of Consumer Green Preference and Capital Opportunity Cost on the Optimal Order Quantity of Green Supply Chain

Figure 2 depicts how the optimal order quantity of the green supply chain varies with consumers’ green preferences under different financing models (retailer financing and manufacturer financing).

The results indicate that the retailer financing model performs better in increasing the optimal order quantity, as it can provide more substantial financial support for green production and procurement. Under both models, the optimal order quantity increases with the level of consumers’ green preference, which corroborates Theorem 2 regarding the positive correlation between the two variables. When consumers’ green preference is low, the difference in order quantities between the two financing models is marginal. However, as consumer green preference strengthens, the gap widens considerably, with the retailer financing model exhibiting a more significant increase in order quantities. Therefore, from the perspective of optimal order quantity, the green supply chain tends to prefer the retailer financing model when consumers’ green preference is higher. This approach more effectively meets and stimulates growing market demand and promotes greener products to unlock further market potential. In contrast, the manufacturer financing model shows relatively limited performance in terms of order quantity expansion.

Figure 3 depicts the variation in optimal order quantity with capital opportunity costs under different financing models. To simplify the analysis, we assume that the horizontal coordinates under different financing models denote the opportunity cost of capital for various supply chain members (see below). As can be seen from the figure, the green supply chain under the retailer financing model shows a higher optimal order quantity, which is attributed to the fact that the retailer is directly facing the market and can respond to changes in consumer demand more agilely. Meanwhile, regardless of the financing model, an increase in the opportunity cost of capital leads to a decrease in the optimal order quantity, which is consistent with the conclusion of Theorem 2. The rising opportunity cost of capital increases the cost of green supply chain members, prompting them to reduce product greenness to cut costs, which in turn will lead to a decrease in the order quantity of green products. Furthermore, in a competitive supply chain environment, when the green supply chain manufacturer faces a capital shortage, cooperating with the retailer becomes a key strategy. By persuading the retailer to bear the cost of green investment and participate in the decision-making regarding product greenness, the order quantity of products can be promoted. Moreover, to increase the order quantity, the manufacturer and the retailer should strive to reduce the opportunity cost of capital during the financing process. Thus, for the green supply chain, the optimal order quantity under the retailer financing model is superior to that under the manufacturer financing model. Therefore, the manufacturer should enhance its credit ratings and actively cooperate with the retailer to promote green product investment and production, thereby increasing the order quantity of green products.

6.2. Analysis of the Impact of Consumer Green Preference and Financial Opportunity Cost on Product Greenness in Green Supply Chain

Figure 4 and Figure 5 depict the trends in product greenness differences with consumer green preferences and capital opportunity costs under the two financing models, respectively. It can be observed that the difference in product greenness under the two models is always greater than 0, indicating that the product greenness achieved under the retailer financing mode is higher compared to the manufacturer financing model. It also means that retailer financing is more conducive to enhancing product greenness, thereby benefiting environmental protection. Because the retailer is directly connected to consumers, it possesses a wealth of consumer information and can accurately capture the growing trend in demand for green products. Therefore, retailer financing can lead to a higher level of product greenness, stimulating market demand and increasing the overall profitability of the supply chain. Hence, when the manufacturer faces financial constraints and struggles to complete the production of green products on its own, it should actively seek cooperation with the retailer. By encouraging the retailer to make green investments, the manufacturer can not only solve their financial problems, but also expand market demand, resulting in a win–win situation with the retailer. Furthermore, the retailer has an incentive to invest in improving environmental performance, and it can use information on market demand to further enhance its competitiveness by requiring specific levels of product greenness from the manufacturer.

Moreover, (a) represents the implicit equity owned by the manufacturer, while (b) represents the implicit equity owned by the retailer (the same applies below). The figure reveals that regardless of how implicit equity is allocated, the retailer financing consistently leads to higher product greenness compared to manufacturer financing. The difference in greenness is positively influenced by green consumers’ preferences and the manufacturer’s cost of capital, but negatively affected by traditional consumers’ preferences and the retailer’s capital opportunity cost. Notably, green performance is strongest when the retailer holds implicit equity, especially when green consumer preference is high and traditional preference is low. Moreover, lower capital opportunity cost for the retailer amplifies the advantage of retailer financing in enhancing environmental performance. These findings underscore the strategic value of retailer financing in promoting green product features and strengthening the supply chain’s ecological and social image.

