Research on Carbon Emission Reduction and Preservation Strategies for Fresh Agricultural Products Under Different Cost-Sharing Mechanisms

Abstract

This study investigates the dynamic carbon emission reduction strategies in a three-tier cold chain supply system consisting of producer, third-party logistics (TPL) provider, and retailer. Using differential game theory, it explores the emission reduction and preservation strategies of supply chain members under different cost-sharing mechanisms. This study finds that when the entity with higher marginal profits shares the costs, the TPL provider increases its efforts in emission reductions and research and development (R&D) investment. The producer and retailer are more willing to enhance their emission reduction efforts when sharing emission reduction costs, which increases carbon emission reduction and decreases overall emissions. Cost-sharing for preservation enhances the TPL provider’s R&D enthusiasm but does not affect the total emission reduction. When the marginal profits of the producer and retailer reach a certain level, sharing both emission reductions and preservation costs can simultaneously improve carbon reduction and preservation quality. An emission reduction cost-sharing contract can increase corporate profits, while a preservation cost-sharing contract further enhances profitability based on emission reduction cost sharing. Furthermore, the carbon emission reduction and preservation quality of fresh products gradually increase over time and eventually stabilize.

1. Introduction

Under the influence of factors such as energy depletion, climate anomalies, and environmental pollution, a low-carbon economy is being advocated globally, which has become the overarching trend in global economic development [1]. China’s carbon emissions rank among the highest in the world, and the pressure to reduce carbon emissions is gradually increasing. Agriculture has substantial carbon emissions; however, it is also the largest carbon sink system. Reducing carbon emissions in the production and transportation of fresh agricultural products aligns with national environmental policies and supports the long-term healthy development of the industry.

On the one hand, agricultural enterprises emit greenhouse gases during the production and processing of agricultural products, which negatively impacts the low-carbon economy. On the other hand, fresh food, due to its seasonal supply and perishable nature, presents significant challenges for the operation and management of fresh cold chains, with refrigeration and transportation being the largest sources of carbon emissions [2]. According to the China Development Network, the greenhouse gases emitted by the transportation industry account for as much as 10.4% of China’s total carbon emissions, with the carbon emissions from fruit and vegetable cold chains being substantial and increasing year by year. Therefore, the production and distribution stages of fresh agricultural products are the most carbon-intensive phases, and their effective operation depends on the efficient functioning of the logistics sector and appropriate carbon reduction investments. Meanwhile, as consumer awareness of low-carbon products increases, these products are gaining popularity. Consumer demand grows in proportion to the carbon reduction levels of the products, and retailer’s low-carbon promotions help communicate the carbon emissions information of products to consumers. Therefore, the carbon reduction in relation to products requires the joint efforts of the producer, third-party logistics provider (TPL), and retailer. The carbon reduction efforts of producer and TPL, along with the retailer’s low-carbon promotions, collectively influence the product’s overall carbon reductions. Based on the above background, this study will examine carbon emission reduction and preservation strategies in the production and transportation stages of fresh agricultural products.

From a theoretical perspective, research that simultaneously considers both carbon emission reductions and preservation quality in the fresh agricultural product supply chain remains relatively scarce. Furthermore, the existing studies on low-carbon supply chains rarely incorporate third-party logistics (TPL) into a three-tier cold chain supply system for fresh agricultural products. This study integrates carbon emission reductions, cold chain preservation quality, and the three-tier fresh agricultural product supply chain to analyze the impact of different cost-sharing contracts on emission reductions, cold chain preservation quality, and corporate profits, providing new insights for research on low-carbon fresh agricultural product supply chains.

From a practical perspective, carbon emissions in the fresh agricultural product supply chain involve not only the producer and retailer but also the TPL provider, which play a role in the emission process. Therefore, incorporating TPL into the low-carbon reduction process better reflects real-world conditions. Analyzing the impact of different cost-sharing contracts on emission reductions, cold chain preservation quality, and corporate profits provides decision-making support for fresh agricultural product supply chain enterprises in selecting appropriate cooperation contracts.

2. Literature Review

2.1. Research on Cold Chain Logistics for Fresh Agricultural Products

Many studies have already examined the involvement of third-party logistics (TPL) in fresh product cold chains [3,4,5,6]. Many scholars have considered the impact of freshness preservation efforts on the fresh cold chain system. For example, Cao Xiaoning et al. [7] studied the impact of supplier preservation efforts, considering the decay of freshness in fresh products, as well as supply chain management and coordination issues in the fresh food supply chain. Xiong Feng et al. [8] studied the optimal preservation efforts and coordination mechanisms between cooperatives and core enterprises. Wang Lei and Dan bin [9] identified that although the same preservation cost is paid, the lower the preservation cost, the lower the sensitivity coefficient of the preservation effect. The retailer will only choose a preservation strategy that focuses solely on controlling single losses. Liu Zheng et al. [10] studied the pricing decisions and system profit changes in a two-tier supply chain composed of suppliers and retailer, considering demand and freshness variations, fairness concerns, and preservation efforts. The research on low-carbon supply chains is still emerging, with carbon emissions in fresh cold chains becoming a key focus of study. Bai et al. [11] studied the collaborative optimization problem of a producer-retailer two-tier supply chain under carbon cap and trade policies, considering fixed decay rates and varying prices, promotional efforts, and carbon reduction levels. Ma et al. [12] assumed that the demand for fresh products is affected by freshness and price and explored the management and coordination issues of a three-tier cold chain supply chain with TPL under carbon quota and trading strategies. Tiwari et al. [13] argued that the carbon emissions of fresh agricultural products during storage are related to inventory and energy consumption, and they studied the carbon-reduction strategies for storing fresh agricultural products, where TPL preservation efforts and carbon reduction levels are endogenous. Ma Xueli et al. [6] investigated the operational decision-making in a three-echelon cold chain system comprising suppliers, third-party logistics (TPL) service providers, and retailers for perishable fresh products characterized by variable shelf lives. The study specifically examined preservation strategies, carbon emission reduction measures, and pricing mechanisms under the scenario where TPL service providers undertake dual responsibilities of product preservation and low-carbon operations.

2.2. Research on Cost-Sharing Contracts in the Fresh Agricultural Supply Chain

In the fresh agricultural supply chain, enterprises typically adopt various coordination contracts to incentivize the retailer to increase order quantities or encourage the sup-plier to provide higher-quality products. These contract mechanisms vary widely, with common types including quantity or price discount contracts, revenue-sharing contracts, and cost-sharing contracts. Feng Ying et al. [14] conducted a Stackelberg game analysis within a VMCL model for a fresh agricultural supply chain consisting of a single producer and a single retailer. Their study revealed that an appropriate combination of contracts can effectively achieve supply chain coordination. Zhang Qinyi et al. [15], considering the perishable nature of agricultural products, developed a freshness utility function that closely links consumer demand with product freshness. Based on this, they examined the decision-making model of the fresh agricultural supply chain and proposed a revenue-sharing contract to effectively enhance product freshness and achieve supply chain coordination. Chen Liuxin et al. [16] established a fresh agricultural supply chain involving a supplier, a TPL, and a retailer. They proposed a logistics cost-sharing contract and con-ducted an in-depth analysis of optimal decisions and profits. Zou Xiao et al. [17] focused on a two-tier fresh agricultural supply chain consisting of a supplier and a retailer. By comparing different decision-making models, they designed a "bilateral revenue-sharing and cost-sharing" contract aimed at achieving supply chain coordination. Qiu Hui et al. [18] examined the coordination of interests in an agricultural supply chain composed of a supermarket and farmers under risk-neutral conditions. They analyzed centralized decision-making, decentralized decision-making without contracts, and different coordination contracts, conducting simulations to validate their findings. Ghosh and Shah [19] analyzed the impact of two types of cost-sharing contracts on green manufacturing costs and consumer environmental preferences in a green supply chain. Song Huan [20] compared de-centralized and centralized decision-making as well as different cost-sharing contracts in terms of supply chain enterprises’ profit-sharing behaviors and outcomes. The study found that cost-sharing contracts facilitated Pareto improvements. Hu Jinsong [21] applied continuous dynamic programming theory to analyze supply chain enterprise decisions under the following four scenarios: centralized decision-making, decentralized decision-making, producer cost sharing, and supplier cost sharing. The study identified the impact of different cost-sharing contracts on the supply chain.

Unlike simple cost-sharing contracts, some scholars have incorporated other contract types into cost-sharing contracts to study supply chain coordination under multiple contractual mechanisms. Zhu Guiju [22] analyzed the profit maximization decisions of food supply chain enterprises under decentralized and centralized decision-making scenarios, ultimately deriving a "bilateral cost-sharing and transfer payment hybrid contract" to maximize overall supply chain profitability. He [23] studied advertising cooperation strategies and designed transfer payment and advertising cost subsidy contracts to achieve supply chain coordination. Tang Run [24] analyzed the preservation efforts and optimal profits of supply chain enterprises under both decentralized and centralized decision-making scenarios and derived revenue-sharing, cost sharing, and price discount con-tracts to coordinate the supply chain. Yue Liuqing et al. [25] examined decision-making in a retailer-led fresh supply chain and designed wholesale price and revenue-sharing contracts. Chen et al. [26] constructed a two-tier supply chain model consisting of a producer and a retailer, designing wholesale price contracts, revenue-sharing contracts, and two-part pricing to mitigate dual-channel conflicts. Wu [27] designed a cost-sharing and wholesale price contract that ensures Pareto improvements for all supply chain members.

The aforementioned studies build upon cost-sharing contracts by incorporating transfer payments, revenue sharing, or price discount contracts, forming more complex hybrid contracts. Some scholars have further analyzed issues such as risk preferences and information asymmetry within cost-sharing contracts, designing option contracts and buyback contracts to examine supply chain coordination under hybrid contract models. Whether a standalone cost-sharing contract or a hybrid contract incorporating transfer payments, revenue sharing, or price discounts, all are fundamentally based on cost-sharing mechanisms, analyzing how fresh agricultural supply chain members maximize profits by distributing costs. This highlights the critical role of cost-sharing contracts in enhancing overall supply chain competitiveness. Most of the existing research focuses on contracts between two parties, with relatively few studies linking producers, third-party logistics providers (TPL), and retailers in the fresh agricultural supply chain to investigate a jointly established cost-sharing contract among the three entities. In practice, supply chain participants are closely interconnected, and constructing a cost-sharing contract among these three parties can further optimize overall supply chain profitability.

In practice, the realization of carbon emission reduction effects and technological investment behaviors is a time-varying process. Therefore, studying the dynamic carbon emission reduction problem in the fresh agricultural cold chain supply chain has practical significance. Zhao Daozhi et al. [28,29] not only investigated the variation curve of product carbon emission reductions but also explored the influence mechanism of the retailer’s low-carbon promotion on dynamic carbon emission reductions. Numerous scholars have explored issues related to dynamic carbon emission reductions [30,31].

Based on the existing literature, (1) most studies on cost-sharing contracts focus on contracts formed between two entities, often using a cost-sharing contract as the foundation. However, fewer studies have considered the simultaneous involvement of a producer, a TPL provider, and a retailer in forming a cost-sharing contract. (2) The research on carbon reduction efforts and preservation efforts has evolved from static to dynamic perspectives, shifting from independent decision-making to joint emission reduction strategies. While studies on preservation efforts in fresh supply chains have mainly focused on fresh food or cold chain logistics, research on carbon reduction efforts remains insufficient. Few studies have integrated both preservation and carbon reduction efforts within a three-tier fresh agricultural supply chain to analyze firms’ emission reductions and preservation decisions.

