Dynamic decision modeling of production and marketing in low-carbon supply chains considering delay effects

Abstract

This article deals with a dynamic decision-making model for a low-carbon supply chain which consists of a manufacturer and a platform retailer. Consideration of delay effects, a delayed differential equation for the effect of low-carbon investment efforts (LIE) in R&D and low-carbon promotional effort (LPE) on low-carbon goodwill (LG) is developed. Moreover, Hamilton's function is applied to solve the decision problem of optimal control. In the model, the differences between the agency selling and reselling patterns are analyzed by comparing LIE, LPE, LG, and net discounted profit. The commission system is a key measure for dynamic decision making on low-carbon products, while the commission rate is also an important reference point for decision making on cooperation patterns. In contrast to the findings of previous studies, this article derives specific thresholds for commissions. Furthermore, this study considers delay effects from a dynamic perspective. The findings show differences in both decentralized and centralized decision-making solutions for supply chains as the delay time changes. The proposed models are analyzed mathematically and numerical examples are illustrated to justify the feasibility of the model in reality. This study provides new insights into the choice of platform sales patterns for firms to develop agency selling and reselling partnership solutions in practice.

Introduction

A consensus that has been endorsed by several countries to reduce carbon emissions, develop a low-carbon economy, etc., with the aim of alleviating the global pressure of environmental pollution and climate change. ¹ Similarly, the Chinese government has set targets for carbon peaking and carbon neutrality. ² As the multiple effects of product greenness and carbon pollution reduction schemes are considered. ³ Some firms intend to deploy green technologies and purchase green products to prevent the environment from being polluted. ⁴ Low-energy consumption and light emissions continue to be a production challenge for Chinese companies. High costs constrain a firm's development, including equipment development, production design and certification, and assessment. In particular, characteristics of low-carbon products are the main constraints to production and marketing promotion, including high prices, expensive costs, and technical difficulties in research and development. ⁵ As a solution to the above dilemma, firms need to balance decarbonization of the process from the production end to the sales end. At the production end, companies can break the technical barriers and reduce the production cost of products, through continuous and stable investment in low-carbon R&D. Ultimately, strong competitiveness is formed in the product market to achieve rapid popularization. In recent years, more and more manufacturers have a strategic awareness to develop a plan for low-carbon development, and increased investment in low-carbon R&D to meet the standards of energy conservation and emission reduction. For instance, GREE, a famous Chinese electrical appliance manufacturer, increased investment in low-carbon technologies to advance low-carbon design and innovation. In 2021, an air conditioning system called “Zero Carbon Source” was produced and accepted quickly by consumers. Emission reduction was its selling point, reducing carbon emissions by 85.7% compared to conventional air conditioners. ⁶ At the sales end, it is difficult to market and promote low-carbon products without promotional efforts. According to a survey conducted by Forbes Insights in partnership with Tum, 66% of senior marketers ($50 million or more in annual revenue) believed that data-driven platform-based marketing had great potential to grow in the business competition. ⁷

Recently, platform sales are drawing more attention from the downstream members of the chain due to traffic hazards and disasters caused by pandemics and epidemic diseases. ⁸ Platform marketing has become the main battleground of the consumer market. It has the ability of data-driven marketing because of the collection of digital technology, artificial intelligence, machine learning, and so on.^9,10 Firms tend to partner with platform retailers to achieve consumer scale for their products. ¹¹ For example, many manufacturers work with Amazon, which uses a recommendation engine to make accurate product matching for users, which not only helps users shorten the time to make decisions but also boosts sales of targeted products. However, different sales patterns are linked to firms’ interests on different platforms. The determination of the sales patterns has also become a key point of decision making in the supply chain. ¹² The agency selling and reselling are two common patterns used by platform retailers in the face of competition from same-tier retailers. ¹³ In the agency selling pattern, platform retailers receive revenue in the form of commissions on transactions and are not responsible for the profit or loss of their products. For example, China's “Tmall” and “Zhihu” generate revenue in the form of commissions. They use intelligent algorithms to analyze customers’ purchasing behaviors, and accurately identify target customers and market with precision. Furthermore, in the second quarter of 2020, 53% of paid units on Amazon were sold through agencies. ¹⁴ In the reselling model, the platform retailer resells the goods from the manufacturer to the consumer at a wholesale level. More importantly, the platform retailers have the full benefits and can mitigate the negative effects of complementary or competitive relationships between products through price coordination. For example, the e-commerce company JD's own stores are a typical reselling model in which the platform makes product acquisitions from partner manufacturers and displays the products on the platform for sale to consumers. The reselling revenue accounted for approximately 88.5% of the total revenue in 2019 on JD.com. ¹⁵ In conclusion, the difference between the two sales approaches is the way in which profits are distributed, which makes it worthwhile to examine the balanced choice of sales model by low-carbon product manufacturers, because of the difference in decision making in the supply chain of low-carbon products under the different models.

Good performance depends on the development and implementation of various strategies, in the supply chain of low-carbon products ¹⁶ low-carbon investment efforts (LIE) and low-carbon promotional effort (LPE) are internally determined variables influencing the market demand for low-carbon products. Manufacturers’ investment in R&D and platform retailers’ efforts in marketing are conducive to building a good brand social image of the company as green, low-carbon, and environmentally friendly, which can also be termed as low-carbon goodwill (LG). The high and low LIE and LPE are not easily perceived by consumers. LG is also a centralized reflection of the brand image of low-carbon products and a comprehensive assessment of the market's ability to achieve sustainable business compliance.^17,18 However, LG is a perception and evaluation of low-carbon products in the consumer market. In management practice, when manufacturers and platforms make improvements to LIE and LPE, it takes time to transfer the flow of commerce, logistics, and information in the supply chain. Therefore, LG becomes a mediating variable between LIE, LPE, and market demand, and a positive LG provides assurance of the quality and reputation of low-carbon products. ¹⁹ Considering the delay effect, this study derives the optimal decision for LIE, LPE, and net discounted profit in the agency selling and reselling patterns, which has some research value.

Environmental benefits, resource efficiency, and economic benefits are being considered by firms in a comprehensive way. In addition, in the context of the outbreak of the epidemic and the continuous development of e-commerce, platform sales have become the preferred channel for product sellers and play an important role in distribution. Decisions on low-carbon products are influenced by factors like investment in R&D and promotion, and the choice of platform sales pattern. Throughout the literature review, some studies still have some deficiencies. Some studies have examined the importance and factors influencing investment in R&D. But low-carbon R&D has not yet been used in dynamic decision-making models of the supply chain. Several scholars provided theoretical foundations for low-carbon marketing, including shared value theory, complex systems theory, corporate sustainability, and green supply chain management. However, fewer scholars considered the decision making in product production and marketing from the low-carbon economic theory. Then, some scholars built low-carbon supply chain models, considering elements such as contract selection, performance improvement, and social preferences. However, fewer studies considered time effects in their models. In addition, some scholars studied the preferences of agency selling and reselling patterns from different contexts, as well as studying the optimal decision conditions of supply chain members. However, no studies considered the differences between agency selling and reselling in terms of production and marketing decisions.

The objectives of this study inquire about the following questions:

Which platform sales pattern should be chosen for the low-carbon supply chain, the agency selling or the reselling?

Are there differences in the manufacturer's LIE, the platform retailer's LPE, the LG of the low-carbon product, and net discount profit under agency selling and reselling patterns?

Are production and marketing decisions in low-carbon supply chains affected by delayed effects?

To concentrate on the above questions, a low-carbon product supply chain is considered where a manufacturer (M) invests in R&D and produces products with low-carbon attributes and a platform retailer sells low-carbon products in two patterns for agency selling and reselling. To reduce the gap between academic research and practical application, a dynamic decision model that takes into account delay effects is developed. Where LIE is the manufacturer's decision variable and LPE is the platform retailer's decision variable. A delay differential equation based on the Nerlove-Arrow model was constructed to analyze the effect of LIE and LPE on LG. This study constructs manufacturer and retailer Hamilton functions separately to solve for the optimal decisions by using the maximum principle. Subsequently, LIE, LPE, LG, and expected profit are compared and analyzed by the manufacturer, the platform retailer and the entire supply chain.

The contribution of this research is threefold. Scholarly contribution is to bridge the gap created between academic research and practical application. Taking into account the delay effect, this study studies the decision making in the production and sale from a dynamic perspective. The practical contribution is dedicated to providing a summary of decision-making ideas and practices for enterprise management. Both the commission system and the control of delay times contribute to promoting the interests of the business. In addition, the basic model is extended to examine the decision problem of the platform sales pattern. This study compares and analyzes LIE, LPE, LG, and net discounted profit from the manufacturer, platform retailer and supply chain perspectives. The scope of the supply chain model is broadened and new insights are provided into platform sales options for low-carbon firms.

The work of this article is as follows: The second section provides a literature review of relevant research topics. The third section performs the model description as well as sets the related assumptions. The fourth section analyzes the decision problems of the supply chain members, solves the manufacturer's optimal production LIE, retailer's optimal LPE, and LG under the two sales models, and conducts a comparative analysis of the profits of the supply chain members. The fifth section analyzes numerically in order to verify the previous theoretical results. The sixth section summarizes the main findings of this study. And the seventh section draws management inspiration.

Literature review

In the 21st century, various environmental problems such as global warming, air pollution, and typical weather changes are directly associated with supply chain behaviors which have destructive effects on human activities. The distribution of low-carbon products in the consumer market is an effective way to avoid further environmental damage. At the same time, it makes more sense to consider these issues at the level of the supply chain. ²⁰ Therefore, many firms have started to invest more in R&D for low-carbon products and implement low-carbon marketing strategies fulfilling consumers’ preferences to facilitate long-term profit in businesses.