6.3. Analysis of the Impacts of Consumer Green Preferences and Financial Opportunity Costs on the Benefits of Green Supply Chain Members

6.3.1. Analysis of the Impact of Financial Opportunity Costs on the Benefits of Green Supply Chain Members

Figure 6 and Figure 7 depict the impact of changes in the opportunity cost of capital on manufacturers’ revenue in the two supply chains under different financing models. It can be observed that under the no-financing model, the manufacturer’s revenue remains constant. In contrast, under the other two financing models, the manufacturer’s revenue decreases as the opportunity costs of capital increase. According to Theorems 4 and 5, there is an inverse relationship between product greenness and the opportunity cost of capital. When the opportunity cost of capital increases, product greenness decreases correspondingly, resulting in a decline in supply chain order quantities and a reduction in the manufacturer’s revenue. Therefore, the manufacturer should strive to reduce the opportunity cost of capital. Moreover, comparing (a) and (b) reveals that when manufacturers possess implicit equity value, they tend to achieve higher returns.

Furthermore, for manufacturers in both traditional and green supply chains, the retailer financing model yields the highest revenue, and financing in general leads to higher manufacturer profits compared to the non-financing model. This indicates that the financing model, particularly when led by retailers, significantly enhances manufacturers’ profitability. Concurrently, financing mechanisms effectively stimulate manufacturers’ revenue growth by increasing the environmental sustainability of products and increasing order volumes. Within green supply chains, financing not only alleviates manufacturers’ capital pressures but also motivates greater focus on green product R&D and production, thereby strengthening market competitiveness. Secondly, the retailer-led financing model emerges as the optimal choice for manufacturers due to retailers’ proximity to the consumer end and their sharper market demand insights. This enables them to significantly boost manufacturers’ revenues through order-driven demand. In contrast, manufacturers’ self-financing faces challenges such as information asymmetry and higher financing costs, resulting in limited profit growth. Therefore, from the perspective of resource allocation and risk sharing, encouraging retailer participation in financing is more conducive to overall supply chain coordination and efficiency improvements.

In summary, in the context of inter-chain competition and implicit equity, when it is difficult for the manufacturer of the green supply chain to obtain financial support, cooperation with the retailer becomes the best choice. The retailer’s decision on product greenness and the bearing of the green investment cost not only promotes the production and sale of green products, but also enhances the manufacturer’s revenue. If the retailer refuses financing, the manufacturer may consider financing itself to achieve higher revenue than the no-financing model. Therefore, the retailer financing model is the optimal choice for manufacturers in both supply chains. Consequently, the manufacturer should strive to improve its credit rating to attract the retailer to invest in green product production.

Figure 8 and Figure 9 depict how retailer revenue in the two supply chains changes as the opportunity costs of capital change under different financing models. Moreover, comparing (a) and (b) reveals that retailers can earn higher returns when they own implicit equity.

For the traditional supply chain retailers depicted in Figure 8, the non-financing model yields the highest revenue, followed by manufacturer financing, with retailer financing resulting in the lowest returns. The underlying reason lies in the fact that although financing enhances order quantity by improving product greenness, the increased green attributes intensify price competition across chains, thereby driving down-selling prices. Since the decline in price outweighs the benefits from a higher sales volume, overall revenue decreases. This effect is particularly pronounced under the retailer financing model, where the highest level of greenness further suppresses the selling price. Hence, traditional retailers tend to avoid financing to maintain price stability and profit margins. Moreover, as the opportunity cost of capital increases, the level of green investment decreases, leading to greater product homogenization and subsequently contributing to price recovery, which in turn improves traditional retailers’ revenue. This suggests that in competitive markets, traditional enterprises must carefully consider strategies for green transformation through financing, balancing green investment against price competition.

In contrast, the green supply chain retailers in Figure 9 achieve the highest revenue under self-financing, followed by manufacturer financing, with the non-financing model performing the worst. This is because under retailer financing, significantly enhanced product greenness not only stimulates market demand but also supports higher sales premiums, thereby increasing both volume and price and ultimately raising revenue. Therefore, green retailers are more inclined to self-finance to leverage their differentiated green advantages. At the same time, reducing the opportunity cost of capital helps further improve green levels, expand market demand, and enhance profitability. This indicates that green supply chain enterprises can achieve a virtuous cycle of competitive advantage and revenue growth by lowering financing costs and increasing green innovation.

In summary, in the context of inter-chain competition and implicit equity, the optimal financing model choice varies for retailers in different supply chains. For the traditional supply chain retailer, the non-financing model is the optimal choice to maximize revenue, while the green supply chain retailer can achieve the highest revenue through self-financing. Moreover, the green supply chain retailer should reduce the opportunity cost of capital, as this not only helps to enhance the revenue, but also effectively promotes both economic and environmental performance.

