Open AccessArticle

Optimal Strategies for E-Commerce Platform Supply Chain: Carbon Emission Reduction and Financing

Yuting Zhang

^* and

Juan Shang

School of Economics and Management, Xidian University, Xi’an 710126, China

Author to whom correspondence should be addressed.

Systems 2024, 12(11), 469; https://doi.org/10.3390/systems12110469

Submission received: 25 September 2024 / Revised: 28 October 2024 / Accepted: 30 October 2024 / Published: 1 November 2024

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Abstract

In the context of global carbon emission reduction (CER) targets and slowing economic growth, it is imperative for suppliers to make informed choices regarding CER and financing strategies. However, limited research has explored the impact of financing strategies on CER. This paper develops a supply chain model that includes a supplier, a manufacturer, an E-commerce platform (E-C platform), and consumers with a preference for low-carbon products. The supplier sets the wholesale price, while the manufacturer controls both the production quantity and the unit amount of CER. We examine whether the manufacturer will invest in CER with sufficient capital or under various financing scenarios, namely (1) traditional production with sufficient capital (Scenario ST); (2) CER implementation with sufficient capital (Scenario SG); (3) CER implementation with E-C platform financing (Scenario EG); (4) CER implementation with bank financing (Scenario BG). Through comparative analysis, the analysis reveals that, regardless of the financing method chosen, the supplier’s profit and the manufacturer’s production quantity increase when the manufacturer invests in CER technology innovation compared to the traditional scenario. Furthermore, in terms of the manufacturer’s profit, if the service cost of bank financing exceeds a certain threshold, the manufacturer should either seek financing from the E-C platform or abandon the CER investment. Additionally, with respect to CER outcomes, Scenario BG outperforms Scenario EG. These findings provide a theoretical foundation and decision-making support for supply chain participants when addressing carbon reduction and financing strategy decisions.

Keywords:

E-commerce platform; supply chain; carbon emission reduction; financing

1. Introduction

As extreme weather events increase worldwide, the environmental climate is becoming increasingly critical, drawing widespread attention to carbon emission reduction (CER) from various countries and organizations [1,2]. The European Commission fulfilled its commitments under the Paris Agreement by proposing to reduce overall EU greenhouse gas emissions by at least 55% by 2030 compared to 1990 levels and has begun developing detailed legislative proposals. Similarly, China has pledged to enhance its nationally determined contributions, aiming for CO₂ emissions to peak by 2030. While governments and national organizations are actively coordinating global efforts to reduce carbon emissions, the implementation of CER faces significant challenges due to growing global economic pressures. Building on these international efforts, our research focuses on identifying the most suitable CER strategies for supply chain members under varying economic conditions.

Numerous studies have shown that a growing number of businesses and consumers prefer low-carbon options, and low-carbon consumption is becoming the choice of an increasing number of consumers [3,4,5,6,7]. Leading companies across various industries are actively contributing to global climate efforts by engaging in CER activities, which not only demonstrate corporate social responsibility but also cater to consumers with a low-carbon preference [8]. For instance, Lenovo Group has committed to reducing its direct and indirect carbon emissions by 50% by 2030. Similarly, American sports brand Nike has proposed “zero carbon emissions” and “zero waste” plans (Move to Zero), proposing to reduce carbon emissions by 30% in its global supply chain by 2030. Do low-carbon consumer preferences enhance profitability for the supply chain? Can manufacturers increase profits by investing in CER technology innovation? These questions carry significant practical implications for supply chain management.

The deep integration of information technology and manufacturing has opened new channels for traditional supply chains through the emergence of E-C platforms [9,10]. Platforms such as Amazon, Alibaba, and JD now offer online services to users worldwide [11]. The rapid development of E-C platforms has provided consumers with convenient services that permeate nearly every aspect of daily life [12]. In response to consumer demand, manufacturers are increasingly investing in CER technology innovation. However, they may face financial constraints in these endeavors.

The development of low-carbon supply chains aligns with global low-carbon development goals [13,14], and enterprises can choose to finance either internally or externally to address the financial constraints of supply chain members [15]. The Ministry of Commerce of the People’s Republic of China has called for the steady promotion of supply chain financing, the diversification of financial service products, and the improvement of financing efficiency. Similarly, the United States Department of State encourages banks and the financial sector to provide financing support for renewable energy and other related enterprises. In China, the People’s Bank of China has issued low-cost funds to promote CER efforts among enterprises via its carbon reduction support tool. However, the challenges of bank financing should not be overlooked, as the stringent approval processes often impose additional costs and resource burdens on enterprises. In practice, the Bank of China (Hong Kong) has launched a green loan incentive plan, but corporate clients must provide green certification documents issued from evaluation agencies recognized by the bank to secure approval.

In addition to banks offering low-carbon financing services to manufacturers, large E-C platforms have also begun providing financing options for small and medium-sized enterprises. For example, JD Finance has introduced service models such as “Jingbaobei” and “Jingxiaodai”, offering various financing solutions for small and medium-sized enterprises that operate on the JD platform. Moreover, JD’s financing business can be applied for online, and the 7 × 24-h service is available for borrowing and returning, providing convenience for enterprises to quickly raise funds while significantly reducing labor and time costs associated with financing.

Conversely, there are also instances of failure in CER financing. For example, the Green Deal launched by the government of the United Kingdom aimed to help households and businesses improve energy efficiency and reduce carbon emissions by offering loans repaid through energy savings. However, the program failed due to a complicated application process and a high interest rate. This highlights the critical role that different low-carbon financing strategies play in supply chain-related enterprises. It also suggests that governments, when formulating relevant low-carbon policies, should thoroughly investigate and understand consumer preferences for low-carbon options and the financing services provided by banks. Furthermore, under the pressure of growing low-carbon demand from governments, markets, and consumers, and faced with several financial instruments for corporate finance, more and more enterprises have started to finance the production of low-carbon products [16]. How to choose the most favorable financing entity and financing plan is a challenge faced by manufacturers. Addressing this issue can provide theoretical support for government departments in formulating relevant policies and offer practical directions for fostering a low-carbon society. Therefore, the CER and financing strategies of the E-C platform supply chain under different financing models are a very worthwhile issue to study. To our knowledge, this is the first study in the field of supply chain operations that integrates E-C platform and bank financing into CER strategies while also considering the impact of consumer preferences.

The research questions addressed in this study, which focus on the decision-making challenges faced by supply chain members, can be summarized as follows:

(1): What is the optimal CER strategy under conditions of sufficient capital or various financing options?
(2): Which financing option has the best CER effect and is more beneficial to supply chain members?
(3): Does a greater consumer preference for low-carbon products increase the profits of supply chain members?
(4): How do parameters such as the commission rate and the interest rate affect the profitability of supply chain?

To address the issues mentioned above, we examine the CER and financing strategies of a platform supply chain, which includes a supplier, a manufacturer, and an E-C platform, across four different scenarios. First, in the case of sufficient capital, the E-C platforms sells traditional products without CER investment. Second, to cater to consumers’ low-carbon preferences and demonstrate the company’s contributions to society and the environment, we establish a model where manufacturers invest in CER technology innovation and the E-C platforms for sale under the condition of sufficient capital. Third, when manufacturers invest in CER technology innovation, a situation of financial constraint may occur, so manufacturers seek financing within the supply chain. In this case, E-C platforms are willing to provide funds to manufacturers to earn profits from them. Therefore, we established a low-carbon supply chain model in which manufacturers seek financing from E-C platforms under financial constraints. Fourth, in addition to seeking financing within the supply chain, manufacturers can also seek external financing. Banks announce interest rate and give certain discount on low-carbon policy interest rate, but financing through banks incurs certain service costs such as time and labor. This paper incorporates these unified bank financing service costs into the supply chain model. It systematically studies the CER and financing strategies of the platform supply chain under these four scenarios, providing a theoretical basis for operational decision-making in supply chain financing.

The main contributions: Firstly, we examine four supply chain models that consider whether or not to invest in CER technology innovation and whether to pursue internal or external financing, and we derive the optimal decision for each model. We derive the optimal decisions for each model and demonstrate that corporate investment in CER technology innovation contributes to reducing carbon emissions and benefits the social environment. Secondly, the supply chain models of the four scenarios are classified into “traditional scenario vs. CER scenario”, “sufficient capital vs. constrained funds” and “E-C platform financing vs. bank financing”, and we make comparative analyses. The results of this analysis provide clear guidance for government carbon reduction policy-making, helping to avoid overly aggressive or ineffective policies. Thirdly, we analyze the influence of key factors, such as consumers’ low-carbon preferences and commission rate, on internal and external financing in low-carbon supply chains, providing theoretical support for the favorable effects of consumer low-carbon preferences. Fourthly, the applicable conditions of internal financing on E-C platforms and external financing from banks are explored, and the CER strategies and optimal financing strategies are given in different scenarios. These findings provide practical recommendations for supply chain members, governments, and consumers.

The remainder of the paper is organized as follows. The “Literature review” section presents the literature review in terms of the low-carbon supply chain, E-C platform supply chain, and supply chain finance, respectively. The “Model description and assumptions” section provides a detailed description of the models and reasonable model assumptions and constructs game models for four different scenarios. The “Model development” section gives the equilibrium results of the four supply chain models. The “Results analysis” section compares and analyzes the optimal decisions of the four scenarios and presents the impacts of CER and financing on each member of the supply chain. Finally, the “Conclusion and future research” section highlights the relevant conclusions, managerial implications, and future research developments of this paper. In addition, the appendix contains all the mathematical proofs.

2. Literature Review

Our research focuses on CER and financing strategies in supply chains. Accordingly, this paper is closely related to the literature on low-carbon supply chains, E-C platform supply chains, and supply chain finance, and we will comprehensively review the studies in these three areas and discuss their differences.

2.1. Low-Carbon Supply Chains

In the field of low-carbon supply chain research, the existing literature focuses on pricing and CER strategies across various channel structures [17,18]. Yang et al. constructed a low-carbon supply chain model with and without remanufacturing and determined the pricing and CER strategies for different channel structures [19]. Xu et al. studied the co-production and pricing issues of multi-product manufacturing firms [20]. Hong et al. studied a two-production-scheme problem under carbon emission constraints and proposed an optimal technology selection strategy [21].

Building on research on pricing and optimal CER strategies, consumers’ low-carbon preferences have gradually become the focus of research. Liu et al. studied the impact of consumer environmental awareness on low-carbon supply chain participants [22]. Ji et al. considered consumer low-carbon preference on a two-channel supply chain model and proposed two CER strategies [23]. Zhang et al. explored the impact of consumer environmental awareness on order quantity and channel coordination within supply chains [24]. Zhou and Wen conducted a systematic literature review on carbon constraint models, offering an in-depth analysis of internal emission reduction, collaborative emission reduction, and carbon offsetting [25].

Most of these studies are based on traditional single channels, and a few scholars have also considered the emission reduction of dual channels. Research on low-carbon supply chains is becoming more comprehensive. However, our focus is on financing strategies for low-carbon supply chains facing financial constraints, a topic that remains underexplored. Additionally, our study not only considers CER and consumers’ low-carbon preferences but also emphasizes the critical role of E-C platforms in supply chain operations.

2.2. E-C Platform Supply Chains

In response to fierce competition in the market and the rapid growth of E-C platforms, the supply chain channel structure has evolved from a single traditional retail channel to a resale channel based on E-C platforms [26,27]. Abhishek et al. compared the agency and resale formats of online retailers, identifying the conditions under which agency sales are preferable and the impact of agency sales on various market participants [28]. Tan and Carrillo created and analyzed a vertically differentiated merchandise model, comparing and analyzing the reselling and wholesale models, and extending the original model to include horizontal differentiation [29]. Their main conclusion demonstrates that the agency model is more efficient than resale and can lead to lower equilibrium prices. Based on this, more scholars have explored on the E-C platform resale supply chain and argue this conclusion. Our study also adopts a resale model and incorporates the carbon reduction and financing strategies of the resale model, which has been rarely studied in previous literature.

Some scholars have also analyzed the problem of choosing between other channels and E-C platform channels under different scenarios. Yi et al. investigated the channel choice problem of a manufacturer selling its products directly or through a reseller channel when consumers are seeking transaction fairness [30]. Tian et al. considered a programmed supply chain and used an E-C platform as an intermediary platform to sell two substitutable products, solving the problem of optimal mode choice under different scenarios [31]. Wei et al. studied the sale of products on an e-retailer’s online platform through either an agency sales format or a resale format [32]. Ha et al. investigated the channel selection problem of an online platform that boosts the demand for its sales channels with its service endeavors [26]. In addition, some other scholars have focused on the coordination study of supply chains in E-C platforms. Zhang et al. studied platform contract selection and manufacturers’ product quality decisions proposing two types of contracts, namely, revenue-sharing contracts and fixed-fee contracts, for coordination among supply chain members [33]. Zhang and Zhang explored two sales modes, namely, e-retailers and brick-and-mortar shops, and devised two types of agreements, namely, agency selling and resellingresale and resale, for coordination among supply chain members [34].

Many of these studies are based on the comparison of basic channels. However, with the rapid rise of E-C platforms, E-C platforms have resulted in significant capital reserves. Some E-C platforms lend their excess capital to supply chain members facing financial constraints, who are short of funds, to earn interest. This form of financing holds great potential, particularly in alleviating funding shortages for CER efforts within supply chains. Therefore, this study conducts a more in-depth exploration of this topic.

2.3. Supply Chain Finance

Supply chain financing is an effective way to address financial constraints [35]. With the deepening of the socialized mode of production, financing can be applied to the core enterprises in the supply chain. Financing methods have also diversified, such as bank financing and deferred payment. Cai et al. studied the choice between bank financing and trade credit financing in a supply chain [36]. Li et al. analyzed the financing strategy problem between partial credit guarantee and trade credit financing [37]. Jena et al. constructed a dual-channel supply chain and a multilevel supply chain in a Stackelberg game model to analyze financing options on supply chain profitability under price-sensitive demand [38]. However, the above literature does not address the E-C platform supply chain model and neglects the E-C platform financing model. Chang et al. studied the financing strategy of an online retailer and the channel sales cooperation contract [39].

Manufacturers in the supply chain will need some funding when investing in CER technology innovation. Financing is an effective way to alleviate financial shortages in low-carbon supply chains [40]. Xu and Fang studied the equilibrium strategy problem of partial-credit guarantee financing and trade credit financing [12]. Cao et al. compared the issue of investment preference between internal supply chain financing and external bank financing [16]. An et al. studied a green supply chain system consisting of a financially constrained manufacturer and a well-financed supplier facing uncertain demand [41]. Shi et al. investigated the financing strategies of secondary low-carbon supply chains facing financial constraints [42]. Lai et al. analyzed green supply chain partial-credit guarantee schemes, deferred payment, bank trade credit financing schemes, and external investment schemes [43]. A few scholars have studied the ways of financing through E-C platforms within supply chains. For example, Qin et al. studied a low-carbon supply chain in which manufacturers obtain funds through E-C platform financing modes, supplier credit financing, and hybrid financing [44]. While the above literature considers either internal or external financing, there are few cases where both E-C platform financing and bank financing are considered and compared.

In summary, few scholars have studied the impact of E-C platform financing as well as bank financing on CER as well as supply chain members, taking CER into account. Although some literature has discussed the CER strategy or financing strategy within E-C platform supply chains, there is a lack of studies comparing financing modes for CER under financial constraints. The primary studies in the related literature are summarized in Table 1 and compared with this study.

As shown in Table 1, previous studies on low-carbon supply chains have primarily focused on low-carbon investments. While some scholars have considered consumers’ low-carbon preferences, they have largely overlooked the funding issues related to low-carbon investments within the supply chain. In particular, the integration of financing strategies for E-C platforms into CER supply chains has received little attention. From the perspective of research topics, most scholars focus on comparing different supply chain channels, providing decision-making strategies in various contexts by analyzing these comparisons. This comparative approach is also employed in this study, where supply chain models were constructed for Scenario ST, Scenario SG, Scenario EG, and Scenario BG, respectively.

In the study of low-carbon supply chains, the innovation cost of CER technology is an indispensable influence factor. The purpose of this research is to provide a feasible strategy for supply chain members when facing financial constraints, helping them decide whether to abandon CER investments or pursue financing, and if financing is chosen, which financing plan to adopt. In addition, consumers are also an important carrier in the supply chain operation process, and some scholars have included consumers’ low-carbon preference in their research. However, few scholars have conducted in-depth research on CER in E-C platform supply chains. Furthermore, there is a lack of research on the need for financing when the members of the supply chain are subject to financial constraints to reduce carbon emissions. Therefore, we analyze the optimal choice of financing from the E-C platform or from the bank when the E-C platform supply chain is subject to financial constraints. We also analyze the impact of consumers’ low-carbon preference, interest rate, and other factors on the decision-making and profitability of each member of the E-C platform supply chain.

3. Model Description and Assumptions

3.1. Model Description

We consider a supply chain in which the supplier provides raw materials to manufacturers. Examples of such suppliers include BASF (Ludwigshafen, Germany) and SinoPec (Beijing, China). The manufacturer processes raw materials from suppliers, and during the manufacturing process, the manufacturer can choose whether to invest in CER technology innovation, for example, Gree and BYD, or an E-C platform with sufficient capital, where traditional or low-carbon products produced and processed by manufacturers are sold through the E-C platform, for example, Amazon and JD.com. When a manufacturer sells traditional or low-carbon products through an E-C platform, the E-C platform charges the manufacturer a percentage of the sales price. When manufacturers have financial constraints, they can obtain financing from well-capitalized E-C platforms or banks. Figure 1 illustrates the sources of CER and financing for the E-C platform supply chains discussed in this paper.

We examine four scenarios based on whether a manufacturer makes an investment in CER and whether it finances it, as shown in Figure 2.

(i): Scenario ST: The manufacturer has sufficient capital but does not invest in CER technology innovation.
(ii): Scenario SG: The manufacturer has sufficient capital and decides to invest in CER technology innovation.
(iii): Scenario EG: The manufacturer, facing financial constraints, seeks financing from a well-funded E-C platform to secure sufficient capital for CER technology innovation. Since E-C platforms are internal members of the supply chain, the lending process is relatively fast.
(iv): Scenario BG: The manufacturer, facing financial constraints, seeks financing from a bank to invest in CER technology innovation. Compared to financing from E-C platforms, bank loans have a slower processing speed, but banks often offer discounted interest rates to manufacturers involved in CER initiatives.

The parameters and definitions presented in the following are summarized in Table 2.

3.2. Assumptions

Assumption 1.

The market demand in the traditional scenario is

D^{S T} = a + ζ

[40]. For simplicity, we assume

a

represents the total market potential (the intercept of the demand function and the amount that would be sold if the price was zero) [22]. Additionally,

ζ

is a random variable representing uncertain market demand [40]. Although supply chain members do not fully understand the potential demand in the market, they can estimate the probability distribution of parameters from the information of the entire industry. Therefore, we assume that

ζ

follows a uniform distribution

(- A, A) (a > 0, A > 0)

[45]. To ensure that the research model is meaningful and that the optimal decision is positive, we assume that

a > A > \frac{a}{3}

[44,46,47]. When manufacturers invest in CER technology innovation, the market demand is

D^{J} = a + η e^{J} + ζ

J \in \{S G, E G, B G\}

Assumption 2.

For production costs, to ensure that the analytical solution for the supply chain model is positive, that is, that all members of the supply chain are profitable, this paper assumes

0 < c < \frac{(3 A - a) p^{T} (1 - λ)}{2 A}

[48,49]. For the CER technology innovation cost, which is considered a one-time investment, it is assumed that the CER technology innovation cost is

C_{e} = \frac{1}{2} k e^{2}

[3,4,22].

Assumption 3.

In line with the commission rate settings of major E-C platforms such as Amazon [50], Taobao [51], and JD [52], and based on the work of Hao and Yang [11], Ghosh and Shah [17], and Qin et al. [40], we assume that commission rate

λ

is an exogenous variable and

0 < λ < \frac{1}{2}

Assumption 4.

To focus on the study of CER and financing, we assume that the retail price of traditional products is

P^{T}

, and the retail price of low-carbon products is

P^{G}

, both of which are exogenous variables [53]. Furthermore, it is assumed that consumers are willing to pay a premium for low-carbon products, i.e.,

P^{G} > P^{T}

[49].

Assumption 5.

We assume that in the capital-constrained scenario, the manufacturer’s initial capital is

M

[41]. In the bank financing scenario, stricter approval processes apply, and labor and time costs are incurred, referred to as the financing service cost

T

[43,54,55]. In addition, the manufacturer is a reputable business that can obtain financing from an E-C platform or a bank and can repay the financing amount and the interest on the financing as required. It is important to note that the interest rate cannot exceed the national policy range. Take China as an example, according to the regulations of the Supreme People’s Court of China on interest rate. We assume that

r < 4 * L P R

is based on the 5-year LPR (

L P R = 3.95 %

) announced by the People’s Bank of China on 20 February 2024 [56].