R&D investment efforts are a key element in the implementation of low-carbon concepts at the production end of the supply chain. Consumers are increasingly aware of environmental protection, and companies are gradually paying attention to the low-carbon operation of their products, such as low-carbon product development and green marketing strategies. ²¹ Meanwhile, Li and Sarkis ²² suggested that there was an increase in eco-sensitive consumers and that manufacturers should seek to capture this growing demand. Energy-efficient processes are increasingly on the radar of manufacturers. They advocate de-internalising low-carbon technologies. Kiss et al. ²³ considered that investment in R&D is beneficial for developing and introducing new technologies, products, and services to enhance market competitiveness for products. Du et al. ²⁴ studied the influence of consumer low-carbon preference attributes on manufacturers’ production R&D decisions under a carbon cap-and-trade policy. Richstein and Neuhoff ²⁵ discovered that first cost, R&D operating cost, and price uncertainty are the main factors affecting R&D investment in low-carbon industries. Wei and Li ²⁶ suggested that manufacturers’ decisions on low-carbon R&D investments are aimed at maximizing profits and that both costs and benefits should be considered when making decisions. Ohlendorf et al. ²⁷ proposed that a carbon trading scheme could stabilize expectations of future carbon prices and thus promote low-carbon R&D investments by firms.

LPEs are the most important reason why low-carbon products are brought to market. Much of the motivation for consumers to buy low-carbon products is due to the marketing strategy, which is inseparable from the promotional efforts of retailers. In terms of theoretical research, some scholars have studied low-carbon promotion from different perspectives. Such as shared value theory, ²⁸ complex systems theory, ²⁹ corporate sustainability, ³⁰ green supply chain management, ³¹ etc. In addition, Sana ³² studied price competition between green and nongreen marketing with the theme of green product marketing. He considered that the demand for products depended on the selling price, carbon emissions and the corporate social responsibility (CSR) index. In terms of modeling research, some scholars have also conducted relevant studies on supply chain models. Zhou ⁵ developed a Stackelberg model based on abatement cost-sharing contracts. His market was to investigate how supply chain contracts can be designed to reduce the price and stimulate demand. Zhou and Ye ³³ constructed a supply chain model. And examined manufacturers’ decisions to invest in low-carbon technologies and retailers’ decisions to place advertisements. Liao et al. ³⁴ compared and analyzed the contract options of manufacturers with different goals, including carbon reduction levels, profits, or partnership periods. Yu et al. ³⁵ proposed that supply chains use incentive contracts to motivate members to make carbon reduction efforts and achieve better performance in the supply chain of low-carbon products. Lin et al. ³⁶ developed a dynamic game model in a low-carbon supply chain that consists of a single manufacturer and a single retailer. Then, he also explored the complex dynamic characteristics of pricing decisions and carbon reduction strategies in the supply chain, focusing on the impact of retailers’ social preferences on pricing decisions. Sana ⁸ constructed a dual-channel inventory model to make decisions about optimal pricing, order lot size, and reorder point for maximizing jointly the average expected profit from offline and online channels. Zhen et al. ³⁷ established a model consisting of a government, a set of heterogeneous consumers and a manufacturer. The aim is to explore government subsidy preferences and pricing decisions for traditional and green products under different government subsidy policies simultaneously.

As e-commerce continues to grow, platform retailers play an important role in distribution as a medium to sell their products. Therefore, more and more manufacturers are choosing their sales models with platform retailers. A review of the existing literature reveals a large number of studies on two sales models, namely agency selling and reselling. ³⁸ To begin with, several scholars have studied platform preferences for agency selling and reselling pattern, considering different influencing factors such as traditional channel competition, ³⁹ revenue-sharing approaches, ⁴⁰ competition from upstream suppliers, ⁴¹ information-sharing strategies, ⁴² and data-driven marketing efficiency. ⁴³ Next, there are also scholars who further focus on different models of online channels, and study the choice of online sales models in supply chain.^44,45 Hagiu and Wright ⁴⁶ suggested that the platform sales model has relatively higher information costs. Zheng et al. ⁴⁷ proposed that suppliers could sell their products directly to customers by paying a fixed entrance fee and a proportional commission to online retailers in a reseller model. Zhang and Hou ⁴⁸ considered that manufacturers should use agency selling (reselling) when the percentage cost is low (high). When the percentage cost is moderate and there is a high brand advantage, the manufacturer prefers reselling. Further, some scholars have given selection conditions and optimization schemes for the optimal sales model under different contexts. Foros et al. ⁴⁹ suggested that when platforms adopt the agency model on a market-by-market basis, it is always anticompetitive (leading to higher retail prices). Lin et al. ⁵⁰ examined how upstream competition and order fulfillment costs affect e-commerce platforms and manufacturers’ decisions in agency selling (reselling) models. Zennyo ⁵¹ studied the contractual options of two suppliers producing different products with the same platform, in the case of symmetrical potential demand. Wei et al. ⁵² studied platform retailers’ decisions on sales models in several cases where channel roles, commissions, and market shares differed. Wang et al. ⁵³ compared the impact of reselling, agency selling, and direct selling models on online retailers, platforms, and overall supply chain profits in a dual-channel supply chain context. Ha et al. ⁵⁴ explored the effects of commission rates and online platform service costs on the choice of sales model.

As shown in Table 1, the existing studies have analyzed R&D investment efforts, low-carbon R&D efforts, low-carbon promotion efforts, and platform sales patterns. The importance and factors of R&D investment had been considered by some scholars. However, there is little existing research on promotion efforts in low-carbon products. Some scholars studied the decision making of low-carbon marketing from the perspectives of shared value theory, complex systems theory, corporate sustainability, and green supply chain management. Then, some scholars constructed low-carbon supply chain models in terms of contract selection, performance improvement, and social preferences. In addition, some scholars studied the preferences for agency selling and reselling, as well as studying the optimal decision conditions of supply chain members for online sales. Yet, there are still some research gaps. One is that the impact of LIE on manufacturers’ production of low-carbon products has not been considered. Two is that no studies have yet considered the LPE of platform retailers as a key variable influencing the market demand for low-carbon products. Three is that the time factor has rarely been considered and that delay effects in low-carbon supply chains can have an impact on supply chain members’ decisions. The fourth is that the difference in decision making between agency selling and reselling patterns has not been studied, from the perspective of low-carbon product supply chains.

Table 1.

Comparison of recent relevant literature research points.

Literature	Time	R&D investment efforts	Low-carbon R&D efforts	Platform sales patterns	Consideration of delay effects
Hagiu and Wright	2015			√
Du et al.	2016	√
Foros et al.	2017			√
Kiss et al.	2018	√
Lin et al.	2018			√
Zhou and Ye	2018	√	√
Sana	2020		√	√
Cao et al.	2020			√
Zhen et al.	2020
Lin et al.	2021				√
Wei et al.	2021			√
Zheng et al.	2021			√
Ohlendorf et al.	2022	√
Richstein and Neuhoff	2022	√
Yu et al.	2022	√		√
Wei and Li	2022	√
Sana	2022			√
Zhang and Hou	2022			√
Wang et al.	2022			√
Albert et al.	2022			√
This article	2022	√	√	√	√

Given the above research gap, this study explores the issue of sales pattern decisions in low-carbon supply chains, considering delay effects. Considering the time delay effect, delay differential equations are constructed. The differences in LIE, LPE, LG, and expected profit of low-carbon products under the two patterns are also analyzed. The aim of this study is to examine the optimal choice of sales patterns, from the perspective of the manufacturer, the platform retailer and the entire supply chain. This not only enriches the research on low-carbon supply chains but also provides new insights into the platform sales options for low-carbon companies.

Delayed differential models and related assumptions

A low-carbon product supply chain system is considered to consist of a manufacturer (M) and a platform-based retailer (I). The manufacturer is responsible for producing low-carbon products at a unit cost of c. And for the purpose of improving the brand reputation and sales volume of low-carbon products, manufacturers enter into sales contracts with platform retailers, agreeing that all products will be sold only through the partner platform-based retailers (0 offline sales volume of low-carbon products). Referring to the assumptions established by Abhishek et al., ⁵⁵ we set the retail price $p_I$ as an exogenous variable. $p_I$ is the uniform selling price printed on the packaging of the low-carbon product. Considering the high cost of self-adjusting the product price because the setting of the markup involves a series of operational processes such as changing the packaging, the retail price once set does not change over a long period of time.

The manufacturer and platform retailer in the low-carbon supply chain primarily choose two sales patterns namely agency selling and reselling.

Agency selling: The platform retailer acts as the agent of low-carbon products. The manufacturer and platform retailer sign a revenue-sharing contract. As required by the contract, the platform retailer will place low-carbon products on the platform for sale at a price $p_I$ . Among them, the platform receives a proportion of sales revenue in the form of commission, and this ratio is noted as $\phi$ . The manufacturer gets a proportion of sales revenue, and this part is noted as $(1-\phi )$ .

Reselling: Unlike the agency selling, the platform retailer has to bear the risk of profit and loss in this pattern. The platform retailer signs a wholesale price contract with the manufacturer. The platform retailer will purchase low-carbon products at wholesale prices w, and then sell the target products on the platform at retail prices $p_I$ . At the same time, the manufacturer's expected profit depends only on the wholesale price and sales volume of the low-carbon product.