Therefore, this study integrates preservation and emission reduction efforts while analyzing the relationship between emission reduction levels and preservation quality, contributing to the literature on supply chain coordination. In summary, existing research on fresh agricultural supply chains either examines the impact of a TPL provider’s preservation efforts on supply chain profits or focuses solely on an individual firm’s carbon reduction behavior. Few studies have considered the impact of a TPL provider’s carbon reduction behavior on the fresh agricultural supply chain. Additionally, limited research studies have explored preservation strategies under carbon emission constraints, despite the crucial role of preservation in the fresh agricultural supply chain. Therefore, this study focuses on an outsourced fresh agricultural supply chain, where a third-party logistics provider offers cold chain logistics services.

3. Research Methodology and Model Construction

This study focuses on a three-tier fresh agricultural product cold chain supply system composed of a producer, third-party logistics (TPL) provider responsible for fresh product transportation, and retailer. The cold chain system is illustrated in Figure 1. The producer plays a dominant role, overseeing the production of fresh agricultural products. To improve distribution quality and reduce transportation losses, retailer outsources the logistics of fresh products to TPL providers. The TPL provider then transports the products via the cold chain from the producer to the retailer, who is responsible for reselling them to the end market.

This study is based on decentralized decision-making without cost sharing and examines corporate decisions under four scenarios: producer and retailer sharing emission reduction costs, and producer and retailer further sharing preservation costs based on emission reduction cost sharing. The objective is to explore the impact of different cost-sharing contracts on carbon emission reductions, preservation quality, and corporate profits in the fresh agricultural product supply chain.

The assumptions in this paper are as follows:

(1) It is assumed that the unit product profits of the producer, TPL provider, and retailer in the fresh supply chain system are ${\pi }_P$, ${\pi }_T$, and ${\pi }_R$, respectively.

(2) The carbon reduction costs of the producer is a quadratic function of the reduction effort level, and the carbon reduction costs increase at an accelerating rate as the reduction effort level increases. $C_P\left(E_P\right(t\left)\right)$ represents the producer’s carbon reduction costs at time t, and it is assumed that the producer’s carbon reduction costs at time t are as follows:

$$C_P\left(E_P\left(t\right)\right)={\displaystyle \frac{1}{2}}{k}_p{E_P}^2\left(t\right)$$

where $k_P>0$ is the cost coefficient of carbon reductions for the producer and $E_P\left(t\right)$ is the carbon reduction effort level of the producer at time t.

(3) TPL provider’s carbon reduction costs are also influenced by its own reduction effort level and increase at an accelerating rate as the TPL provider’s carbon reduction effort level increases. $C_T(E_T\left(t\right)$) represents TPL provider’s carbon reduction costs at time t, and it is assumed that TPL provider’s carbon reduction costs at time t can be expressed as follows:

$$C_T\left(E_T\left(t\right)\right)={\displaystyle \frac{1}{2}}{k}_T{E_T}^2\left(t\right)$$

where $k_T>0$ is the cost coefficient of carbon reductions for the TPL provider and $E_T\left(t\right)$ is the carbon reduction effort level of TPL provider at time t.

(4) Similar to the reduction effort and costs, TPL provider’s research and development (R&D) costs are a convex function of the investment level in cold chain preservation technology. The higher the level of R&D investment in cold chain preservation technology, the higher the costs of further improving the cold chain preservation technology. $C_F(E_F\left(t\right)$) represents the R&D costs of TPL provider’s cold chain preservation technology at time t, and it is assumed that the R&D costs at time t are expressed as follows:

$$C_F\left(E_F\left(t\right)\right)={\displaystyle \frac{1}{2}}{k}_F{E_F}^2\left(t\right)$$

where $k_F>0$ is the cost coefficient of TPL’s cold chain preservation technology R&D investment, and $E_F\left(t\right)$ is the cold chain preservation technology R&D investment level of TPL provider at time t.

(5) The retailer’s low-carbon promotion costs increase with the increase in the level of low-carbon promotional efforts. $C_R\left(E_R\left(t\right)\right)$ represents the retailer’s low-carbon promotion costs at time t, and it is assumed that the retailer’s low-carbon promotion costs at time t are as follows:

$$C_R\left(E_R\left(t\right)\right)={\displaystyle \frac{1}{2}}{k}_R{E_R}^2\left(t\right)$$

where $k_R>0$ is the cost coefficient of the retailer’s low-carbon promotion and $E_R\left(t\right)$ is the level of promotional and marketing efforts for low-carbon products by the retailer at time t.

(6) Since carbon emissions in the agricultural supply chain primarily come from production and transportation, to explore the impact of the carbon reduction behavior of producer and TPL provider on the operation of the low-carbon fresh agricultural product supply chain, it is assumed that the carbon reduction efforts of producer and TPL provider are key factors in determining the final carbon reduction amount of the product. The differential equation for the change in the carbon reduction amount of fresh agricultural products is as follows:

$$\dot{\tau }\left(t\right)={\beta }_P{E}_P\left(t\right)+{\beta }_T{E}_T\left(t\right)-\lambda \tau \left(t\right)$$

(1)

$\tau \left(t\right)$ is the carbon reduction amount of the product at time t, with an initial carbon reduction amount of $\mathrm{\tau }\left(0\right)={\mathrm{\tau }}_0\ge 0$ and $\lambda$ represents the natural decay rate of carbon reduction in relation to fresh agricultural products over time.

(7) Cold chain logistics are responsible for the transportation, distribution, and other tasks involved in moving fresh agricultural products from production sites to retailer and then to consumers, meeting the consumer demand for fresh agricultural products and cold chain logistics quality. Therefore, the differential equation for the cold chain preservation quality of fresh agricultural products, which is influenced by the TPL provider’s cold chain preservation technology R&D investment level, is set as follows:

$$\dot{\chi }\left(t\right)={\beta }_{\mathrm{F}}{E}_F\left(t\right)-\xi \chi \left(t\right)$$

(2)

$\chi \left(t\right)$ is the cold chain preservation quality of the product at time t, with an initial preservation quality of $\chi \left(0\right)={\chi }_0\ge 0$. $\xi$ represents the natural decay rate of cold chain preservation quality of fresh agricultural products over time.

(8) It is assumed that carbon reductions, cold chain preservation quality, and low-carbon promotion jointly influence the market demand for fresh agricultural products at time t, and they are linearly related, as follows:

$$D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)=D_0+\eta \tau \left(t\right)+\theta \chi \left(t\right)+\delta E_R\left(t\right)$$

(3)

$\eta$ represents the demand sensitivity to the carbon reduction level, i.e., the environmental awareness of consumers; $\theta$ represents the demand sensitivity to freshness, i.e., consumer preference for freshness; and $\delta$ represents the sensitivity of consumers to low-carbon promotion.

(9) Retailer’s promotion of low-carbon products can increase consumer purchasing behavior, making consumers more willing to buy. As a result, producer can gain greater benefits. To encourage retailer to engage in low-carbon promotion, it is assumed that producer will bear a certain proportion of the promotional costs, denoted as ${\varphi }_R\left(t\right)$, where $0<{\varphi }_R\left(t\right)<1$. As consumer demand for environmentally friendly and sustainable products increases, the producer that reduces its carbon footprint to meet green market demands may gain a larger market share. By jointly assuming responsibility for carbon reductions, the producer can optimize the supply chain, reduces carbon emissions in the logistics process, and enhances overall supply chain efficiency and responsiveness. Therefore, it is assumed that producer is willing to share part of the TPL provider’s emission reduction costs, denoted as ${\varphi }_T\left(t\right)$, where $0<{\varphi }_T\left(t\right)<1$. Furthermore, TPL provider’s cold chain preservation technology affects the freshness of fresh agricultural products. Since consumers are highly sensitive to freshness, their willingness to purchase increases for products with higher preservation quality. Given this, the producer may have an incentive to share the R&D costs of TPL provider’s cold chain preservation technology. It is assumed that the producer’s cost-sharing ratio is ${\varphi }_F\left(t\right)$, where $0<{\varphi }_F\left(t\right)<1$.

(10) As the level of product carbon reductions increases and consumers become more willing to adopt low-carbon consumption, the retailer can achieve higher profits. To encourage the producer and TPL provider to increase their carbon reduction investments, it is assumed that the retailer is willing to share a certain proportion of the producer’s and TPL provider’s carbon reduction costs, denoted as ${\epsilon }_P\left(t\right)$,${\epsilon }_T\left(t\right)$, respectively, where $0<{\epsilon }_P\left(t\right)<1$, $0<{\epsilon }_T\left(t\right)<1$. Additionally, due to the unique nature of fresh agricultural products, consumer preferences for freshness significantly influence their purchasing decisions. To incentivize the TPL provider to increase its investment in preservation technology R&D, it is assumed that the retailer is willing to share the R&D costs of the TPL provider’s preservation technology, denoted as ${\epsilon }_F\left(t\right)$, where $0<{\epsilon }_F\left(t\right)<1$.

Table 1. Description of notations.

Notation	Description
$E_P\left(t\right)$	The carbon reduction effort level of the producer at time t.
$E_T\left(t\right){,E}_F\left(t\right)$	The carbon reduction effort level of the TPL provider at time t and the cold chain preservation technology R&D investment level.
$E_R\left(t\right)$	The promotional and marketing effort level of the retailer for low-carbon products at time t.
$C_P\left(E_P\right(t\left)\right)$	The carbon reduction costs of the producer at time t.
$C_T(E_T\left(t\right)$$), C_F(E_F\left(t\right)$)	The carbon reduction costs of TPL provider at time t and the cold chain preservation technology R&D costs.
$C_R\left(E_R\left(t\right)\right)$	The low-carbon promotion costs of the retailer at time t.
$k_P,k_T$	The cost coefficients for the carbon reduction costs of the producer and TPL provider, where $k_P>0$ and $k_T>0$.
$k_F$	The cost coefficient for the TPL provider’s cold chain preservation technology R&D investment, where $k_F>0$.
$k_R$	The cost coefficient for the retailer’s low-carbon promotion, where $k_R>0$.
$\tau \left(t\right)$	The carbon reduction amount of fresh agricultural products at time t.
$\chi \left(t\right)$	The cold chain preservation quality of fresh agricultural products at time t.
${\beta }_P$$,{\beta }_T$	The influence coefficients of the producer’s and TPL provider’s carbon reduction efforts on the product’s carbon reduction amount, where $0 < 0<{\beta }_P<1$ and $0<{\beta }_T<1$.
${\beta }_F$	The influence coefficient of TPL provider’s cold chain preservation technology R&D investment on the product’s cold chain preservation quality, where $0<{\beta }_F<1$.
$\lambda$$,\xi$	The decay coefficients for the product’s carbon reduction amount and cold chain preservation quality, where $\lambda >0$$\mathrm{and} \xi >0$.
$D(\tau \left(t\right),\chi \left(t\right),E_R(t\left)\right)$	The market demand for fresh agricultural products at time t.
$\eta$	The demand sensitivity to the carbon reduction level, i.e., consumer environmental awareness.
$\theta$	The demand sensitivity to freshness, i.e., consumer preference for freshness.
$\delta$	The sensitivity of consumers to low-carbon promotion.
${\varphi }_R\left(t\right)$$,{\varphi }_T\left(t\right)$$,{\varphi }_F\left(t\right)$	The proportion coefficients of the costs borne by the producer at time t for the retailer’s promotional costs, the TPL provider’s emission reduction costs, and the TPL provider’s technology R&D investment, where $0<{\varphi }_i\left(t\right)<1$$,i=R,T,F$
${\epsilon }_P\left(t\right)$$,{\epsilon }_T\left(t\right)$$,{\epsilon }_F\left(t\right)$	The proportion coefficients of the costs borne by the retailer at time t for the producer’s emission reduction costs, TPL provider’s emission reduction costs, and technology R&D costs, where $0<{\epsilon }_i\left(t\right)<1$$,i=R,T,F$
$D_0$	The market demand for fresh agricultural products at the initial time point.
${\pi }_P$$,{\pi }_T$$,{\pi }_R$	The marginal profit obtained from the production, sales, and transportation of a unit product by the producer, retailer, and TPL provider.
Superscript N	No cost sharing.
Superscript NP	The producer bearing the emission reduction costs.
Superscript NR	The retailer bearing the emission reduction costs.
Superscript NPR	The producer shares emission reduction costs while the retailer bears freshness-keeping costs
Superscript NRP	The retailer bearing the emission reduction costs and the producer bearing the preservation costs.

lists the notations used in this paper and their explanations.