4. Model Development

4.1. Scenario ST: Traditional Production with Sufficient Capital

Consider the study of optimal decision-making of each member in the sufficient capital and traditional production scenario. The event sequence of the supplier, the manufacturer, and the E-C platform is shown in Figure 3. Stage 1: The supplier decides on the wholesale price

w^{S T}

. Stage 2: The manufacturer decides on the production quantity

q^{S T}

. Stage 3: The E-C platform sells at retail price

p^{T}

and receives commissions from the manufacturer. Then, we derive the profit function of supply chain members in Scenario ST.

The supplier’s profit function is as follows:

\prod_{s}^{S T} = (w^{S T} - c) q^{S T}

(1)

The manufacturer’s profit function is as follows:

\begin{array}{l} \max \prod_{m}^{S T} & = (1 - λ) p^{T} E [\min (D^{S T}, q^{S T})] - w^{S T} q^{S T} \\ = (1 - λ) p^{T} (q^{S T} - \int_{- A}^{q^{S T} - a} F (ζ) d ζ) - w^{S T} q^{S T} \end{array}

(2)

The E-C platform’s profit function is as follows:

\begin{array}{l} \max \prod_{e}^{S T} & = λ p^{T} E [\min (D^{T}, q^{T})] \\ = λ p^{T} (q^{T} - \int_{- A}^{q^{T} - a} F (ζ) d ζ) \end{array}

(3)

The equilibrium outcomes of Scenario ST are obtained by solving the problem by backward induction.

Lemma 1.

In Scenario ST, there exist equilibrium solutions and profits:

(1): Wholesale price and production quantity

$w^{S T *} = \frac{(a + A) (1 - λ) p^{T}}{4 A} + \frac{c}{2}$

$q^{S T *} = \frac{a + A}{2} - \frac{A c}{(1 - λ) p^{T}}$
(2): Profitability of supply chain members

$\prod_{s}^{S T *} = \frac{{((a + A) (1 - λ) p^{T} + 2 A c)}^{2}}{8 A (1 - λ) p^{T}}$

$\prod_{e}^{S T *} = \frac{λ (a^{2} + 14 A a - A^{2}) p^{T}}{16 A} - \frac{λ (a + A) c}{4 (1 - λ)} - \frac{λ A c^{2}}{4 {(1 - λ)}^{2} p^{T}}$

$\prod_{m}^{S T *} = \frac{A c^{2}}{4 (1 - λ) p^{T}} + \frac{(3 A - a) (3 a - A) (1 - λ) p^{T}}{16 A} - \frac{a c + A c}{4}$

See Appendix A for proof. From Lemma 1, we observe that under Scenario ST, the commission rate of E-C platform not only affects the profits of both the E-C platform and the manufacturer but also influences the profits of supplier. Building on Lemma 1, Proposition 1 examines the impact of commission rate on optimal decision-making. When manufacturers have sufficient funds, they may consider whether to invest in CER. We conduct a detailed analysis of Scenario SG.

4.2. Scenario SG: CER Implementation with Sufficient Capital

Consider the study of optimal decision-making of each member in the sufficient capital and investment in CER scenario. The event sequence of the supplier, the manufacturer, and the E-C platform is shown in Figure 4. Stage 1: The supplier determines the wholesale price

w^{S G}

. Stage 2: The manufacturer inputs the CER technology innovation cost

C_{e}

and determines the production quantity

q^{S G}

and the unit amount of CER

e^{S G}

. Stage 3: The E-C platform sells at retail price

p^{G}

and receives commissions from the manufacturer. Then, we have the profit function of supply chain members in Scenario SG.

The supplier’s profit function is as follows:

\prod_{s}^{S G} = (w^{S G} - c) q^{S G}

(4)

The manufacturer’s profit function is as follows:

\begin{array}{l} \max \prod_{m}^{S G} & = (1 - λ) p^{G} E [\min (D^{S G}, q^{S G})] - w^{S G} q^{S G} - \frac{1}{2} k {(e^{S G})}^{2} \\ = (1 - λ) p^{G} (q^{S G} - \int_{- A}^{q^{S G} - a - η e^{S G}} F (ζ) d ζ) - w^{S G} q^{S G} - \frac{1}{2} k {(e^{S G})}^{2} \end{array}

(5)

The E-C platform’s profit function is as follows:

\begin{array}{l} \max \prod_{e}^{S G} & = λ p^{G} E [\min (D^{S G}, q^{S G})] \\ = λ p^{G} (q^{S G} - \int_{- A}^{q^{S G} - a - η e^{S G}} F (ζ) d ζ) \end{array}

(6)

The equilibrium outcomes of Scenario SG are obtained by solving the problem by backward induction.

Lemma 2.

In Scenario SG, there exist equilibrium solutions and profits:

(1): Wholesale price and production quantity:

$w^{S G *} = \frac{{(η (1 - λ) p^{G})}^{2} + (1 - λ) (c η^{2} + A k + a k) p^{G} + 2 A c k}{2 η^{2} (1 - λ) p^{G} + 4 A k}$

$q^{S G *} = \frac{(a k + A k - c η^{2}) (1 - λ) p^{G} + {(η (1 - λ) p^{G})}^{2} - 2 A c k}{2 k p^{G} (1 - λ)}$
(2): Unit amount of CER

$e^{S G *} = \frac{η ({(η (1 - λ) p^{G})}^{2} + (1 - λ) (3 A k - a k - c η^{2}) p^{G} - 2 A c k)}{2 (2 A k + (1 - λ) p^{G} η^{2}) k}$
(3): Profitability of supply chain members

$\prod_{s}^{S G *} = \frac{{({(η (1 - λ) p^{G})}^{2} - (A k + a k - c η^{2}) (1 - λ) p^{G} + 2 A c k)}^{2}}{4 (η^{2} (1 - λ) p^{G} + 2 A k) k (1 - λ) p^{G}}$

$\prod_{m}^{S G *} = \frac{({(η (1 - λ) p^{G})}^{2} - (c η^{2} + A k - 3 a k) (1 - λ) p^{G} - 2 A c k) e^{S G *}}{4 (1 - λ) p^{G} η}$

$\prod_{e}^{S G *} = \frac{p^{G} λ}{4} (2 a + A + \frac{2 η^{2} ((1 - λ) p^{G} - c)}{k} + \frac{(a - A) (c η^{2} + 2 A k)}{η^{2} (1 - λ) p^{G} + 2 A k} - \frac{{(a - A)}^{2} A k^{2}}{{(η^{2} (1 - λ) p^{G} + 2 A k)}^{2}} - \frac{A c^{2}}{{((1 - λ) p^{G})}^{2}} - \frac{A c + a c}{p^{G} (1 - λ)})$

See Appendix A for proof. From Lemma 2, we observe that under Scenario SG, after a manufacturer invests in CER, consumers’ low-carbon preference not only affects manufacturer’s profits but also impacts the profits of the supplier and the E-C platform. Building upon Lemma 2, Proposition 8 examines the impact of consumers’ low-carbon preference on the profits of supply chain members.

Previous research has shown that consumers are willing to pay higher prices for low-carbon products [22]. Whether it is profitable for manufacturer to finance CER under financial constraints is a pressing research question. We conducted a detailed study on E-C platform financing in Scenario EG.

4.3. Scenario EG: CER Implementation with E-C Platform Financing

Consider the study of optimal decision-making of each member in the E-C platform financing and investment in CER scenario. The event sequence of the supplier, the manufacturer, and the E-C platform is shown in Figure 5. Stage 1: The E-C platform announces interest rate

r

. Stage 2: The supplier determines the wholesale price

w^{E G}

. Stage 3: The manufacturer inputs the CER technology innovation cost

C_{e}

and determines the production quantity

q^{E G}

and the unit amount of CER

e^{E G}

. Stage 4: The E-C platform sells at retail price

p^{G}

and receives a commission. The manufacturer returns the principal and pays interest for the E-C platform. Then, we have the profit function of supply chain members in Scenario EG.

The supplier’s profit function is as follows:

\prod_{s}^{E G} = (w^{E G} - c) q^{E G}

(7)

The manufacturer’s profit function is as follows:

\begin{array}{l} \max \prod_{m}^{E G} & = (1 - λ) p^{G} E [\min (D^{E G}, q^{E G})] - (1 + r) (w^{E G} q^{E G} + \frac{1}{2} k {(e^{E G})}^{2} - M) - M \\ = (1 - λ) p^{G} (q^{E G} - \int_{- A}^{q^{E G} - a - η e^{E G}} F (ζ) d ζ) - (1 + r) (w^{E G} q^{E G} + \frac{1}{2} k {(e^{E G})}^{2} - M) - M \end{array}

(8)

The E-C platform’s profit function is as follows:

\begin{array}{l} \max \prod_{e}^{E G} & = λ p^{G} E [\min (D^{E G}, q^{E G})] + r (w^{E G} q^{E G} + \frac{1}{2} k {(e^{E G})}^{2} - M) \\ = λ p^{G} (q^{E G} - \int_{- A}^{q^{E G} - a - η e^{E G}} F (ζ) d ζ) + r (w^{E G} q^{E G} + \frac{1}{2} k {(e^{E G})}^{2} - M) \end{array}

(9)

The equilibrium outcomes of Scenario EG are obtained by solving the problem by backward induction.

Lemma 3.

In Scenario EG, there exist equilibrium solutions and profits:

(1): Wholesale price and production quantity

$w^{E G *} = \frac{{(η (1 - λ) p^{G})}^{2} + (1 - λ) (A k + a k + c η^{2}) (1 + r) p^{G} + 2 A c k {(1 + r)}^{2}}{2 (1 + r) (2 A k (1 + r) + p^{G} η^{2} (1 - λ))}$

$q^{E G *} = \frac{(1 + r) (a k + A k - c η^{2}) (1 - λ) p^{G} + {(η (1 - λ) p^{G})}^{2} - 2 A c k {(1 + r)}^{2}}{2 k (1 + r) (1 - λ) p^{G}}$
(2): Unit amount of CER

$e^{E G *} = \frac{η ((1 + r) (3 A k - a k - c η^{2}) (1 - λ) p^{G} + {(η (1 - λ) p^{G})}^{2} - 2 A c k {(1 + r)}^{2})}{2 k (1 + r) (2 A k (1 + r) + η^{2} (1 - λ) p^{G})}$
(3): Profitability of supply chain members

$\prod_{s}^{E G *} = \frac{{({(η (1 - λ) p^{G})}^{2} + (1 + r) (1 - λ) (A k + a k - c η^{2}) p^{G} - 2 A c k {(1 + r)}^{2})}^{2}}{4 p^{G} (1 - λ) (p^{G} η^{2} (1 - λ) + 2 A k r (1 + r)) {(1 + r)}^{2} k}$

$\prod_{m}^{E G *} = \frac{({(η (1 - λ) p^{G})}^{2} - (1 + r) (1 - λ) (A k - 3 a k + c η^{2}) p^{G} - 2 A c k {(1 + r)}^{2}) e^{E G *}}{4 p^{G} (1 - λ) η} + M r$

$\begin{array}{l} \prod_{e}^{E G *} & = \frac{λ ((1 - λ) p^{G} A k {(1 + r)}^{2} (3 a k + A k - 3 c η^{2}) - {(η (1 - λ) p^{G})}^{2} ((1 + r) (a k + 3 A k - c η^{2}) + η^{2} (1 - λ) p^{G}) - 2 A^{2} c k^{2} {(1 + r)}^{3})}{2 k (1 + r) (1 - λ) (2 A k (1 + r) + η^{2} (1 - λ) p^{G})} \\ - \frac{A {(1 + r)}^{2} λ {(2 A c k (1 + r) + (a k - A k + c η^{2}) (1 - λ) p^{G})}^{2}}{4 {(1 - λ)}^{2} p^{G} {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} - M r - \frac{A λ p^{G}}{4} + \frac{r k {(e^{E G *})}^{2}}{2} + r q^{E G *} w^{E G *} \end{array}$

See Appendix A for proof. From Lemma 3, we observe that after the manufacturer secures financing from the E-C platform, the financing interest rate not only affects the profit of each member of the supply chain but also impacts the unit amount of CER. Based on Lemma 3, Proposition 6 examines the effect of financing interest rate on unit amount of CER. Manufacturers can receive financing from banks in addition to financing from E-C platform. We provide an in-depth analysis of bank financing in Scenario BG.

4.4. Scenario BG: CER Implementation with Bank Financing

Consider the study of optimal decision-making of each member of the supply chain in bank financing and investment in the CER scenario. Compared to financing from E-C platforms, the bank financing process requires stricter approval and incurs some labor and time costs, which we refer to as financing service cost

T

. To compensate for this disadvantage, banks offer interest rate discounts. Therefore, the actual interest rate for bank financing is

β r

[57]. The event sequence of the supplier, the manufacturer, the E-C platform, and the bank is shown in Figure 6. Stage 1: The bank announces interest rate

r

and discount

β

. Stage 2: The supplier determines the wholesale price

w^{B G}

. Stage 3: The manufacturer inputs the CER technology innovation cost

C_{e}

and determines the production quantity

q^{B G}

and the unit amount of CER

e^{B G}

. Stage 4: The E-C platform sells at retail price

p^{G}

. The manufacturer repays the principal interest to the bank. Then, we derive the profit function of supply chain members in Scenario BG.

The supplier’s profit function is as follows:

\prod_{s}^{B G} = (w^{B G} - c) q^{B G}

(10)

The manufacturer’s profit function is as follows:

\begin{array}{l} \max \prod_{m}^{B G} & = (1 - λ) p^{G} E [\min (D^{B G}, q^{B G})] - (1 + β r) (w^{B G} q^{B G} + \frac{1}{2} k {(e^{B G})}^{2} + T - M) - M \\ = (1 - λ) p^{G} (q^{B G} - \int_{- A}^{q^{B G} - a - η e^{B G}} F (ζ) d ζ) - (1 + β r) (w^{B G} q^{B G} + \frac{1}{2} k {(e^{B G})}^{2} + T - M) - M \end{array}

(11)

The E-C platform’s profit function is as follows:

\begin{array}{l} \max \prod_{e}^{B G} & = λ p^{G} E [\min (D^{B G}, q^{B G})] \\ = λ p^{G} (q^{B G} - \int_{- A}^{q^{B G} - a - η e^{B G}} F (ζ) d ζ) \end{array}

(12)

The bank’s profit function is as follows:

\prod_{b}^{B G} = β r (w^{B G} q^{B G} + \frac{1}{2} k {(e^{B G})}^{2} + T - M)

(13)

Equilibrium outcomes of Scenario BG are obtained by solving the problem by backward induction.

Lemma 4.

In Scenario BG, there exist equilibrium solutions and profits:

(1): Wholesale price and production quantity

$w^{B G *} = \frac{{(η (1 - λ) p^{G})}^{2} + (1 - λ) (1 + β r) (A k + a k + c η^{2}) p^{G} + 2 A c k {(1 + β r)}^{2}}{2 ((1 - λ) p^{G} η^{2} + 2 k (1 + β r) A) (1 + β r)}$

$q^{B G *} = \frac{(1 + r β) (a k + A k - c η^{2}) (1 - λ) p^{G} + {(η (1 - λ) p^{G})}^{2} - 2 A c k {(1 + r β)}^{2}}{2 k (1 + r β) (1 - λ) p^{G}}$
(2): Unit amount of CER

$e^{B G *} = \frac{η ((1 + r β) (3 A k - a k - c η^{2}) (1 - λ) p^{G} - 2 A c k {(1 + r β)}^{2} + {(η (1 - λ) p^{G})}^{2})}{2 k (1 + r β) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}$
(3): Profitability of supply chain members

$\prod_{s}^{B G *} = \frac{{({(η (1 - λ) p^{G})}^{2} + (1 - λ) (1 + β r) (A k + a k - c η^{2}) p^{G} - 2 A c k {(1 + β r)}^{2})}^{2}}{4 (η^{2} (1 - λ) p^{G} + 2 k (1 + β r) A) {(1 + β r)}^{2} k (1 - λ) p^{G}}$

$\prod_{m}^{B G *} = \frac{({(η (1 - λ) p^{G})}^{2} - 2 A c k {(1 + β r)}^{2} - (1 - λ) (1 + β r) (A k - 3 a k + c η^{2}) p^{G}) e^{B G *}}{4 (1 - λ) p^{G} η} + r (M - T) β - T$

$\prod_{e}^{B G *} = \frac{p^{G} (4 a - A) λ}{4} - \frac{p^{G} A (a - A) k (1 + β r) λ}{4 A k (1 + β r) + 2 η^{2} (1 - λ) p^{G}} - \frac{A c (1 + β r) λ}{2 - 2 λ} - \frac{A {(1 + r β)}^{2} λ {(2 A c k (1 + r β) + (a k - A k + c η^{2}) (1 - λ) p^{G})}^{2}}{4 p^{G} {(2 A k (1 + β r) + η^{2} (1 - λ) p^{G})}^{2} {(1 - λ)}^{2}} + p^{G} η λ e^{B G *}$

$\prod_{b}^{B G *} = \frac{r β η^{2} {({(η (1 - λ) p^{G})}^{2} + (1 - λ) (3 A k - a k - c η^{2}) (1 + β r) p^{G} - 2 A c k {(1 + β r)}^{2})}^{2}}{8 k {(1 + β r)}^{2} {(2 A k (1 + β r) + η^{2} (1 - λ) p^{G})}^{2}} + r β (q^{B G *} w^{B G *} + T - M)$

See Appendix A for proof. From Lemma 4, we observe that after the manufacturer obtains financing from a bank, the impact of the interest rate discount on the profits of the supply chain members and the unit of CER is straightforward. Meanwhile, the financing service costs directly affect the profits of manufacturer. Building on Lemma 4, Propositions 8 and 12 provide a detailed analysis of the effects of interest rate discounts and financing service costs on the profits of supply chain members.

5. Results Analysis

We analyzed the sensitivity of the commission rate, consumers’ low-carbon preference coefficient, CER technology innovation cost coefficient, interest rate, and interest rate discount to equilibrium solutions and profits in Scenario ST, Scenario SG, Scenario EG, and Scenario BG, respectively. In addition, the equilibrium solutions and profits under four scenarios were compared.

Proposition 1.

The sensitivity analysis of relevant parameters to wholesale prices is as follows:

(1): (i) $\frac{\partial w^{S T *}}{\partial λ} < 0$ , $\frac{\partial w^{S G *}}{\partial λ} < 0$ , $\frac{\partial w^{E G *}}{\partial λ} < 0$ , $\frac{\partial w^{B G *}}{\partial λ} < 0$ ; (ii) $\frac{\partial w^{S G *}}{\partial η} < 0$ , $\frac{\partial w^{E G *}}{\partial η} < 0$ , $\frac{\partial w^{B G *}}{\partial η} < 0$ ; (iii) $\frac{\partial w^{S G *}}{\partial k} > 0$ , $\frac{\partial w^{E G *}}{\partial k} > 0$ , $\frac{\partial w^{B G *}}{\partial k} > 0$ ; (iv) $\frac{\partial w^{E G *}}{\partial r} < 0$ , $\frac{\partial w^{B G *}}{\partial r} < 0$ ; (v) $\frac{\partial w^{B G *}}{\partial β} < 0$ .
(2): (i) $|\frac{\partial w^{S G *}}{\partial λ}| > |\frac{\partial w^{B G *}}{\partial λ}| > |\frac{\partial w^{E G *}}{\partial λ}|$ ; (ii) $|\frac{\partial w^{S G *}}{\partial η}| > |\frac{\partial w^{B G *}}{\partial η}| > |\frac{\partial w^{E G *}}{\partial η}|$ ; (iii) $|\frac{\partial w^{S G *}}{\partial k}| > |\frac{\partial w^{B G *}}{\partial k}| > |\frac{\partial w^{E G *}}{\partial k}|$ .

See Appendix A for proof.

Proposition 1 (1) demonstrates the impact of commission rate, consumers’ low-carbon preference coefficient, and CER technology innovation cost coefficient on wholesale prices in four scenarios. Case (1) (i) indicates that the wholesale prices in Scenario ST, Scenario SG, Scenario EG, and Scenario BG decrease as the commission rate increases (see Figure 7a). For suppliers, regardless of the scenario chosen by the manufacturer, a higher commission rate can enhance better sales on E-C platforms. Case (1) (ii) indicates that when manufacturers invest in CER technology innovation, wholesale prices decrease as the consumers’ low-carbon preference coefficient increases. This indicates that suppliers are willing to lower wholesale prices to meet consumers’ low-carbon needs. Case (1) (iii) reflects that wholesale prices increase with the rise of the CER technology innovation cost coefficient, indicating that suppliers are attempting to compensate for the losses caused by CER by raising wholesale prices. Cases (1) (iv) and (v) show that when the interest rate and the interest rate discount increase, the supplier will reduce the wholesale price. The supplier concedes a portion of its profit to the manufacturer to avoid a situation where the manufacturer would experience a significant reduction in the quantity produced. Proposition 1 (2) reveals that in the three CER scenarios, the commission rate, the consumers’ low-carbon preference coefficient, and the CER technology innovation cost coefficient have a more consistent trend in influencing suppliers’ wholesale prices. As shown in Figure 7, these parameters have a greater impact on Scenario SG and a smaller impact on Scenario EG. This highlights that investing in CER technology innovation is a preferable option when funds are sufficient. In 2024, Amazon increased the commission rate for some product categories [50]. At this time, Amazon’s suppliers could reduce their wholesale prices based on the sensitivity analysis of the commission rate discussed above.