The platform will take advantage of the consumer big data information collected by the channel advantage. After that, the platform analyzes consumers’ browsing traces, shopping records and other big data through cloud computing, text analysis and other data analysis technologies, so as to accurately portray users’ portraits. This then achieves the purpose of accurate promotion of low-carbon products.^56–58 From this, we identify LPE as a decision variable for platform retailers and set it as $f_2(t)$ . In addition, some major barriers are shown such as high production costs, high technological barriers, etc., in the production of low-carbon products. Therefore, LIE is the main decision variable for manufacturers in the production of low-carbon products, which is set as $f_1(t)$ . However, in this article, we consider the delayed effect of LIE and LPE on LG, and the impact of the decision of manufacturer's LIE and retailer's LPE on market demand is mainly through two paths. The essential difference between the two paths is the different delay times. One is that LIE (LPE) directly affects market demand. If the manufacturer increases (decreases) the level of LIE investment effort, the demand will be reflected in a short period of time accordingly, and the same is true for the platform retailer's decision. The second is that the impact of LIE (LPE) on market demand will be prolonged due to the dynamic delayed effect of LIE (LPE) on LG, which exhibits a delayed effect. As shown in Figure 1.

Figure 1.

Delayed differential models. LIE: low-carbon investment efforts; LPE: low-carbon promotional effort.

The following assumptions related to this article are given next in this context. Hypothesis 1

LG is a comprehensive evaluation of low-carbon products, which largely determines consumers’ willingness to purchase the brand's products. It is mainly influenced by the LIE and LPE of low-carbon products and shows dynamic variability over time. Based on Nerlove-Arrow, ⁵⁸ LIE and LPE jointly and positively affect LG, therefore, in order to describe the dynamic change of product brand reputation, we can establish the delay differential equation as shown in equation (1).

$$\{\begin{array}{c}\stackrel{\bullet }{G(t)}=\gamma f_1(t-d_M)+\beta f_2(t-d_I)-\sigma G(t)\\ G(0)=G_0\end{array}$$

(1)

where $G(t)$ denotes the LG of low-carbon product at moment t, and $G(0)=G_0>0$ the initial LG of the low-carbon product. $d_M$ indicates the time of LG change due to the manufacturer's improvement of LIE. Similarly, $d_I$ indicates the time of LG change due to the manufacturer's improvement of LPE. $\gamma$ ( $\gamma >0$ ) indicates the extent to which the manufacturer's improvement in LIE affects the rate of change of LG. $\beta$ ( $\beta >0$ ) indicates the extent to which the platform retailer's improvements to LPE affect the rate of change of LG. $\sigma$ ( $\sigma >0$ ) means the attenuation coefficient of LG. With reference to Ma et al., ⁵⁹ the reasons for LG's decay can be summarized as consumers forgetting the brand and switching to other competitors for product purchases, etc. Hence, the delayed differential equation for LG shows that the more willing manufacturers are to work with to improve LIE, the more significantly the growth rate of LG increases. Similarly, a similar phenomenon occurs for LG as platform retailers increase their willingness to improve LPE. However, the growth rate of LG decreases as the attenuation coefficient increases.

Hypothesis 2

The demand for low-carbon products is influenced by a combination of manufacturers’ LIE, LG, and platform retailers’ LPE, which is denoted as $D_I(t)$ . LIE and LPE indirectly influence the market demand through LG, and at the same time, the improvement of LIE and LPE can directly promote the product sales volume. ⁶⁰ In addition, the demand for low-carbon products is also affected by LG, and the higher brand effect can make consumers believe that the product has a high-quality guarantee, which in turn increases the sales of the product whether to meet consumer preferences. ⁶¹ Therefore, with reference to the model assumptions of Basak et al. ⁶² and Chen et al., ⁶³ we give the demand function under the platform retailing model as equation (2).

$$D_I(t)=a+bG(t)+{\theta }_1{f}_1(t)+{\theta }_2{f}_2(t)$$

(2)

where $D_I(t)$ denotes the market demand in the online channel and varies with the dynamics at moment t. a is the impact of the platform retailer's brand effect on product sales, and a is set as a constant value. b indicates the sensitivity factor of LG to product market demand ( $b>0$ ). ${\theta }_1$ ( ${\theta }_1>0$ ) indicates the sensitivity factor of the manufacturer's LIE acting on the product market demand. Similarly, ${\theta }_2$ ( ${\theta }_2>0$ ) indicates the sensitivity factor of the manufacturer's LPE acting on the product market demand.

Hypothesis 3

The cost for the manufacturer to make improvements to the LIE is $C_M$ . The LPE enhancement costs for platform retailers are $C_I$ . According to Liu et al., ⁶⁴ The improvement costs of LIE and LPE are their own incremental functions, and their second-order derivatives are >0. This relationship is denoted as $C_M^{\mathrm{″}}(f_1)>0$ , $C_I^{\mathrm{″}}(f_2)>0$ . Therefore, the improved cost functions of LIE and LPE are shown for equations (3) and (4).

$$C_M(t)=\frac{1}{2}{k}_M{f}_1^2(t)$$

(3)

$$C_I(t)=\frac{1}{2}{k}_I{f}_2^2(t)$$

(4)

where $k_M$ ( $k_M>0$ ) represents the cost factor of LIE and $k_I$ ( $k_I>0$ ) represents the cost factor of LPE.

Hypothesis 4

In this article, we assume that the manufacturer and the platform retailer are making profit-optimal decisions in an infinite time zone scenario. And there are different expressions for the net discounted profit function of the manufacturer, the retailer, and the whole supply chain under different sales patterns.

When a platform retailer chooses the agency reselling model, the manufacturer sells the low-carbon product on the platform at a price $p_I$ .And who shares a percentage of the total sales revenue with the platform retailer in the form of a commission. Typically, the revenue-sharing ratio is specified in the contract between the manufacturer and the platform retailer and will not change easily over a longer period of time. For example, the platform retail giants, Amazon and JD, have contracts with manufacturers based on product categories and do not change the model for the duration of the partnership. Then, the differential game model of supply chain members is shown below.

$$\begin{array}{rl}max{J}_M^A=& \underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}\{[(1-\phi )p_I-c][a+bG(t)+{\theta }_1{f}_1(t)+{\theta }_2{f}_2(t)]\\ & -\frac{1}{2}{k}_M{f}_1^2(t)\}dt\end{array}$$

(5)

$$max{J}_{{}_I}^A=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}\{\phi p_I[a+bG(t)+{\theta }_1{f}_1(t)+{\theta }_2{f}_2(t)]-\frac{1}{2}{k}_I{f}_2^2(t)\}dt$$

(6)

$$\begin{array}{rl}max{J}_{{}_{MI}}^A=& \underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}\{(p_I-c)[a+bG(t)+{\theta }_1{f}_1(t)+{\theta }_2{f}_2(t)]-\frac{1}{2}{k}_I{f}_2^2(t)\\ & -\frac{1}{2}{k}_M{f}_1^2(t)\}dt\end{array}$$

(7)

where the discount rate is denoted by parameter $\lambda$ . In the agency selling pattern, $max{J}_M^A$ , $max{J}_{{}_I}^A$ , and $max{J}_{{}_{MI}}^A$ are defined as the optimal control function for the net discounted profit for the manufacturer, the platform retailer, and the entire supply chain, respectively. They are equivalent to a definite integral over a multiplicative equation which is the expected profit multiplied by the net discounted exponential function.

When the platform retailer chooses the reselling model, who purchases goods from the manufacturer at wholesale price and sells them on the platform at retail price. At this time, the differential game model of the supply chain members can be expressed as below.

$$max{J}_{{}_M}^R=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}\{(w-c)[a+bG(t)+{\theta }_1{f}_1(t)+{\theta }_2{f}_2(t)]-\frac{1}{2}{k}_M{f}_1^2(t)\}dt$$

(8)

$$max{J}_{{}_I}^R=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}\{(p_I-w)[a+bG(t)+{\theta }_1{f}_1(t)+{\theta }_2{f}_2(t)]-\frac{1}{2}{k}_I{f}_1^2(t)\}dt$$

(9)

$$\begin{array}{rl}max{J}_{{}_{MI}}^R=& \underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}\{(p_I-c)[a+bG(t)+{\theta }_1{f}_1(t)+{\theta }_2{f}_2(t)]-\frac{1}{2}{k}_M{f}_1^2(t)\\ & -\frac{1}{2}{k}_I{f}_2^2(t)\}dt\end{array}$$

(10)

In the rese lling pattern, $max{J}_M^R$ , $max{J}_{{}_I}^R$ , and $max{J}_{{}_{MI}}^R$ are defined as the optimal control function for the net discounted profit for the manufacturer, the platform retailer, and the entire supply chain, respectively.