4. Decision Model Without Cost Sharing

In the case of no cost sharing, the producer, TPL provider, and retailer have equal status, forming a Nash differential game. At this point, the producer, TPL provider, and retailer only need to make decisions aimed at maximizing their own profits. The objective function under no cost sharing is as follows: ρ represents the discount factor, and under infinite time, the decision objectives of the three parties are as follows:

$$J_P\left(\tau ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left[{\pi }_{P}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_P{E_P}^2\left(t\right)\right]dt$$

(4)

$$J_T\left(\tau ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left[{\pi }_{T}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_T{E_T}^2\left(t\right)-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2\left(t\right)\right]dt$$

(5)

$$J_R\left(\tau ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left\{{\pi }_{R}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_R{E_R}^2\left(t\right)\right\}dt$$

(6)

The solution is obtained using the HJB equation. Assuming that the parameters in the model are independent of time, for convenience, t is omitted in the subsequent calculations. The results under decentralized decision-making are given in Proposition 1.

Proposition 1.

Under non-cooperative control, the results of the decentralized decision model are as follows:

(1) The optimal decisions of the producer, TPL provider, and retailer are as follows:

$$E_T^{N*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }\pi T}{k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}},E_F^{N*}={\displaystyle \frac{{\beta }_F\xi {\pi }_T}{k_T\left(\theta +\mathrm{\rho }\right)}},E_P^{N*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P{\pi }_P}{(\lambda +\rho )k_P}},E_R^N={\displaystyle \frac{\delta {\pi }_R}{k_R}}$$

(2) The optimal trajectory of carbon reductions in relation to fresh agricultural products is as follows:

$$\begin{array}{c}{\tau }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{A}}^{\mathrm{N}}}{\lambda }}+\left({\tau }_0-{\displaystyle \frac{{\mathrm{A}}^{\mathrm{N}}}{\lambda }}\right)e^{-\lambda t},{\chi }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{A}}^{\mathrm{N}}}{\theta }}+\left({\chi }_0-{\displaystyle \frac{{\mathrm{B}}^{\mathrm{N}}}{\theta }}\right)e^{-\theta t}\hfill \\ A^N={\displaystyle \frac{{{\beta }_P}^2\mathrm{\eta }{k}_T{\pi }_P+{{\beta }_T}^2\eta k_P{\pi }_T}{k_P{k}_T\left(\lambda +\rho \right)}},B^N={\displaystyle \frac{{{\beta }_F}^2\theta {\pi }_T}{k_F\left(\theta +\rho \right)}}\hfill \end{array}$$

(3) The optimal profits of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}\hfill {\mathrm{J}}_P^{N*}(\tau ,\chi ,t)=e^{-\rho t}(a_{13}^{*}\tau +{\mathrm{b}}_{13}^{*}\chi +c_{13}^{*})\hfill \\ \hfill {\mathrm{J}}_T^{N*}(\tau ,\chi ,t)=e^{-\rho t}(a_{14}^{*}\tau +{\mathrm{b}}_{14}^{*}\chi +c_{14}^{*})\hfill \\ \hfill {\mathrm{J}}_R^{N*}(\tau ,\chi ,t)=e^{-\rho t}(a_{15}^{*}\tau +{\mathrm{b}}_{15}^{*}\chi +c_{15}^{*})\hfill \\ \mathrm{where}, a_{13}^{*}={\displaystyle \frac{{\pi }_P\eta }{(\lambda +\rho )}},b_{13}^{*}={\displaystyle \frac{{\pi }_P\theta }{(\xi +\rho )}},c_{13}^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_P{D}_0+{\displaystyle \frac{{\pi }_P\delta {\pi }_R}{k_R}}+{\displaystyle \frac{{a_{13}}^2{{\beta }_P}^2}{2k_P}}+{\displaystyle \frac{a_{13}{a}_{14}{{\beta }_T}^2}{k_T}}+{\displaystyle \frac{b_{13}{b}_{14}{{\beta }_F}^2}{k_F}}\right]\\ a_{14}^{*}={\displaystyle \frac{{\pi }_T\eta }{(\lambda +\rho )}},b_{14}^{*}={\displaystyle \frac{{\pi }_T\theta }{(\xi +\rho )}},c_{14}^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_T{D}_0+{\displaystyle \frac{\delta {\pi }_T{\pi }_R}{k_R}}+{\displaystyle \frac{a_{13}{a}_{14}{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{{a_{14}}^2{{\beta }_T}^2}{2k_T}}+{\displaystyle \frac{{b_{14}}^2{{\beta }_F}^2}{2k_F}}\right]\\ a_{15}^{*}={\displaystyle \frac{{\pi }_R\eta }{(\lambda +\rho )}},b_{15}^{*}={\displaystyle \frac{{\pi }_R\theta }{(\xi +\rho )}},c_{15}^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_R{D}_0+{\displaystyle \frac{\delta {{\pi }_R}^2}{k_R}}-{\displaystyle \frac{{(\delta {\pi }_R)}^2}{2k_R}}+{\displaystyle \frac{a_{13}{a}_{15}{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{a_{14}{a}_{15}{{\beta }_T}^2}{k_T}}+{\displaystyle \frac{b_{14}{b}_{15}{{\beta }_F}^2}{k_F}}\right]\end{array}$$

Proof of Proposition 1.

The optimal decision problem for the retailer at time t is represented as follows:

$$J_R^{N*}\left(\tau ,\chi ,t\right)=e^{-\rho t}\underset{E_R}{\max }{{\displaystyle \int }}_0^{\infty }{e}^{-\rho \left(s-t\right)}\left\{{\pi }_{R}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_R{E_R}^2\left(t\right)\right\}ds$$

(7)

Let

$$V_R^N\left(\tau ,\chi \right)=\underset{E_R}{\max }{{\displaystyle \int }}_0^{\infty }{e}^{-\rho \left(s-t\right)}\left\{{\pi }_{R}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_R{E_R}^2\left(t\right)\right\}ds$$

(8)

Thus, the result is: $U_R^{N^{*}}(\tau ,\chi ,t)=e^{-\rho t}{V}_R^N(\tau ,\chi )$.

For all $\tau \ge 0$, the HJB equation is satisfied as follows:

$$\rho V_R^N\left(\tau ,\chi \right)=\underset{E_R}{max}\left\{{\pi }_{R}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_R{E_R}^2\left(t\right)+{V^{N^{\prime }}}_{R\tau }\dot{\tau }\left(t\right)+{V^{N^{\prime }}}_{R\chi }\dot{\chi }\left(t\right)\right\}$$

(9)

Substitute Equations (1), (2), and (8) into Equation (9) to obtain the following:

$$\rho V_R^N\left(\tau ,\chi \right)=\underset{E_R}{max}\left\{{\pi }_R\left(D_0+\eta \tau +\theta \chi +\delta E_R\right)-{\displaystyle \frac{1}{2}}{k}_R{E_R}^2+{V^{N^{\prime }}}_{R\tau }\left({\beta }_P{E}_P+{\beta }_T{E}_T-\lambda \tau \right)+{V^{N^{\prime }}}_{R\chi }\left({\beta }_{\mathrm{F}}{E}_F-\xi \chi \right)\right\}$$

(10)

Since $\rho {V^{″}}_R\left(\tau \right)<0$, an optimal solution exists. By taking the partial derivative of $E_R$, we obtain the following:

$$E_R^N={\displaystyle \frac{\delta {\pi }_R}{k_R}}$$

(11)

The optimal decision problem for TPL at time t is represented as follows:

$$J_T^{N*}\left(\tau ,\chi ,t\right)=e^{-\rho t}\underset{E_T}{\max }{{\displaystyle \int }}_0^{\infty }{e}^{-\rho \left(s-t\right)}\left[{\pi }_{T}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_T{E_T}^2-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2\right]ds$$

(12)

Let

$$V_T^N\left(\tau ,\chi ,t\right)=\underset{E_T}{\max }{{\displaystyle \int }}_0^{\infty }{e}^{-\rho \left(s-t\right)}\left[{\pi }_{T}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_T{E_T}^2-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2\right]ds$$

(13)

Thus, the optimal decision problem for TPL can be transformed as follows:

$$J_T^{N*}\left(\tau ,\chi ,t\right)=e^{-\rho t}{V}_T^N\left(\tau ,\chi \right)$$

(14)

For any $\tau \ge 0$, the HJB equation is satisfied as follows:

$$\rho V_T^N\left(\tau ,\chi \right)=\underset{E_T,E_F}{\max }\left[{\pi }_{T}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_T{E_T}^2-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2+{V^{N^{\prime }}}_{T\tau }\dot{\tau }\left(t\right)+{V^{N^{\prime }}}_{T\chi }\dot{\chi }\left(t\right)\right]$$

(15)

Substitute Equations (1), (2), and (13) into Equation (15) to obtain the following:

$$\rho V_T^N\left(\tau ,\chi \right)=\underset{E_T,E_F}{\max }\left[\begin{array}{l}{\pi }_T\left(D_0+\eta \tau +\theta \chi +\delta E_R\right)-{\displaystyle \frac{1}{2}}{k}_T{E_T}^2-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2\\ +{V^{N^{\prime }}}_{T\tau }\left({\beta }_P{E}_P+{\beta }_T{E}_T-\lambda \tau \right)+{V^{N^{\prime }}}_{T\chi }\left({\beta }_{\mathrm{F}}{E}_F-\xi \chi \right)\end{array}\right]$$

(16)

Substitute Equation (11) into Equation (16) to obtain the following:

$$\rho V_T^N\left(\tau ,\chi \right)=\underset{E_T,E_F}{\max }\left\{{\pi }_T\left[D_0+\eta \tau ++\theta \chi +\delta {\displaystyle \frac{\delta {\pi }_R}{k_R}}\right]-{\displaystyle \frac{1}{2}}{k}_T{E_T}^2-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2+{V^{N^{\prime }}}_{T\tau }\left({\beta }_P{E}_P+{\beta }_T{E}_T-\lambda \tau \right)+{V^{N^{\prime }}}_{T\chi }\left({\beta }_{\mathrm{F}}{E}_F-\xi \chi \right)\right\}$$

(17)

Take the partial derivatives of ${\mathrm{E}}_T$, ${\mathrm{E}}_F$, set the derivatives to 0, and solve as follows:

$$E_T^N={\displaystyle \frac{{V^{N^{\prime }}}_{T\tau }{\beta }_T}{k_T}}$$

(18)

$$E_F^N={\displaystyle \frac{{V^{N^{\prime }}}_{T\chi }\beta F}{k_F}}$$

(19)

The optimal decision problem for the producer at time t is represented as follows:

$$J_P^{N*}\left(\tau ,\chi ,t\right)=e^{-\rho t}\underset{E_P,E_S}{\max }{{\displaystyle \int }}_0^{\infty }{e}^{-\rho \left(s-t\right)}\left[{\pi }_{P}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_P{E_P}^2\right]ds$$

(20)

$$V_P^N\left(\tau ,\chi \right)=\underset{E_P,E_S}{\max }{{\displaystyle \int }}_0^{\infty }{e}^{-\rho \left(s-t\right)}\left[{\pi }_{P}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_P{E_P}^2\right]ds$$

(21)

Thus, the optimal decision problem for the producer can be transformed as follows:

$$J_P^{N*}\left(\tau ,\chi ,t\right)=e^{-\rho t}{V}_T^N\left(\tau ,\chi \right)$$