Proposition 2.

The comparative results of wholesale prices under the three CER scenarios and traditional scenario satisfy the relationship

(1): (i) if $p^{G} > \frac{(A + a) p^{T}}{2 A}$ , $w^{S G *} > w^{S T *}$ ; (ii) if $p^{G} < \frac{(A + a) p^{T}}{2 A}$ and $k > k_{1}$ , $w^{S G *} > w^{S T *}$ ; (iii) if $p^{G} < \frac{(A + a) p^{T}}{2 A}$ and $k < k_{1}$ , $w^{S G *} < w^{S T *}$ .
(2): (i) if $p^{G} > \frac{(1 + r) (A + a) p^{T}}{2 A}$ , $w^{E G *} > w^{S T *}$ ; (ii) if $p^{G} < \frac{(1 + r) (A + a) p^{T}}{2 A}$ and $k > k_{2}$ , $w^{E G *} > w^{S T *}$ ; (iii) if $p^{G} < \frac{(1 + r) (A + a) p^{T}}{2 A}$ and $k < k_{2}$ , $w^{E G *} < w^{S T *}$ .
(3): (i) if $p^{G} > \frac{(a + A) p^{T} (1 + r β)}{2 A}$ , $w^{B G *} > w^{S T *}$ ; (ii) if $p^{G} < \frac{(a + A) p^{T} (1 + r β)}{2 A}$ and $k > k_{3}$ , $w^{B G *} > w^{S T *}$ ; (iii) if $p^{G} < \frac{(a + A) p^{T} (1 + r β)}{2 A}$ and $k < k_{3}$ , $w^{B G *} < w^{S T *}$ .

See Appendix A for proof.

Proposition 2 compares wholesale prices across three CER scenarios and traditional scenario. Proposition 2 (1) compares the wholesale prices of Scenario SG with Scenario ST. It is evident that when the retail price of low-carbon products is higher, suppliers aim to increase the wholesale price to obtain a greater price advantage. However, when the price of low-carbon products is low, the wholesale price of Scenario SG and Scenario ST depends on the cost coefficient of CER technology innovation. When the cost coefficient of CER technology innovation is large, the suppliers in Scenario SG increase the wholesale price to alleviate the cost pressure caused by CER technology innovation. The logic behind Proposition 2 (2) and Proposition 2 (3) is similar to that behind Proposition 2 (1); therefore, details are omitted for brevity. Figure 8 illustrates the trend of the impact of retail prices and CER technology innovation cost on wholesale prices in the four scenarios. In reality, when the cost of CER is low, the wholesale price set by suppliers in CER scenarios should be lower than that in traditional production scenario.

Proposition 3.

The comparative results of wholesale prices under the three CER scenarios satisfy the relationship

w^{S G *} > w^{B G *} > w^{E G *}

See Appendix A for proof.

Proposition 3 provides comparative results regarding wholesale prices across the three scenarios where CER is considered. The results show that the wholesale price is highest for CER with sufficient capital, while it is lowest in Scenario EG. According to Proposition 1, in the three CER scenarios, the increase in the CER technology innovation coefficient leads to higher wholesale price. The commission rate, consumers’ low-carbon preference coefficient, and the CER technology innovation cost coefficient have a more significant impact on the wholesale price of Scenario SG. This suggests that the supplier in the SG scenario is more inclined to raise wholesale prices to compensate for the CER technology innovation cost. In practice, suppliers avoid set wholesale prices uniformly but set lower wholesale prices in the EG scenario.

Proposition 4.

The sensitivity analysis of relevant parameters to production quantity is as follows:

(1): (i) $\frac{\partial q^{S T *}}{\partial λ} < 0$ , $\frac{\partial q^{S G *}}{\partial λ} < 0$ , $\frac{\partial q^{E G *}}{\partial λ} < 0$ , $\frac{\partial q^{B G *}}{\partial λ} < 0$ ; (ii) $\frac{\partial q^{S G *}}{\partial η} > 0$ , $\frac{\partial q^{E G *}}{\partial η} > 0$ , $\frac{\partial q^{B G *}}{\partial η} > 0$ ; (iii) $\frac{\partial q^{S G *}}{\partial k} < 0$ , $\frac{\partial q^{E G *}}{\partial k} < 0$ , $\frac{\partial q^{B G *}}{\partial k} < 0$ ; (iv) $\frac{\partial q^{E G *}}{\partial r} < 0$ , $\frac{\partial q^{B G *}}{\partial r} < 0$ ; (v) $\frac{\partial q^{B G *}}{\partial β} < 0$ ;
(2): (i) if $k < k_{4}$ , $|\frac{\partial q^{S G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}| > |\frac{\partial q^{E G *}}{\partial λ}|$ ; (ii) if $k_{4} < k < k_{5}$ , $|\frac{\partial q^{S G *}}{\partial λ}| > |\frac{\partial q^{E G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}|$ ; (iii) if $k_{5} < k < k_{6}$ , $|\frac{\partial q^{E G *}}{\partial λ}| > |\frac{\partial q^{S G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}|$ ; (iv) if $k > k_{6}$ , $|\frac{\partial q^{E G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}| > |\frac{\partial q^{S G *}}{\partial λ}|$ ;
(3): $|\frac{\partial q^{S G *}}{\partial η}| > |\frac{\partial q^{B G *}}{\partial η}| > |\frac{\partial q^{E G *}}{\partial η}|$ ;
(4): $|\frac{\partial q^{S G *}}{\partial k}| > |\frac{\partial q^{B G *}}{\partial k}| > |\frac{\partial q^{E G *}}{\partial k}|$ .

See Appendix A for proof.

Proposition 4 (1) demonstrates the effects of commission rate, consumers’ low-carbon preference coefficient, and CER technology innovation cost coefficient on the quantity of production in each of the four scenarios. Case (1) (i) shows the quantity of production in Scenario ST, Scenario SG, Scenario EG, and Scenario BG as the commission rate increases. This is shown in Figure 9a. A higher commission rate directly raises the manufacturer’s cost, leading to a reduction in production. Case (1) (ii) shows that the amount of manufacturers’ production increases with the rise of the consumers’ low-carbon preference coefficient in the three CER scenarios (Scenario SG, EG, and BG). It indicates that the more consumers are interested in low-carbon products, the more demand increases, which drives the quantity produced by manufacturers. This is shown in Figure 9b. Case (1) (iii) reflects that the higher the cost coefficient of CER technology innovation, the lower the production quantity of manufacturers, as shown in Figure 9c. This is similar to the reason in Case (1) (i), where increased CER technology innovation costs lead manufacturers to reduce production to avoid profit losses. Cases (1) (iv) and (v) indicate that increases in interest rate and interest rate discounts cause manufacturers to reduce production. Higher interest rates indirectly raise manufacturers’ costs, resulting in lower production, similar to the reasoning in Case (1) (i).

Proposition 4 (2) shows that in the three CER scenarios (Scenarios SG, EG, and BG), the extent to which the amount of manufacturers’ production varies with the commission rate depends on the CER technology innovation cost coefficient. When the cost coefficient is low, production in Scenario SG decreases the fastest with an increasing commission rate, compared to Scenarios EG and BG. However, when the cost coefficient is high, the rate of production decline in Scenario SG is smaller than in the other two scenarios, as shown in Figure 9a. According to Proposition 4 (1), it is stated that both the commission rate and the CER technology innovation cost coefficient negatively correlate with the production quantity. When

k

is large, it indicates that the manufacturer is bearing a higher cost for CER. For manufacturers with sufficient capital, the increase in commission rate has a greater impact on the production quantity of manufacturers under financial constraints.

The conclusions of Proposition 4 (3) and (4) are more intuitive. Both the consumers’ low-carbon preference coefficient and CER technology innovation cost coefficient have a greater impact on Scenario SG and a smaller impact on Scenario EG. This is shown in Figure 9b,c. This emphasizes the positive effect of consumers’ low-carbon preference in terms of the number of products produced and the negative effect of the CER technology innovation cost. Thus encouraging supply chain members to strive to enhance the appeal of low-carbon products to consumers and help manufacturers to overcome the difficulties of CER technology innovation.

Proposition 5.

The comparative results for production quantities under the four scenarios satisfy the relationship

q^{S G *} > q^{B G *} > q^{E G *} > q^{S T *}

See Appendix A for proof.

Proposition 5 presents the results of the comparison among the four scenarios regarding production quantity. The findings indicate that the quantity of production with CER carried out with sufficient capital is consistently higher than in the other two financed scenarios. The production quantity of Scenario BG is always greater than that of Scenario EG solutions, but the production quantity of Scenario ST is always the smallest. This is because, according to Proposition 4, production increases with consumers’ low-carbon preference. Consumers’ low-carbon preference increases the quantity produced by manufacturers for low-carbon products. Although there is a negative correlation between CER costs and production quantity, the results show that the benefits of consumers’ low-carbon preference outweigh the burden of increased CER costs. Therefore, in practice, whether manufacturers are financed by an E-C platform or a bank, from the perspective of production quantity, manufacturers should invest in CER.

Proposition 6.

The sensitivity analysis of relevant parameters to the unit amount of CER is as follows:

(1): (i) $\frac{\partial e^{S G *}}{\partial λ} < 0$ , $\frac{\partial e^{E G *}}{\partial λ} < 0$ , $\frac{\partial e^{B G *}}{\partial λ} < 0$ ; (ii) $\frac{\partial e^{S G *}}{\partial η} > 0$ , $\frac{\partial e^{E G *}}{\partial η} > 0$ , $\frac{\partial e^{B G *}}{\partial η} > 0$ ; (iii) $\frac{\partial e^{S G *}}{\partial k} < 0$ , $\frac{\partial e^{E G *}}{\partial k} < 0$ , $\frac{\partial e^{B G *}}{\partial k} < 0$ ; (iv) $\frac{\partial e^{E G *}}{\partial r} < 0$ , $\frac{\partial e^{B G *}}{\partial r} < 0$ ; (v) $\frac{\partial e^{B G *}}{\partial β} < 0$ ;
(2): (i) $|\frac{\partial e^{S G *}}{\partial λ}| > |\frac{\partial e^{B G *}}{\partial λ}| > |\frac{\partial e^{E G *}}{\partial λ}|$ ; (ii) $|\frac{\partial e^{S G *}}{\partial η}| > |\frac{\partial e^{B G *}}{\partial η}| > |\frac{\partial e^{E G *}}{\partial η}|$ ; (iii) $|\frac{\partial e^{S G *}}{\partial k}| > |\frac{\partial e^{B G *}}{\partial k}| > |\frac{\partial e^{E G *}}{\partial k}|$ .

See Appendix A for proof.

Proposition 6 demonstrates the effects of commission rate, consumers’ low-carbon preference coefficient, and CER technology innovation cost coefficient on the unit amount of CER in three CER scenarios. Case (1) (i) shows that the unit amount of CER decreases as the commission rate increases. The reason for this phenomenon is that higher commission to the E-C platform increases the cost to the manufacturer and will inevitably capture the funds invested by the manufacturer for CER technology innovation. Case (1) (ii) shows that the unit amount of CER increases with a rise in the commission rate. It shows that when the consumer’s low-carbon preference is greater, it can directly increase the unit amount of CER of the manufacturer. Case (1) (iii) shows that the unit amount of CER decreases with the increase of the CER technology innovation cost coefficient. The reason for this situation is roughly the same as that of Case (1) (i), where the increase in manufacturer costs reduces the unit amount of CER. At the same time, similar reasons also indicate that in Case (1) (iv) and (v), the increase in the interest rate and interest rate discount reduced the unit amount of CER.

Proposition 6 (2) indicates that in the three CER scenarios, the commission rate, consumer’s low-carbon preference coefficient, and CER technology innovation cost coefficient have a relatively consistent impact trend on CER. As shown in Figure 10. This is similar to Proposition 1 (2). Once again, it highlights that carbon emissions reduction scenarios have certain requirements for manufacturers’ financial reserves. Scenario SG is conducive to improving the efficiency of CER.

Proposition 7.

The comparative results of unit amount of CER under the three scenarios satisfy the relationship

e^{S G *} > e^{B G *} > e^{E G *}

See Appendix A for proof.

Proposition 7 indicates that after manufacturers invest in CER technology innovation, the optimal unit amount of CER in the case of sufficient capital is greater than that with funding constraints requiring financing. Among the two financing strategies, the optimal unit amount of CER in the bank financing Scenario BG is higher than that for the E-C platform financing Scenario EG. This is because, in both financing scenarios, the interest cost and service cost that manufacturers need to pay is the manufacturer’s investment cost in CER technology innovation. However, in the process of financing from banks, the government subsidizes the interest rate, and the service costs consumed by the bank financing process do not directly affect the unit amount of CER. Therefore, the optimal unit amount of CER in the case of bank financing exceeds that in the case of E-C platform financing, as illustrated in Figure 10. In practice, when external conditions demand high CER capabilities from manufacturers, they should prioritize CER technology investment with sufficient capital. Moreover, when choosing between E-C platform financing and bank financing, bank financing leads to better CER outcomes. Figure 10 clearly illustrates this conclusion.

Proposition 8.

The sensitivity analysis of relevant parameters to the profitability of suppliers in four scenarios is as follows:

(1): $\frac{\partial \prod_{s}^{S T *}}{\partial λ} < 0$ , $\frac{\partial \prod_{s}^{S G *}}{\partial λ} < 0$ , $\frac{\partial \prod_{s}^{E G *}}{\partial λ} < 0$ , $\frac{\partial \prod_{s}^{B G *}}{\partial λ} < 0$ .
(2): $\frac{\partial \prod_{s}^{S G *}}{\partial η} > 0$ , $\frac{\partial \prod_{s}^{E G *}}{\partial η} > 0$ , $\frac{\partial \prod_{s}^{B G *}}{\partial η} > 0$ .
(3): $\frac{\partial \prod_{s}^{S G *}}{\partial k} < 0$ , $\frac{\partial \prod_{s}^{E G *}}{\partial k} < 0$ , $\frac{\partial \prod_{s}^{B G *}}{\partial k} < 0$ .
(4): $\frac{\partial \prod_{s}^{E G *}}{\partial r} < 0$ , $\frac{\partial \prod_{s}^{B G *}}{\partial r} < 0$ .
(5): $\frac{\partial \prod_{s}^{B G *}}{\partial β} < 0$ .

See Appendix A for proof.

Proposition 8 demonstrates the effects of commission rate, consumers’ low-carbon preference coefficient, CER technology innovation cost coefficient, interest rate, and interest rate discount on suppliers’ profits across different scenarios. Case (1) (i) shows that suppliers’ profit in all four scenarios decreases as the reseller’s commission rate increases, indicating that a higher commission rate inevitably reduces suppliers’ profits (see Figure 11). This suggests that as the E-C platform matures, its development will gradually slow down. Case (1) (ii) shows that the increase in consumer’s low-carbon preference coefficient can raise suppliers’ profit. Even when entity in the supply chain, suppliers can benefit from increased consumer preference for low-carbon products, as also demonstrated in the experiments of Qin et al. [40]. Case (1) (iii), (iv), and (v) show that the suppliers’ profit increases with decreases in CER technology innovation cost coefficient, interest rate, and interest rate discount. These reflect the interconnectedness of the supply chain, as the cost of manufacturers downstream of the supply chain increases, the profit of suppliers will also decrease.

Proposition 9.

The comparative results for the supplier’s profit under the three scenarios are given as follows:

(1): $\prod_{s}^{S G *} > \prod_{s}^{S T *}$ ;
(2): $\prod_{s}^{E G *} > \prod_{s}^{S T *}$ ;
(3): $\prod_{s}^{B G *} > \prod_{s}^{S T *}$ .

See Appendix A for proof.

Proposition 9 compares the profits of suppliers in three CER scenarios (Scenario SG, EG, and BG) with those in traditional scenarios (Scenario ST). Proposition 9. (1) indicates that the supplier’s profit in Scenario SG is greater than the profit in Scenario ST. This is because consumers have a preference for low-carbon products, and manufacturers who invest in CER technology innovation are favored by these preferences. As demand for low-carbon products increases, manufacturers’ production quantity rise accordingly. This leads to an increase in profits for manufacturer, and the increased profits from selling low-carbon products outweigh the CER technology innovation cost. As a result, the supplier’s profit in Scenario SG is greater than that in Scenario ST. These fully illustrate that investing in CER technology innovation can increase suppliers’ profits.

Proposition 9 (2) shows that the supplier’s profit in Scenario EG is greater than in Scenario ST. Proposition 9 (3) suggests that the supplier’s profit in Scenario BG is greater than the profit in Scenario ST. Figure 11. visually represents. The reasons for the emergence of Proposition 9 (2) and Proposition 9 (3) are quite similar to those of Proposition 9 (1). However, it is important to note that the cost incurred in both financing scenarios (Scenario EG and Scenario BG) will not result in lower profits for suppliers compared to the traditional scenario (Scenario ST). Therefore, in the supply chain, suppliers should actively encourage manufacturers to invest in CER technology innovation. For example, China’s ZX Packing Group raised CNY 1.538 billion to invest in an environmentally friendly tableware project and research and development center, resulting in increased overall profits [58].

Proposition 10.

The sensitivity analysis of relevant parameters to the profitability of manufacturers in four scenarios is as follows:

(1): $\frac{\partial \prod_{m}^{S T *}}{\partial λ} < 0$ , $\frac{\partial \prod_{m}^{S G *}}{\partial λ} < 0$ , $\frac{\partial \prod_{m}^{E G *}}{\partial λ} < 0$ , $\frac{\partial \prod_{m}^{B G *}}{\partial λ} < 0$ .
(2): $\frac{\partial \prod_{m}^{S G *}}{\partial η} > 0$ , $\frac{\partial \prod_{m}^{E G *}}{\partial η} > 0$ , $\frac{\partial \prod_{m}^{B G *}}{\partial η} > 0$ .
(3): $\frac{\partial \prod_{m}^{S G *}}{\partial k} < 0$ , $\frac{\partial \prod_{m}^{E G *}}{\partial k} < 0$ , $\frac{\partial \prod_{m}^{B G *}}{\partial k} < 0$ .

See Appendix A for proof.

Proposition 10 demonstrates the impact of the commission rate, consumers’ low-carbon preference coefficient, and the CER technology innovation cost coefficient on manufacturer profits across different scenarios. Case (1) (i) shows that the manufacturer’s profit decrease as the commission rate increases in all four scenarios. The rise in the commission rate coefficient for consignment sales inevitably affects manufacturers’ profits. When combined with Proposition 8 Case (1) (i), it is clear that the more commission a manufacturer pays to an E-C platform, the lower the profits of both upstream suppliers and manufacturers. This conclusion aligns with the real-world example of DouyinEC’s recent reduction of platform commissions [59]. Case (1) (ii) indicates that an increase in consumers’ low-carbon preference can increase manufacturers’ profits. This is also the primary reason manufacturers are willing to invest in CER technology innovation. Case (1) (iii) shows that as the CER technology innovation cost coefficient increases, manufacturers’ profits decline. Reducing CER technology innovation costs is a goal that all members of the supply chain should strive to achieve.

Proposition 11.

The comparative results for the manufacturer’s profit under the three scenarios are given as follows:

\prod_{m}^{S G *} > \prod_{m}^{S T *}

See Appendix A for proof.

Proposition 11 compares the profits of the manufacturer in the two well-capitalized scenarios (Scenario ST and Scenario SG). The manufacturer’s profit in Scenario SG is higher than in Scenario ST. This is because consumer demand for low-carbon products increases, leading to a corresponding increase in the manufacturer’s production quantity. This conclusion is fully supported by Proposition 4 (3). Therefore, investing in CER technology innovation is profitable for manufacturers.

Proposition 12.

The comparative results of manufacturer’s profit under the two financing scenarios and traditional scenario satisfy the following relationship:

(1): $\prod_{m}^{E G *} > \prod_{m}^{S T *}$ ;
(2): (i) if $T > T_{1}$ , $\prod_{m}^{B G *} < \prod_{m}^{S T *}$ ; (ii) if $T < T_{1}$ , $\prod_{m}^{B G *} > \prod_{m}^{S T *}$ ;
(3): (i) if $T > T_{2}$ , $\prod_{m}^{B G *} < \prod_{m}^{E G *}$ ; (ii) if $T < T_{2}$ , $\prod_{m}^{B G *} > \prod_{m}^{E G *}$ .

See Appendix A for proof.

Proposition 12 (1) shows that the manufacturer’s profit in Scenario EG is greater than in Scenario ST. This is because consumer preference for low-carbon products increases demand, leading to a rise in the manufacturer’s production quantity. However, the increase in profit from selling low-carbon products compensates for the cost of CER and financing, so the profit of the manufacturer in Scenario EG is greater than that in Scenario ST, where the manufacturer is well capitalized but has not reduced its carbon emissions, and even if the manufacturer is faced with financial constraints and needs to pay the interest rate cost of financing on the E-C platform, it should still actively adopt the E-C platform financing strategy.