Model solution and analysis

Decision making in agency selling

In the agency pattern, the platform retailer exhibits low-carbon products to consumers through certain promotional programs. The manufacturer would share the sales revenue with the platform retailer on a proportional basis. The percentage of revenue is called the commission rate which is an important parameter in the optimal decision of the manufacturer and the platform retailer. Proposition 1

In the agency selling, the manufacturer's optimal LIE and the platform retailer's optimal LPE are expressed as follows:

$$f_{1A}^{*}(t)={\displaystyle \frac{(1-\phi )p_I-c}{k_M}({\theta }_1+\frac{b\beta e^{\sigma d_I}}{\sigma +\lambda })}$$

(11)

$$f_{2A}^{*}(t)={\displaystyle \frac{\phi p_I}{k_I}\left({\theta }_2+\frac{b\beta e^{\sigma d_I}}{\sigma +\lambda }\right)}$$

(12)

The optimal LG for low-carbon products is as follows:

$$G_A^{*}(t)={\displaystyle \frac{(1-e^{-\delta t})}{\delta }\left[(\gamma {\theta }_1+\frac{b{\gamma }^2{e}^{\sigma d_M}}{\sigma +\lambda }){\displaystyle \frac{(1-\phi )p_I-c}{k_M}+\left(\beta {\theta }_2+\frac{b{\beta }^2{e}^{\sigma d_I}}{\sigma +\lambda }\right){\displaystyle \frac{\phi p_I}{k_I}}}\right]+G_0{e}^{{}^{{}^{{}^{-\sigma t}}}}}$$

(13)

Proof:

Combined with the constraint equation (1), the manufacturer's optimal control problem can be denoted by the net discount function, noted as $\underset{f_1>0}{max}{J}_M^A$ . In a finite interval, the problem of optimal control of the manufacturer's profit can be determined by the principles provided by extreme value to determine the regularity of the optimal control. Therefore, according to the principle of extreme value, the Hamiltonian function of the manufacturer's profit is constructed in this study as follows:

$$H_M^A=e^{-\lambda t}\{[(1-\phi )p_I-c][a+b{G}_A(t)+{\theta }_1{f}_{1A}(t)+{\theta }_2{f}_{2A}(t)]-\frac{1}{2}{k}_M{f}_{1A}^2(t)\}+{\chi }_M(t)[\gamma f_{1A}(t-d_M)+\beta f_{2A}(t-d_I)-\sigma G_A(t)]$$

(14)

Following the steps of the extreme value solution, the Hamilton function is solved for the first-order derivative with respect to the LIE. And ordering $\frac{d{H}_M^A}{d{f}_{1A}(t)}=0$ , equation (15) can be obtained.

$$e^{-\lambda t}\{{\theta }_1[(1-\phi )p_I-c]-k_M{f}_{1A}(t)\}+\gamma {\chi }_M(t)\frac{d{f}_{1A}(t-d_z)}{d{f}_{1A}(t)}=0$$

(15)

Order $\frac{d{f}_{1A}(t-d_z)}{d{f}_{1A}(t)}=M(t)$ , by shifting the terms, the manufacturer's decision variable LIE is expressed as equation (16).

$$f_{1A}(t)=\frac{1}{k_M}\{{\theta }_1[(1-\phi )p_I-c]+\gamma {\chi }_M(t)M(t)e^{\lambda t}\}$$

(16)

Since ${\chi }_M(t)$ is unknown, the Hamilton function to find the first order derivative with respect to $G_A(t)$ is sought. The formula for the manufacturer's optimal decision quantity is as in equation (17).

$${\displaystyle \frac{d{H}_M^A}{d{G}_A(t)}=e^{-\lambda t}b[(1-\phi )p_I-c]-\sigma {\chi }_M(t)}$$

(17)

Referring to Basin and Gonzalez, ⁶⁵ the following equation is got.

$$\stackrel{\bullet }{{\chi }_M}(t)=-{\displaystyle \frac{d{H}_M^A}{d{G}_A(t)}=\sigma {\chi }_M(t)-e^{-\lambda t}b[(1-\phi )p_I-c]}$$

(18)

According to the solution of the first-order linear differential equation, it is obtained that

$${\chi }_M(t)=\mathrm{C}{e}^{\sigma t}+\frac{b[(1-\phi )p_I-c]}{\sigma +\lambda }{e}^{-\lambda t}$$

(19)

where $M(t)=\mathrm{\Phi }(t-d_M,t)=e^{\sigma d_M}$ , then, ${\chi }_M(t)=\mathrm{C}{e}^{\sigma t}+\frac{b[(1-\phi )p_I-c]}{\sigma +\lambda }{e}^{-\lambda t}$ is substituted into equation (16).

$$f_{1A}^{*}(t)={\displaystyle \frac{1}{k_M}\left\{{\theta }_1[(1-\phi )p_I-c]+\gamma e^{\sigma d_M}\left[\mathrm{C}{e}^{(\sigma +\lambda )t}+\frac{b[(1-\phi )p_I-c]}{\sigma +\lambda }\right]\right\}}$$

(20)

Considering the reality of business operations, there is a limit to the amount of investment in R&D for low-carbon products. Therefore, when $t\to \mathrm{\infty }$ , the manufacturer's LIE is limited, and mathematically it is denoted as $\underset{t\to \mathrm{\infty }}{lim}z(t)<\mathrm{\infty }$ . Then, when time is taken to be infinite, $f_{1A}^{*}(t)$ is finite only if the condition $C=0$ is satisfied. Ultimately, $C=0$ is brought into equation (20), and the manufacturer's optimal decision quantity is expressed as

$$f_{1A}^{*}(t)={\displaystyle \frac{(1-\phi )p_I-c}{k_M}({\theta }_1+\frac{\gamma b{e}^{\sigma d_M}}{\sigma +\lambda })}$$

(21)

Likewise, combining the constraint equation (1), the optimal control problem for a platform retailer can be represented by a net discount function, denoted as $\underset{f_{1A}>0}{max}{J}_{{}_I}^A$ . Using the extreme value principle, the Hamiltonian function is constructed.

$$H_I^A=e^{-\lambda t}\{\phi p_I[a+b{G}_A(t)+{\theta }_1{f}_{1A}(t)+{\theta }_2{f}_{2A}(t)]-\frac{1}{2}{k}_I{f}_{2A}^2(t)\}+{\chi }_I(t)[\gamma f_{1A}(t-d_M)+\beta f_{2A}(t-d_I)-\sigma G_A(t)]$$

(22)

To obtain the extreme value point, the first-order derivative of the Hamilton function with respect to LPE is found. And the first-order derivative is made to be 0. A system of equations is then obtained along the lines of equations (15–18), as follows.

$$\{\begin{array}{c}{\displaystyle \frac{d{H}_I}{d{f}_2(t)}=e^{-\lambda t}(\phi {\theta }_2{p}_I-k_I{f}_{2A}(t))+\beta {\chi }_I(t)\frac{d{f}_{2A}(t-d_I)}{d{f}_{2A}(t)}=0}\\ \stackrel{\bullet }{{\chi }_I}(t)=-{\displaystyle \frac{d{H}_{{}_I}^A}{d{G}_A(t)}=\delta {\chi }_I(t)-b\phi p_I{e}^{-\lambda t}}\end{array}$$

(23)

Similarly, the optimal LPE is found as

$$f_{2A}^{*}(t)={\displaystyle \frac{\phi p_I}{k_I}\left({\theta }_2+\frac{b\beta e^{\sigma d_I}}{\sigma +\lambda }\right)}$$

(24)

The delay differential equation for LG has been given in equation (1), and LG's decision is related to the decision of the manufacturer and the platform retailer. Further, substituting equations (21) and (24) into equation (1) yields.

$$\{\begin{array}{c}\stackrel{\bullet }{G_A^{*}(t)}=\gamma f_{1A}^{*}(t-d_M)+\beta f_{2A}^{*}(t-d_I)-\sigma G_A^{*}(t)\\ G_A(0)=G_0\end{array}$$

(25)

Then, according to the solution of the first-order differential equation, the optimal decision for LG is derived as in equation (26).

$$G_A^{*}(t)={\displaystyle \frac{(1-e^{-\delta t})}{\delta }[\gamma f_{1A}^{*}(t-d_M)+\beta f_{2A}^{*}(t-d_I)]+G_0{e}^{-\sigma t}}$$

(26)

From Proposition 1, the optimal decision of LIE, LPE, and LG when the platform retailer chooses the agency selling pattern to cooperate with the manufacturer is denoted as $f_{1A}^{*}(t)$ , $f_{2A}^{*}(t)$ , and $G_A^{*}(t)$ . Since they are known algebraically, the differential game equation (5) according to the agency selling pattern can be further simplified as

$$max{J}_M^A={\displaystyle \frac{1}{\lambda }\{[(1-\phi )p_I-c][a+b{G}_A^{*}(t)+{\theta }_1{f}_{1A}^{*}(t)+{\theta }_2{f}_{2A}^{*}(t)]-\frac{1}{2}{k}_M(f_{1A}^{*}(t){)}^2\}}$$

(27)

Next, substituting equations (11), (12), and (13) into equation (14), respectively, yields equation (28).

$$J_M^A=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}\{[(1-\phi )p_I-c][a+{\theta }_1{f}_{1A}^{*}(t)+{\theta }_2{f}_{2A}^{*}(t)]-\frac{1}{2}{k}_M{(f_{1A}^{*}(t))}^2\}dt+\underset{0}{\overset{\mathrm{\infty }}{\int }}b[(1-\phi )p_I-c]G_A^{*}(t)e^{-\lambda t}dt$$

(28)

Further, $order\mathrm{L}=(\gamma {\theta }_1+\frac{b{\gamma }^2{e}^{\sigma d_M}}{\sigma +\lambda })\frac{(1-\phi )p_I-c}{\sigma k_M}+\left(\beta {\theta }_2+\frac{b{\beta }^2{e}^{\sigma d_I}}{\sigma +\lambda }\right)\frac{\phi p_I}{\sigma k_I}$ , according to the calculation of the definite integral, it can be divided into two parts, denoted as A and B.