(22)

Satisfy the HJB equation as follows:

$$\rho V_P^N\left(\tau ,\chi \right)=\underset{E_P,E_S}{\max }\left[{\pi }_{P}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_P{E_P}^2+{V^{N^{\prime }}}_{p\tau }\dot{\tau }\left(t\right)+{V^{N^{\prime }}}_{p\chi }\dot{\chi }\left(t\right)\right]$$

(23)

Substitute Equations (1), (2), and (21) into Equation (23) and simplify to obtain the following:

$$\rho V_P^N\left(\tau \right)=\underset{E_P,E_S}{\max }\left[\pi P\left(D_0+\eta \tau +\theta \chi +\delta E_R\right)-{\displaystyle \frac{1}{2}}{k}_P{E_P}^2+{V^{N^{\prime }}}_{p\tau }\left({\beta }_P{E}_P+{\beta }_T{E}_T-\lambda \tau \right)+{V^{N^{\prime }}}_{p\chi }\left({\beta }_{\mathrm{F}}{E}_F-\xi \chi \right)\right]$$

(24)

Substitute Equations (14), (18), and (19) into Equation (24) to obtain the following:

$$\rho V_P^N\left(\tau \right)=\underset{E_P,E_S}{\max }\left[{\pi }_P\left(D_0+\eta \tau +\theta \chi +\delta {\displaystyle \frac{\delta {\pi }_R}{k_R}}\right)-{\displaystyle \frac{1}{2}}{k}_P{E_P}^2+{V^{N^{\prime }}}_{p\tau }\left({\beta }_P{E}_P+{\beta }_T{\displaystyle \frac{{V^{N^{\prime }}}_{T\tau }{\beta }_T}{k_T}}-\lambda \tau \right)+{V^{N^{\prime }}}_{p\chi }\left({\beta }_{\mathrm{F}}{\displaystyle \frac{{V^{N^{\prime }}}_{T\chi }\beta F}{k_F}}-\xi \chi \right)\right]$$

(25)

Simultaneously, take the partial derivatives of $E_P$, $\varphi$, set the derivatives to 0, and solve as follows:

$$E_P^N={\displaystyle \frac{{V^{N^{\prime }}}_{P\tau }{\beta }_P}{k_P}}$$

(26)

Substitute Equations (14), (18), (19) and (26) into Equations (13), (17), and (24) to obtain the following:

$$\rho V_P^N\left(\tau \right)=\underset{E_P,E_S}{\max }\left\{\pi P\left(D_0+\eta \tau +\theta \chi +{\displaystyle \frac{{\delta }^2{\pi }_R}{k_R}}\right)-{\displaystyle \frac{1}{2}}{k}_P{({\displaystyle \frac{{V^{N^{\prime }}}_{P\tau }{\beta }_P}{k_P}})}^2+{V^{N^{\prime }}}_{P\tau }\left({\beta }_P{\displaystyle \frac{{V^{N^{\prime }}}_{P\tau }{\beta }_P}{k_P}}+{\beta }_T{\displaystyle \frac{{V^{N^{\prime }}}_{T\tau }{\beta }_T}{k_T}}-\lambda \tau \right)+{V^{N^{\prime }}}_{P\chi }\left({\beta }_{\mathrm{F}}{\displaystyle \frac{{V^{N^{\prime }}}_{T\chi }{\beta }_F}{k_F}}-\xi \chi \right)\right\}$$

(27)

$$\rho V_T^N\left(\tau ,\chi \right)=\underset{E_T,E_F}{\max }\left\{\begin{array}{l}{\pi }_T\left[D_0+\eta \tau ++\theta \chi +\delta {\displaystyle \frac{\delta {\pi }_R}{k_R}}\right]-{\displaystyle \frac{1}{2}}{k}_T{({\displaystyle \frac{{V^{N^{\prime }}}_{T\tau }{\beta }_T}{k_T}})}^2-{\displaystyle \frac{1}{2}}{k}_F{({\displaystyle \frac{{V^{N^{\prime }}}_{T\chi }{\beta }_F}{k_F}})}^2\\ +{V^{N^{\prime }}}_{T\tau }\left({\beta }_P{\displaystyle \frac{{V^{N^{\prime }}}_{P\tau }{\beta }_P}{k_P}}+{\beta }_T{\displaystyle \frac{{V^{N^{\prime }}}_{T\tau }{\beta }_T}{k_T}}-\lambda \tau \right)+{V^{N^{\prime }}}_{T\chi }\left({\beta }_{\mathrm{F}}{\displaystyle \frac{{V^{N^{\prime }}}_{T\chi }{\beta }_F}{k_F}}-\xi \chi \right)\end{array}\right\}$$

(28)

$$\rho V_R^N\left(\tau ,\chi \right)=\underset{E_R}{max}\left\{\begin{array}{l}(1+\sigma )[{\pi }_R\left[D_0+\eta \tau ++\theta \chi +\delta {\displaystyle \frac{\delta {\pi }_R}{k_R}}\right]-{\displaystyle \frac{1}{2}}{k}_R{({\displaystyle \frac{\delta {\pi }_R}{k_R}})}^2]-\sigma [{\pi }_P(D_0+\eta \tau +\theta \chi +\delta {\displaystyle \frac{\delta {\pi }_R}{k_R}})\\ -{\displaystyle \frac{1}{2}}{k}_P{({\displaystyle \frac{{V^{N^{\prime }}}_{P\tau }{\beta }_P}{k_P}})}^2]+{V^{N^{\prime }}}_{R\tau }\left({\beta }_P{\displaystyle \frac{{V^{N^{\prime }}}_{P\tau }{\beta }_P}{k_P}}+{\beta }_T{\displaystyle \frac{{V^{N^{\prime }}}_{T\tau }{\beta }_T}{k_T}}-\lambda \tau \right)+{V^{N^{\prime }}}_{R\chi }\left({\beta }_{\mathrm{F}}{\displaystyle \frac{{V^{N^{\prime }}}_{T\chi }{\beta }_F}{k_F}}-\xi \chi \right)\end{array}\right\}$$

(29)

Based on Equations (27), (28), and (29), the characteristics of the equations can be inferred, let

$$V_P^N\left(\tau ,\chi \right)=a_{13}\tau +b_{13}\chi +c_{13},V_T^N\left(\tau ,\chi \right)=a_{14}\tau +b_{14}\chi +c_{14},V_R^N\left(\tau ,\chi \right)=a_{15}\tau +b_{15}\chi +c_{15}$$

(30)

Substituting Equation (30) into Equations (27), (28), and (29) gives the following:

$$\rho (a_{13}\tau +b_{13}\chi +c_{13})=\left\{(\pi P\eta -a_{13}\lambda )\tau +({\pi }_P\theta -b_{13}\xi )\chi +{\pi }_P\left(D_0+{\displaystyle \frac{{\delta }^2\delta {\pi }_R}{k_R}}\right)+{\displaystyle \frac{{a_{13}}^2{{\beta }_P}^2}{2k_P}}+{\displaystyle \frac{a_{13}{a}_{14}{{\beta }_T}^2}{k_T}}+{\displaystyle \frac{b_{13}{b}_{14}{{\beta }_F}^2}{k_F}}\right\}$$

(31)

$$\rho (a_{14}\tau +b_{14}\chi +c_{14})=\left\{({\pi }_T\eta -a_{14}\lambda )\tau +({\pi }_T\theta -b_{14}\xi )\chi +{\pi }_T\left[D_0+\delta {\displaystyle \frac{\delta {\pi }_R}{k_R}}\right]+{\displaystyle \frac{a_{13}{a}_{14}{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{{a_{14}}^2{{\beta }_T}^2}{2k_T}}+{\displaystyle \frac{{b_{14}}^2{{\beta }_F}^2}{2k_F}}\right\}$$

(32)

$$\rho (a_{15}\tau +b_{15}\chi +c_{15})=\left\{\begin{array}{l}({\pi }_R\eta -a_{15}\lambda )\tau +({\pi }_R\theta -b_{15}\xi )\chi +{\pi }_R\left[D_0+{\displaystyle \frac{{\delta }^2\delta {\pi }_R}{k_R}}\right]-{\displaystyle \frac{{(\delta {\pi }_R)}^2}{2k_R}}+{\displaystyle \frac{\sigma {a_{13}}^2{{\beta }_P}^2}{2k_P}}\\ +{\displaystyle \frac{a_{13}{a}_{14}{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{a_{14}{a}_{15}{{\beta }_T}^2}{k_T}}+{\displaystyle \frac{b_{14}{b}_{15}{{\beta }_F}^2}{k_F}}\end{array}\right\}$$

(33)

From this, we observe

$$a_{13}^{*}={\displaystyle \frac{{\pi }_P\eta }{(\lambda +\rho )}},b_{13}^{*}={\displaystyle \frac{{\pi }_P\theta }{(\xi +\rho )}},c_{13}^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_P{D}_0+{\displaystyle \frac{{\pi }_P\delta {\pi }_R}{k_R}}+{\displaystyle \frac{{a_{13}}^2{{\beta }_P}^2}{2k_P}}+{\displaystyle \frac{a_{13}{a}_{14}{{\beta }_T}^2}{k_T}}+{\displaystyle \frac{b_{13}{b}_{14}{{\beta }_F}^2}{k_F}}\right]$$

(34)

$$a_{14}^{*}={\displaystyle \frac{{\pi }_T\eta }{(\lambda +\rho )}},b_{14}^{*}={\displaystyle \frac{{\pi }_T\theta }{(\xi +\rho )}},c_{14}^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_T{D}_0+{\displaystyle \frac{\delta {\pi }_T{\pi }_R}{k_R}}+{\displaystyle \frac{a_{13}{a}_{14}{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{{a_{14}}^2{{\beta }_T}^2}{2k_T}}+{\displaystyle \frac{{b_{14}}^2{{\beta }_F}^2}{2k_F}}\right]$$

(35)

$$a_{15}^{*}={\displaystyle \frac{{\pi }_R\eta }{(\lambda +\rho )}},b_{15}^{*}={\displaystyle \frac{{\pi }_R\theta }{(\xi +\rho )}},c_{15}^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_R{D}_0+{\displaystyle \frac{\delta {{\pi }_R}^2}{k_R}}-{\displaystyle \frac{{(\delta {\pi }_R)}^2}{2k_R}}+{\displaystyle \frac{a_{13}{a}_{15}{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{a_{14}{a}_{15}{{\beta }_T}^2}{k_T}}+{\displaystyle \frac{b_{14}{b}_{15}{{\beta }_F}^2}{k_F}}\right]$$

(36)

Substituting ($a_{13},a_{14},a_{15},b_{13},b_{14},b_{15},c_{13},c_{14},c_{15}$) into Equation (30) gives the following:

$$V_P^{N*}\left(\tau ,\chi \right)=a_{13}^{*}\tau +b_{13}^{*}\chi +c_{13}^{*},V_T^{N*}\left(\tau ,\chi \right)=a_{14}^{*}\tau +b_{14}^{*}\chi +c_{14}^{*},V_R^{N*}\left(\tau ,\chi \right)=a_{15}^{*}\tau +b_{15}^{*}\chi +c_{15}^{*}$$

(37)

Substituting Equation (37) into Equations (14), (18), (19), and (26) yields the following:

$$E_T^{N*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }\pi T}{k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}} ,E_F^{N*}={\displaystyle \frac{{\beta }_F\xi {\pi }_T}{k_T\left(\theta +\mathrm{\rho }\right)}},E_P^{N*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P{\pi }_P}{(\lambda +\rho )k_P}},E_R^N={\displaystyle \frac{\delta {\pi }_R}{k_R}}$$

(38)