Proposition 12 (2) demonstrates when manufacturers should finance and invest in CER technology innovation and when they should abandon it. Even if the manufacturer pays the interest to finance the E-C platform, production quantity and profitability of the manufacturer increases with the consumer’s low-carbon preference. This is an interesting finding, as it contradicts the findings in previous studies [16]. The difference is that our study extends to the effect of financing service cost on bank financing. When the service cost of bank financing is large (i.e.,

T > T_{1}

), Scenario ST is more profitable and the manufacturer should choose to give up financing from the bank. In a practical example, Barclays announced its intention to stop directly financing new oil and gas projects.

Proposition 12 (3) shows that when the service cost of bank financing is small (i.e.,

T < T_{2}

), the manufacturer’s profit in Scenario BG is greater than in Scenario EG. Due to the manufacturer’s capital constraint on bank financing, although the interest rate discount reduces the interest rate, it is affected by the service costs such as time and labor, and when service cost

T

falls below a certain threshold, the manufacturer chooses to finance from the bank, and vice versa to finance from the E-C platform. Figure 12 illustrates the combined effect of bank service costs and other different parameters on the manufacturers’ profit for both financing options and traditional scenarios. Specific CER implementation and financing strategies are provided for each member of the supply chain.

In this paper, we comprehensively analyze how the sensitivity of equilibrium solutions and profits of supply chain members changes in response to factors such as commission rate and consumers’ low-carbon preferences under different financial structures. Additionally, we compare the CER and financing strategies of suppliers, manufacturers, and E-C platforms across various scenarios. Finally, we provide recommendations applicable to supply chain operations in real-world situations.

6. Conclusions and Further Research

6.1. Conclusions

This paper investigates CER and financing strategies in a platform supply chain comprising one supplier, one manufacturer, one E-C platform, and consumers with low-carbon preference across four scenarios (Scenario ST, SG, EG, and BG), respectively. Unlike previous studies, Qin et al. treated the wholesale price as an exogenous variable in their analysis of low-carbon supply chain finance [40], whereas in reality, the wholesale price is endogenous. We innovatively model the wholesale price as an endogenous variable to examine the impact of CER and financing changes on pricing. In addition, Fu et al. did not consider demand uncertainty in their study of corporate CER [1]. However, in practice, demand may be unstable and deterministic demand will no longer be appropriate. Moreover, our study innovatively considers the problem of uncertain market demand. Our study enriches the research on low-carbon financing decision-making of supply chains by establishing four different scenarios of supply chain models for the scenarios “traditional scenario vs. CER scenario”, “sufficient capital vs. constrained funds”, and “E-C platform financing vs. bank financing”. We explore the impacts of interest rate, consumers’ low-carbon preference, commission rate, and CER technology innovation cost on the CER and profits of supply chain members with different financing strategies and give the equilibrium results of each supply chain member under the four scenarios. The equilibrium outcomes for each supply chain member across the four scenarios yield several important conclusions and managerial insights, as outlined below:

(1): Supply chain CER: First, the unit amount of CER is higher in the sufficient capital scenario than in the financing scenario. Second, when the manufacturers’ capital is constrained, the unit amount of CER decreases as interest rate rise. The unit amount of CER is greater in Scenario BG than in Scenario EG. Finally, the impact of consumers’ low-carbon preference on the unit amount of CER depends on the CER technology innovation cost. When the CER technology innovation cost is greater than a certain threshold, the unit amount of CER always increases with the increase of consumers’ low-carbon preference.
(2): Supply chain profit: On the one hand, whether manufacturers invest in CER technology innovation, supplier profits and manufacturer profits decrease as the commission rate increases. In three CER scenarios (Scenario SG, EG, and BG), the increase in consumer low-carbon preference proposes to increase the profitability of suppliers and manufacturers. On the other hand, the manufacturer’s profit in Scenario SG is greater than in Scenario ST. In addition, although there are financing costs for the manufacturer in Scenario EG and Scenario BG, the suppliers’ profit is still greater than in Scenario ST.
(3): Financing strategy: When the manufacturer is constrained by the capital for CER, the manufacturer should choose to carry out financing rather than give up CER for the production of traditional products. If the service cost of bank financing is less than a certain threshold, bank financing is chosen, and vice versa, E-C platform financing is chosen.

6.2. Recommendations

Firstly, for suppliers, if manufacturers invest in CER technology innovation, their profits will increase, and carbon emissions during production will decrease, thereby significantly reducing environmental pressure on society. When consumers’ low-carbon preference increases, we recommend that suppliers should lower wholesale prices to incentivize manufacturers to increase production. In addition, suppliers can take the initiative to help manufacturers overcome difficulties when they have financial constraints. If manufacturers abandon investment in CER technology innovation, suppliers will lose the opportunity to increase their profits, and it will also hinder improvements in the social environment.

Secondly, manufacturers should not abandon investing in CER technology innovation when they are subject to financial constraints. We suggest that manufacturers carefully assess financing costs and actively seek the optimal financing model. In addition, when consumers’ low-carbon preference increases, we suggest that manufacturers increase the quantity of production. Since consumer preferences can enhance the profits of both suppliers and manufacturers, supply chain members should leverage their strengths to promote low-carbon initiatives and further stimulate consumer demand for low-carbon products. For example, manufacturers can include low-carbon messaging on product packaging, while E-C platforms can promote low-carbon products to consumers. In practice, P&G Group prints carbon reduction slogans on product packaging to highlight the company’s commitment to carbon reduction and enhance consumer preference [60]. Similarly, Gold Hongye Paper Group has organized environmental flash mobs to promote carbon reduction activities and attract more consumers [61]. Consumers with a strong preference for low-carbon products are more likely to engage in corporate carbon reduction initiatives, while for those less sensitive to low-carbon products, opting for traditional production scenarios may be more cost-effective.

Thirdly, for E-C platforms and banks, when E-C platforms have sufficient funds, they should consider providing financial support to manufacturers to encourage them to invest in CER technology innovation. In practice, China Guangfa Bank has targeted CER goals by launching aviation leasing financing projects, which serve as valuable examples of promoting low-carbon development and green transformation in the aviation industry [62]. E-C platforms and banks should widely adopt “green finance” tools to bridge CER information gaps among supply chain members and reduce additional costs incurred during the financing process.

Fourthly, for governments, when manufacturers choose to invest in CER, the social environment improves, helping the government achieve carbon neutrality targets more quickly and efficiently, thereby fulfilling international environmental commitments. Furthermore, by encouraging manufacturers to invest in carbon reduction, the government accelerates industrial greening while reducing environmental management costs, contributing to the development of a sustainable society.

6.3. Limitations and Further Research

This paper provides an in-depth analysis of supply chain operation models across four scenarios, in which financing from E-C platforms and financing from banks are compared, respectively. However, practical supply chain operation models are more complex. There are several limitations to our study. We describe these limitations and discuss possible directions for future research.

First, while our model accounts for uncertain market demand and changes in interest rate, studying policy-related factors such as inflation would be valuable. Future research could incorporate dynamic elements, such as financing policies, into the strategic analysis of CER in supply chains.

Second, we briefly compared our findings with real-world examples, such as XZ Packing, Amazon, and DouyinEC. In future research, empirical data from companies could be used to validate the accuracy of the low-carbon supply chain financing model and improve its applicability in practical settings.

Finally, with the development of financial technology, E-C platforms and banks may adopt technologies like blockchain to streamline the financing process and reduce the cost. These developments represent promising future research directions and priorities.

Author Contributions

Conceptualization, J.S.; methodology, Y.Z.; software, Y.Z.; validation, Y.Z.; formal analysis, Y.Z. and J.S.; investigation, J.S.; resources, J.S.; data curation, Y.Z.; writing-original draft, Y.Z. and J.S.; writing-review and editing, Y.Z. and J.S.; visualization, J.S.; supervision, Y.Z. and J.S.; project administration, J.S.; funding acquisition, J.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Shaanxi Provincial Social Science Foundation Project (No. 2021D044); Xi’an Social Science Planning Fund Project (No. 23JX61); The Fundamental Research Funds for the Central Universities (No. GLZX23042).

Data Availability Statement

All data generated or analyzed during this study are included in this published article.

Conflicts of Interest

The authors declare no competing interests.

Appendix A

Proof of Lemma 1.

Solving the problems by backward induction, we obtain the equilibrium outcomes of Scenario EG. Because

\frac{\partial^{2} \prod_{m}^{S T}}{\partial {(q^{T})}^{2}} = - \frac{(1 - λ) p^{T}}{2 A} < 0

q^{S T}

has a maximum value. Let

\frac{\partial \prod_{m}^{S T}}{\partial q^{T}} = 0

, we can get

q^{S T} = a + A - \frac{2 A w^{T}}{(1 - λ) p^{T}}

. We substitute

q^{S T} = a + A - \frac{2 A w^{T}}{(1 - λ) p^{T}}

into Equation (1). Subject to

\frac{\partial^{2} \prod_{s}^{S T}}{\partial {(w^{T})}^{2}} = - \frac{4 A}{(1 - λ) p^{T}} < 0

and let

\frac{\partial \prod_{s}^{S T}}{\partial w^{T}} = 0

, we can derive

w^{S T *} = \frac{(a + A) (1 - λ) p^{T}}{4 A} + \frac{c}{2}

. Substituting

w^{S T *} = \frac{(a + A) (1 - λ) p^{T}}{4 A} + \frac{c}{2}

into

q^{S T} = a + A - \frac{2 A w^{T}}{(1 - λ) p^{T}}

, we can obtain

q^{S T *} = \frac{a + A}{2} - \frac{A c}{(1 - λ) p^{T}}

. □

Proof of Lemma 2.

The proof of Lemma 2 is similar to that of Lemma 1. Thus, we omit it. □

Proof of Lemma 3.

The proof of Lemma 3 is similar to that of Lemma 1. Thus, we omit it. □

Proof of Lemma 4.

The proof of Lemma 4 is similar to that of Lemma 1. Thus, we omit it. □

Proof of Proposition 1.

(1)

(i): $\frac{\partial w^{S T *}}{\partial λ} = - \frac{p^{T} (A + a)}{4 A} < 0$ , $\frac{\partial w^{S G *}}{\partial λ} = - \frac{p^{G} ({(η^{2} (1 - λ) p^{G})}^{2} + 4 A k η^{2} (1 - λ) p^{G} + 2 A k^{2} (A + a))}{2 {(η^{2} p^{G} (1 - λ) + 2 k A)}^{2}} < 0$ , $\frac{\partial w^{E G *}}{\partial λ} = - \frac{p^{G}}{2 + 2 r} - \frac{(a - A) A k^{2} p^{G} (1 + r)}{{(2 A k (1 + r) + p^{G} η^{2} (1 - λ))}^{2}} < 0$ , $\frac{\partial w^{B G *}}{\partial λ} = - \frac{p^{G}}{2 r β + 2} - \frac{(a - A) A k^{2} p^{G} (1 + r β)}{{(2 A k (r β + 1) + (1 - λ) p^{G} η^{2})}^{2}} < 0$ ;
(ii): $\frac{\partial w^{S G *}}{\partial η} = - \frac{{((1 - λ) p^{G})}^{2} η k (a - A)}{{(η^{2} (1 - λ) p^{G} + 2 A k)}^{2}} < 0$ , $\frac{\partial w^{E G *}}{\partial η} = - \frac{(a - A) k {(p^{G})}^{2} η {(1 - λ)}^{2}}{{(2 A k (1 + r) + p^{G} η^{2} (1 - λ))}^{2}} < 0$ , $\frac{\partial w^{B G *}}{\partial η} = - \frac{(a - A) k {((1 - λ) p^{G})}^{2} η}{{(2 A k (1 + r β) + (1 - λ) p^{G} η^{2})}^{2}} < 0$ .
(iii): $\frac{\partial w^{S G *}}{\partial k} = \frac{(a - A) {(p^{G} η (1 - λ))}^{2}}{2 {(η^{2} (1 - λ) p^{G} - 2 A k)}^{2}} > 0$ , $\frac{\partial w^{E G *}}{\partial k} = \frac{(a - A) {(η (1 - λ) p^{G})}^{2}}{2 {(2 A k (1 + r) - η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $\frac{\partial w^{B G *}}{\partial k} = \frac{(a - A) {(η (1 - λ) p^{G})}^{2}}{2 {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ .
(iv): $\frac{\partial w^{E G *}}{\partial r} = - \frac{(1 - λ) p^{G} (2 A (a + A) k^{2} {(1 + r)}^{2} + η^{2} (1 - λ) p^{G} (4 A k (1 + r) + η^{2} (1 - λ) p^{G}))}{2 {(1 + r)}^{2} {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial w^{B G *}}{\partial r} = - \frac{β (1 - λ) p^{G} (2 A (a + A) {(k + k r β)}^{2} + η^{2} (1 + λ) p^{G} (4 A k (1 + r β) + η^{2} (1 - λ) p^{G}))}{2 {(1 + r β)}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ .
(v): $\frac{\partial w^{B G *}}{\partial β} = - \frac{r (1 - λ) p^{G} (2 A (a + A) {(k + k r β)}^{2} + η^{2} (1 - λ) p^{G} (4 A k (1 + r β) + η^{2} (1 - λ) p^{G}))}{2 {(1 + r β)}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ .

(2)

(i): $|\frac{\partial w^{S G *}}{\partial λ}| - |\frac{\partial w^{B G *}}{\partial λ}| = \frac{r β p^{G} (\begin{array}{l} 8 A^{3} (a + A) k^{4} {(1 + r β)}^{2} + 16 A^{3} k^{3} (2 + 3 r β + r^{2} β^{2}) η^{2} (1 - λ) p^{G} + 4 A k (2 + r β) η^{6} {((1 - λ) p^{G})}^{3} \\ - 2 A k^{2} (a + a r β - A (13 + 13 r β + 2 r^{2} β^{2})) η^{4} {((1 - λ) p^{G})}^{2} + η^{8} {((1 - λ) p^{G})}^{4} \end{array})}{2 (1 + r β) {(2 A k + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $|\frac{\partial w^{B G *}}{\partial λ}| - |\frac{\partial w^{E G *}}{\partial λ}| = \frac{(r (1 - β) p^{G} (\begin{array}{l} 8 A^{3} (a + A) k^{4} {(1 + r)}^{2} {(1 + r β)}^{2} + 4 A k (2 + r + r β) η^{2} (1 - λ) p^{G} (4 A^{2} k^{2} (1 + r) (1 + r β) + η^{4} {((1 - λ) p^{G})}^{2}) \\ - 2 A k^{2} (a (1 + r) (1 + r β) - A (13 + 13 r (1 + β) + r^{2} (2 + 9 β + 2 β^{2}))) η^{4} {((1 - λ) p^{G})}^{2} + η^{8} {((1 - λ) p^{G})}^{4} \end{array}))}{2 (1 + r) (1 + r β) {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ ; Thus, we can obtain $|\frac{\partial w^{S G *}}{\partial λ}| > |\frac{\partial w^{B G *}}{\partial λ}| > |\frac{\partial w^{E G *}}{\partial λ}|$ .
(ii): $|\frac{\partial w^{S G *}}{\partial η}| - |\frac{\partial w^{B G *}}{\partial η}| = \frac{4 (a - A) A r β η {(k (1 - λ) p^{G})}^{2} (A k (2 + r β) + η^{2} (1 - λ) p^{G})}{{(2 A k + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $|\frac{\partial w^{B G *}}{\partial η}| - |\frac{\partial w^{E G *}}{\partial η}| = \frac{4 (a - A) A r (1 - β) η {(k (1 - λ) p^{G})}^{2} (A k (2 + r + r β) + η^{2} (1 - λ) p^{G})}{{(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ . Thus, we can obtain $|\frac{\partial w^{S G *}}{\partial η}| > |\frac{\partial w^{B G *}}{\partial η}| > |\frac{\partial w^{E G *}}{\partial η}|$ .
(iii): $|\frac{\partial w^{S G *}}{\partial k}| - |\frac{\partial w^{B G *}}{\partial k}| = \frac{2 (a - A) A k r β {(η (1 - λ) p^{G})}^{2} (A k (2 + r β) + η^{2} (1 - λ) p^{G})}{{(2 A k + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $|\frac{\partial w^{B G *}}{\partial k}| - |\frac{\partial w^{E G *}}{\partial k}| = \frac{2 (a - A) A k r (1 - β) {(η (1 - λ) p^{G})}^{2} (A k (2 + r + r β) + η^{2} (1 - λ) p^{G})}{{(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ . Thus, we can obtain $|\frac{\partial w^{S G *}}{\partial k}| > |\frac{\partial w^{B G *}}{\partial k}| > |\frac{\partial w^{E G *}}{\partial k}|$ . □

Proof of Proposition 2.

(1): $w^{S G *} - w^{S T *} = \frac{(1 - λ) (2 A (p^{G} - p^{T}) (A + a) k - ((p^{T} - 2 p^{G}) A + a p^{T}) p^{G} (1 - λ) η^{2})}{4 (p^{G} η^{2} (1 - λ) + 2 A k) A}$ . Subject to $4 (p^{G} η^{2} (1 - λ) + 2 A k) A > 0$ and $2 A (1 - λ) (p^{G} - p^{T}) (A + a) > 0$ , $(1 - λ) (2 A (p^{G} - p^{T}) (A + a) k - ((p^{T} - 2 p^{G}) A + a p^{T}) p^{G} (1 - λ) η^{2})$ increases monotonically with $k$ . Solving $(1 - λ) (2 A (p^{G} - p^{T}) (A + a) k - ((p^{T} - 2 p^{G}) A + a p^{T}) p^{G} (1 - λ) η^{2}) = 0$ , we have $k_{1} = \frac{((p^{T} - 2 p^{G}) A + a p^{T}) p^{G} (1 - λ) η^{2}}{2 A (p^{G} - p^{T}) (A + a)}$ . Similarly, subject to $2 A (p^{G} - p^{T}) (A + a) > 0$ and $- 2 A η^{2} (1 - λ) < 0$ , $((p^{T} - 2 p^{G}) A + a p^{T}) p^{G} (1 - λ) η^{2}$ is a quadratic function of $p^{G}$ . That is, the function $((p^{T} - 2 p^{G}) A + a p^{T}) p^{G} (1 - λ) η^{2} = 0$ has two different real roots. Solving $((p^{T} - 2 p^{G}) A + a p^{T}) p^{G} (1 - λ) η^{2} = 0$ , we obtain $p_{1}^{G^{'}} = \frac{(A + a) p^{T}}{2 A}$ and $p^{G^{″}} = 0$ . Since $p^{G^{″}} > 0$ , we just need to discuss $p_{1}^{G^{'}} = \frac{(A + a) p^{T}}{2 A}$ . Thus, we ensure that (i) if $p^{G} > \frac{(A + a) p^{T}}{2 A}$ , then $w^{S G *} > w^{S T *}$ ; (ii) if $p^{G} < \frac{(A + a) p^{T}}{2 A}$ and $k > k_{1}$ , then $w^{S G *} > w^{S T *}$ ; (iii) if $p^{G} < \frac{(A + a) p^{T}}{2 A}$ and $k < k_{1}$ , then $w^{S G *} < w^{S T *}$ .
(2): $w^{E G *} - w^{S T *} = \frac{(1 - λ) (2 (A + a) (1 + r) (p^{G} - p^{T} (1 + r)) A k - p^{G} η^{2} (1 - λ) ((1 + r) (A + a) p^{T} - 2 A p^{G}))}{4 (p^{G} η^{2} (1 - λ) + 2 k A (1 + r)) (1 + r) A}$ . Subject to $4 (p^{G} η^{2} (1 - λ) + 2 k A (1 + r)) (1 + r) A > 0$ and $2 A (1 - λ) (A + a) (1 + r) (p^{G} - p^{T} (1 + r)) > 0$ , $(1 - λ) (2 (A + a) (1 + r) (p^{G} - p^{T} (1 + r)) A k - p^{G} η^{2} (1 - λ) ((1 + r) (A + a) p^{T} - 2 A p^{G}))$ increases monotonically with $k$ . Solving $(1 - λ) (2 (A + a) (1 + r) (p^{G} - p^{T} (1 + r)) A k - p^{G} η^{2} (1 - λ) ((1 + r) (A + a) p^{T} - 2 A p^{G})) = 0$ , we have $k_{2} = \frac{p^{G} η^{2} (1 - λ) ((1 + r) (A + a) p^{T} - 2 A p^{G})}{2 A (A + a) (1 + r) (p^{G} - p^{T} (1 + r))}$ . Similarly, subject to $2 A (A + a) (1 + r) (p^{G} - p^{T} (1 + r)) > 0$ and $- 2 η^{2} (1 - λ) A < 0$ , $p^{G} η^{2} (1 - λ) ((1 + r) (A + a) p^{T} - 2 A p^{G})$ is a quadratic function of $p^{G}$ . That is, the function $p^{G} η^{2} (1 - λ) ((1 + r) (A + a) p^{T} - 2 A p^{G}) = 0$ has two different real roots. Solving $p^{G} η^{2} (1 - λ) ((1 + r) (A + a) p^{T} - 2 A p^{G}) = 0$ , we obtain $p_{2}^{G^{'}} = \frac{(1 + r) (A + a) p^{T}}{2 A}$ and $p_{2}^{G^{″}} = 0$ . Since $p^{G^{″}} > 0$ , we just need to discuss $p_{2}^{G^{'}} = \frac{(1 + r) (A + a) p^{T}}{2 A}$ . Thus, we ensure that: (i) if $p^{G} > \frac{(1 + r) (A + a) p^{T}}{2 A}$ , then $w^{E G *} > w^{S T *}$ ; (ii) if $p^{G} < \frac{(1 + r) (A + a) p^{T}}{2 A}$ and $k > k_{2}$ , then $w^{E G *} > w^{S T *}$ ; (iii) if $p^{G} < \frac{(1 + r) (A + a) p^{T}}{2 A}$ and $k < k_{2}$ , then $w^{E G *} < w^{S T *}$ .
(3): $w^{B G *} - w^{S T *} = \frac{2 A k (a + A) (1 + r β) (1 - λ) (p^{G} - p^{T} - r β p^{T}) + η^{2} {(1 - λ)}^{2} p^{G} (2 A p^{G} - a (1 + r β) p^{T} - A (1 + r β) p^{T})}{4 A (1 + r β) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}$ . Subject to $4 A (1 + r β) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G}) > 0$ and $2 A (a + A) (1 + r β) (1 - λ) (p^{G} - p^{T} - r β p^{T}) > 0$ , $2 A k (a + A) (1 + r β) (1 - λ) (p^{G} - p^{T} - r β p^{T}) + η^{2} {(1 - λ)}^{2} p^{G} (2 A p^{G} - a (1 + r β) p^{T} - A (1 + r β) p^{T})$ increases monotonically with $k$ . Solving $2 A k (a + A) (1 + r β) (1 - λ) (p^{G} - p^{T} - r β p^{T}) + η^{2} {(1 - λ)}^{2} p^{G} (2 A p^{G} - a (1 + r β) p^{T} - A (1 + r β) p^{T}) = 0$ , we have $k_{3} = \frac{- 2 A η^{2} (1 - λ) {(p^{G})}^{2} + η^{2} (1 - λ) p^{G} ((a + A) (1 + r β) p^{T})}{2 A (a + A) (1 + r β) (p^{G} - p^{T} - r β p^{T})}$ . Similarly, subject to $2 A (a + A) (1 + r β) (p^{G} - p^{T} - r β p^{T}) > 0$ and $- 2 A η^{2} (1 - λ) < 0$ , $- 2 A η^{2} (1 - λ) {(p^{G})}^{2} + η^{2} (1 - λ) p^{G} ((a + A) (1 + r β) p^{T})$ is a quadratic function of $p^{G}$ . That is, the function $- 2 A η^{2} (1 - λ) {(p^{G})}^{2} + η^{2} (1 - λ) p^{G} ((a + A) (1 + r β) p^{T}) = 0$ has two different real roots. Solving $- 2 A η^{2} (1 - λ) {(p^{G})}^{2} + η^{2} (1 - λ) p^{G} ((a + A) (1 + r β) p^{T}) = 0$ , we obtain $p_{3}^{G^{'}} = \frac{(a + A) p^{T} (1 + r β)}{2 A}$ and $p_{3}^{G^{″}} = 0$ . Since $p^{G^{″}} > 0$ , we just need to discuss $p_{3}^{G^{'}} = \frac{(a + A) p^{T} (1 + r β)}{2 A}$ . Thus, we ensure that: (i) if $p^{G} > \frac{(a + A) p^{T} (1 + r β)}{2 A}$ , then $w^{B G *} > w^{S T *}$ ; (ii) if $p^{G} < \frac{(a + A) p^{T} (1 + r β)}{2 A}$ and $k > k_{3}$ , then $w^{B G *} > w^{S T *}$ ; (iii) if $p^{G} < \frac{(a + A) p^{T} (1 + r β)}{2 A}$ and $k < k_{3}$ , then $w^{B G *} < w^{S T *}$ . □