By solving for the definite integral, A follows that

$$\begin{array}{rl}& A=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}\{[(1-\phi )p_I-c][a+{\theta }_1{f}_{1A}^{*}(t)+{\theta }_2{f}_{2A}^{*}(t)]-\frac{1}{2}{k}_M{(f_{1A}^{*}(t))}^2\}dt\\ & ={\displaystyle \frac{1}{\lambda }\{[(1-\phi )p_I-c][a+{\theta }_1{f}_{1A}^{*}(t)+{\theta }_2{f}_{2A}^{*}(t)]-\frac{1}{2}{k}_M(f_{1A}^{*}(t){)}^2\}}\end{array}$$

(29)

Substituting 1 into B, equation (30) can be found.

$$B=\underset{0}{\overset{\mathrm{\infty }}{\int }}b[(1-\phi )p_I-c]G_A^{*}(t)e^{-\lambda t}dt=b[(1-\phi )p_I-c]\left[{\displaystyle \frac{\mathrm{L}}{\lambda }+{\displaystyle \frac{G_0-L}{\lambda +\sigma }}}\right]$$

(30)

Therefore, the manufacturer's net discounted profit in the reseller model can be expressed as equation (31).

$$max{J}_M^A={\displaystyle \frac{b[(1-\phi )p_I-c]}{\lambda }\left[(a+{\theta }_1{f}_{1A}^{*}(t)+{\theta }_2{f}_{2A}^{*}(t))-{\displaystyle \frac{k_M{(f_{1A}^{*}(t))}^2}{2b[(1-\phi )p_I-c]}+\mathrm{L}+{\displaystyle \frac{\lambda (G_0-L)}{\lambda +\sigma }}}\right]}$$

(31)

Similarly, the net discounted profit of the platform retailer under the agency selling model can be expressed as equation (32).

$$max{J}_I^A={\displaystyle \frac{b\phi p_I}{\lambda }\left[(a+{\theta }_1{f}_{1A}^{*}(t{)}_A^{*}(t)+{\theta }_2{f}_{2A}^{*}(t))-{\displaystyle \frac{k_{I}f{{}_{2A}^{*}}^2(t)}{2b\phi p_I}+\mathrm{L}+{\displaystyle \frac{\lambda (G_0-L)}{\lambda +\sigma }}}\right]}$$

(32)

In the agency selling pattern, the LIE is related to parameters such as $k_M$ , ${\theta }_1$ , $\gamma$ , $\sigma$ , and $\lambda$ . And the LPE is related to parameters such as $k_I$ , ${\theta }_2$ , b, $\beta$ , and $\sigma$ . Several inferences can be drawn. Corollary 1.

LIE, LPE, and LG are positively correlated with the sensitivity of market demand, respectively. The higher the sensitivity of market demand ( ${\theta }_1$ , ${\theta }_2$ , b,), the more willing manufacturers and platform retailers are to increase the level of investment in LIE and LPE. Because for every unit of LIE and LPE that is added, there is a corresponding increase in LG for low-carbon products and a further increase in revenue.

Corollary 2.

The discount rate $\lambda$ is negatively related to the manufacturer's LIE and the platform retailer's LPE, and also negatively related to their profit.

Corollary 3.

The commission rate is an important factor that influences the decision of manufacturers and platform retailers. When $\phi \in (0,1-\frac{c}{P_I}]$ , manufacturers are willing to put effort into R&D. Conversely, when $\phi \in (1-\frac{c}{P_I},1]$ , manufacturers will be resistant to invest in LIE and even unwilling to do R&D. For platform manufacturers, the higher the commission rate is, the more aggressive they are in promoting their products.

Corollary 4.

Delay time from LIE and LPE to LG keep positively correlated with the manufacturer and the retailer's optimal decisions, respectively. If the delay time is longer, the corresponding optimal LIE and LPE will be higher. However, the profits of the manufacturer and the platform retailer have different directions of variation with the delay time. For example, as the delay time for improving the LIE increases, the manufacturer's profit function decreases. However, as the delay time of the improved LPE increases, the manufacturer's profit function also increases. Conversely, the retailer's optimal profit function decreases as the delay time of the improved LPE increases, but increases as the delay time of the improved LIE increases.

Corollary 4 illustrates that platform retailers and manufacturers will be willing to improve their level of effort when certain conditions are met. Because the improvement of LIE and LPE is also affected by time, the longer the delay in affecting LG, the higher the cost of product investment in R&D, technology introduction and advertising and promotion. That ultimately reduces the profits that manufacturers and retailers receive. Therefore, manufacturers and platform retailers are willing to make R&D and promotional improvements only if the increased revenue from the improvements can offset the higher costs incurred and if there is a surplus. That is, when the condition $k_M\mathrm{\Delta }{f}_1(t)<2{\theta }_{1}b[(1-\phi )p_I-c]$ is satisfied, manufacturers are willing to make improvements to the LIE. When the condition $k_I\mathrm{\Delta }{f}_2(t)<2{\theta }_{1}b\phi p_I$ is satisfied, platform retailers are willing to make improvements to LPE to ultimately enhance their own profit.

Decision making in reselling

In reselling pattern, the platform retailer purchases the low-carbon product at the wholesale price w and sells the product to the platform consumer at the sales price $p_I$ . Proposition 2

In the reselling pattern, the manufacturer's optimal LIE and the platform retailer's optimal LPE are as follow:

$$f_{{}^{1R}}^{*}(t)={\displaystyle \frac{w-c}{k_M}({\theta }_1+\frac{\gamma b{e}^{\delta d_M}}{\sigma +\lambda })}$$

(33)

$$f_{2R}^{*}(t)={\displaystyle \frac{p_I-w}{k_I}\left({\theta }_2+\frac{b\beta e^{\delta d_I}}{\sigma +\lambda }\right)}$$

(34)

The optimal LG for low-carbon products is

$$G_R^{*}(t)={\displaystyle \frac{(1-e^{-\delta t})}{\delta }\left[{\displaystyle \frac{w-c}{k_M}(\gamma {\theta }_1+\frac{{\gamma }^{2}b{e}^{\delta d_M}}{\sigma +\lambda })+{\displaystyle \frac{p_I-w}{k_I}\left(\beta {\theta }_2+\frac{b{\beta }^2{e}^{\delta d_I}}{\sigma +\lambda }\right)}}\right]+G_0{e}^{{}^{{}^{{}^{-\sigma t}}}}}$$

(35)

Proof:

According to the constraint equation (1), the manufacturer's optimal control problem in the reselling pattern can be represented by the net discount function, denoted as $\underset{f_1>0}{max}{J}_M^R$ . Analogously, the manufacturer's Hamilton function is constructed as follows:

$$H_M^R=e^{-\lambda t}\{(w-c)[a+b{G}_R(t)+{\theta }_1{f}_{1R}(t)+{\theta }_2{f}_{2R}(t)]-\frac{1}{2}{k}_I{f}_{2R}^2(t)\}+{\chi }_M(t)[\gamma f_{1R}(t-d_M)+\beta f_{2R}(t-d_I)-\delta G_R(t)]$$

(36)

Similar to the procedure for the agency selling pattern, the first-order derivatives of the Hamilton function with respect to LIE and $G_R(t)$ are derived, and ordered $\frac{d{H}_M^R}{d{f}_{1R}(t)}=0$ , $\frac{d{f}_{1R}(t-d_z)}{d{f}_{1R}(t)}=N(t)$ .

$$\{\begin{array}{c}{e}^{-\lambda t}[{\theta }_1(w-c)-k_M{f}_{1R}(t)]+\gamma {\pi }_M(t)N(t)=0\\ {\displaystyle \frac{d{H}_M^R}{d{G}_R(t)}=b(w-c)e^{-\lambda t}-\delta {\pi }_M(t)}\end{array}$$

(37)

Referring to Basin and Gonzalez, ⁶⁵ the following equation is got.

$$\stackrel{\bullet }{{\pi }_M}(t)=-{\displaystyle \frac{d{H}_M^R}{d{G}_R(t)}=\delta {\pi }_M(t)-b(w-c)e^{-\lambda t}}$$

(38)

According to the solution of the first-order linear differential equation, ${\pi }_M(t)$ is obtained that

$${\pi }_M(t)=\mathrm{C}{e}^{\sigma t}+\frac{b(w-c)}{\sigma +\lambda }{e}^{-\lambda t}$$

(39)

Because of $N(t)=\mathrm{T}(t-d_M,t)=e^{\delta d_M}$ , equation (39) is substituted into $f_{1R}(t)$ . The optimal decision for LIE can be expressed as

$$f_{1R}^{*}(t)={\displaystyle \frac{w-c}{k_M}\left[{\theta }_1+\gamma e^{\delta d_M}(\frac{\mathrm{C}}{w-c}{e}^{(\sigma +\lambda )t}+\frac{b}{\sigma +\lambda })\right]}$$

(40)

Because R&D investment in low-carbon products is not infinite, when $t\to \mathrm{\infty }$ , the manufacturer's LIE is also finite, namely $\underset{t\to \mathrm{\infty }}{lim}{f}_{1R}(t)<\mathrm{\infty }$ . From equation (39), it is possible to determine $C=0$ . Then, the manufacturer's optimal decision quantity is expressed as

$$f_{{}^{1R}}^{*}(t)={\displaystyle \frac{w-c}{k_M}({\theta }_1+\frac{\gamma b{e}^{\delta d_M}}{\sigma +\lambda })}$$

(41)

The optimal control problem for a platform retailer can be represented by a net discount function, denoted as $\underset{f_{2R}>0}{max}{J}_{{}_I}^R$ . The Hamilton function is denoted as

$$H_I^R=e^{-\lambda t}\{(p_I-w)[a+b{G}_R(t)+{\theta }_1{f}_{1R}(t)+{\theta }_2{f}_{2R}(t)]-\frac{1}{2}{k}_I{f}_{2R}^2(t)\}+{\chi }_I(t)[\gamma f_1(t-d_M)+\beta f_{2R}(t-d_I)-\delta G_R(t)]$$

(42)

Similarly, the optimal LPE for the platform retailer is found.

$$f_{2R}^{*}(t)={\displaystyle \frac{p_I-w}{k_I}({\theta }_2+b\beta e^{\delta d_I})}$$

(43)

Further, we substitute equations (41) and (43) into equation (1) to obtain the solution of the delay differential equation (1) with respect to LG as shown in equation (44).