Substituting Equation (38) into Equations (1) and (2) yields the following:

$$\begin{array}{l}{\tau }^{*}\left(t\right)=-\lambda \tau +{\displaystyle \frac{{{\beta }_P}^2\mathrm{\eta }{k}_T{\pi }_P}{k_P{k}_T\left(\lambda +\rho \right)}}+{\displaystyle \frac{{{\beta }_T}^2\eta k_P{\pi }_T}{k_P{k}_T\left(\lambda +\rho \right)}}\\ {\chi }^{*}(t)=-\xi \chi +{\displaystyle \frac{{{\beta }_F}^2\theta {\pi }_T}{k_F\left(\theta +\rho \right)}}\end{array}$$

(39)

$$\mathrm{let} A^N={\displaystyle \frac{{{\beta }_P}^2\mathrm{\eta }{k}_T{\pi }_P+{{\beta }_T}^2\eta k_P{\pi }_T}{k_P{k}_T\left(\lambda +\rho \right)}},B^N={\displaystyle \frac{{{\beta }_F}^2\theta {\pi }_T}{k_F\left(\theta +\rho \right)}}$$

With $\tau \left(0\right)={\tau }_0$ and $\chi \left(0\right)={\chi }_0$, the optimal trajectory for the carbon reductions in relation to fresh agricultural products is the following:

$${\tau }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{A}}^N}{\lambda }}+\left({\tau }_0-{\displaystyle \frac{{\mathrm{A}}^{\mathrm{N}}}{\lambda }}\right)e^{-\lambda t}$$

(40)

The optimal trajectory for the cold chain preservation quality of fresh agricultural products is the following:

$${\chi }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{B}}^{\mathrm{N}}}{\theta }}+\left({\chi }_0-{\displaystyle \frac{{\mathrm{B}}^{\mathrm{N}}}{\theta }}\right)e^{-\theta t}$$

(41)

The profit functions of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}{\mathrm{J}}_P^{N*}(\tau ,\chi ,t)=e^{-\rho t}(a_{13}^{*}\tau +{\mathrm{b}}_{13}^{*}\chi +c_{13}^{*})\\ {\mathrm{J}}_T^{N*}(\tau ,\chi ,t)=e^{-\rho t}(a_{14}^{*}\tau +{\mathrm{b}}_{14}^{*}\chi +c_{14}^{*})\\ {\mathrm{J}}_R^{N*}(\tau ,\chi ,t)=e^{-\rho t}(a_{15}^{*}\tau +{\mathrm{b}}_{15}^{*}\chi +c_{15}^{*})\end{array}$$

(42)

Proof completed. □

5. Decision Model Considering the Sharing of Emission Reduction Costs

5.1. Decision Model for the Producer Sharing Emission Reduction Costs

This study considers the scenario where the producer leads, and TPL provider and the retailer follow. When the producer shoulders the main emission reduction costs, its control over production technology, equipment upgrades, and financial resources naturally positions it as the decision-maker. The retailer and third-party logistics (TPL) provider, as supporting players, must adjust their strategies (e.g., adopting low-carbon packaging or transport) to comply with the producer’s sustainability requirements. From a theoretical perspective, many scholars have explored similar power structures in the fresh supply chain management field involving TPL providers [32,33]. Based on the previous literature, the Stackelberg game process among the participants is as follows: ① the producer decides its optimal carbon emission reduction effort level $E_P\left(t\right)$ and the emission reduction cost-sharing ratios ${\varphi }_R\left(t\right)$,${\varphi }_T\left(t\right)$; ② the third-party logistics (TPL) service provider decides its optimal carbon emission reduction effort level $E_T\left(t\right)$ and preservation effort level $E_F\left(t\right)$; ③ the retailer decides its optimal low-carbon promotion effort level $E_R(t$).

Based on the above assumptions, the objective functions of the supply chain members are as follows:

$$J_P^{NP}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left[{\pi }_{P}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_P{E_P}^2\left(t\right)-{\displaystyle \frac{1}{2}}{\varphi }_R{k}_R{E_R}^2\left(t\right)-{\displaystyle \frac{1}{2}}{\varphi }_T{k}_T{E_T}^2\left(t\right)\right]dt$$

(43)

$$J_T^{NP}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[{\pi }_{T}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}(1-{\varphi }_T)k_T{E_T}^2\left(t\right)-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2\left(t\right)]dt$$

(44)

$$J_R^{NP}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left\{{\pi }_{R}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}\left(1-{\varphi }_R\right)k_R{E_R}^2\left(t\right)\right\}dt$$

(45)

The solution is obtained using the HJB equation. The results under decentralized decision-making are given in Proposition 2.

Proposition 2.

Under non-cooperative control, the results of the decentralized decision model are as follows:

(1) The optimal decisions of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}{E}_T^{NP*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }(2\pi P+\pi T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}},E_F^{NP*}={\displaystyle \frac{{\beta }_F\xi {\pi }_T}{kF\left(\theta +\mathrm{\rho }\right)}},E_P^{NP*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P{\pi }_P}{(\lambda +\rho )k_P}}\\ E_R^{NP*}={\displaystyle \frac{\delta (2\pi P+\pi R)}{2k_R}},{\varphi }_T^{NP*}={\displaystyle \frac{2\pi P-\pi T}{2\pi P+\pi T}},{\varphi }_R^{NP*}={\displaystyle \frac{2\pi P-\pi R}{2\pi P+\pi R}}\end{array}$$

(2) The optimal trajectory of carbon reductions in relation to fresh agricultural products is as follows:

$$\begin{array}{c}{\tau }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{A}}^{\mathrm{NP}}}{\lambda }}+\left({\tau }_0-{\displaystyle \frac{{\mathrm{A}}^{\mathrm{NP}}}{\lambda }}\right)e^{-\lambda t},{\chi }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{B}}^{\mathrm{NP}}}{\theta }}+\left({\chi }_0-{\displaystyle \frac{{\mathrm{B}}^{\mathrm{NP}}}{\theta }}\right)e^{-\theta t}\hfill \\ A^{NP}={\displaystyle \frac{{{\beta }_P}^2\mathrm{\eta }{\pi }_P}{(\lambda +\rho )k_P}}+{\displaystyle \frac{{{\beta }_T}^2\mathrm{\eta }(2\pi P+\pi T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}},B^{NP}={\displaystyle \frac{{\pi }_T{{\beta }_F}^2\theta }{k_F\left(\theta +\rho \right)}}\hfill \end{array}$$

(3) The optimal profits of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}\hfill {\mathrm{J}}_P^{NP*}(\tau ,\chi ,t)=e^{-\rho t}(a_1^{*}\tau +{\mathrm{b}}_1^{*}\chi +c_1^{*})\hfill \\ \hfill {\mathrm{J}}_T^{NP*}(\tau ,\chi ,t)=e^{-\rho t}(a_2^{*}\tau +{\mathrm{b}}_2^{*}\chi +c_2^{*})\hfill \\ \hfill {\mathrm{J}}_R^{NP*}(\tau ,\chi ,t)=e^{-\rho t}(a_3^{*}\tau +{\mathrm{b}}_3^{*}\chi +c_3^{*})\hfill \\ \mathrm{where}, a_1^{*}={\displaystyle \frac{{\pi }_P\eta }{(\lambda +\rho )}},b_1^{*}={\displaystyle \frac{{\pi }_P\theta }{(\xi +\rho )}},\hfill \\ c_1^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_P\left(D_0+{\displaystyle \frac{{\delta }^2(2{\pi }_P-{\pi }_R)}{2k_R}}\right)-{\displaystyle \frac{{a_1}^2{{\beta }_P}^2}{2k_P}}-{\displaystyle \frac{{\delta }^2(4{\pi }_P^2-{\pi }_R^2)}{8k_R}}-{\displaystyle \frac{{a_2}^2{{\beta }_T}^2(4{\pi }_P^2-{\pi }_T^2)}{8k_T{\pi }_T^2}}+{\displaystyle \frac{{a_1}^2{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{a_1{a}_2{{\beta }_T}^2(2{\pi }_P-\pi T)}{2\pi T{k}_T}}+{\displaystyle \frac{b_1{b}_3{{\beta }_{\mathrm{F}}}^2}{k_F}}\right]\hfill \\ a_2^{*}={\displaystyle \frac{{\pi }_T\eta }{(\lambda +\rho )}},b_2^{*}={\displaystyle \frac{{\pi }_T\theta }{(\xi +\rho )}},\hfill \\ c_2^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_T\left(D_0+{\displaystyle \frac{{\delta }^2(2{\pi }_P-{\pi }_R)}{2k_R}}\right)-{\displaystyle \frac{{a_2}^2{{\beta }_T}^2(2{\pi }_P-\pi T)}{2k_T\pi T}}-{\displaystyle \frac{{b_2}^2{{\beta }_{\mathrm{F}}}^2}{2k_F}}+{\displaystyle \frac{a_1{a}_2{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{{a_2}^2{{\beta }_T}^2(2{\pi }_P-\pi T)}{2k_T\pi T}}+{\displaystyle \frac{{b_2}^2{{\beta }_{\mathrm{F}}}^2}{k_F}}\right]\hfill \\ a_3^{*}={\displaystyle \frac{{\pi }_R\eta }{(\lambda +\rho )}},b_3^{*}={\displaystyle \frac{{\pi }_R\theta }{(\xi +\rho )}},\hfill \\ c_3^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_R\left(D_0+{\displaystyle \frac{{\delta }^2(2{\pi }_P-{\pi }_R)}{2k_R}}\right)-{\displaystyle \frac{{\delta }^2(4{\pi }_P^2-{\pi }_R^2)}{8k_R}}+{\displaystyle \frac{a_1{a}_3{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{a_2{a}_3{{\beta }_T}^2(2{\pi }_P-\pi T)}{2k_T\pi T}}+{\displaystyle \frac{b_2{b}_3{{\beta }_{\mathrm{F}}}^2}{k_F}}\right]\hfill \end{array}$$

The proof is similar to the above and is omitted.

5.2. Decision Model for Retailer Sharing Emission Reduction Costs

If the retailer takes on emission reduction costs (e.g., green packaging, low-carbon delivery), its direct access to consumers and influence over pricing/marketing grants it decision-making power. Manufacturers and TPL providers must adapt to the retailer’s sustainability standards, such as modifying product designs or optimizing logistics routes. Under this cost-sharing arrangement, the retailer assumes responsibility for the emission abatement costs of both the producer and third-party logistics provider, thereby attaining a leadership position in the supply chain hierarchy, with the producer and TPL provider occupying follower roles. The decision sequence is as follows: first, solve for the emission reduction decisions and preservation technology R&D of the producer and TPL provider, then the retailer, based on the optimal decisions of the suppliers and producer, determines its own low-carbon promotion decisions and cost-sharing coefficients.

$$J_P^{NR}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left[{\pi }_{P}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}(1-{\epsilon }_p)k_P{E_P}^2\left(t\right)\right]dt$$

(46)

$$J_T^{NR}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[{\pi }_{T}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}(1-\epsilon T)k_T{E_T}^2\left(t\right)-{\displaystyle \frac{1}{2}}{k}_F{E_F}^2\left(t\right)]dt$$

(47)

$$J_R^{NR}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left\{{\pi }_{R}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_R{E_R}^2\left(t\right)-{\displaystyle \frac{1}{2}}{\epsilon }_p{k}_P{E_P}^2\left(t\right)-{\displaystyle \frac{1}{2}}\epsilon T{k}_T{E_T}^2\left(t\right)\right\}dt$$

(48)

The solution is obtained using the HJB equation. The results under decentralized decision-making are given in Proposition 3.

Proposition 3.