Proof of Proposition 3.

w^{S G *} - w^{B G *} = \frac{r β (1 - λ) p^{G} (2 A (a + A) k^{2} (1 + r β) + η^{2} (1 - λ) p^{G} (2 A k (2 + r β) + η^{2} (1 - λ) p^{G}))}{2 (1 + r β) (2 A k + η^{2} (1 - λ) p^{G}) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})} > 0

w^{B G *} - w^{E G *} = \frac{r (1 - β) (1 - λ) p^{G} (2 A (a + A) k^{2} (1 + r) (1 + r β) + η^{2} (1 - λ) p^{G} (2 A k (2 + r + r β) + η^{2} (1 - λ) p^{G}))}{2 (1 + r) (1 + r β) (2 A k (1 + r) + η^{2} (1 - λ) p^{G}) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})} > 0

. Thus, we can obtain

w^{S G *} > w^{B G *} > w^{E G *}

. □

Proof of Proposition 4.

(1)

(i): $\frac{\partial q^{S T *}}{\partial λ} = - \frac{c A}{{(1 - λ)}^{2} p^{T}} < 0$ , $\frac{\partial q^{S G *}}{\partial λ} = - \frac{η^{2} p^{G}}{2 k} - \frac{A c}{{(1 - λ)}^{2} p^{G}} < 0$ , $\frac{\partial q^{E G *}}{\partial λ} = - \frac{p^{G} η^{2}}{2 k (1 + r)} - \frac{A c (1 + r)}{p^{G} {(1 - λ)}^{2}} < 0$ , $\frac{\partial q^{B G *}}{\partial λ} = - \frac{p^{G} η^{2}}{2 k r β + 2 k} - \frac{A c (1 + r β)}{p^{G} {(1 - λ)}^{2}} < 0$ .
(ii): $\frac{\partial q^{S G *}}{\partial η} = \frac{(p^{G} (1 - λ) - c) η}{k} > 0$ , $\frac{\partial q^{E G *}}{\partial η} = \frac{η (p^{G} (1 - λ) - c (1 + r))}{k (1 + r)} > 0$ , $\frac{\partial q^{B G *}}{\partial η} = \frac{η (1 - λ) p^{G}}{k (1 - r β)} - \frac{η c}{k} > 0$ .
(iii): $\frac{\partial q^{S G *}}{\partial k} = - \frac{η^{2} (p^{G} (1 - λ) - c)}{2 k^{2}} < 0$ , $\frac{\partial q^{E G *}}{\partial k} = - \frac{η^{2} ((1 - λ) p^{G} - c (1 + r))}{2 k^{2} (1 + r)} < 0$ , $\frac{\partial q^{B G *}}{\partial k} = - \frac{η^{2} ((1 - λ) p^{G} - c (1 + r β))}{2 k^{2} (1 + r β)} < 0$ .
(iv): $\frac{\partial q^{E G *}}{\partial r} = - \frac{2 A c k {(1 + r)}^{2} + {(η (1 - λ) p^{G})}^{2}}{2 k {(1 + r)}^{2} (1 - λ) p^{G}} < 0$ , $\frac{\partial q^{B G *}}{\partial r} = - \frac{A c β}{(1 - λ) p^{G}} - \frac{β η^{2} (1 - λ) p^{G}}{2 k {(1 + r β)}^{2}} < 0$ .
(v): $\frac{\partial q^{B G *}}{\partial β} = - \frac{A c r}{(1 - λ) p^{G}} - \frac{r η^{2} (1 - λ) p^{G}}{2 k {(1 + r β)}^{2}} < 0$ .

(2)

|\frac{\partial q^{B G *}}{\partial λ}| - |\frac{\partial q^{E G *}}{\partial λ}| = \frac{r (β - 1) (2 A c k (1 + r) (1 + r β) - {(η (1 - λ) p^{G})}^{2})}{2 k (1 + r) (1 + r β) {(1 - λ)}^{2} p^{G}}

. Subject to

2 k (1 + r) (1 + r β) {(1 - λ)}^{2} p^{G} > 0

and

2 k (1 + r) (1 + r β) {(1 - λ)}^{2} p^{G} > 0

r (β - 1) (2 A c k (1 + r) (1 + r β) - {(η (1 - λ) p^{G})}^{2})

decreases monotonically with

k

. Solving

r (β - 1) (2 A c k (1 + r) (1 + r β) - {(η (1 - λ) p^{G})}^{2}) = 0

, we have

k_{4} = \frac{{(η (1 - λ) p^{G})}^{2}}{2 A c (1 + r) (1 + r β)}

. That is,

|\frac{\partial q^{B G *}}{\partial λ}| < |\frac{\partial q^{E G *}}{\partial λ}|

k > k_{4}

; otherwise,

|\frac{\partial q^{B G *}}{\partial λ}| > |\frac{\partial q^{E G *}}{\partial λ}|

|\frac{\partial q^{S G *}}{\partial λ}| - |\frac{\partial q^{E G *}}{\partial λ}| = \frac{r ({(η (1 - λ) p^{G})}^{2} - 2 A c k (1 + r))}{2 k p^{G} (1 + r) {(1 - λ)}^{2}}

. Subject to

2 k p^{G} (1 + r) {(1 - λ)}^{2} > 0

and

- 2 A c r (1 + r) < 0

r ({(η (1 - λ) p^{G})}^{2} - 2 A c k (1 + r))

decreases monotonically with

k

. Solving

r ({(η (1 - λ) p^{G})}^{2} - 2 A c k (1 + r)) = 0

, we have

k_{5} = \frac{{(η (1 - λ) p^{G})}^{2}}{2 A c (1 + r)}

. That is,

|\frac{\partial q^{S G *}}{\partial λ}| < |\frac{\partial q^{E G *}}{\partial λ}|

k > k_{5}

; otherwise,

|\frac{\partial q^{S G *}}{\partial λ}| > |\frac{\partial q^{E G *}}{\partial λ}|

|\frac{\partial q^{S G *}}{\partial λ}| - |\frac{\partial q^{B G *}}{\partial λ}| = \frac{r β ({(η (1 - λ) p^{G})}^{2} - 2 A c k (1 + r β))}{2 k (1 + r β) {(1 - λ)}^{2} p^{G}}

. Subject to

2 k (1 + r β) {(1 - λ)}^{2} p^{G} > 0

and

- 2 A c k (1 + r β) r β < 0

r β ({(η (1 - λ) p^{G})}^{2} - 2 A c k (1 + r β))

decreases monotonically with

k

. Solving

r β ({(η (1 - λ) p^{G})}^{2} - 2 A c k (1 + r β)) = 0

, we have

k_{6} = \frac{{(η (1 - λ) p^{G})}^{2}}{2 A c (1 + r β)}

. That is,

|\frac{\partial q^{S G *}}{\partial λ}| < |\frac{\partial q^{B G *}}{\partial λ}|

k > k_{6}

; otherwise,

|\frac{\partial q^{S G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}|

. Comparing

k_{4}

k_{5}

and

k_{6}

, we have

k_{6} - k_{5} = \frac{r (1 - β) {(η (1 - λ) p^{G})}^{2}}{2 A c (1 + r) (1 + r β)} > 0

k_{5} - k_{4} = \frac{r β {(η (1 - λ) p^{G})}^{2}}{2 A c (1 + r) (1 + r β)} > 0

. Therefore, we conclude that

k_{6} > k_{5} > k_{4}

holds.

Thus, we can obtain (i) if

k < k_{4}

, then

|\frac{\partial q^{S G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}| > |\frac{\partial q^{E G *}}{\partial λ}|

, (ii) if

k_{4} < k < k_{5}

, then

|\frac{\partial q^{S G *}}{\partial λ}| > |\frac{\partial q^{E G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}|

, (iii) if

k_{5} < k < k_{6}

, then

|\frac{\partial q^{E G *}}{\partial λ}| > |\frac{\partial q^{S G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}|

, (iv) if

k > k_{6}

, then

|\frac{\partial q^{E G *}}{\partial λ}| > |\frac{\partial q^{B G *}}{\partial λ}| > |\frac{\partial q^{S G *}}{\partial λ}|

(see Figure A1).

Figure A1. Illustration of

q^{J *}

cut interval.

Figure A1. Illustration of

q^{J *}

cut interval.

(3): $|\frac{\partial q^{S G *}}{\partial η}| - |\frac{\partial q^{B G *}}{\partial η}| = \frac{r β η (1 - λ) p^{G}}{k + k r β} > 0$ , $|\frac{\partial q^{B G *}}{\partial η}| - |\frac{\partial q^{E G *}}{\partial η}| = \frac{r (1 - β) η (1 - λ) p^{G}}{k (1 + r) (1 + r β)} > 0$ . Thus, we can obtain $|\frac{\partial q^{S G *}}{\partial η}| > |\frac{\partial q^{B G *}}{\partial η}| > |\frac{\partial q^{E G *}}{\partial η}|$ .
(4): $|\frac{\partial q^{S G *}}{\partial k}| - |\frac{\partial q^{B G *}}{\partial k}| = \frac{r β η^{2} (1 - λ) p^{G}}{2 k^{2} (1 + r β)} > 0$ , $|\frac{\partial q^{B G *}}{\partial k}| - |\frac{\partial q^{E G *}}{\partial k}| = \frac{r (1 - β) η^{2} (1 - λ) p^{G}}{2 k^{2} (1 + r) (1 + r β)} > 0$ . Thus, we can obtain $|\frac{\partial q^{S G *}}{\partial k}| > |\frac{\partial q^{B G *}}{\partial k}| > |\frac{\partial q^{E G *}}{\partial k}|$ . □

Proof of Proposition 5.

q^{S G *} - q^{B G *} = \frac{r β η^{2} (1 - λ) p^{G}}{2 k (1 + r β)} + \frac{A c r β}{p^{G} - λ p^{G}} > 0

q^{B G *} - q^{E G *} = \frac{r (1 - β) (2 A c k (1 + r) (1 + r β) + {(η (1 - λ) p^{G})}^{2})}{2 k (1 + r) (1 + r β) (1 - λ) p^{G}} > 0

q^{B G *} - q^{S T *} = \frac{η^{2} ((1 - λ) p^{G} - c - c r β)}{2 k (1 + r β)} + \frac{A c (p^{G} - p^{T} - r β p^{T})}{(1 - λ) p^{G} p^{T}} > 0

. Thus, we can obtain

q^{S G *} > q^{B G *} > q^{E G *} > q^{S T *}

. □

Proof of Proposition 6.

(1)

(i): $\frac{\partial e^{S G *}}{\partial λ} = - \frac{η p^{G}}{2 k} + \frac{(a - A) A k η p^{G}}{{(2 A k + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial e^{E G *}}{\partial λ} = - \frac{η p^{G}}{2 k (1 + r)} + \frac{(a - A) A k (1 + r) η p^{G}}{{(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial e^{B G *}}{\partial λ} = - \frac{η p^{G}}{2 k (1 + r β)} + \frac{(a - A) A k (1 + r β) η p^{G}}{{(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ .
(ii): $\frac{\partial e^{S G *}}{\partial η} = \frac{2 A^{2} k^{2} (3 p^{G} (1 - λ) - 2 c) - A k p^{G} (2 a k + η^{2} (4 c - 3 p^{G} (1 - λ))) (1 - λ) + {(η p^{G})}^{2} (a k + η^{2} (p^{G} (1 - λ) - c)) {(1 - λ)}^{2}}{2 k {(2 A k - η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $\frac{\partial e^{E G *}}{\partial η} = \frac{η^{4} {((1 - λ) p^{G})}^{3} + (3 A k + a k - c η^{2}) (1 + r) {(η (1 - λ) p^{G})}^{2} + 2 {(1 + r)}^{2} (1 - λ) A (3 A k - a k - 2 c η^{2}) k p^{G} - 4 A^{2} c k^{2} {(1 + r)}^{3}}{2 k (1 + r) {(2 A k (1 + r) + p^{G} η^{2} (1 - λ))}^{2}} > 0$ , $\frac{\partial e^{B G *}}{\partial η} = \frac{η^{4} {((1 - λ) p^{G})}^{3} + η^{2} (1 + r β) (3 k A + k a - c η^{2}) {((1 - λ) p^{G})}^{2} + 2 {(1 + r β)}^{2} k (1 - λ) A (3 k A - k a - 2 c η^{2}) p^{G} - 4 A^{2} c k^{2} {(1 + r β)}^{3}}{2 k (1 + r β) {(p^{G} η^{2} (1 - λ) + 2 k A (1 + r β))}^{2}} > 0$ .
(iii): $\frac{\partial e^{S G *}}{\partial k} = - \frac{η ((1 - λ) p^{G} (2 A k (3 A k - a k - 2 c η^{2}) - η^{2} (1 - λ) p^{G} (c η^{2} - η^{2} (1 - λ) p^{G} - 4 A k)) - 4 A^{2} c k^{2})}{2 k^{2} {(2 A k + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial e^{E G *}}{\partial k} = - \frac{η ((1 - λ) p^{G} (2 A k {(1 + r)}^{2} (3 A k - a k - 2 c η^{2}) - η^{2} (1 - λ) p^{G} ((1 + r) (c η^{2} - 4 A k) - η^{2} (1 - λ) p^{G})) - 4 A^{2} c k^{2} {(1 + r)}^{3})}{2 k^{2} (1 + r) {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial e^{B G *}}{\partial k} = - \frac{η ((1 - λ) p^{G} (2 A k {(1 + r β)}^{2} (3 A k - 2 c η^{2} - a k) - η^{2} (1 - λ) p^{G} ((1 + r β) (c η^{2} - 4 A k) - η^{2} (1 - λ) p^{G})) - 4 A^{2} c k^{2} {(1 + r β)}^{3})}{2 k^{2} (1 + r β) {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ .
(iv): $\frac{\partial e^{E G *}}{\partial r} = - \frac{η (1 - λ) p^{G} (2 (3 A - a) A k^{2} {(1 + r)}^{2} + 4 A k (1 + r) η^{2} (1 - λ) p^{G} + η^{4} {((1 - λ) p^{G})}^{2})}{2 k {(1 + r)}^{2} {(- 2 A k (1 + r) + η^{2} (- 1 + λ) 〈 p^{G} 〉)}^{2}} < 0$ , $\frac{\partial e^{B G *}}{\partial r} = - \frac{β η (1 - λ) p^{G} (2 (3 A - a) A {(k + k r β)}^{2} + 4 A k (1 + r β) η^{2} (1 - λ) p^{G} + η^{4} {((1 - λ) p^{G})}^{2})}{2 k {(1 + r β)}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ .
(v): $\frac{\partial e^{B G *}}{\partial β} = - \frac{r η (1 - λ) p^{G} (2 (3 A - a) A {(k + k r β)}^{2} + 4 A k (1 + r β) η^{2} (1 - λ) p^{G} + η^{4} {((1 - λ) p^{G})}^{2})}{2 k {(1 + r β)}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ .

(2)

(i): $|\frac{\partial e^{S G *}}{\partial λ}| - |\frac{\partial e^{B G *}}{\partial λ}| = \frac{η p^{G}}{2} (\frac{1}{k} - \frac{1}{k (1 + r β)} - \frac{2 A (a - A) k}{{(2 A k + η^{2} (1 - λ) p^{G})}^{2}} + \frac{2 (a - A) A k (1 + r β)}{{(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}}) > 0$ , $|\frac{\partial e^{B G *}}{\partial λ}| - |\frac{\partial e^{E G *}}{\partial λ}| = \frac{η p^{G}}{2 k (1 + r β)} - \frac{(a - A) A k (1 + r β) η p^{G}}{{(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} - \frac{η p^{G}}{2 k (1 + r)} + \frac{(a - A) A k (1 + r) η p^{G}}{{(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $|\frac{\partial e^{S G *}}{\partial λ}| > |\frac{\partial e^{B G *}}{\partial λ}| > |\frac{\partial e^{E G *}}{\partial λ}|$ .
(ii): $|\frac{\partial e^{B G *}}{\partial η}| - |\frac{\partial e^{E G *}}{\partial η}| = \frac{(r (1 - β) (1 - λ) p^{G} (\begin{array}{l} {(η^{2} (1 - λ) p^{G})}^{3} (4 A k (2 + r + r β) + η^{2} (1 - λ) p^{G}) + 8 (3 A - a) A^{3} k^{4} {(1 + r)}^{2} {(1 + r β)}^{2} \\ + η^{4} {((1 - λ) p^{G})}^{2} (2 A k^{2} (3 a (1 + r) (1 + r β) + A (3 + r + 2 r β) (3 + 2 r + β r))) \end{array}))}{2 k (1 + r) (1 + r β) {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) - η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $|\frac{\partial e^{S G *}}{\partial η}| - |\frac{\partial e^{B G *}}{\partial η}| = \frac{(r β (1 - λ) p^{G} (\begin{array}{l} η^{4} {((1 - λ) p^{G})}^{2} (2 A k^{2} (3 a (1 + r β) + A (3 + r β) (3 + 2 r β)) + η^{2} (1 - λ) p^{G} (4 A k (2 + r β) + η^{2} (1 - λ) p^{G})) \\ + 8 (3 A - a) A^{3} k^{4} {(1 + r β)}^{2} + 4 A^{2} (a + 3 A) k^{3} (1 + r β) (2 + r β) η^{2} (1 - λ) p^{G} \end{array}))}{2 k (1 + r β) {(2 A k + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , Thus, we can obtain $|\frac{\partial e^{S G *}}{\partial η}| > |\frac{\partial e^{B G *}}{\partial η}| > |\frac{\partial e^{E G *}}{\partial η}|$ .
(iii): $|\frac{\partial e^{S G *}}{\partial k}| - |\frac{\partial e^{B G *}}{\partial k}| = \frac{(r β η (1 - λ) p^{G} (\begin{array}{l} 24 A^{4} k^{4} {(1 + r β)}^{2} + 16 A^{3} k^{3} (2 + 3 r β + r^{2} β^{2}) η^{2} (1 - λ) p^{G} + 2 A^{2} k^{2} (11 + 11 r β + 2 r^{2} β^{2}) η^{4} {((1 - λ) p^{G})}^{2} \\ + 4 A k (2 + r β) η^{6} {((1 - λ) p^{G})}^{3} + η^{8} {((1 - λ) p^{G})}^{4} - 2 a A k^{2} (1 + r β) (4 A^{2} k^{2} (1 + r β) - η^{4} {((1 - λ) p^{G})}^{2}) \end{array}))}{2 k^{2} (1 + r β) {(2 A k + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $|\frac{\partial e^{B G *}}{\partial k}| - |\frac{\partial e^{E G *}}{\partial k}| = \frac{(r (1 - β) η (1 - λ) p^{G} (\begin{array}{l} 8 A^{3} k^{3} (1 + r) (1 + r β) (3 A k (1 + r) (1 + r β) + 2 (2 + r + r β) η^{2} (1 - λ) p^{G}) \\ + 2 A^{2} k^{2} (11 + 11 r (1 + β) + r^{2} (2 + 7 β + 2 β^{2})) η^{4} {((1 - λ) p^{G})}^{2} \\ + 4 A k (2 + r + r β) η^{6} {((1 - λ) p^{G})}^{3} + η^{8} {((1 - λ) p^{G})}^{4} \\ - 2 a A k^{2} (1 + r) (1 + r β) (4 A^{2} k^{2} (1 + r) (1 + r β) - η^{4} {((1 - λ) p^{G})}^{2}) \end{array}))}{2 k^{2} (1 + r) (1 + r β) {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , Thus, we can obtain $|\frac{\partial e^{S G *}}{\partial k}| > |\frac{\partial e^{B G *}}{\partial k}| > |\frac{\partial e^{E G *}}{\partial k}|$ . □

Proof of Proposition 7.

e^{S G *} - e^{B G *} = \frac{r β η (1 - λ) p^{G} (2 (3 A - a) A k^{2} (1 + r β) + 2 A k (2 + r β) η^{2} (1 - λ) p^{G} + η^{4} {((1 - λ) p^{G})}^{2})}{2 k (1 + r β) (2 A k + η^{2} (1 - λ) p^{G}) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})} > 0

e^{B G *} - e^{E G *} = \frac{r η p^{G} (1 - β) (1 - λ) (2 (3 A - a) A k^{2} (1 + r) (1 + r β) + 2 A k (2 + r + r β) η^{2} (1 - λ) p^{G} + η^{4} {((1 - λ) p^{G})}^{2})}{2 k (1 + r) (1 + r β) (2 A k (1 + r) + η^{2} (1 - λ) p^{G}) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})} > 0

, Thus, we can obtain

e^{S G *} > e^{B G *} > e^{E G *}

. □

Proof of Proposition 8.

(1): $\frac{\partial \prod_{s}^{S T *}}{\partial λ} = \frac{4 A^{2} c^{2} - {(1 - λ)}^{2} {(p^{T})}^{2} {(A + a)}^{2}}{8 A {(1 - λ)}^{2} p^{T}} < 0$ , $\frac{\partial \prod_{s}^{S G *}}{\partial λ} = - \frac{1}{2 p^{G}} (\frac{{(p^{G})}^{2} (k a + (p^{G} - c) η^{2})}{k} + \frac{{(a - A)}^{2} A k^{2} {(p^{G})}^{2}}{{(η^{2} (1 - λ) p^{G} + 2 k A)}^{2}} - \frac{A c^{2}}{{(1 - λ)}^{2}} - \frac{{(p^{G})}^{3} η^{2} λ}{k}) < 0$ , $\frac{\partial \prod_{s}^{E G *}}{\partial λ} = \frac{(\begin{array}{l} (η^{4} {((1 - λ) p^{G})}^{3} + A^{2} k^{2} {(1 + r)}^{2} (2 c (1 + r) + (1 - λ) p^{G}) + A k (1 + r) (1 - λ) p^{G} (a k (1 + r) + η^{2} (c + c r + 3 (1 - λ) p^{G}))) \\ \times (A k (1 + r) (2 c (1 + r) - (1 - λ) p^{G}) + (1 - λ) p^{G} (η^{2} (c + c r - (1 - λ) p^{G}) - a k (1 + r))) \end{array})}{2 k {(1 + r)}^{2} {(1 - λ)}^{2} p^{G} {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial \prod_{s}^{B G *}}{\partial λ} = \frac{(\begin{array}{l} (k^{2} (2 c (1 + β r) + p^{G} (1 - λ)) {(1 + β r)}^{2} A^{2} - k p^{G} (a (1 + β r) k + η^{2} (3 (1 - λ) p^{G} + c (1 + β r))) (1 + β r) (λ - 1) A + η^{4} {((1 - λ) p^{G})}^{3}) \\ \times (2 A c k {(1 + β r)}^{2} - {(η (1 - λ) p^{G})}^{2} - (1 - λ) (1 + β r) (A k + a k - c η^{2}) p^{G}) \end{array})}{2 {(1 - λ)}^{2} k {(1 + β r)}^{2} p^{G} {(p^{G} η^{2} (1 - λ) + 2 k A (1 + β r))}^{2}} < 0$ .
(2): $\frac{\partial \prod_{s}^{S G *}}{\partial η} = \frac{η (A k (3 p^{G} (1 - λ) - 2 c) - p^{G} (a k + η^{2} (c - p^{G} (1 - λ))) (1 - λ)) (p^{G} (a k - η^{2} (c - p^{G} (1 - λ))) (1 - λ) + A k (p^{G} (1 - λ) - 2 c))}{2 k {(2 A k + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $\frac{\partial \prod_{s}^{E G *}}{\partial η} = \frac{η (2 A k {(1 + r)}^{2} c - {(η (1 - λ) p^{G})}^{2} - (1 - λ) (3 A k - a k - η^{2} c) (1 + r) p^{G}) (2 A k {(1 + r)}^{2} c - {(η (1 - λ) p^{G})}^{2} - (1 - λ) (1 + r) (A k + a k - η^{2} c) p^{G})}{2 k {(1 + r)}^{2} {((1 - λ) p^{G} η^{2} + 2 k (1 + r) A)}^{2}} > 0$ , $\frac{\partial \prod_{s}^{B G *}}{\partial η} = \frac{(\begin{array}{l} (A k η (1 + r β) (2 c (1 + r β) - (1 - λ) p^{G}) + η (1 - λ) p^{G} (η^{2} (c + c r β - (1 - λ) p^{G}) - a k (1 + r β))) \\ \times (A k η (1 + r β) (2 c (1 + r β) - 3 (1 - λ) p^{G}) + η (1 - λ) p^{G} (a k (1 + r β) + η^{2} (c + c r β - (1 - λ) p^{G}))) \end{array})}{2 k {(1 + r β)}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} > 0$ .
(3): $\frac{\partial \prod_{s}^{S G *}}{\partial k} = \frac{η^{2} (A k (2 c - (1 - λ) p^{G}) + (1 - λ) p^{G} (η^{2} (c - (1 - λ) p^{G}) - a k)) (A k (3 (1 - λ) p^{G} - 2 c) - (1 - λ) p^{G} (a k + η^{2} (c - (1 - λ) p^{G})))}{4 k^{2} {(2 A k + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial \prod_{s}^{E G *}}{\partial k} = - \frac{(\begin{array}{l} η^{2} (A k (1 + r) (2 c (1 + r) - (1 - λ) p^{G}) + (1 - λ) p^{G} (η^{2} (c + c r - (1 - λ) p^{G}) - a k (1 + r))) \\ \times (A k (1 + r) (2 c (1 + r) - 3 (1 - λ) p^{G}) + (1 - λ) p^{G} (a k (1 + r) + η^{2} (c + c r - (1 - λ) p^{G}))) \end{array})}{4 k^{2} {(1 + r)}^{2} {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial \prod_{s}^{B G *}}{\partial k} = - \frac{(\begin{array}{l} η^{2} (A k (1 + r β) (2 c (1 + r β) - (1 - λ) p^{G}) + (1 - λ) p^{G} (η^{2} (c (1 + r β) - (1 - λ) p^{G}) - a k (1 + r β))) \\ \times (A k (1 + r β) (2 c (1 + r β) - 3 (1 - λ) p^{G}) + (1 - λ) p^{G} (a k (1 + r β) + η^{2} (c (1 + r β) - (1 - λ) p^{G}))) \end{array})}{4 k^{2} {(1 + r β)}^{2} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ .
(4): $\frac{\partial \prod_{s}^{E G *}}{\partial r} = - \frac{(\begin{array}{l} (2 A^{2} c k^{2} {(1 + r)}^{3} + (1 - λ) p^{G} (A k {(1 + r)}^{2} (a k + A k + c η^{2}) + η^{2} (1 - λ) p^{G} (3 A k (1 + r) + η^{2} (1 - λ) p^{G}))) \\ \times ((1 - λ) p^{G} ((1 + r) (a k + A k - c η^{2}) + η^{2} (1 - λ) p^{G}) - 2 A c k {(1 + r)}^{2}) \end{array})}{2 k {(1 + r)}^{3} (1 - λ) p^{G} {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial \prod_{s}^{B G *}}{\partial r} = - \frac{(\begin{array}{l} β (2 A^{2} c k^{2} {(1 + r β)}^{3} + (1 - λ) p^{G} (A k {(1 + r β)}^{2} (a k + A k + c η^{2}) + η^{2} (1 - λ) p^{G} (3 A k (1 + r β) + η^{2} (1 - λ) p^{G}))) \\ \times ((1 - λ) p^{G} ((1 + r β) (a k + A k - c η^{2}) + η^{2} (1 - λ) p^{G}) - 2 A c k {(1 + r β)}^{2}) \end{array})}{2 k {(1 + r β)}^{3} (1 - λ) p^{G} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ .
(5): $\frac{\partial \prod_{s}^{B G *}}{\partial β} = - \frac{(\begin{array}{l} r (2 A^{2} c k^{2} {(1 + r β)}^{3} + (1 - λ) p^{G} (A k {(1 + r β)}^{2} (a k + A k + c η^{2}) + η^{2} (1 - λ) p^{G} (3 A k (1 + r β) + η^{2} (1 - λ) p^{G}))) \\ \times ((1 - λ) p^{G} ((1 + r β) (a k + A k - c η^{2}) + η^{2} (1 - λ) p^{G}) - 2 A c k {(1 + r β)}^{2}) \end{array})}{2 k {(1 + r β)}^{3} (1 - λ) p^{G} {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ . □

Proof of Proposition 9.

(1): $\prod_{s}^{S G *} - \prod_{s}^{S T *} = \frac{2 {(A k ((1 - λ) p^{G} - 2 c) - (1 - λ) p^{G} (η^{2} (c - (1 - λ) p^{G}) - a k))}^{2}}{8 (1 - λ) k p^{G} (2 A k + η^{2} (1 - λ) p^{G})} - \frac{{(2 A c - (a + A) (1 - λ) p^{T})}^{2}}{8 (1 - λ) A p^{T}}$ . Subject to $8 A k p^{G} p^{T} (1 - λ) (2 A k + η^{2} (1 - λ) p^{G}) > 0$ and $2 A (p^{G} - p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} ({(1 - λ)}^{2} p^{G} p^{T} - 4 c^{2})) > 0$ , $\begin{array}{l} 2 A k^{2} (p^{G} - p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} ({(1 - λ)}^{2} p^{G} p^{T} - 4 c^{2})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c - (1 - λ) p^{G})}^{2} p^{T} \\ - k η^{2} (1 - λ) p^{G} (- 2 a A {(1 - λ)}^{2} p^{G} (2 p^{G} - p^{T}) p^{T} + p^{G} {(a (1 - λ) p^{T})}^{2} + A^{2} (4 c^{2} (p^{G} - 2 p^{T}) + 8 c (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} (4 p^{G} - p^{T}) p^{T})) \end{array}$ is a quadratic function of $k$ . That is, the function $\begin{array}{l} 2 A k^{2} (p^{G} - p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} ({(1 - λ)}^{2} p^{G} p^{T} - 4 c^{2})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c - (1 - λ) p^{G})}^{2} p^{T} \\ - k η^{2} (1 - λ) p^{G} (- 2 a A {(1 - λ)}^{2} p^{G} (2 p^{G} - p^{T}) p^{T} + p^{G} {(a (1 - λ) p^{T})}^{2} + A^{2} (4 c^{2} (p^{G} - 2 p^{T}) + 8 c (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} (4 p^{G} - p^{T}) p^{T})) = 0 \end{array}$ has two different real roots. Solving $\begin{array}{l} 2 A k^{2} (p^{G} - p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} ({(1 - λ)}^{2} p^{G} p^{T} - 4 c^{2})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c - (1 - λ) p^{G})}^{2} p^{T} \\ - k η^{2} (1 - λ) p^{G} (- 2 a A {(1 - λ)}^{2} p^{G} (2 p^{G} - p^{T}) p^{T} + p^{G} {(a (1 - λ) p^{T})}^{2} + A^{2} (4 c^{2} (p^{G} - 2 p^{T}) + 8 c (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} (4 p^{G} - p^{T}) p^{T})) = 0 \end{array}$ we obtain $k_{s 1}^{'} = - \frac{(η^{2} (1 - λ) p^{G} (\begin{array}{l} (2 a A {(1 - λ)}^{2} p^{G} (2 p^{G} - p^{T}) p^{T} - a^{2} {(1 - λ)}^{2} p^{G} {(p^{T})}^{2} - A^{2} (4 c^{2} (p^{G} - 2 p^{T}) - 8 c (- 1 + λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} (4 p^{G} - p^{T}) p^{T})) \\ + p^{G} (2 A c - (a + A) (1 - λ) p^{T}) \sqrt{2 a A (1 - λ) (2 c - (1 - λ) (4 p^{G} - p^{T})) p^{T} + {(a (1 - λ) p^{T})}^{2} + A^{2} (4 c^{2} - 12 c (1 - λ) p^{T} + {(1 - λ)}^{2} p^{T} (8 p^{G} + p^{T}))} \end{array}))}{4 A (p^{G} - p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} ({(1 - λ)}^{2} p^{G} p^{T} - 4 c^{2}))} < 0$ and $k_{s 1}^{″} = - \frac{(η^{2} (1 - λ) p^{G} (\begin{array}{l} (2 a A {(1 - λ)}^{2} p^{G} (2 p^{G} - p^{T}) p^{T} - a^{2} {(1 - λ)}^{2} p^{G} {(p^{T})}^{2} - A^{2} (4 c^{2} (p^{G} - 2 p^{T}) - 8 c (- 1 + λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} (4 p^{G} - p^{T}) p^{T})) \\ - p^{G} (2 A c - (a + A) (1 - λ) p^{T}) \sqrt{2 a A (1 - λ) (2 c - (1 - λ) (4 p^{G} - p^{T})) p^{T} + {(a (1 - λ) p^{T})}^{2} + A^{2} (4 c^{2} - 12 c (1 - λ) p^{T} + {(1 - λ)}^{2} p^{T} (8 p^{G} + p^{T}))} \end{array}))}{4 A (p^{G} - p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} ({(1 - λ)}^{2} p^{G} p^{T} - 4 c^{2}))} < 0$ . Since $k > 0$ , we conclude that $\prod_{s}^{S G *} > \prod_{s}^{S T *}$ holds.
(2): $\prod_{s}^{E G *} - \prod_{s}^{S T *} = \frac{2 {(A k (1 + r) (2 c (1 + r) - (1 - λ) p^{G}) - (1 - λ) p^{G} (a k (1 + r) - η^{2} (c + c r - (1 - λ) p^{G})))}^{2}}{8 k p^{G} (1 - λ) {(1 + r)}^{2} (2 A k (1 + r) + η^{2} (1 - λ) p^{G})} - \frac{{(2 A c - (a + A) (1 - λ) p^{T})}^{2}}{8 (1 - λ) A p^{T}}$ . Subject to $8 A k p^{G} p^{T} {(1 + r)}^{2} (1 - λ) (2 A k (1 + r) + η^{2} (1 - λ) p^{G}) > 0$ and $2 A {(1 + r)}^{2} (p^{T} + r p^{T} - p^{G}) (- a^{2} {(1 - λ)}^{2} p^{G} p^{T} - 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} (4 c^{2} (1 + r) - {(1 - λ)}^{2} p^{G} p^{T})) > 0$ , $\begin{array}{l} 2 A k^{2} {(1 + r)}^{2} (p^{T} + r p^{T} - p^{G}) (- a^{2} {(1 - λ)}^{2} p^{G} p^{T} - 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} (4 c^{2} (1 + r) - {(1 - λ)}^{2} p^{G} p^{T})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c (1 + r) - (1 - λ) p^{G})}^{2} p^{T} \\ - k (1 + r) η^{2} (1 - λ) p^{G} (a {(1 - λ)}^{2} p^{G} p^{T} (a (1 + r) p^{T} + 2 A (p^{T} + r p^{T} - 2 p^{G})) + A^{2} (8 c (1 + r) (1 - λ) p^{G} p^{T} + 4 c^{2} (1 + r) (p^{G} - 2 (1 + r) p^{T}) - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} - (1 + r) p^{T}))) \end{array}$ is a quadratic function of $k$ . That is, the function $\begin{array}{l} 2 A k^{2} {(1 + r)}^{2} (p^{T} + r p^{T} - p^{G}) (- a^{2} {(1 - λ)}^{2} p^{G} p^{T} - 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} (4 c^{2} (1 + r) - {(1 - λ)}^{2} p^{G} p^{T})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c (1 + r) - (1 - λ) p^{G})}^{2} p^{T} \\ - k (1 + r) η^{2} (1 - λ) p^{G} (a {(1 - λ)}^{2} p^{G} p^{T} (a (1 + r) p^{T} + 2 A (p^{T} + r p^{T} - 2 p^{G})) + A^{2} (8 c (1 + r) (1 - λ) p^{G} p^{T} + 4 c^{2} (1 + r) (p^{G} - 2 (1 + r) p^{T}) - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} - (1 + r) p^{T}))) \end{array}$ has two different real roots. Solving $\begin{array}{l} 2 A k^{2} {(1 + r)}^{2} (p^{T} + r p^{T} - p^{G}) (- a^{2} {(1 - λ)}^{2} p^{G} p^{T} - 2 a A {(1 - λ)}^{2} p^{G} p^{T} + A^{2} (4 c^{2} (1 + r) - {(1 - λ)}^{2} p^{G} p^{T})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c (1 + r) - (1 - λ) p^{G})}^{2} p^{T} \\ - k (1 + r) η^{2} (1 - λ) p^{G} (a {(1 - λ)}^{2} p^{G} p^{T} (a (1 + r) p^{T} + 2 A (p^{T} + r p^{T} - 2 p^{G})) + A^{2} (8 c (1 + r) (1 - λ) p^{G} p^{T} + 4 c^{2} (1 + r) (p^{G} - 2 (1 + r) p^{T}) - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} - (1 + r) p^{T}))) \end{array}$ we obtain $k_{s 2}^{'} = - \frac{(\begin{array}{l} (1 + r) η^{2} (- 1 + λ) p^{G} ((a + A) {(1 - λ)}^{2} p^{G} p^{T} (a (1 + r) p^{T} + A (- 4 p^{G} + p^{T} + r p^{T})) + 4 A^{2} c (1 + r) (2 (1 - λ) p^{G} p^{T} + c (p^{G} - 2 (1 + r) p^{T}))) + (2 A c - (a - A) (1 - λ) p^{T}) \\ \times \sqrt{{(1 - λ)}^{2} {(1 + r)}^{3} {(η p^{G})}^{4} ({(1 - λ)}^{2} p^{T} (a^{2} (1 + r) p^{T} + A^{2} (8 p^{G} + p^{T} + r p^{T})) + A (4 A c (1 + r) (c - 3 (1 - λ) p^{T}) + 2 a (1 - λ) p^{T} (2 c (1 + r) - (1 - λ) (4 p^{G} - (1 + r) p^{T}))))} \end{array})}{4 A {(1 + r)}^{2} (p^{G} - p^{T} - r p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} - A^{2} (4 c^{2} (1 + r) - {(1 - λ)}^{2} p^{G} p^{T}))} < 0$ and $k_{s 2}^{″} = - \frac{(\begin{array}{l} (1 + r) η^{2} (- 1 + λ) p^{G} ((a + A) {(1 - λ)}^{2} p^{G} p^{T} (a (1 + r) p^{T} + A (- 4 p^{G} + p^{T} + r p^{T})) + 4 A^{2} c (1 + r) (2 (1 - λ) p^{G} p^{T} + c (p^{G} - 2 (1 + r) p^{T}))) - (2 A c - (a - A) (1 - λ) p^{T}) \\ \times \sqrt{{(1 - λ)}^{2} {(1 + r)}^{3} {(η p^{G})}^{4} ({(1 - λ)}^{2} p^{T} (a^{2} (1 + r) p^{T} + A^{2} (8 p^{G} + p^{T} + r p^{T})) + A (4 A c (1 + r) (c - 3 (1 - λ) p^{T}) + 2 a (1 - λ) p^{T} (2 c (1 + r) - (1 - λ) (4 p^{G} - (1 + r) p^{T}))))} \end{array})}{4 A {(1 + r)}^{2} (p^{G} - p^{T} - r p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} - A^{2} (4 c^{2} (1 + r) - {(1 - λ)}^{2} p^{G} p^{T}))} < 0$ . Since $k > 0$ , we conclude that $\prod_{s}^{E G *} > \prod_{s}^{S T *}$ holds.
(3): $\prod_{s}^{B G *} - \prod_{s}^{S T *} = \frac{2 {(A k (1 + r β) (2 c (1 + r β) - (1 - λ) p^{G}) + (1 - λ) p^{G} (η^{2} (c + c r β - (1 - λ) p^{G}) - a k (1 + r β)))}^{2}}{8 (1 - λ) k {(1 + r β)}^{2} p^{G} (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})} - \frac{{(2 A c - a (1 - λ) p^{T} - A (1 - λ) p^{T})}^{2}}{8 (1 - λ) A p^{T}}$ . Subject to $8 A k p^{G} p^{T} {(1 + r β)}^{2} (1 - λ) (2 A k + 2 A k r β + η^{2} p^{G} - η^{2} λ p^{G}) > 0$ and $2 A {(1 + r β)}^{2} (p^{G} - p^{T} - r β p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} - A^{2} (4 c^{2} (1 + r β) - {(1 - λ)}^{2} p^{G} p^{T})) > 0$ , $\begin{array}{l} 2 A k^{2} {(1 + r β)}^{2} (p^{G} - p^{T} - r β p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} - A^{2} (4 c^{2} (1 + r β) - {(1 - λ)}^{2} p^{G} p^{T})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c + c r β - (1 - λ) p^{G})}^{2} p^{T} - k (1 + r β) η^{2} (1 - λ) p^{G} \\ (a^{2} (1 + r β) p^{G} {((1 - λ) p^{T})}^{2} - 2 a A {(1 - λ)}^{2} p^{G} p^{T} (2 p^{G} - (1 + r β) p^{T}) A^{2} (8 c (1 + r β) (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} - (1 + r β) p^{T}) + 4 c^{2} (1 + r β) (p^{G} - 2 p^{T} (1 + r β)))) \end{array}$ is a quadratic function of $k$ . That is, the function $\begin{array}{l} 2 A k^{2} {(1 + r β)}^{2} (p^{G} - p^{T} - r β p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} - A^{2} (4 c^{2} (1 + r β) - {(1 - λ)}^{2} p^{G} p^{T})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c + c r β - (1 - λ) p^{G})}^{2} p^{T} - k (1 + r β) η^{2} (1 - λ) p^{G} \\ (a^{2} (1 + r β) p^{G} {((1 - λ) p^{T})}^{2} - 2 a A {(1 - λ)}^{2} p^{G} p^{T} (2 p^{G} - (1 + r β) p^{T}) A^{2} (8 c (1 + r β) (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} - (1 + r β) p^{T}) + 4 c^{2} (1 + r β) (p^{G} - 2 p^{T} (1 + r β)))) \end{array}$ has two different real roots. Solving $\begin{array}{l} 2 A k^{2} {(1 + r β)}^{2} (p^{G} - p^{T} - r β p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} - A^{2} (4 c^{2} (1 + r β) - {(1 - λ)}^{2} p^{G} p^{T})) + 2 A η^{4} {((1 - λ) p^{G})}^{2} {(c + c r β - (1 - λ) p^{G})}^{2} p^{T} - k (1 + r β) η^{2} (1 - λ) p^{G} \\ (a^{2} (1 + r β) p^{G} {((1 - λ) p^{T})}^{2} - 2 a A {(1 - λ)}^{2} p^{G} p^{T} (2 p^{G} - (1 + r β) p^{T}) A^{2} (8 c (1 + r β) (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} - (1 + r β) p^{T}) + 4 c^{2} (1 + r β) (p^{G} - 2 p^{T} (1 + r β)))) \end{array}$ we obtain $k_{s 3}^{'} = - \frac{(\begin{array}{l} (1 + r) η^{2} (- 1 + λ) p^{G} ((a + A) {(1 - λ)}^{2} p^{G} p^{T} (a (1 + r) p^{T} + A (- 4 p^{G} + p^{T} + r p^{T})) + 4 A^{2} c (1 + r) (2 (1 - λ) p^{G} p^{T} + c (p^{G} - 2 (1 + r) p^{T}))) + (2 A c - (a - A) (1 - λ) p^{T}) \\ \times \sqrt{{(1 - λ)}^{2} {(1 + r)}^{3} {(η p^{G})}^{4} ({(1 - λ)}^{2} p^{T} (a^{2} (1 + r) p^{T} + A^{2} (8 p^{G} + p^{T} + r p^{T})) + A (4 A c (1 + r) (c - 3 (1 - λ) p^{T}) + 2 a (1 - λ) p^{T} (2 c (1 + r) - (1 - λ) (4 p^{G} - (1 + r) p^{T}))))} \end{array})}{4 A {(1 + r)}^{2} (p^{G} - p^{T} - r p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} - A^{2} (4 c^{2} (1 + r) - {(1 - λ)}^{2} p^{G} p^{T}))} < 0$ and $k_{s 3}^{″} = - \frac{(\begin{array}{l} (1 + r) η^{2} (- 1 + λ) p^{G} ((a + A) {(1 - λ)}^{2} p^{G} p^{T} (a (1 + r) p^{T} + A (- 4 p^{G} + p^{T} + r p^{T})) + 4 A^{2} c (1 + r) (2 (1 - λ) p^{G} p^{T} + c (p^{G} - 2 (1 + r) p^{T}))) - (2 A c - (a - A) (1 - λ) p^{T}) \\ \times \sqrt{{(1 - λ)}^{2} {(1 + r)}^{3} {(η p^{G})}^{4} ({(1 - λ)}^{2} p^{T} (a^{2} (1 + r) p^{T} + A^{2} (8 p^{G} + p^{T} + r p^{T})) + A (4 A c (1 + r) (c - 3 (1 - λ) p^{T}) + 2 a (1 - λ) p^{T} (2 c (1 + r) - (1 - λ) (4 p^{G} - (1 + r) p^{T}))))} \end{array})}{4 A {(1 + r)}^{2} (p^{G} - p^{T} - r p^{T}) (a^{2} {(1 - λ)}^{2} p^{G} p^{T} + 2 a A {(1 - λ)}^{2} p^{G} p^{T} - A^{2} (4 c^{2} (1 + r) - {(1 - λ)}^{2} p^{G} p^{T}))} < 0$ . Since $k > 0$ , we conclude that $\prod_{s}^{E G *} > \prod_{s}^{S T *}$ holds. □