$$G_R^{*}(t)={\displaystyle \frac{(1-e^{-\delta t})}{\delta }[\gamma f_{1R}^{*}(t-d_M)+\beta f_{2R}^{*}(t-d_I)](1-e^{{}^{{}^{{}^{-\sigma t}}}})+G_0{e}^{{}^{{}^{{}^{-\sigma t}}}}}$$

(44)

Following Proposition 2, when the platform retailer and the manufacturer adopt the reselling pattern, the optimal decisions for LIE, LPE, and LG are $f_{1R}^{*}(t)$ , $f_{2R}^{*}(t)$ , and $G_R^{*}(t)$ . According to the differential game model, the net discounted profit of the manufacturer and the platform retailer can be expressed as equations (44) and (45), respectively.

$$max{J}_M^R={\displaystyle \frac{b(w-c)}{\lambda }\left[(a+{\theta }_1{f}_{1R}^{*}(t)+{\theta }_2{f}_{2R}^{*}(t))-{\displaystyle \frac{k_{M}f{{}_{1R}^{*}}^2(t)}{2b(w-c)}+\mathrm{L}+{\displaystyle \frac{\lambda (G_0-L)}{\lambda +\sigma }}}\right]}$$

(44)

$$max{J}_I^R={\displaystyle \frac{b(p_I-w)}{\lambda }\left[(a+{\theta }_1{f}_{1R}^{*}(t)+{\theta }_2{f}_{2R}^{*}(t))-{\displaystyle \frac{k_{I}f{{}_{2R}^{*}}^2(t)}{2b(p_I-w)}+\mathrm{L}+{\displaystyle \frac{\lambda (G_0-L)}{\lambda +\sigma }}}\right]}$$

(45)

Moreover, similar to Corollaries 1, 2, and 4, the optimal decision of LIE and LPE, is positively related to the sensitivity of market demand and negatively related to the discount rate, respectively. Moreover, the longer the delay affecting LG, the higher the amount of the corresponding optimal LIE, LPE, and LG.

Comparative analyses

Proposition 3.

The commission rate is an important parameter in the choice of sales pattern. When the commission rate is taken at different values, the LIE, LPE, LG, and net discounted profit also differ.

When $\phi <1-\frac{w}{P_I}$ , $f_{1A}^{*}(t)>f_{{}^{1R}}^{*}(t)$ , $G_A^{*}(t)>G_{{}^R}^{*}(t)$ , $max{J}_M^A>max{J}_{{}^M}^R$ .

When $\phi \ge 1-\frac{w}{P_I}$ , $f_{2A}^{*}(t)\ge f_{{}^{2R}}^{*}(t)$ , $G_A^{*}(t)\le G_{{}^R}^{*}(t)$ , $max{J}_I^A\ge max{J}_{{}^I}^R$ .

Proposition 3 illustrates that there are certain thresholds for commission rates, which have different effects on the decisions of supply chain members when they are within and outside the threshold range. The corresponding values are higher in agency selling than in reselling, when the commission rate satisfies $\phi \in \left[0,1-\frac{w}{P_I}\right)$ , such as the optimal LIE, net discounted profit, and LG of low-carbon products. However, when $\phi \in \left[1-\frac{w}{P_I},1\right]$ , the commission system incentivizes platform retailers to invest more in promotions in agency selling patterns, which in turn boosts product sales and increases the platform retailers’ expected profits. Therefore, as the commission rate was made in this condition, the platform is more willing to choose the agency reselling model to sell low-carbon products. And the platform retailer's optimal LIE, net discount profit, and LG of low-carbon products under the agency selling are better than the other.

Corollary 5.

Price setting is a critical determinant of the manufacturer's and platform retailer's decisions. Wholesale price is positively related to the manufacturer's LIE optimal decision, but negatively related to the platform retailer's LPE optimal decision.

When the condition $k_I\gamma ({\theta }_1(\sigma +\lambda )+\gamma b{e}^{\sigma d_M})>k_M\beta [{\theta }_2(\sigma +\lambda )+\beta b{e}^{\sigma d_M}]$ (considered as condition CW1) is satisfied, there are $G^{\mathrm{″}}(t)-G^{\mathrm{\prime }}(t)=\frac{(1-e^{-\delta t})\mathrm{\Delta }w}{\delta }\left[\frac{\gamma {\theta }_1(\sigma +\lambda )+{\gamma }^{2}b{e}^{\delta d_M}}{k_M(\sigma +\lambda )}-\frac{\beta {\theta }_2(\sigma +\lambda )+{\beta }^2{e}^{\delta d_I}}{k_I(\sigma +\lambda )}\right]>0$ .

Therefore, wholesale price are positively correlated with LG of low-carbon products, and manufacturers increasing wholesale prices will increase LG of their products.

When the condition $k_I\gamma ({\theta }_1(\sigma +\lambda )+\gamma b{e}^{\sigma d_M})<k_M\beta [{\theta }_2(\sigma +\lambda )+\beta b{e}^{\sigma d_M}]$ (considered as condition CW2) is satisfied, there are $G^{\mathrm{″}}(t)-G^{\mathrm{\prime }}(t)=\frac{(1-e^{-\delta t})\mathrm{\Delta }w}{\delta }\left[\frac{\gamma {\theta }_1(\sigma +\lambda )+{\gamma }^{2}b{e}^{\delta d_M}}{k_M(\sigma +\lambda )}-\frac{\beta {\theta }_2(\sigma +\lambda )+{\beta }^2{e}^{\delta d_I}}{k_I(\sigma +\lambda )}\right]<0$ .

Hence, the wholesale price is negatively related to the LG of low-carbon products, and an increase in the wholesale price by the manufacturer reduces the LG of the product.

When the condition $k_I\gamma ({\theta }_1(\sigma +\lambda )+\gamma b{e}^{\sigma d_M})=k_M\beta [{\theta }_2(\sigma +\lambda )+\beta b{e}^{\sigma d_M}]$ (considered as condition CW3) is satisfied, there are $G^{\mathrm{″}}(t)-G^{\mathrm{\prime }}(t)=\frac{(1-e^{-\delta t})\mathrm{\Delta }w}{\delta }\left[\frac{\gamma {\theta }_1(\sigma +\lambda )+{\gamma }^{2}b{e}^{\delta d_M}}{k_M(\sigma +\lambda )}-\frac{\beta {\theta }_2(\sigma +\lambda )+{\beta }^2{e}^{\delta d_I}}{k_I(\sigma +\lambda )}\right]=0$ .

Thus, the change in wholesale price is not correlated with LG.

Proposition 4

In terms of centralized decision making, there are differences in supply chain decisions when the range for delay time takes different. The main patterns are shown below.

When $J_2\le \frac{A_1^2}{4B_1}$ , and $d_M\in MI1\cap \{\phi |w>(1-\phi )P_I>c\}$ (considered as CD1). The supply chain profit for reselling is greater than the corresponding value under agency selling, indicating that satisfying CD1 is that the optimal decision model for low-carbon supply chains is reselling.

When $J_2\le \frac{A_1^2}{4B_1}$ , and $d_M\in MI2\cap \{\phi |w>(1-\phi )P_I>c\}$ or $d_M\in MI3\cap \{\phi |w>(1-\phi )P_I>c\}$ (considered as CD2). Supply chain profits are higher under the agency selling model than another one. From the perspective of the supply chain as a whole, supply chain members should choose the agency selling model.

When $J_2\le \frac{A_1^2}{4B_1}$ , and $d_M\in MI4\cap \{\phi |(1-\phi )P_I>w>c\}$ (considered as CD3). There keeps $J_{MI}(d_M,d_I)=J_{MI}^A-J_{MI}^R>0$ . Thus, the supply chain profit under agency selling is higher than that under reselling. The former should be chosen from the supply chain as a whole.

When $J_2\le \frac{A_1^2}{4B_1}$ , and $d_M\in MI5\cap \{\phi |(1-\phi )P_I>w>c\}$ or $d_M\in MI6\cap \{\phi |(1-\phi )P_I>w>c\}$ (considered as CD4). The low-carbon supply chain should choose the reselling model. Because supply chains are more profitable when manufacturers work with platforms through a reselling model.

Proof

According to equations (7) and (10), the profit in both models can be expressed as the sum of the profit of each member. As follow:

$$\{\begin{array}{c}{J}_{MI}^A=J_M^A+J_I^A\\ J_{MI}^R=J_M^R+J_I^R\end{array}$$

(46)

According to equation (46), in the reselling model, the net discount for the supply chain is expressed as equation (47). Similarly, the expected profit of the supply chain as a whole in the reselling pattern is simplified to equation (48).

$$J_{MI}^A={\displaystyle \frac{b(p_I-c)}{\lambda }\left[(a+{\theta }_1{f}_{1A}^{*}(t)+{\theta }_2{f}_{2A}^{*}(t))+\mathrm{L}+{\displaystyle \frac{\lambda (G_0-L)}{\lambda +\sigma }}\right]-{\displaystyle \frac{1}{2\lambda }(k_{M}f{{}_{1A}^{*}}^2(t)+k_{I}f{{}_{2A}^{*}}^2(t))}}$$

(47)

$$J_{MI}^R={\displaystyle \frac{b(p_I-c)}{\lambda }\left[(a+{\theta }_1{f}_{1R}^{*}(t)+{\theta }_2{f}_{2R}^{*}(t))+\mathrm{L}+{\displaystyle \frac{\lambda (G_0-L)}{\lambda +\sigma }}\right]-{\displaystyle \frac{1}{2\lambda }(k_{M}f{{}_{1R}^{*}}^2(t)+k_{I}f{{}_{2R}^{*}}^2(t))}}$$

(48)