Under non-cooperative control, the results of the decentralized decision model are as follows:

(1) The optimal decisions of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}{E}_T^{NR*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }(2{\pi }_R+{\pi }_T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}},E_F^{NR*}={\displaystyle \frac{{\beta }_F\xi {\pi }_T}{kF\left(\theta +\mathrm{\rho }\right)}},E_P^{NR*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P(2{\pi }_R+\pi P)}{2(\lambda +\rho )k_P}}\\ E_R^{NR*}={\displaystyle \frac{\delta {\pi }_R}{k_R}},{\epsilon }_T^{NR*}={\displaystyle \frac{2\pi R-\pi T}{2\pi R+\pi T}},{\epsilon }_P^{NR*}{\displaystyle \frac{-\pi P+2\pi R}{\pi P+2\pi R}}\end{array}$$

(2) The optimal trajectory of carbon reductions in relation to fresh agricultural products is as follows:

$$\begin{array}{c}{\tau }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{A}}^{NR}}{\lambda }}+\left({\tau }_0-{\displaystyle \frac{{\mathrm{A}}^{\mathrm{NR}}}{\lambda }}\right)e^{-\lambda t},{\chi }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{B}}^{\mathrm{NR}}}{\theta }}+\left({\chi }_0-{\displaystyle \frac{{\mathrm{B}}^{\mathrm{NR}}}{\theta }}\right)e^{-\theta t}\hfill \\ A^{NR}={\displaystyle \frac{{{\beta }_T}^2\mathrm{\eta }(2{\pi }_R+{\pi }_T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}+{\displaystyle \frac{\mathrm{\eta }{{\beta }_P}^2(2{\pi }_R+\pi P)}{2(\lambda +\rho )k_P}},B^{NR}={\displaystyle \frac{{\pi }_T{{\beta }_F}^2\theta }{k_F\left(\theta +\rho \right)}}\hfill \end{array}$$

(3) The optimal profits of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{c}\hfill {\mathrm{J}}_T^{NR*}(\tau ,\chi ,t)=e^{-\rho t}(a_4^{*}\tau +{\mathrm{b}}_4^{*}\chi +c_4^{*})\hfill \\ \hfill {\mathrm{J}}_P^{NP*}(\tau ,\chi ,t)=e^{-\rho t}(a_5^{*}\tau +{\mathrm{b}}_5^{*}\chi +c_5^{*})\hfill \\ \hfill {\mathrm{J}}_R^{NR*}(\tau ,\chi ,t)=e^{-\rho t}(a_6^{*}\tau +{\mathrm{b}}_6^{*}\chi +c_6^{*})\hfill \\ \mathrm{where},a_4^{*}={\displaystyle \frac{{\pi }_T\eta }{(\lambda +\rho )}},b_4^{*}={\displaystyle \frac{{\pi }_T\theta }{(\xi +\rho )}},\hfill \\ c_4^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_T\left(D_0+{\delta }^2{\displaystyle \frac{{\pi }_R}{k_R}}\right)-{\displaystyle \frac{{a_4}^2{{\beta }_T}^2(\pi T+2\pi R)}{4\pi T{k}_T}}-{{\displaystyle \frac{{b_4}^2{\beta }_{\mathrm{F}}}{2k_F}}}^2+{\displaystyle \frac{a_4{a}_5{{\beta }_P}^2(\pi P+2\pi R)}{2\pi P{k}_P}}+{\displaystyle \frac{{a_4}^2{{\beta }_T}^2(\pi T+2\pi R)}{2\pi T{k}_T}}+{\displaystyle \frac{{b_4}^2{{\beta }_{\mathrm{F}}}^2}{k_F}}\right]\hfill \\ a_5^{*}={\displaystyle \frac{{\pi }_P\eta }{(\lambda +\rho )}},b_5^{*}={\displaystyle \frac{{\pi }_P\theta }{(\xi +\rho )}},\hfill \\ c_5^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_P\left(D_0+{\delta }^2{\displaystyle \frac{{\pi }_R}{k_R}}\right)-{\displaystyle \frac{{a_5}^2{{\beta }_P}^2(\pi P+2\pi R)}{4\pi P{k}_P}}+{\displaystyle \frac{{a_5}^2{{\beta }_P}^2(\pi P+2\pi R)}{2\pi P{k}_P}}+{\displaystyle \frac{a_5{a}_4{{\beta }_T}^2(\pi T+2\pi R)}{2\pi T{k}_T}}+{\displaystyle \frac{b_4{b}_5{{\beta }_{\mathrm{F}}}^2}{k_F}}\right]\hfill \\ a_6^{*}={\displaystyle \frac{{\pi }_R\eta }{(\lambda +\rho )}},b_6^{*}={\displaystyle \frac{{\pi }_R\theta }{(\xi +\rho )}},\hfill \\ c_6^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_R{D}_0+{\pi }_R{\delta }^2{\displaystyle \frac{{\pi }_R}{k_R}}-{{\displaystyle \frac{(\delta {\pi }_R)}{2}}}^2-{\displaystyle \frac{{a_5}^2{{\beta }_p}^2(4\pi R^2-\pi P^2)}{8\pi P^2{k}_p}}-{\displaystyle \frac{{a_4}^2{{\beta }_T}^2(4\pi R^2-\pi T^2)}{8\pi T^2{k}_T}}+{\displaystyle \frac{a_5{a}_6{{\beta }_P}^2(\pi P+2\pi R)}{2\pi P{k}_P}}+{\displaystyle \frac{a_4{a}_6{{\beta }_T}^2(\pi T+2\pi R)}{2\pi T{k}_T}}+{\displaystyle \frac{b_4{b}_6{{\beta }_{\mathrm{F}}}^2}{k_F}}\right]\hfill \end{array}$$

The proof is similar to the above and is omitted.

6. Considering the Cost Sharing of Emission Reduction and Preservation

6.1. Decision Model for Producer Sharing Emission Reduction Costs and Retailer Sharing Preservation Costs

When the producer shares the emission reduction costs, the retailer shares the preservation costs with the TPL provider. In this case, part of the emission reduction and preservation costs of the TPL provider are shared by the producer and retailer; the producer shares the emission reduction costs of other parties, and the retailer shares part of the emission reduction costs borne by the producer but also shares the additional preservation costs of the TPL provider. Here, the manufacturer leads due to its responsibility for emissions (requiring larger investments and long-term planning), while the retailer’s role in freshness control (e.g., cold chain storage) strengthens its influence. TPL providers meet both parties’ requirements. The decision sequence is as follows: the producer first makes the emission reduction decision and the cost-sharing coefficient for emission reductions, the retailer decides on promotion efforts and the preservation cost-sharing coefficient, and finally, the TPL provider makes the emission reduction decision and preservation technology R&D. The objective functions in this case are as follows:

$$J_P^{NPR}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left[{\pi }_{P}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_P{E_P}^2\left(t\right)-{\displaystyle \frac{1}{2}}{\varphi }_R{k}_R{E_R}^2\left(t\right)-{\displaystyle \frac{1}{2}}{\varphi }_T{k}_T{E_T}^2\left(t\right)\right]dt$$

(49)

$$J_T^{NPR}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left[{\pi }_{T}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}(1-{\varphi }_T)k_T{E_T}^2\left(t\right)-{\displaystyle \frac{1}{2}}(1-\epsilon F)k_F{E_F}^2\left(t\right)\right]dt$$

(50)

$$J_R^{NPR}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left\{{\pi }_{R}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}\left(1-{\varphi }_R\right)k_R{E_R}^2\left(t\right)-{\displaystyle \frac{1}{2}}\epsilon F{k}_F{E_F}^2\left(t\right)\right\}dt$$

(51)

The solution is obtained using the HJB equation. The results under decentralized decision-making are given in Proposition 4.

Proposition 4.

Under non-cooperative control, the results of the decentralized decision model are as follows:

(1) The optimal decisions of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}{E}_T^{NPR*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }(2\pi P+\pi T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}},E_F^{NPR*}={\displaystyle \frac{{\beta }_F\xi (2\pi R+\pi T)}{2kF\left(\theta +\mathrm{\rho }\right)}},E_P^{NPR*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P{\pi }_P}{(\lambda +\rho )k_P}}\\ E_R^{NPR*}={\displaystyle \frac{\delta (2\pi P+\pi R)}{2k_R}},{\varphi }_T^{NPR*}={\displaystyle \frac{2\pi P-\pi T}{2\pi P+\pi T}},{\varphi }_R^{NPR*}={\displaystyle \frac{2\pi P-\pi R}{2\pi P+\pi R}},{\epsilon }_F^{N\mathrm{P}R*}={\displaystyle \frac{2\pi R-\pi T}{2\pi R+\pi T}}\end{array}$$

(2) The optimal trajectory of carbon reductions in relation to fresh agricultural products is as follows:

$$\begin{array}{c}{\tau }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{A}}^{\mathrm{NPR}}}{\lambda }}+\left({\tau }_0-{\displaystyle \frac{{\mathrm{A}}^{\mathrm{NPR}}}{\lambda }}\right)e^{-\lambda t},{\chi }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{B}}^{\mathrm{NPR}}}{\theta }}+\left({\chi }_0-{\displaystyle \frac{{\mathrm{B}}^{\mathrm{NPR}}}{\theta }}\right)e^{-\theta t}\hfill \\ A^{NPR}={\displaystyle \frac{{{\beta }_P}^2\mathrm{\eta }{\pi }_P}{(\lambda +\rho )k_P}}+{\displaystyle \frac{{{\beta }_T}^2\mathrm{\eta }(2\pi P+\pi T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}},B^{NPR}={\displaystyle \frac{{{\beta }_F}^2\theta (2\pi R+\pi T)}{2k_F\left(\theta +\rho \right)}}\hfill \end{array}$$

(3) The optimal profits of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}\hfill {\mathrm{J}}_T^{NPR*}(\tau ,\chi ,t)=e^{-\rho t}(a_7^{*}\tau +{\mathrm{b}}_7^{*}\chi +c_7^{*})\hfill \\ \hfill {\mathrm{J}}_R^{NPR*}(\tau ,\chi ,t)=e^{-\rho t}(a_8^{*}\tau +{\mathrm{b}}_8^{*}\chi +c_8^{*})\hfill \\ \hfill {\mathrm{J}}_P^{NPR*}(\tau ,\chi ,t)=e^{-\rho t}(a_9^{*}\tau +{\mathrm{b}}_9^{*}\chi +c_9^{*})\hfill \\ \mathrm{where},a_7^{*}={\displaystyle \frac{{\pi }_T\eta }{(\lambda +\rho )}},b_7^{*}={\displaystyle \frac{{\pi }_T\theta }{(\xi +\rho )}},\hfill \\ c_7^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_T\left(D_0+{\displaystyle \frac{{\delta }^2(2{\pi }_P+{\pi }_R)}{2k_R}}\right)-{\displaystyle \frac{{a_7}^2{{\beta }_T}^2(2{\pi }_P+{\pi }_T)}{4k_T{\pi }_T}}-{\displaystyle \frac{{b_7}^2{{\beta }_{\mathrm{F}}}^2(2{\pi }_R+{\pi }_T)}{4k_F{\pi }_T}}+{\displaystyle \frac{a_9{a}_7{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{{a_7}^2{{\beta }_T}^2(2{\pi }_P+{\pi }_T)}{2k_T{\pi }_T}}+{\displaystyle \frac{{b_7}^2{{\beta }_{\mathrm{F}}}^2(2{\pi }_R+{\pi }_T)}{2k_F{\pi }_T}}\right]\hfill \\ a_8^{*}={\displaystyle \frac{{\pi }_R\eta }{(\lambda +\rho )}},b_8^{*}={\displaystyle \frac{{\pi }_R\theta }{(\xi +\rho )}},\hfill \\ c_8^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_R\left(D_0+{\displaystyle \frac{{\delta }^2(2{\pi }_P+{\pi }_R)}{k_R\left(1-{\varphi }_R\right)}}\right)-{\displaystyle \frac{{\delta }^2(2{\pi }_P-{\pi }_R)}{4k_R}}+{\displaystyle \frac{a_9{a}_8{{\beta }_P}^2}{k_P}}+{\displaystyle \frac{a_7{a}_8{{\beta }_T}^2(2{\pi }_P+{\pi }_T)}{2k_T{\pi }_T}}+{\displaystyle \frac{b_7{b}_8{{\beta }_{\mathrm{F}}}^2(2{\pi }_R+{\pi }_T)}{2k_F{\pi }_T}}\right]\hfill \\ a_9^{*}={\displaystyle \frac{{\pi }_P\eta }{(\lambda +\rho )}},b_9^{*}={\displaystyle \frac{{\pi }_P\theta }{(\xi +\rho )}},\hfill \\ c_9^{*}={\displaystyle \frac{1}{\rho }}\left[\begin{array}{l}{\pi }_P\left(D_0+{\displaystyle \frac{{\delta }^2(2{\pi }_P+{\pi }_R)}{2k_R}}\right)-{\displaystyle \frac{{a_9}^2{{\beta }_P}^2}{2k_P}}-{\displaystyle \frac{{\delta }^2(4{\pi }_P^2-{\pi }_R^2)}{8{\pi }_R^2{k}_R}}-{\displaystyle \frac{{a_7}^2{{\beta }_T}^2(4{\pi }_P^2-{\pi }_T^2)}{8k_T{\pi }_T^2}}+{\displaystyle \frac{{a_9}^2{{\beta }_P}^2}{k_P}}\\ +{\displaystyle \frac{a_9{a}_7{{\beta }_T}^2(2{\pi }_P+{\pi }_T)}{2k_T{\pi }_T}}+{\displaystyle \frac{b_9{b}_7{{\beta }_{\mathrm{F}}}^2(2{\pi }_R+{\pi }_T)}{2k_F{\pi }_T}}\end{array}\right]\hfill \end{array}$$