Proof of Proposition 10.

(1): $\frac{\partial \prod_{m}^{S T *}}{\partial λ} = - \frac{(3 A - a) {((1 - λ) p^{T})}^{2} (3 a - A) - 4 A^{2} c^{2}}{16 A {(1 - λ)}^{2} p^{T}} < 0$ , $\frac{\partial \prod_{m}^{S G *}}{\partial λ} = \frac{1}{4 p^{G}} (\frac{3 A {((a - A) k p^{G})}^{2}}{{(2 A k + η^{2} (1 - λ) p^{G})}^{2}} + \frac{A c^{2}}{{(1 - λ)}^{2}} + \frac{{(p^{G})}^{3} η^{2} λ}{k} - \frac{{(p^{G})}^{2} (a k + (p^{G} - c) η^{2})}{k}) < 0$ , $\frac{\partial \prod_{m}^{E G *}}{\partial λ} = \frac{(\begin{array}{l} 4 A^{3} c^{2} k^{3} {(1 + r)}^{5} - η^{6} {((1 - λ) p^{G})}^{5} - (1 + r) η^{4} (4 A k + a k - c η^{2}) {((1 - λ) p^{G})}^{4} - 4 (A k + a k - c η^{2}) {(1 + r)}^{2} η^{2} A k {((1 - λ) p^{G})}^{3} \\ + ((A - 3 a) (3 A - a) k^{2} + 4 A c η^{2} k + c^{2} η^{4}) {(1 + r)}^{3} A k {((1 - λ) p^{G})}^{2} + 4 A^{2} c^{2} k^{2} η^{2} {(1 + r)}^{4} (1 - λ) p^{G} \end{array})}{4 p^{G} k (1 + r) {(p^{G} η^{2} (1 - λ) + 2 A k (1 + r))}^{2} {(1 - λ)}^{2}} < 0$ , $\frac{\partial \prod_{m}^{B G *}}{\partial λ} = \frac{(\begin{array}{l} A k ((3 A - a) (A - 3 a) k^{2} + 4 A c k η^{2} + c^{2} η^{4}) {(1 + β r)}^{3} {((1 - λ) p^{G})}^{2} + 4 A^{2} c^{2} k^{2} η^{2} {(1 + β r)}^{4} (1 - λ) p^{G} + 4 A^{3} c^{2} k^{3} {(1 + β r)}^{5} \\ - η^{6} {((1 - λ) p^{G})}^{5} - (1 + β r) (4 A k + k a - c η^{2}) {(η (1 - λ) p^{G})}^{4} - 4 η^{2} A k {(1 + β r)}^{2} (A k + a k - c η^{2}) {((1 - λ) p^{G})}^{3} \end{array})}{4 k p^{G} (1 + β r) {(η^{2} p^{G} (1 - λ) + 2 A k (1 + β r))}^{2} {(1 - λ)}^{2}} < 0$ .
(2): $\frac{\partial \prod_{m}^{S G *}}{\partial η} = \frac{(\begin{array}{l} 2 A k p^{G} η (2 η^{2} {(c + p^{G} (1 - λ))}^{2} - 3 a k p^{G} (1 - λ)) (1 - λ) + η {((1 - λ) p^{G})}^{2} (3 a^{2} k^{2} + η^{4} {(c + p^{G} (1 - λ))}^{2}) \\ + A^{2} k^{2} η (4 c^{2} - 8 c (1 - λ) p^{G} + 7 {((1 - λ) p^{G})}^{2}) \end{array})}{4 k {(2 A k + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $\frac{\partial \prod_{m}^{S G *}}{\partial η} = \frac{(\begin{array}{l} 2 A k p^{G} η (2 η^{2} {(c + p^{G} (1 - λ))}^{2} - 3 a k p^{G} (1 - λ)) (1 - λ) + η {((1 - λ) p^{G})}^{2} (3 a^{2} k^{2} + η^{4} {(c + p^{G} (1 - λ))}^{2}) \\ + A^{2} k^{2} η (4 c^{2} - 8 c (1 - λ) p^{G} + 7 {((1 - λ) p^{G})}^{2}) \end{array})}{4 k {(2 A k + η^{2} (1 - λ) p^{G})}^{2}} > 0$ , $\frac{\partial \prod_{m}^{E G *}}{\partial η} = \frac{(\begin{array}{l} η {(η (1 - λ) p^{G})}^{4} + 2 η^{3} (1 + r) (2 k A - η^{2} c) {((1 - λ) p^{G})}^{3} - 4 A c η {(1 + r)}^{3} (2 k A - η^{2} c) k (1 - λ) p^{G} + 4 η A^{2} c^{2} k^{2} {(1 + r)}^{4} \\ + η ((7 A^{2} - 6 A a + 3 a^{2}) k^{2} - 8 A c k η^{2} + c^{2} η^{4}) {((1 + r) (1 - λ) p^{G})}^{2} \end{array})}{4 (1 + r) {(2 k (1 + r) A + (1 - λ) p^{G} η^{2})}^{2} k} > 0$ , $\frac{\partial \prod_{m}^{B G *}}{\partial η} = \frac{(\begin{array}{l} {(η (1 - λ) p^{G})}^{4} + 2 η^{2} (1 + β r) (2 A k - c η^{2}) {((1 - λ) p^{G})}^{3} + ((7 A^{2} - 6 A a + 3 a^{2}) k^{2} - 8 A c k η^{2} + c^{2} η^{4}) {((1 + β r) (1 - λ) p^{G})}^{2} \\ - 4 c A {(1 + β r)}^{3} (1 - λ) (2 A k - c η^{2}) k p^{G} + 4 A^{2} c^{2} k^{2} {(1 + β r)}^{4} \end{array})}{4 k (1 + β r) {(η^{2} p^{G} (1 - λ) + 2 A k (1 + β r))}^{2}} > 0$ .
(3): $\frac{\partial \prod_{m}^{S G *}}{\partial k} = - \frac{(\begin{array}{l} A^{2} k^{2} η^{2} (4 c^{2} - 8 c (1 - λ) p^{G} + 7 {((1 - λ) p^{G})}^{2}) + 2 A k η^{2} (1 - λ) p^{G} (2 η^{2} {(c - (1 - λ) p^{G})}^{2} - 3 a k (1 - λ) p^{G}) \\ + {(η (1 - λ) p^{G})}^{2} (3 a^{2} k^{2} + η^{4} {(c - (1 - λ) p^{G})}^{2}) \end{array})}{8 k^{2} {(2 A k + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial \prod_{m}^{E G *}}{\partial k} = - \frac{(\begin{array}{l} A^{2} k^{2} η^{2} {(1 + r)}^{2} (4 c^{2} {(1 + r)}^{2} - 8 c (1 + r) (1 - λ) p^{G} + 7 {((1 - λ) p^{G})}^{2}) + {(η (1 - λ) p^{G})}^{2} (3 a^{2} k^{2} {(1 + r)}^{2} + η^{4} {(c + c r - (1 - λ) p^{G})}^{2}) \\ + 2 A k η^{2} (1 + r) (1 - λ) p^{G} (2 η^{2} {(c + c r - (1 - λ) p^{G})}^{2} - 3 a k (1 + r) (1 - λ) p^{G}) \end{array})}{8 k^{2} (1 + r) {(2 A k (1 + r) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ , $\frac{\partial \prod_{m}^{B G *}}{\partial k} = - \frac{(\begin{array}{l} A^{2} k η^{2} {(1 + r β)}^{2} (4 c^{2} {(1 + r β)}^{2} - 8 c (1 + r β) (1 - λ) p^{G} + 7 {((1 - λ) p^{G})}^{2}) + {(η (1 - λ) p^{G})}^{2} (3 a^{2} k^{2} {(1 + r β)}^{2} + η^{4} {(c (1 + r β) - (1 - λ) p^{G})}^{2}) \\ + 2 A k η^{2} (1 + r β) (1 - λ) p^{G} (2 η^{2} {(c (1 + r β) - (1 - λ) p^{G})}^{2} - 3 a k (1 + r β) (1 - λ) p^{G}) \end{array})}{8 k^{2} (1 + r β) {(2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}^{2}} < 0$ . □

Proof of Proposition 11.

\prod_{m}^{S G *} - \prod_{m}^{S T *} = \frac{(\begin{array}{l} k^{2} (2 a A (10 A - 3 a) p^{G} (p^{G} - p^{T}) p^{T} {(1 - λ)}^{2} - 2 A^{3} (p^{G} - p^{T}) (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T})) + 2 A {(η^{2} (1 - λ) p^{G})}^{2} {(c - (1 - λ) p^{G})}^{2} p^{T} \\ + k (a {(1 - λ)}^{3} {(η p^{G})}^{2} p^{T} (4 A p^{G} + (3 a - 10 A) p^{T}) - A^{2} η^{2} (1 - λ) p^{G} (4 c^{2} (p^{G} - 2 p^{T}) + 8 c (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}))) \end{array})}{16 A k (1 - λ) p^{G} (2 A k + η^{2} (1 - λ) p^{G}) p^{T}}

. Subject to

16 A k (1 - λ) p^{G} (2 A k + η^{2} (1 - λ) p^{G}) p^{T} > 0

and

2 a A (- 3 a + 10 A) p^{G} (p^{G} - p^{T}) p^{T} {(- 1 + λ)}^{2} - 2 A^{3} (p^{G} - p^{T}) (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T}) > 0

\begin{array}{l} k^{2} (2 a A (- 3 a + 10 A) p^{G} (p^{G} - p^{T}) p^{T} {(- 1 + λ)}^{2} - 2 A^{3} (p^{G} - p^{T}) (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T})) + 2 A {(η^{2} (1 - λ) p^{G})}^{2} {(c - (1 - λ) p^{G})}^{2} p^{T} \\ + k (a {(1 - λ)}^{3} {(η p^{G})}^{2} p^{T} (4 A p^{G} + (3 a - 10 A) p^{T}) - A^{2} η^{2} (1 - λ) p^{G} (4 c^{2} (p^{G} - 2 p^{T}) + 8 c (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}))) \end{array}

is a quadratic function of

k

. That is, the function

\begin{array}{l} k^{2} (2 a A (- 3 a + 10 A) p^{G} (p^{G} - p^{T}) p^{T} {(- 1 + λ)}^{2} - 2 A^{3} (p^{G} - p^{T}) (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T})) + 2 A {(η^{2} (1 - λ) p^{G})}^{2} {(c - (1 - λ) p^{G})}^{2} p^{T} \\ + k (a {(1 - λ)}^{3} {(η p^{G})}^{2} p^{T} (4 A p^{G} + (3 a - 10 A) p^{T}) - A^{2} η^{2} (1 - λ) p^{G} (4 c^{2} (p^{G} - 2 p^{T}) + 8 c (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}))) \end{array}

has two different real roots. Solving

\begin{array}{l} k^{2} (2 a A (- 3 a + 10 A) p^{G} (p^{G} - p^{T}) p^{T} {(- 1 + λ)}^{2} - 2 A^{3} (p^{G} - p^{T}) (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T})) + 2 A {(η^{2} (1 - λ) p^{G})}^{2} {(c - (1 - λ) p^{G})}^{2} p^{T} \\ + k (a {(1 - λ)}^{3} {(η p^{G})}^{2} p^{T} (4 A p^{G} + (3 a - 10 A) p^{T}) - A^{2} η^{2} (1 - λ) p^{G} (4 c^{2} (p^{G} - 2 p^{T}) + 8 c (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}))) = 0 \end{array}

, we obtain

k_{m 1}^{'} = - \frac{(\begin{array}{l} η^{2} (1 - λ) p^{G} (p^{G} (a (10 A - 3 a) {((1 - λ) p^{T})}^{2} + A^{2} (4 c^{2} + 8 c (1 - λ) p^{T} - 3 {((1 - λ) p^{T})}^{2})) - 8 A^{2} c^{2} p^{T} - 4 A (a + A) {((1 - λ) p^{G})}^{2} p^{T}) \\ + η^{2} (1 - λ) p^{G} \sqrt{\begin{array}{l} {(2 a {(1 - λ)}^{2} p^{G} p^{T} (4 A p^{G} + 3 a p^{T} - 10 A p^{T}) + A^{2} (4 c^{2} (2 p^{T} - p^{G}) - 8 c (1 - λ) p^{G} p^{T} + {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T})))}^{2} \\ + 16 A^{2} {(c - (1 - λ) p^{G})}^{2} (p^{G} - p^{T}) p^{T} (a (3 a - 10 A) {(1 - λ)}^{2} p^{G} p^{T} + A^{2} (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T})) \end{array}} \end{array})}{4 A (p^{G} - p^{T}) (a (3 a - 10 A) {(1 - λ)}^{2} p^{G} p^{T} + A^{2} (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T}))} < 0

and

k_{m 1}^{″} = - \frac{(\begin{array}{l} η^{2} (1 - λ) p^{G} (p^{G} (a (10 A - 3 a) {((1 - λ) p^{T})}^{2} + A^{2} (4 c^{2} + 8 c (1 - λ) p^{T} - 3 {((1 - λ) p^{T})}^{2})) - 8 A^{2} c^{2} p^{T} - 4 A (a + A) {((1 - λ) p^{G})}^{2} p^{T}) \\ - η^{2} (1 - λ) p^{G} \sqrt{\begin{array}{l} {(2 a {(1 - λ)}^{2} p^{G} p^{T} (4 A p^{G} + 3 a p^{T} - 10 A p^{T}) + A^{2} (4 c^{2} (2 p^{T} - p^{G}) - 8 c (1 - λ) p^{G} p^{T} + {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T})))}^{2} \\ + 16 A^{2} {(c - (1 - λ) p^{G})}^{2} (p^{G} - p^{T}) p^{T} (a (3 a - 10 A) {(1 - λ)}^{2} p^{G} p^{T} + A^{2} (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T})) \end{array}} \end{array})}{4 A (p^{G} - p^{T}) (a (3 a - 10 A) {(1 - λ)}^{2} p^{G} p^{T} + A^{2} (4 c^{2} + 3 {(1 - λ)}^{2} p^{G} p^{T}))} < 0

. Since

k > 0

, we conclude that

\prod_{m}^{S G *} > \prod_{m}^{S T *}

holds. □

Proof of Proposition 12.