To better compare the relationship between the magnitude of expected supply chain profits, we constructed the function $J_{MI}(d_M,d_I)=J_{MI}^A-J_{MI}^R$ , as follows:

$$\begin{array}{rl}& J_{MI}(d_M,d_I)={\displaystyle \frac{b{\theta }_1(p_I-c)[w-(1-\phi )p_I]}{\lambda k_M}{g}_1(d_M)+{\displaystyle \frac{[w+(1-\phi )p_I-2c][w-(1-\phi )p_I]}{2\lambda k_M}}}\\ & \times g_1^2(d_M)+{\displaystyle \frac{[(1-\phi )p_I-w]b{\theta }_2(p_I-c)}{\lambda k_I}{g}_2(d_I)+{\displaystyle \frac{[(1+\phi )p_I-w][(1-\phi )p_I-w]}{2\lambda k_I}{g}_2^2(d_I)}}\end{array}$$

(49)

Further, to simplify equation (49), the four equations are set:

$$\{\begin{array}{c}{\mathrm{A}}_1={\displaystyle \frac{b{\theta }_1(p_I-c)[w-(1-\phi )p_I]}{\lambda k_M}}\\ {\mathrm{A}}_2={\displaystyle \frac{[(1-\phi )p_I-w]b{\theta }_2(p_I-c)}{\lambda k_I}}\\ B_1={\displaystyle \frac{[w+(1-\phi )p_I-2c][w-(1-\phi )p_I]}{2\lambda k_M}}\\ B_2={\displaystyle \frac{[(1+\phi )p_I-w][(1-\phi )p_I-w]}{2\lambda k_I}}\end{array}$$

(50)

Then, we let the two functions be denoted, respectively, as $g_1(d_M)={\theta }_1+\frac{\gamma b{e}^{\sigma d_M}}{\sigma +\lambda }$ and $g_2(d_I)={\theta }_2+\frac{\gamma b{e}^{\sigma d_I}}{\sigma +\lambda }$ . Thus, $J_{MI}(d_M,d_I)=J_{MI}^A-J_{MI}^R$ can be simplified as:

$$J_{MI}(d_M,d_I)={\mathrm{A}}_1{g}_1(d_M)+B_1{g}_1^2(d_M)+{\mathrm{A}}_2{g}_2(d_I)+B_2{g}_2^2(d_I)$$

(51)

Further, let $J_1={\mathrm{A}}_1{g}_1(d_M)+B_1{g}_1^2(d_M)$ and $J_2={\mathrm{A}}_2{g}_2(d_I)+B_2{g}_2^2(d_I)$ . There are two ideas for discussion based on the discriminant of roots.

When $J_2>\frac{A_1^2}{4B_1}$ , equation (51) has no real roots. However, if $w>(1-\phi )P_I>c$ , several patterns exist: $B_1>0$ , $A_1>0$ , $A_2<0$ , $B_2<0$ . This shows that equation (51) cannot have any real roots. If $(1-\phi )P_I>w>c$ , equation (51) must have real roots. Thus, taking the above analysis together, $J_2>\frac{A_1^2}{4B_1}$ is an impossible condition and equation (51) must have real roots.

When $J_2\le \frac{A_1^2}{4B_1}$ , $J_{MI}(d_M,d_I)=0$ has real roots. By the equation solving step, in the valid interval, $J_{MI}(d_M,d_I)=B_1{g}_1^2(d_M)+{\mathrm{A}}_1{g}_1(d_M)+B_2{g}_2^2(d_I)+{\mathrm{A}}_2{g}_2(d_I)$ has two real roots denoted by $d_1$ and $d_2$ .

$$\{\begin{array}{c}{d}_1={\displaystyle \frac{-A_1+\sqrt{A_1^2-4B_1{J}_2}}{2B_1}}\\ d_2=-{\displaystyle \frac{A_1+\sqrt{A_1^2-4B_1{J}_2}}{2B_1}}\end{array}.$$

(52)

Subsequently, two intervals are divided to further discuss the variability of supply chain profitability under the two patterns. In each interval, three ranges of delay time are discussed.

1. If $w>(1-\phi )P_I>c$ , the range of the coefficients in the function equation $J_2={\mathrm{A}}_2{g}_2(d_I)+B_2{g}_2^2(d_I)<0$ is denoted as $B_1>0$ , $A_1>0$ , $A_2<0$ , $B_2<0$ , then, $J_2<0$ , according to the law of distribution of the roots of the function, $d_2>d_1$ is drawn. Three ranges of values for the time delay are discussed.

2. If $g(d_M)>d_2$ , $d_M$ satisfies MI2, as shown in equation (54). $J_{MI}(d_M,d_I)>0$ is got. Then, the supply chain net discounted profit is higher in the agency selling, compared to the reselling pattern. The low-carbon supply chain should choose the agency selling.

   $$d_M\in MI2=\{d_M|\frac{1}{\sigma }[\ln (g_1(d_M)-{\theta }_1+d_2)+\ln (\lambda +\delta )-\ln rb]>0\}$$

   (54)

3. If $g(d_M)<d_1$ , $d_M$ satisfies MI3, as shown in equation (55). $J_{MI}(d_M,d_I)>0$ is got. Similar to the conclusion of (b), the low-carbon supply chain should choose the agency selling model.

   $$d_M\in MI3=\{d_M|\frac{1}{\sigma }[\ln (g_1(d_M)-{\theta }_1+d_1)+\ln (\lambda +\delta )-\ln rb]<0\}$$

   (55)

If $(1-\phi )P_I>w>c$ , there are $B_1<0$ , $A_1>0$ , $A_2>0$ , $B_2>0$ . Then $J_2>0$ , $d_2>d_1$ is still satisfied. Likewise, three ranges for the delay time are discussed.

If $d_1<g(d_M)<d_2$ , $d_M$ satisfies MI4, as shown in equation (56). $J_{MI}(d_M,d_I)>0$ is got. Therefore, from a profit perspective, the low-carbon supply chain should choose the reselling pattern.

$$d_M\in MI4=\{d_M|\frac{1}{\sigma }[\ln (g_1(d_M)-{\theta }_1+d_2)+\ln (\lambda +\delta )-\ln rb]>0\}\cup \{d_M|\frac{1}{\sigma }[\ln (g_1(d_M)-{\theta }_1+d_1)+\ln (\lambda +\delta )-\ln rb]<0\}$$

(56)

If $g(d_M)>d_2$ , $d_M$ satisfies MI5, as shown in equation (57). $J_{MI}(d_M,d_I)<0$ is got. Compared to reselling, the supply chain is less profitable when an agency selling pattern is chosen.

$$d_M\in MI5=\{d_M|\frac{1}{\sigma }[\ln (g_1(d_M)-{\theta }_1+d_2)+\ln (\lambda +\delta )-\ln rb]<0\}$$

(57)

If $g(d_M)<d_1$ , $d_M$ satisfies MI6, as shown in equation (58). $J_{MI}(d_M,d_I)<0$ is got. Therefore, the low-carbon supply chain should choose the agency selling pattern, which is more profitable than reselling.

$$d_M\in MI6=\{d_M|\frac{1}{\sigma }[\ln (g_1(d_M)-{\theta }_1+d_1)+\ln (\lambda +\delta )-\ln rb]>0\}$$

(58)

Simulation analysis

To analyze the results derived from the above mathematical model, this article uses Matlab 2016a software for numerical simulation. The relationship between LIE, LPE, LG, and supply chain profit in the two models is discussed.

Referring to Chen et al. ⁶³ and Basak et al., ⁶² we assume that the values of each parameter are $\lambda =0.1$ , $\delta =0.01$ , $b=3$ , $a=1000$ , $c=1$ , ${\theta }_1=2$ , ${\theta }_2=3$ , $\gamma =2$ , $\beta =2$ , $k_M=2$ , $k_I=2$ , $p_I=4$ , $w=2$ , $\phi =0.2$ , $G(0)=2$ .

First of all, according to the above basic parameters, take $d_M=1$ , $d_I=1$ , $t\in [0,200]$ . After that, the optimal trajectory of the product LG in the agency selling and the reselling is plotted, as shown in Figure 2.

Figure 2.

Trend of LG with time t. LG: low-carbon goodwill.

Following Figure 2, the finding shows that the LG is positively correlated with time t in both patterns. As time increases, LG will gradually be accumulated. However, LG is higher in the agency selling pattern compared to the reselling, and the comparative difference of LG in both modes first gradually increases and then slowly stabilizes as time increases.

Then, the relationship between delay time and decision variables is considered. Based on the above basic parameters, we plot LIE with $d_M$ , and LPE with delay time $d_I$ ( $d_I=1$ , $d_M\in [0,100]$ ), as in Figures 3 and 4, respectively.

Figure 3.

Trend of LIE with a delay time. LIE: low-carbon investment efforts

Figure 4.

Trend of LPE with a delay time. LPE: low-carbon promotional effort.

As can be seen in Figure 3, LIE is positively correlated with delay time $d_M$ , and the manufacturer pays more for LIE as delay time increases. However, the manufacturer's LIE moves more flatly in reselling than in agency selling. More importantly, for the same delay time, LIE of the agency selling is higher than the corresponding value of reselling. It shows that the increase in delay time can create a hindrance to manufacturers’ low-carbon technology development. When the delay time of low-carbon products is lower, the lower the low-carbon cost for manufacturers. More importantly, this effect is more pronounced in agency selling.

As seen in Figure 4, LPE is positively correlated with the delay time $d_I$ , and the platform retailer pays more for LPE as $d_I$ increases. Contrary to the trend for manufacturers, LPE of platform retailers has a flatter trend in agency selling than in reselling. Moreover, for the same delay time, LPE in reselling is higher than the corresponding value in agency selling. It shows that the increase in delay time creates negative sentiment for platform retailers’ willingness to promote LPE, and the manufacturer's willingness to promote LPE is stronger when the delay time of LPE is lower. Furthermore, this effect is more pronounced in the reselling mode.