The proof is similar to the above and is omitted.

6.2. Decision Model for Retailer Sharing Emission Reduction Costs and Producer Sharing Preservation Costs

When the retailer controls both emission reductions and delegates freshness costs to the manufacturer (e.g., fresh-food e-commerce demanding suppliers’ refrigeration tech), its market dominance dictates supply chain decisions. The manufacturer, though investing in freshness tech, remains subordinate to the retailer, with TPL providers at the bottom. Under this cost-sharing framework, the retailer bears the emission reduction costs for both the producer and the TPL provider, while the producer assumes the freshness technology R&D costs for the TPL provider. The decision-making sequence is as follows: the retailer first makes the emission reduction decision and the cost-sharing coefficient for emission reductions, the producer makes the emission reduction decision and the preservation cost-sharing coefficient decision, and finally, the TPL provider makes the emission reduction decision and preservation technology R&D decision.

The objective functions in this case are as follows:

$$J_P^{NRP}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left[{\pi }_{P}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}(1-\epsilon P)k_P{E_P}^2\left(t\right)-{\displaystyle \frac{1}{2}}{\varphi }_F{k}_F{E_F}^2(t)\right]dt$$

(52)

$$J_T^{NRP}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}[{\pi }_{T}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}(1-\epsilon T)k_T{E_T}^2\left(t\right)-{\displaystyle \frac{1}{2}}(1-{\varphi }_F)k_F{E_F}^2\left(t\right)]dt$$

(53)

$$J_R^{NRP}\left(\tau ,\chi ,t\right)={{\displaystyle \int }}_0^{\infty }{e}^{-\rho t}\left\{{\pi }_{R}D\left(\tau \left(t\right),\chi \left(t\right),E_R\left(t\right)\right)-{\displaystyle \frac{1}{2}}{k}_R{E_R}^2\left(t\right)-{\displaystyle \frac{1}{2}}\epsilon p{k}_P{E_P}^2\left(t\right)-{\displaystyle \frac{1}{2}}\epsilon T{k}_T{E_T}^2\left(t\right)\right\}dt$$

(54)

The solution is obtained using the HJB equation. The results under decentralized decision-making are given in Proposition 5.

Proposition 5.

Under non-cooperative control, the results of the decentralized decision model are as follows:

(1) The optimal decisions of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}{E}_T^{NRP*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }(2{\pi }_R+{\pi }_T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}},E_F^{NRP*}={\displaystyle \frac{{\beta }_F\xi (2\pi P+{\pi }_T)}{2kF\left(\theta +\mathrm{\rho }\right)}},E_P^{NRP*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P(2{\pi }_R+\pi P)}{2(\lambda +\rho )k_P}}\\ E_R^{NRP*}={\displaystyle \frac{\delta {\pi }_R}{k_R}},{\epsilon }_T^{NRP*}={\displaystyle \frac{2\pi R-\pi T}{2\pi R+\pi T}},{\epsilon }_P^{NRP*}{\displaystyle \frac{-\pi P+2\pi R}{\pi P+2\pi R}},{\varphi }_F^{NRP*}={\displaystyle \frac{2\pi P-\pi T}{2\pi P+\pi T}}\end{array}$$

(2) The optimal trajectory of carbon reductions in relation to fresh agricultural products is as follows:

$$\begin{array}{c}{\tau }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{A}}^{NRP}}{\lambda }}+\left({\tau }_0-{\displaystyle \frac{{\mathrm{A}}^{\mathrm{NRP}}}{\lambda }}\right)e^{-\lambda t},{\chi }^{*}\left(t\right)={\displaystyle \frac{{\mathrm{B}}^{\mathrm{NRP}}}{\theta }}+\left({\chi }_0-{\displaystyle \frac{{\mathrm{B}}^{\mathrm{NRP}}}{\theta }}\right)e^{-\theta t}\hfill \\ A^{NRP}={\displaystyle \frac{{{\beta }_T}^2\mathrm{\eta }(2{\pi }_R+{\pi }_T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}+{\displaystyle \frac{\mathrm{\eta }{{\beta }_P}^2(2{\pi }_R+\pi P)}{2(\lambda +\rho )k_P}},B^{NRP}={\displaystyle \frac{{{\beta }_F}^2\theta (2\pi P+{\pi }_T)}{2k_F\left(\theta +\rho \right)}}\hfill \end{array}$$

(3) The optimal profits of the producer, TPL provider, and retailer are as follows:

$$\begin{array}{l}\hfill {\mathrm{J}}_T^{NRP*}(\tau ,\chi ,t)=e^{-\rho t}(a_{10}^{*}\tau +{\mathrm{b}}_{10}^{*}\chi +c_{10}^{*}),\hfill \\ \hfill {\mathrm{J}}_P^{NRP*}(\tau ,\chi ,t)=e^{-\rho t}(a_{11}^{*}\tau +{\mathrm{b}}_{11}^{*}\chi +c_{11}^{*})\hfill \\ \hfill {\mathrm{J}}_R^{NRP*}(\tau ,\chi ,t)=e^{-\rho t}(a_{12}^{*}\tau +{\mathrm{b}}_{12}^{*}\chi +c_{12}^{*})\hfill \\ \mathrm{where},a_{10}^{*}={\displaystyle \frac{{\pi }_T\eta }{(\lambda +\rho )}},b_{10}^{*}={\displaystyle \frac{{\pi }_T\theta }{(\xi +\rho )}},\hfill \\ c_{10}^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_T\left(D_0+{\delta }^2{\displaystyle \frac{{\pi }_R}{k_R}}\right)-{\displaystyle \frac{{a_{10}}^2{{\beta }_T}^2(\pi T+2\pi R)}{4\pi T{k}_T}}-{\displaystyle \frac{{b_{10}}^2{{\beta }_{\mathrm{F}}}^2(\pi T+2\pi P)}{4\pi T{k}_F}}+{\displaystyle \frac{a_{10}{a}_{11}{{\beta }_P}^2(\pi P+2\pi R)}{2\pi P{k}_P}}+{\displaystyle \frac{{a_{10}}^2{{\beta }_T}^2(\pi T+2\pi R)}{2\pi T{k}_T}}+{\displaystyle \frac{{b_{10}}^2{{\beta }_{\mathrm{F}}}^2(\pi T+2\pi P)}{2\pi T{k}_F}}\right]\hfill \\ a_{11}^{*}={\displaystyle \frac{{\pi }_P\eta }{(\lambda +\rho )}},b_{11}^{*}={\displaystyle \frac{{\pi }_P\theta }{(\xi +\rho )}},\hfill \\ c_{11}^{*}={\displaystyle \frac{1}{\rho }}\left[{\pi }_P\left(D_0+{\delta }^2{\displaystyle \frac{{\pi }_R}{k_R}}\right)-{\displaystyle \frac{{b_{10}}^2{{\beta }_{\mathrm{F}}}^2(4\pi P^2-\pi T^2)}{8k_F\pi T^2}}-{\displaystyle \frac{{a_{11}}^2{{\beta }_P}^2(\pi P+2\pi R)}{4\pi P{k}_P}}+{\displaystyle \frac{{a_{11}}^2{{\beta }_P}^2(\pi P+2\pi R)}{2\pi P{k}_P}}+{\displaystyle \frac{a_{11}{a}_{10}{{\beta }_T}^2(\pi T+2\pi R)}{2\pi T{k}_T}}+{\displaystyle \frac{b_{10}{b}_{11}{{\beta }_{\mathrm{F}}}^2(\pi T+2\pi P)}{2\pi T{k}_F}}\right]\hfill \\ a_{12}^{*}={\displaystyle \frac{{\pi }_R\eta }{(\lambda +\rho )}},b_{12}^{*}={\displaystyle \frac{{\pi }_R\theta }{(\xi +\rho )}},\hfill \\ c_{12}^{*}={\displaystyle \frac{1}{\rho }}\left[\begin{array}{l}{\pi }_R{D}_0+{\pi }_R{\delta }^2{\displaystyle \frac{{\pi }_R}{k_R}}-{{\displaystyle \frac{(\delta {\pi }_R)}{2}}}^2-{\displaystyle \frac{{a_5}^2{{\beta }_p}^2(4\pi R^2-\pi P^2)}{8\pi P^2{k}_p}}-{\displaystyle \frac{{a_4}^2{{\beta }_T}^2(4\pi R^2-\pi T^2)}{8\pi T^2{k}_T}}+{\displaystyle \frac{a_5{a}_6{{\beta }_P}^2(\pi P+2\pi R)}{2\pi P{k}_P}}\\ +{\displaystyle \frac{a_4{a}_6{{\beta }_T}^2(\pi T+2\pi R)}{2\pi T{k}_T}}+{\displaystyle \frac{b_4{b}_6{{\beta }_{\mathrm{F}}}^2(\pi T+2\pi P)}{2\pi T{k}_F}}\end{array}\right]\hfill \end{array}$$

The proof is similar to the above and is omitted.

7. Model Comparison and Analysis

Based on the model construction and solution, this section compares the emission reductions and preservation decisions under five different cooperation models and further explores the impact of different cost-sharing contracts on emission reductions, fresh agricultural product preservation quality, and corporate profits. The decision comparisons under the five scenarios are presented in

Table 2. Comparison of decisions under five scenarios.