6.3.2. Analysis of the Impact of Consumers’ Green Preferences on the Benefits of Green Supply Chain Members

Figure 10 and Figure 11 depict how the revenue of each member of the green supply chain changes as the green preferences of consumers in the traditional supply chain change under different financing models. Moreover, comparing (a) and (b) reveals that different members of the green supply chain can achieve higher revenue when they own implicit equity.

Firstly, for the retailer in the green supply chain depicted in Figure 10, self-financing is the optimal choice when the green preference of consumers in the traditional supply chain is below a specific threshold. Compared to the no-financing model, the retailer’s revenue is significantly higher under the other two financing models. However, as the green preference of consumers in the traditional supply chain increases, the retailer’s revenue decreases. This trend is attributable to an inverse relationship between product greenness and the level of green preference in the traditional channel. Increased consumer environmental awareness actually leads to reduced product greenness, which in turn lowers sales and profitability. Hence, when green preference is relatively low in the traditional segment, the retailer achieves higher returns through self-financing.

Secondly, from the perspective of green supply chain manufacturers, as depicted in Figure 11, retailer financing emerges as the most beneficial option. While self-financing by the manufacturer yields a modest revenue increase compared to the no-financing case, it offers limited incentive. In contrast, the retailer financing enhances product greenness, resulting in larger order quantities and higher revenue for the manufacturer. Consequently, manufacturers show a marked preference for retailer-led financing, which better aligns with their economic interests and supports greater value creation through improved green performance.

In summary, in the presence of both inter-supply chain competition and implicit equity, for different members of the green supply chain, their optimal financing models are different, with the optimal financing model for the manufacturer being the retailer financing model, and the retailer’s choice of the optimal financing model being related to the magnitude of the traditional supply chain’s consumer green preferences. Moreover, their revenues will all decrease as the green preferences of consumers in the traditional supply chain increase. Therefore, when developing financing strategies, green supply chain members should analyze the traditional supply chain consumers’ information in depth, especially to have a clear understanding of the extent of their green product preferences, so that they can choose financing models more effectively.

Figure 12 and Figure 13 depict how the revenue of members changes as the green preferences of green supply chain consumers change under different financing models, respectively. Moreover, comparing (a) and (b) reveals that different members of the green supply chain can obtain higher revenue when they own implicit equity.

Firstly, for the retailer in the green supply chain depicted in Figure 12, self-financing is the optimal choice. As consumers’ green preferences increase, retailers show a significantly stronger motivation to seek financing. Compared with the non-financing model, retailers consistently demonstrate a relatively high willingness to obtain financing. Even in a competitive environment, producing green products can enhance profitability and market competitiveness. When compared to manufacturer financing, retailers show a greater inclination toward self-financing, particularly when consumers’ green preference is high. The reason is that higher green preference drives sales and revenue growth for green products, thereby strengthening the economic advantage of self-financing. Conversely, when consumers’ green preferences are low, retailers’ financing motivation decreases accordingly. Therefore, when making financing decisions, retailers need to fully assess the level of consumers’ green preferences to select the most appropriate financing method and improve capital allocation efficiency and economic returns.

On the other hand, the manufacturer in the green supply chain depicted in Figure 13 achieves maximum revenue under the retailer financing model. Although self-financing can increase manufacturers’ revenue, the growth is relatively limited, resulting in a weaker motivation for self-financing and a greater preference for retailer-led financing. Retailer financing not only effectively enhances product greenness but also expands the order scale for green products, thereby significantly increasing manufacturers’ income. At the same time, manufacturers’ revenue is highly dependent on the intensity of consumers’ green preferences: the stronger the green preference, the higher the manufacturers’ revenue. This suggests that manufacturers should actively collaborate with retailers to conduct market research and data analysis on consumers’ green preferences, especially when green preferences are high. By working together to improve product greenness, it is possible to increase the overall benefits of the supply chain.

In summary, in the presence of both inter-supply chain competition and implicit equity, for different members of the green supply chain, their optimal equilibrium financing models are all retailer financing models. Meanwhile, their revenue rises with an increase in the green preferences of consumers in the green supply chain. Therefore, members of the green supply chain should work closely together to raise consumer awareness and demand for green products through extensive publicity and promotion, educational activities, and other initiatives to stimulate retailers’ motivation to finance green products, thereby realizing a win–win situation for supply chain members.