Further, to compare and analyze the impact of commission rate on profit under agency selling and reselling patterns, we plotted the correlation between profit and commission rate for each member of the supply chain ( $\phi \in [0,1]$ ) in Figures 5 and 6.

Figure 5.

Analysis of the relationship between the manufacturer's profit and commission rate.

Figure 6.

Analysis of the relationship between profit and commission rate of platform retailers.

As seen in Figure 5, the manufacturer's profit is a parabola that goes up and then down, under the agency selling model. The manufacturer's profit is a horizontal straight line because under reselling, the profit of low-carbon products is not affected by the commission rate. However, the manufacturer's profit curve for both sales patterns intersects at a point whose horizontal coordinate is denoted as M. When the commission rate is less than M, the manufacturer prefers the agency selling model to reselling, because the manufacturer's profit under the agency selling is higher than the other. When the commission rate is greater than M, manufacturers can earn higher profits by working with the platform through reselling compared to agency selling. And as the commission rate increases, manufacturers have a more significant advantage in reselling.

As seen in Figure 6, under the agency selling model, the platform retailer's profit is positively correlated with the commission rate, and the rate of change of the curve gradually decreases as the commission rate increases. Similar to Figure 5, the profit function under the reselling model is independent of the commission rate, and whose profit seems a horizontal straight line. Nevertheless, the platform profit curve for both sales patterns also has a crossover point with the horizontal coordinate of N. When the commission rate is less than N, the platform retailer's profit is higher under reselling compared to the other one, conversely. When the commission rate is higher than N, the platform prefers the agency selling model to sell low-carbon products, because the profit he gets is higher than the corresponding value under reselling. And as the commission rate increases, the platform's advantage keeps more obvious.

Conclusion

In recent years, low-carbon development has driven firms to optimize their production and operation models and production processes in light of the energy issue and the “double carbon” target. The rapid growth of mobile communications has driven the boom in internet retail platforms and made online shopping one of the main forms of shopping for consumers. Consumers are the decision-makers on online platforms. The more touchpoints they have to access information, the more effective the marketing of low-carbon products becomes. From this perspective, platform sales attract upstream and downstream members of the supply chain. Firms are gradually gravitating toward the platform sales channel, which consists mainly of agency selling and reselling. However, if low-carbon products are to be recognized by the consumer market, it requires a low-carbon strategic awareness of the supply chain and a focus on production and promotion. At the production end, manufacturers strive to meet energy-saving and emission-reduction standards through LIE. At the sales end, platform retailers invest in LPE. They use technological tools such as big data, cloud computing, artificial intelligence, and machine learning to efficiently match their target customers. Unfortunately, in management practice, the utility value of a product's LIE and LPE is not easily perceived by firms due to the lag and uncertainty of market feedback. Generally, the consumer market evaluates low-carbon products mainly through LG, and positive LG provides assurance for quality and reputation. Therefore, distinguishing the results from previous studies as shown in Table 1. This study considers the delay effect from a dynamic perspective and explores the dynamic decision making in the production and marketing of low-carbon products in the supply chain.

Considering the dynamics of decision making, a differential game model based on delay effects is applied to the low-carbon supply chain. In addition, Hamilton functions are constructed to solve for the optimal LIE, LPE, and LG for agency selling and reselling patterns, as well as to explore the differences in decision making. Finally, numerical simulations are used to demonstrate the soundness of the proposed model. The model helps firms identify the best platform sales pattern and develop reasonable contractual terms covering sales price, commission system, LIE, LPE, etc. to maximize profitability in decentralized and centralized decision making in the low-carbon supply chain.

As a whole, a significant conclusion and a most counterintuitive one can be concluded.

The commission mechanism is a key measure of dynamic decision making for low-carbon products, while the commission rate is also an important reference for the choice of marketing pattern. In contrast to the findings of previous studies, this article derives specific thresholds for commissions. When the commission is below the threshold, the net discounted profit of manufacturers, LIE, and LG are higher in agency selling, compared to the reselling pattern. This illustrates that, within the threshold, manufacturers are more willing to contract with platform retailers in an agency selling pattern. Moreover, when commissions are above a threshold, platform retailers are more likely to choose the agency selling pattern. Because high commissions will incentivize platform retailers to invest more in promotions, which in turn boosts product sales and increases the platform retailer's net discounted profit.

There are differences in both decentralized and centralized decision making for the supply chain as the delay time varies. The delay time is positively related to the manufacturer's optimal LIE and the retailer's optimal LPE, respectively. If the delay time is longer, the corresponding LIE and LPE will be higher. However, the net discounted profit differs, along with the different delay times of LIE and LPE to LG. From a decentralized perspective, although improvements in LIE and LPE can lead to increased sales revenue. The longer the time between improvement and feedback, the higher the costs in terms of R&D investment, technology introduction, and promotion. Profits to manufacturers and retailers are reduced. They will only be willing to improve if the increase in revenue from the improvement is greater than the increase in cost. From a centralized decision, as the delay time varies, firms choose different sales models. When the delay time satisfies CD1 and CD4, the net discounted profit of the supply chain is more beneficial under the reselling pattern, which is the optimal decision pattern. Conversely, when the delay time satisfies CD2 and CD3, the agency selling pattern should be chosen.

The proposed model has some limitations. The negative impact of offline sales channels is not considered. The model can be further extended to take into account that both online and offline channels are running simultaneously. In addition, the time value of money and inflation may also be considered in the future to extend the current model. Finally, the impact of consumer behavior on decision making is not considered. Many low-carbon products are directly eliminated from the consumer market, resulting in irreversible product choices. Firms are unable to compensate for demand through LIE and LPE improvements. Therefore, future research also needs to pay more attention to behavioral factors such as consumer preferences.

Management inspiration

Management inspiration for manufacturers: When choosing a sales pattern, firm leadership needs to take advantage of the commission system and the laws between wholesale prices and LG. If they prefer agency sales, they need to ensure that the commission rate in the contract is kept within a threshold. Because within the threshold, firms increase investment in R&D can facilitate the accumulation of LG. And firms’ profits keep higher in the agency selling than the other. Conversely, if the commission rate is above the threshold, firm leadership should choose the reselling pattern. Furthermore, wholesale prices can create strategic advantages and economic benefits for manufacturers. And it is a key element in a contract setting for both manufacturers and platform retailers. The wholesale price of the product is set, following the relationship between the wholesale price and LG. The findings show that wholesale price is positively related to the optimal LIE. Therefore, when CW1 is satisfied, the firm's revenue is increased. Because the wholesale price of the low-carbon product is increased to stimulate market demand for the product. When CW2 is satisfied, manufacturers should not increase the wholesale price. Because the wholesale price is negatively related to the LG, increasing the wholesale price will instead reduce the LG.

Management inspiration for platform retailers: The aim of the marketing efforts invested by platform retailers is to increase the LG of the product. It stimulates both the consumer's desire to buy and increases the sales of the product. As soon as a certain level of information sharing is reached, the market will weaken the barriers of information asymmetry. Accordingly, price competition cannot create strategic advantages and economic benefits for platform retailers. Price wars will only cause both sides to lose, and the competitive dimension shifts to service strategy. ⁶² Platform retailers are also focusing more on the advantages of big data to carry out big data marketing, and it can recommend low-carbon products to consumers in an intelligent, accurate, and personalized way that matches consumers’ preferences. However, when platform retailers choose a sales model, they need to pay attention to two aspects. On the one hand, commission rates are used as the primary measure of platform revenue. If a platform works with manufacturers by agency selling, who should ensure that the commission rate is above the threshold? A high commission system can incentivize platform retailers to invest more in promotions, which in turn will boost product sales and increase the platform companies’ expected profits. On the other hand, a company cannot do without the ability to reinvent itself for long-term development. Even a platform retailer is no exception, it needs the ability to innovate its own marketing. While improvements in promotions by platform retailers can cause an increase in sales revenue. Gradually increasing investment in promotions will frustrate the incentive of innovative marketing for platforms. Nevertheless, improved marketing does not quickly lead to increased brand goodwill for low-carbon products and can easily cause them to undermine innovative marketing efforts. The promotion of low-carbon products will increasingly move in a positive direction, only if innovative marketing yields two results. One is a positive net profit after innovative marketing, and the other is that innovative marketing will occur in a reasonable time to positively influence the brand goodwill of the low-carbon product, ultimately resulting in additional profits for the platform retailer.

Management inspiration in low-carbon supply chains. As people pay more attention to the issue of climate change, many companies are trying to find a balance between energy saving and long-term development. However, relying on the power of individual companies is too thin, and the supply chain is one of the important ways to promote the development of a low-carbon economy and solve environmental pollution. ⁶⁶ However, at present, many aspects still encounter bottlenecks, such as production and sales, and the production and marketing of low-carbon products only through traditional methods are not effective in time and space. According to the data published by the China Bureau of Statistics, from 515.56 billion yuan in 2016 to 117.601 billion yuan in 2020, China's platform retail sales are increasing year by year at an average annual compound growth rate of 22.89%. ⁶⁷ As a result, the platform sales model has become the main battleground for low-carbon supply chain. ⁶⁸ According to the conclusion of the model in this article, the delay time is a key factor in the low-carbon supply chain. Business managers can set up appropriate decision options based on the conditions of the model. ⁶⁵ If the delay time is satisfied (following condition CD2 and condition CD3), the supply chain manager should set up an agency selling contract. Conversely, if the business managers want to cooperate through reselling model, they need to estimate the profitability and set reasonable contract terms according to conditions CD1 and CD4.