(1): Comparing the $\prod_{m}^{E G *}$ with $\prod_{m}^{S T *}$ , we have $\prod_{m}^{E G *} - \prod_{m}^{S T *} = △_{1} + r M - \prod_{m}^{S T *}$ , where $△_{1} - \prod_{m}^{S T *} = \frac{(\begin{array}{l} 2 A^{3} k^{2} {(1 + r)}^{2} (4 c r (- 1 + λ) p^{G} p^{T} - 3 {(1 - λ)}^{2} p^{G} (p^{G} - p^{T}) p^{T} + 4 c^{2} (- p^{G} + {(1 + r)}^{2} p^{T})) - 3 a^{2} k (1 + r) η^{2} {(- 1 + λ)}^{3} {(p^{G} p^{T})}^{2} \\ + 2 A {(1 - λ)}^{2} p^{G} p^{T} (η^{4} p^{G} {(c (1 + r) - (1 - λ) p^{G})}^{2} - 3 a^{2} k^{2} {(1 + r)}^{2} (p^{G} - p^{T}) - a k (1 + r) η^{2} p^{G} (2 c r - (1 - λ) (2 p^{G} - 5 p^{T}))) \\ - A^{2} k (1 + r) (1 - λ) p^{G} (4 a k (1 + r) p^{T} (2 c r - 5 (1 - λ) (p^{G} - p^{T})) + η^{2} (4 c (2 + 3 r) (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}) + 4 c^{2} (p^{G} - 2 {(1 + r)}^{2} p^{T}))) \end{array})}{16 A k (1 + r) (1 - λ) p^{G} (2 A k (1 + r) + η^{2} (1 - λ) p^{G}) p^{T}}$ Subject to $16 A k (1 + r) (1 - λ) p^{G} (2 A k (1 + r) + η^{2} (1 - λ) p^{G}) p^{T} > 0$ and $2 A {(1 + r)}^{2} (4 A^{2} c^{2} {(1 + r)}^{2} p^{T} - (a - 3 A) (3 a - A) {((1 - λ) p^{G})}^{2} p^{T} - p^{G} (4 A^{2} c^{2} + (1 - λ) p^{T} (4 A (a + A) c r - (a - 3 A) (3 a - A) (1 - λ) p^{T}))) > 0$ , $\begin{array}{l} 2 A^{3} k^{2} {(1 + r)}^{2} (4 c r (- 1 + λ) p^{G} p^{T} - 3 {(1 - λ)}^{2} p^{G} (p^{G} - p^{T}) p^{T} + 4 c^{2} (- p^{G} + {(1 + r)}^{2} p^{T})) - 3 a^{2} k (1 + r) η^{2} {(- 1 + λ)}^{3} {(p^{G} p^{T})}^{2} \\ + 2 A {(1 - λ)}^{2} p^{G} p^{T} (η^{4} p^{G} {(c (1 + r) - (1 - λ) p^{G})}^{2} - 3 a^{2} k^{2} {(1 + r)}^{2} (p^{G} - p^{T}) - a k (1 + r) η^{2} p^{G} (2 c r - (1 - λ) (2 p^{G} - 5 p^{T}))) \\ - A^{2} k (1 + r) (1 - λ) p^{G} (4 a k (1 + r) p^{T} (2 c r - 5 (1 - λ) (p^{G} - p^{T})) + η^{2} (4 c (2 + 3 r) (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}) + 4 c^{2} (p^{G} - 2 {(1 + r)}^{2} p^{T}))) \end{array}$ is a quadratic function of $k$ . That is, the function $\begin{array}{l} 2 A^{3} k^{2} {(1 + r)}^{2} (4 c r (- 1 + λ) p^{G} p^{T} - 3 {(1 - λ)}^{2} p^{G} (p^{G} - p^{T}) p^{T} + 4 c^{2} (- p^{G} + {(1 + r)}^{2} p^{T})) - 3 a^{2} k (1 + r) η^{2} {(- 1 + λ)}^{3} {(p^{G} p^{T})}^{2} \\ + 2 A {(1 - λ)}^{2} p^{G} p^{T} (η^{4} p^{G} {(c (1 + r) - (1 - λ) p^{G})}^{2} - 3 a^{2} k^{2} {(1 + r)}^{2} (p^{G} - p^{T}) - a k (1 + r) η^{2} p^{G} (2 c r - (1 - λ) (2 p^{G} - 5 p^{T}))) \\ - A^{2} k (1 + r) (1 - λ) p^{G} (4 a k (1 + r) p^{T} (2 c r - 5 (1 - λ) (p^{G} - p^{T})) + η^{2} (4 c (2 + 3 r) (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}) + 4 c^{2} (p^{G} - 2 {(1 + r)}^{2} p^{T}))) \end{array}$ has two different real roots. Solving $\begin{array}{l} 2 A^{3} k^{2} {(1 + r)}^{2} (4 c r (- 1 + λ) p^{G} p^{T} - 3 {(1 - λ)}^{2} p^{G} (p^{G} - p^{T}) p^{T} + 4 c^{2} (- p^{G} + {(1 + r)}^{2} p^{T})) - 3 a^{2} k (1 + r) η^{2} {(- 1 + λ)}^{3} {(p^{G} p^{T})}^{2} \\ + 2 A {(1 - λ)}^{2} p^{G} p^{T} (η^{4} p^{G} {(c (1 + r) - (1 - λ) p^{G})}^{2} - 3 a^{2} k^{2} {(1 + r)}^{2} (p^{G} - p^{T}) - a k (1 + r) η^{2} p^{G} (2 c r - (1 - λ) (2 p^{G} - 5 p^{T}))) \\ - A^{2} k (1 + r) (1 - λ) p^{G} (4 a k (1 + r) p^{T} (2 c r - 5 (1 - λ) (p^{G} - p^{T})) + η^{2} (4 c (2 + 3 r) (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}) + 4 c^{2} (p^{G} - 2 {(1 + r)}^{2} p^{T}))) = 0 \end{array}$ we obtain $k_{m 2}^{'} = \frac{(\begin{array}{l} - (1 + r) η^{2} (1 - λ) p^{G} (8 A^{2} c^{2} {(1 + r)}^{2} p^{T} + 4 A (a + A) {((1 - λ) p^{G})}^{2} p^{T} - p^{G} (4 A^{2} c^{2} + (1 - λ) p^{T} (4 A c (2 A + a r + 3 A r) - (a - 3 A) (3 a - A) (1 - λ) p^{T}))) \\ + (1 + r) (1 - λ) {(η p^{G})}^{2} \sqrt{16 A^{4} c^{4} - (1 - λ) p^{T} (\begin{array}{l} 32 A^{3} c^{2} (- a c r + A c (2 + r) + (a - A) (1 - λ) p^{G}) - 8 A^{2} (1 - λ) p^{T} φ_{1} - {(a - 3 A)}^{2} {(A - 3 a)}^{2} {((1 - λ) p^{T})}^{3} \\ - 8 A (3 a - A) (a - 3 A) (A c (2 + r) - a c r + (a - A) (1 - λ) p^{G}) {((1 - λ) p^{T})}^{2} \end{array})} \end{array})}{4 A {(1 + r)}^{2} (4 A^{2} c^{2} {(1 + r)}^{2} p^{T} - (a - 3 A) (3 a - A) {((1 - λ) p^{G})}^{2} p^{T} - p^{G} (4 A^{2} c^{2} + (1 - λ) p^{T} (4 A (a + A) c r - (a - 3 A) (3 a - A) (1 - λ) p^{T})))} < 0$ and $k_{m 2}^{″} = \frac{(\begin{array}{l} - (1 + r) η^{2} (1 - λ) p^{G} (8 A^{2} c^{2} {(1 + r)}^{2} p^{T} + 4 A (a + A) {((1 - λ) p^{G})}^{2} p^{T} - p^{G} (4 A^{2} c^{2} + (1 - λ) p^{T} (4 A c (2 A + a r + 3 A r) - (a - 3 A) (3 a - A) (1 - λ) p^{T}))) \\ - (1 + r) (1 - λ) {(η p^{G})}^{2} \sqrt{16 A^{4} c^{4} - (1 - λ) p^{T} (\begin{array}{l} 32 A^{3} c^{2} (- a c r + A c (2 + r) + (a - A) (1 - λ) p^{G}) - 8 A^{2} (1 - λ) p^{T} φ_{1} - {(a - 3 A)}^{2} {(A - 3 a)}^{2} {((1 - λ) p^{T})}^{3} \\ - 8 A (3 a - A) (a - 3 A) (A c (2 + r) - a c r + (a - A) (1 - λ) p^{G}) {((1 - λ) p^{T})}^{2} \end{array})} \end{array})}{4 A {(1 + r)}^{2} (4 A^{2} c^{2} {(1 + r)}^{2} p^{T} - (a - 3 A) (3 a - A) {((1 - λ) p^{G})}^{2} p^{T} - p^{G} (4 A^{2} c^{2} + (1 - λ) p^{T} (4 A (a + A) c r - (a - 3 A) (3 a - A) (1 - λ) p^{T})))} < 0$ , where $φ_{1} = c^{2} (- 2 a A (1 + 8 r (2 + r)) + a^{2} (3 + 4 r (3 + 2 r)) + A^{2} (11 + 4 r (5 + 2 r))) - 4 (a - A) (1 - λ) p^{G} (a c (3 + 4 r) - A c (5 + 4 r) - 2 (a - A) (1 - λ) p^{G})$ . Since $k > 0$ , we conclude that $△_{1} - \prod_{m}^{S T *} > 0$ holds. Here, $\prod_{m}^{E G *} > \prod_{m}^{S T *}$ is proved.
(2): $\begin{array}{l} \prod_{m}^{B G *} - \prod_{m}^{S T *} = \frac{(2 A c + (a - 3 A) (1 - λ) p^{T}) (3 a (1 - λ) p^{T} - A (2 c + p^{T} - λ p^{T}))}{16 A (1 - λ) p^{T}} + \frac{A^{2} k^{2} {(1 + r β)}^{2} (2 c (1 + r β) - 3 (1 - λ) p^{G}) (2 c (1 + r β) + p^{G} - λ p^{G})}{8 k (1 + r β) (1 - λ) p^{G} (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})} \\ + \frac{(\begin{array}{l} 2 A k (1 + r β) (- 1 + λ) p^{G} (8 k (1 + r β) (T - r (M - T) β) - η^{2} (c + c r β - (1 - λ) p^{G}) (2 (c + c r β) - (1 - λ) p^{G}) + a k (1 + r β) (2 c (1 + r β) - 5 (1 - λ) p^{G})) \\ + {((1 - λ) p^{G})}^{2} (- 3 a^{2} {(k + k r β)}^{2} - 8 k (1 + r β) (T - r (M - T) β) η^{2} - 2 a k (1 + r β) η^{2} (c + c r β - (1 - λ) p^{G}) + η^{4} {(c + c r β - (1 - λ) p^{G})}^{2}) + \end{array})}{8 k (1 + r β) (1 - λ) p^{G} (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})} \end{array}$ decreases monotonically with $T$ . Solving $\prod_{m}^{B G *} - \prod_{m}^{S T *} = 0$ , we have $T_{1} = \frac{(\begin{array}{l} A^{2} k (1 + r β) (1 - λ) p^{G} (32 k M r β (1 + r β) p^{T} - 4 a k (1 + r β) p^{T} (2 c r β - 5 (1 - λ) (p^{G} - p^{T})) - η^{2} (4 c (2 + 3 r β) (1 - λ) p^{G} p^{T} - {(1 - λ)}^{2} p^{G} p^{T} (4 p^{G} + 3 p^{T}) + 4 c^{2} (p^{G} - 2 {(1 + r β)}^{2} p^{T}))) \\ + 2 A {(1 - λ)}^{2} p^{G} p^{T} (η^{2} p^{G} (8 k M r β (1 + r β) + η^{2} {(c + c r β - (1 - λ) p^{G})}^{2}) - 3 a^{2} {(k + k r β)}^{2} (p^{G} - p^{T}) - a k (1 + r β) η^{2} p^{G} (2 c r β - 2 (1 - λ) p^{G} + 5 (1 - λ) p^{T})) \\ + 3 a^{2} k (1 + r β) η^{2} {(1 - λ)}^{3} {(p^{G} p^{T})}^{2} + 2 A^{3} {(k + k r β)}^{2} (4 c r β (- 1 + λ) p^{G} p^{T} - 3 {(1 - λ)}^{2} p^{G} (p^{G} - p^{T}) p^{T} + 4 c^{2} (- p^{G} + {(1 + r β)}^{2} p^{T})) \end{array})}{(16 A k {(1 + r β)}^{2} (1 - λ) p^{G} (2 A k (1 + r β) + η^{2} (1 - λ) p^{G}) p^{T})}$ That is, $\prod_{m}^{B G *} < \prod_{m}^{S T *}$ if $T > T_{1}$ ; otherwise, $\prod_{m}^{B G *} > \prod_{m}^{S T *}$ .
(3): $\prod_{m}^{E G *} - \prod_{m}^{B G *} = \frac{(\begin{array}{l} 8 A^{3} c k^{3} r {(1 + r)}^{2} (β - 1) {(1 + r β)}^{2} (c (2 + r + r β) - (1 - λ) p^{G}) - A^{2} k^{2} (1 + r) (1 + r β) (1 - λ) p^{G} (8 k (1 + r) (1 + r β) (4 (T + r (M - M β + T β)) - a c r (1 - β)) + r (1 - β) φ_{1}) \\ - η^{2} {((1 - λ) p^{G})}^{3} (a k r (1 + r) (β - 1) (1 + r β) (3 a k + 2 c η^{2}) - 8 k (1 + r) (1 + r β) (- M r (1 - β) - T (1 + r β)) η^{2} + r (1 - β) η^{4} (c^{2} (1 + r) (1 + r β) - {((1 - λ) p^{G})}^{2})) \\ + 2 A k {(η (1 - λ) p^{G})}^{2} (k (1 + r) (1 + r β) (2 (2 + r + r β) (- a c r (1 - β) + 4 (T + r (M - M β + T β))) + 3 a r (1 - β) (1 - λ) p^{G}) + r (1 - β) η^{2} φ_{2}) \end{array})}{(8 k (1 + r) (1 + r β) (λ - 1) p^{G} (2 A k (1 + r) + η^{2} (1 - λ) p^{G}) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G}))}$ increases monotonically with $T$ . Solving $\prod_{m}^{E G *} - \prod_{m}^{B G *} = 0$ , we have $T_{2} = \frac{(r (1 - β) (\begin{array}{l} A^{2} k^{2} (1 + r) (1 + r β) (1 - λ) p^{G} (- 8 k (a c - 4 M) (1 + r) (1 + r β) + φ_{1}) - 2 A k {(η (1 - λ) p^{G})}^{2} (k (1 + r) (1 + r β) (2 (a c - 4 M) (2 + r + r β) - 3 a (1 - λ) p^{G}) - η^{2} φ_{2}) \\ + 8 A^{3} c k^{3} {(1 + r)}^{2} {(1 + r β)}^{2} (c (2 + r + r β) - (1 - λ) p^{G}) - η^{2} {((1 - λ) p^{G})}^{3} ((1 + r) (1 + r β) (3 a^{2} k^{2} + 2 k (a c - 4 M) η^{2} - c^{2} η^{4}) + η^{4} {((1 - λ) p^{G})}^{2}) \end{array}))}{8 k (1 + r) {(1 + r β)}^{2} (λ - 1) p^{G} (2 A k (1 + r) + η^{2} (1 - λ) p^{G}) (2 A k (1 + r β) + η^{2} (1 - λ) p^{G})}$ , where $φ_{1} = η^{2} (4 c^{2} (5 + 5 r (1 + β) + r^{2} (1 + 3 β + β^{2})) - 4 c (2 + r + r β) (1 - λ) p^{G} - 7 {((1 - λ) p^{G})}^{2})$ and $φ_{2} = 2 c^{2} (1 + r) (2 + r + 3 r β + r^{2} β (1 + β)) - c (1 + r) (1 + r β) (1 - λ) p^{G} - (2 + r + r β) {((1 - λ) p^{G})}^{2}$ . That is, $\prod_{m}^{E G *} < \prod_{m}^{B G *}$ if $T > T_{2}$ ; otherwise, $\prod_{m}^{E G *} > \prod_{m}^{B G *}$ . □

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Figure 1. E-C platform supply chain CER and financing model.

Figure 2. Supply chain structures.

Figure 3. Event sequence of Scenario ST.

Figure 4. Event sequence of Scenario SG.

Figure 5. Event sequence of Scenario EG.

Figure 6. Event sequence of Scenario BG.

Figure 7. Impact of

λ

and

η

w^{J *}

Figure 7. Impact of

λ

and

η

w^{J *}

Figure 8. Impact of

k

w^{J *}

Figure 8. Impact of

k

w^{J *}

Figure 9. Impact of

λ

η

, and

k

q^{J *}

Figure 9. Impact of

λ

η

, and

k

q^{J *}

Figure 10. Impact of

λ

η

, and

k

e^{J *}

Figure 10. Impact of

λ

η

, and

k

e^{J *}

Figure 11. Impact of

λ

\prod_{s}^{J *}

Figure 11. Impact of

λ

\prod_{s}^{J *}

Figure 12. The manufacturers’ preference under different parameters.

Table 1. Comparisons between this study and related literature.

Representative Paper	Channel Selection	Low-Carbon Supply Chains		E-C Platform Supply Chains		Supply Chain Finance
Representative Paper	Channel Selection	Low-Carbon Innovation	Consumers’ Low-Carbon Preference	E-C Platform	Commission Rate	E-C Platform Finance	Bank Finance
Yang et al. [3]	√	√
Liu et al. [22]		√	√
Hsiao et al. [27]				√	√
Chang et al. [39]	√			√	√	√
An et al. [41]	√	√					√
Shi et al. [42]	√	√	√				√
Lai et al. [43]	√	√	√				√
Qin et al. [44]	√	√	√	√	√	√
This study	√	√	√	√	√	√	√

Table 2. Notations.

Parameters	Definitions
$a$	Total market potential
$ζ$	Random variable that represents uncertain market demand
$M$	Manufacturer’s initial capital
$w^{J}$	Wholesale price in Strategy $J$ , where $J \in \{S T, S G, E G, B G\}$
$P^{T}$	Retail price in Strategy $S T$
$P^{G}$	Retail price in Strategy $J$ , where $J \in \{S G, E G, B G\}$
$λ$	The commission rate, where $λ \in (0, 1 / 2)$
$η$	The coefficient of consumers’ low-carbon preference
$r$	Interest rate
$β$	The discount of interest rate, where $β \in (0, 1)$
$e^{J}$	Unit amount of CER in Scenario $J$ , where $J \in \{S G, E G, B G\}$
$c$	Manufacturer’s variable production cost
$k$	The coefficient of CER technology innovation cost
$T$	The service cost of bank financing
$C_{e}$	The CER technology innovation cost
$q^{J}$	Production quantity in Scenario $J$ , where $J \in \{S T, S G, E G, B G\}$
$D^{J}$	Market demand in Scenario $J$ , where $J \in \{S T, S G, E G, B G\}$
$\prod_{i}^{J}$	The profit of $i$ in Scenario $J$ , where $i \in \{s, m, e, b\}$ , $s$ (supplier), $m$ (manufacturer), $e$ (E-C platform), $b$ (bank), $J \in \{S T, S G, E G, B G\}$

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Zhang, Y.; Shang, J. Optimal Strategies for E-Commerce Platform Supply Chain: Carbon Emission Reduction and Financing. Systems 2024, 12, 469. https://doi.org/10.3390/systems12110469

AMA Style

Zhang Y, Shang J. Optimal Strategies for E-Commerce Platform Supply Chain: Carbon Emission Reduction and Financing. Systems. 2024; 12(11):469. https://doi.org/10.3390/systems12110469

Chicago/Turabian Style

Zhang, Yuting, and Juan Shang. 2024. "Optimal Strategies for E-Commerce Platform Supply Chain: Carbon Emission Reduction and Financing" Systems 12, no. 11: 469. https://doi.org/10.3390/systems12110469

APA Style

Zhang, Y., & Shang, J. (2024). Optimal Strategies for E-Commerce Platform Supply Chain: Carbon Emission Reduction and Financing. Systems, 12(11), 469. https://doi.org/10.3390/systems12110469

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Optimal Strategies for E-Commerce Platform Supply Chain: Carbon Emission Reduction and Financing

Abstract

1. Introduction

2. Literature Review

2.1. Low-Carbon Supply Chains

2.2. E-C Platform Supply Chains

2.3. Supply Chain Finance

3. Model Description and Assumptions

3.1. Model Description

3.2. Assumptions

4. Model Development

4.1. Scenario ST: Traditional Production with Sufficient Capital

4.2. Scenario SG: CER Implementation with Sufficient Capital

4.3. Scenario EG: CER Implementation with E-C Platform Financing

4.4. Scenario BG: CER Implementation with Bank Financing

5. Results Analysis

6. Conclusions and Further Research

6.1. Conclusions

6.2. Recommendations

6.3. Limitations and Further Research

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Appendix A

References

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