6.4. Analysis of Green Supply Chain Equilibrium Financing Model

Through the simulation analysis based on the above values, it can be concluded that in the environment of supply chain competition and implicit equity, the manufacturers and retailers of the two supply chains choose their respective equilibrium financing models and implicit equity holding methods. The conclusions are as follows.

(i) In an environment where inter-chain competition and implicit equity coexist, for the traditional supply chain manufacturer, choosing the retailer financing model in the green supply chain is the equilibrium financing model. Specifically, when the green supply chain adopts the retailer financing model, the traditional supply chain manufacturer can achieve higher revenue. Meanwhile, its implicit equity holding is more beneficial to its revenue. This can be attributed to the intense competition in product greenness among different supply chain retailers, which drives an increase in product greenness. For the manufacturer, although the wholesale price of products remains relatively stable, the enhancement of product greenness leads to increased order quantities within the traditional supply chain, thereby boosting the revenue of the traditional supply chain manufacturer. Therefore, under inter-chain competition and implicit equity environments, the retailer financing model emerges as the optimal financing strategy for the traditional supply chain manufacturer.

(ii) In an environment where inter-chain competition and implicit equity coexist, for the traditional supply chain retailer, choosing the no-financing model in the green supply chain represents the equilibrium financing model. Specifically, the traditional supply chain retailer should implement measures to prevent the green supply chain from securing financing, thereby encouraging it to produce traditional products to maximize revenue. Meanwhile, it is more beneficial for the retailer to hold implicit equity. This is because, in the current market environment, green products enjoy a broader consumer base, and their production can lead to higher selling prices and market shares for the retailer, ultimately boosting its revenue. However, since the traditional supply chain is clearly focused on the production of traditional products, if the financial constraints of the green supply chain can also shift to the production of traditional products, it can reduce the competition they face from the green product market, thereby maximizing the revenue of the traditional supply chain manufacturer. Therefore, under inter-chain competition and implicit equity environments, the no-financing model is the optimal financing model for the traditional supply chain retailer.

(iii) In an environment where inter-chain competition and implicit equity coexist, for the green supply chain manufacturer, choosing a retailer financing model in the green supply chain is the equilibrium financing model. Specifically, the green supply chain manufacturer should actively promote retailer financing to cover the costs of green investments, which is optimal for the manufacturer. Meanwhile, it is more beneficial for the manufacturer to hold implicit equity. This is because, in a competitive market environment, if the green supply chain wants to gain more market share and enhance its competitiveness, it should choose to finance the production of green products. However, the main difference between the two financing models is the significant difference in the greenness of the products decided upon. The retailer has a clearer understanding of demand information due to its direct access to consumers and the availability of capital. Therefore, the green supply chain manufacturer adopting the retailer financing model is conducive to lowering the cost of green investment and realizing a more lucrative return from the continuous growth of consumer demand for green products. Therefore, under inter-chain competition and implicit equity environments, the retailer financing model is the optimal financing model for the green supply chain manufacturer.

(iv) In an environment where inter-chain competition and implicit equity coexist, for the green supply chain retailer, the choice of financing model for the green supply chain is a balanced financing model only when certain conditions are met regarding consumer preferences for green products. Specifically, the green supply chain retailer is willing to finance and bear the cost of green investments only under certain conditions. Meanwhile, the retailer holding implicit equity is more favorable to their returns. Although the retailer’s costs increase due to the green investment, this can significantly enhance product greenness, which leads to an increase in the selling price of the product and stimulates consumer demand for greener products, which in turn leads to higher revenue for the retailer. However, the retailer will finance only when consumers’ green preferences are below a specific threshold; otherwise, it prefers to let the manufacturer finance.

7. Conclusions

This paper addresses green supply chain financing decisions based on inter-chain competition and implicit equity considerations. First, it establishes a Cournot competitive supply chain revenue model with no-financing, manufacturer financing, and retailer financing modes for the green supply chain while considering two implicit equity holding models. It calculates the optimal order quantities and optimal product greenness for both supply chains (traditional supply chain and green supply chain) under different financing models and implicit equity holding models. Subsequently, for the green supply chain under the scenario of inter-chain competition and implicit equity consideration, the influence mechanism of related variables on the optimal order quantity and product greenness of the green supply chain is deeply analyzed through mathematical derivation, and we obtained the following conclusions.

(i) When inter-chain competition and implicit equity are considered, an increase in consumer green preferences in both supply chains leads to an increase in the optimal order quantity of the green supply chain, while an increase in the opportunity cost of financing leads to a decrease in the optimal order quantity.

(ii) When green supply chain financing is successful, an increase in the opportunity cost of funds will lead to a reduction in product greenness, while an increase in consumer preference for green products will enhance product greenness. Moreover, the higher the product greenness, the greater the order quantity will be.

(iii) In the context of inter-chain competition and implicit equity, the green supply chains have higher optimal order quantities and product greenness in the retailer financing model than in the manufacturer financing model.

Further, the effects of consumer green preference, capital opportunity cost, and other factors on supply chain members’ revenue, optimal order quantity, and product greenness were analyzed through numerical simulation, and we obtained the following conclusions.

(i) When the traditional supply chain competes with the green supply chain, the manufacturers or retailers of the two supply chains can achieve higher revenue when they own implicit equity.

(ii) When green supply chain financing is successful, an increase in the opportunity costs of funds leads to a decrease in the benefits of green supply chain members. An increase in the green preference of traditional supply chain consumers leads to a reduction in the benefits of green supply chain members, while an increase in the green preference of green supply chain consumers leads to an increase in the benefits of green supply chain members.

(iii) When considering inter-chain competition and implicit equity, the green supply chain manufacturer has relatively little motivation for financing, and it prefers the retailer for financing, while the green supply chain retailer only finances when the green preferences of consumers in the traditional supply chain are low.

The above conclusions lead to the following insights. First, considering the impact of the incentive competition between traditional and green supply chains and implicit equity, increasing investment in green product production significantly improves the performance of green supply chains. This indicates that, as consumers have paid more attention to green products, the production of green products has become an essential way to enhance the market competitiveness of the green supply chain. This not only meets market demand but also promotes the sustainable development of the supply chain. Second, for members of the green supply chain, the retailer financing model can significantly improve environmental and supply chain performance. Therefore, the manufacturer should aim to improve its production capacity and credit rating and establish closer cooperation with the retailer to increase the chances of successful financing. Furthermore, lowering the opportunity cost of funds is also crucial, as it can increase the incentive to finance to a certain extent. If the retailer’s willingness to finance is low, the manufacturer can motivate its participation through preferential policies or risk-sharing mechanisms. Finally, consumers’ preferences for green products have a significant impact on the green supply chain product’s greenness and financing motivation. Therefore, to realize the goal of green policy, the government and social institutions should strengthen publicity and promotional activities to reinforce consumers’ environmental awareness, effectively guide them to buy green products, and lay the market foundation for the sustainable development of the green supply chain.

At the same time, the findings of this study provide new focal points for the government to promote green finance. Policymakers should not only focus on the traditional binary relationship between banks and enterprises, but should also encourage capital to flow through supply chain relationships to green innovation areas. (i) Promote green supply chain finance models: Drawing on China’s experience in building a green financial system, green supply chain finance should be incorporated into the scope of green credit statistics and incentive mechanisms. Core enterprises (retailers) that provide financing for upstream green projects should be granted interest subsidies, tax incentives, or regulatory assessment bonuses. (ii) Establish platforms and set standards: Companies should support or collaborate with financial institutions to build digital green supply chain finance platforms, leveraging technologies such as blockchain to address information asymmetry and trust issues. Simultaneously, they should establish certification standards for green supply chains to ensure that financing is genuinely directed toward environmental improvement projects. (iii) Cultivating the green consumption market: As the conclusion indicates, consumer green preferences are the core driving force. Governments and social institutions should continue to strengthen environmental protection, publicity, and education, using measures such as green labels and environmental subsidies to guide and expand green consumption, thereby creating market conditions for the sustainable development of green supply chains from the ground up.

Moreover, this study examines green supply chain financing decisions under inter-chain competition and implicit equity considerations, focusing on scenarios involving two supply chains that produce differentiated products. It does not incorporate the impact of information asymmetry on financing choices. Future research could fruitfully investigate the implications of information asymmetry (e.g., private manufacturer costs or market demand information) for optimal financing model selection by green supply chain members, particularly capital-constrained manufacturers. Furthermore, while the current analysis assumes competing parties produce differentiated goods, a valuable extension would be to examine the effect of intense market competition arising when both supply chains produce homogeneous or highly substitutable green products, analyzing its impact on the financing strategy choices and performance of capital-constrained manufacturers. Finally, this study employs a Cournot (quantity-based Nash equilibrium) model to characterize inter-chain competition. Future work could consider more general asymmetric power structures, such as Stackelberg competition models featuring dominant (leader) and follower supply chains to analyze how power differentials influence green investment incentives, financing model selection, and overall supply chain efficiency.