Equilibrium Results	No Cost Sharing	Emission Reduction Cost Sharing		Emission Reduction and Preservation Cost Sharing
		Producer Shares Emission Reduction Costs	Retailer Shares Emission Reduction Costs	Producer Shares Emission Reduction Costs, Retailer Shares Preservation Costs	Retailer Shares Emission Reduction Costs, Producer Shares Preservation Costs
$E_P$	$E_P^{N*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P{\pi }_P}{(\lambda +\rho )k_P}}$	$E_P^{NP*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P{\pi }_P}{(\lambda +\rho )k_P}}$	$E_P^{NR*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P(2{\pi }_R+\pi P)}{2(\lambda +\rho )k_P}}$	$E_P^{NPR*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P{\pi }_P}{(\lambda +\rho )k_P}}$	$E_P^{NRP*}={\displaystyle \frac{\mathrm{\eta }{\beta }_P(2{\pi }_R+\pi P)}{2(\lambda +\rho )k_P}}$
$E_R$	$E_R^N={\displaystyle \frac{\delta {\pi }_R}{k_R}}$	$E_R^{NP*}={\displaystyle \frac{\delta (2\pi P+\pi R)}{2k_R}}$	$E_R^{NR*}={\displaystyle \frac{\delta {\pi }_R}{k_R}}$	$E_R^{NPR*}={\displaystyle \frac{\delta (2\pi P+\pi R)}{2k_R}}$	$E_R^{NRP*}={\displaystyle \frac{\delta {\pi }_R}{k_R}}$
$E_T$	$E_T^{N*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }\pi T}{k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}$	$E_T^{NP*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }(2\pi P+\pi T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}$	$E_T^{NR*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }(2{\pi }_R+{\pi }_T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}$	$E_T^{NPR*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }(2\pi P+\pi T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}$	$E_T^{NRP*}={\displaystyle \frac{{\beta }_T\mathrm{\eta }(2{\pi }_R+{\pi }_T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}$
$E_F$	$E_F^{N*}={\displaystyle \frac{{\beta }_F\xi {\pi }_T}{k_T\left(\theta +\mathrm{\rho }\right)}}$	$E_F^{NP*}={\displaystyle \frac{{\beta }_F\xi {\pi }_T}{kF\left(\theta +\mathrm{\rho }\right)}}$	$E_F^{NR*}={\displaystyle \frac{{\beta }_F\xi {\pi }_T}{kF\left(\theta +\mathrm{\rho }\right)}}$	$E_F^{NPR*}={\displaystyle \frac{{\beta }_F\xi (2\pi R+\pi T)}{2kF\left(\theta +\mathrm{\rho }\right)}}$	$E_F^{NRP*}={\displaystyle \frac{{\beta }_F\xi (2\pi P+{\pi }_T)}{2kF\left(\theta +\mathrm{\rho }\right)}}$
$A$	$\begin{array}{l}{A}^N={\displaystyle \frac{{{\beta }_P}^2\mathrm{\eta }{\pi }_P}{k_P\left(\lambda +\rho \right)}}\\ +{\displaystyle \frac{{{\beta }_T}^2\eta {\pi }_T}{k_T\left(\lambda +\rho \right)}}\end{array}$	$\begin{array}{l}{A}^{NP}={\displaystyle \frac{{{\beta }_P}^2\mathrm{\eta }{\pi }_P}{(\lambda +\rho )k_P}}\\ +{\displaystyle \frac{{{\beta }_T}^2\mathrm{\eta }(2\pi P+\pi T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}\end{array}$	$\begin{array}{l}{A}^{NR}={\displaystyle \frac{{{\beta }_T}^2\mathrm{\eta }(2{\pi }_R+{\pi }_T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}\\ +{\displaystyle \frac{\mathrm{\eta }{{\beta }_P}^2(2{\pi }_R+\pi P)}{2(\lambda +\rho )k_P}}\end{array}$	$\begin{array}{l}{A}^{NPR}={\displaystyle \frac{{{\beta }_P}^2\mathrm{\eta }{\pi }_P}{(\lambda +\rho )k_P}}\\ +{\displaystyle \frac{{{\beta }_T}^2\mathrm{\eta }(2\pi P+\pi T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}\end{array}$	$\begin{array}{l}{A}^{NRP}={\displaystyle \frac{{{\beta }_T}^2\mathrm{\eta }(2{\pi }_R+{\pi }_T)}{2k_T\left(\mathrm{\lambda }+\mathrm{\rho }\right)}}\\ +{\displaystyle \frac{\mathrm{\eta }{P_{\beta }}^2(2{\pi }_R+\pi P)}{2(\lambda +\rho )k_P}}\end{array}$
$B$	$B^N={\displaystyle \frac{{{\beta }_F}^2\theta {\pi }_T}{k_F\left(\lambda +\rho \right)}}$	$B^{NP}={\displaystyle \frac{{\pi }_T{{\beta }_F}^2\theta }{k_F\left(\lambda +\rho \right)}}$	$B^{NR}={\displaystyle \frac{{\pi }_T{{\beta }_F}^2\theta }{k_F\left(\lambda +\rho \right)}}$	$B^{NPR}={\displaystyle \frac{{{\beta }_F}^2\theta (2\pi R+\pi T)}{2k_F\left(\theta +\rho \right)}}$	$B^{NRP}={\displaystyle \frac{{{\beta }_F}^2\theta (2\pi P+{\pi }_T)}{2k_F\left(\theta +\rho \right)}}$

. By comparing the emission reduction efforts, preservation efforts, and related expressions in Table 2, calculations lead to conclusions regarding emission reductions, preservation quality, and profit, as summarized in Propositions 1 to 4.

Proposition 6.

For the TPL provider, when an entity with higher marginal profit shares its costs, the TPL provider’s emission reduction efforts and research and development (R&D) investments increase. For the producer, regardless of the magnitude of marginal profit, the producer is more willing to enhance its emission reduction efforts when the retailer shares its emission reduction costs. Similarly, for the retailer, regardless of marginal profit size, the retailer is more inclined to exert greater low-carbon promotional efforts when the producer shares its promotional costs.

Proof Proposition 6.

From Table 2, it can be observed that for the producer, the emission reduction effort is given by $E_P^{NRP\mathrm{*}}-E_P^{N\mathrm{*}}={\displaystyle \frac{\eta {\beta }_P(2{\pi }_R-{\pi }_P)}{2k_P(\lambda +\rho )}},$ Since the cost-sharing coefficient is greater than zero, it follows that $\left(2{\pi }_R-{\pi }_P\right)>0$, which implies $E_P^{NRP\mathrm{*}}>E_P^{N\mathrm{*}}$, Thus, we have $E_P^{NR\mathrm{*}}=E_P^{NRP\mathrm{*}}>E_P^{N\mathrm{*}}=E_P^{NP\mathrm{*}}=E_P^{NPR\mathrm{*}}$. Similarly, for the retailer’s low-carbon promotional efforts, we find that $E_R^{NP\mathrm{*}}=E_P^{NPR\mathrm{*}}>E_P^{N\mathrm{*}}=E_P^{NR\mathrm{*}}=E_P^{NRP\mathrm{*}}.$

For the TPL provider’s emission reduction effort, it holds that $E_P^{NRP\mathrm{*}}-E_P^{N\mathrm{*}}={\displaystyle \frac{\eta {\beta }_P(2{\pi }_R-{\pi }_P)}{2k_P(\lambda +\rho )}}$, Likewise, for the TPL provider’s preservation technology development, we have ${E_F^{NRP\mathrm{*}}>E}_F^{NR\mathrm{*}}=E_F^{N\mathrm{*}}=E_F^{NP\mathrm{*}},E_F^{NPR\mathrm{*}}>E_F^{NR\mathrm{*}}=E_F^{N\mathrm{*}}=E_F^{NP\mathrm{*}}$. Further comparing the emission reduction efforts when the producer and the retailer separately bear emission reduction costs, we obtain $E_T^{NR\mathrm{*}}-E_T^{NP\mathrm{*}}={\displaystyle \frac{{\beta }_T\eta ({\pi }_R-{\pi }_P)}{k_T(\lambda +\rho )}}$,$E_F^{NRP\mathrm{*}}-E_F^{NPR\mathrm{*}}={\displaystyle \frac{{\beta }_F\xi \left({\pi }_R-{\pi }_P\right)}{k_F\left(\theta +\rho \right)}};$ therefore, for the TPL provider, when ${\pi }_R>{\pi }_P$, it follows that $E_T^{NR\mathrm{*}}=E_T^{NRP\mathrm{*}}>E_T^{NP\mathrm{*}}=E_T^{NPR\mathrm{*}}>E_T^{N\mathrm{*}}$, $E_F^{NRP\mathrm{*}}>E_F^{NPR\mathrm{*}}{>E}_F^{NR\mathrm{*}}=E_F^{N\mathrm{*}}=E_F^{NP\mathrm{*}}$. From the above comparison of emission reductions and preservation decisions, we derive Proposition 6. Proof Completed. □

Proposition 7.

The sharing of carbon reduction costs increases the carbon reduction level of fresh agricultural products, while the sharing of preservation costs enhances the preservation quality. However, adding a preservation cost-sharing contract on top of a carbon reduction cost-sharing contract does not further improve the unit carbon reduction level of fresh agricultural products. Similarly, adding a carbon reduction cost-sharing contract to a preservation cost-sharing agreement does not further enhance the preservation quality of individual products. When the producer’s unit product’s marginal profit is lower than that of the retailer, the unit carbon reductions resulting from the producer’s cost sharing are less than that achieved through the retailer’s cos sharing. Under these conditions, the improvement in preservation quality from the producer’s cost sharing is also lower than that from the retailer’s cost sharing.

Proof Proposition 7.

Since $E_P^{NR\mathrm{*}}=E_P^{NRP\mathrm{*}}>E_P^{N\mathrm{*}}=E_P^{NP\mathrm{*}}=E_P^{NPR\mathrm{*}}$, it follows that ${\mathrm{A}}^{NR}={\mathrm{A}}^{NRP}>{\mathrm{A}}^N$, and thus, the carbon reduction per unit product satisfies ${\mathrm{\tau }}^{NR}={\mathrm{\tau }}^{NRP}>{\mathrm{\tau }}^N$. Since $E_T^{NP\mathrm{*}}=E_T^{NPR\mathrm{*}}>E_T^{N\mathrm{*}}$, it follows that ${\mathrm{A}}^{NP}={\mathrm{A}}^{NPR}>{\mathrm{A}}^N$, and thus, the carbon reduction per unit product satisfies ${\mathrm{\tau }}^{NP}={\mathrm{\tau }}^{NPR}>{\mathrm{\tau }}^N$. Further comparing ${\mathrm{\tau }}^{NP}$ and ${\mathrm{\tau }}^{NR}$, ${\mathrm{A}}^{NR}-{\mathrm{A}}^{NP}={\displaystyle \frac{2{\eta \beta }_T^2{k}_P\left({\pi }_R-{\pi }_P\right)+{\eta \beta }_P^2{k}_T(2{\pi }_R-{\pi }_P)}{2k_T{k}_P(\lambda +\rho )}}$. From this equation, it follows that when ${\displaystyle \frac{{\pi }_P}{{\pi }_R}}<{\displaystyle \frac{{2\eta \beta }_T^2{k}_P+2{\eta \beta }_P^2{k}_T}{{2\eta \beta }_T^2{k}_P+{\eta \beta }_P^2{k}_T}}$,${\mathrm{\tau }}^{NP}<{\mathrm{\tau }}^{NR}$, then ${\mathrm{\tau }}^{NPR}<{\mathrm{\tau }}^{NRP}$, the reverse also holds. Furthermore, if, ${\pi }_P<{\pi }_R$, ${\mathrm{\tau }}^{NPR}<{\mathrm{\tau }}^{NRP}$ always holds.

Similarly, regarding the preservation quality of fresh agricultural products, since, ${E_F^{NRP\mathrm{*}}>E}_F^{NR\mathrm{*}}=E_F^{N\mathrm{*}}=E_F^{NP\mathrm{*}},E_F^{NPR\mathrm{*}}>E_F^{NR\mathrm{*}}=E_F^{N\mathrm{*}}=E_F^{NP\mathrm{*}}$, it follows that ${\mathrm{B}}^{NRP}>{\mathrm{B}}^{NR}={\mathrm{B}}^N={\mathrm{B}}^{NP}$, ${\mathrm{B}}^{NPR}>{\mathrm{B}}^{NR}={\mathrm{B}}^N={\mathrm{B}}^{NP}$, leading to ${\chi }^{NPR}>{\chi }^{NP}>{\chi }^{NR}>{\chi }^N$, ${\chi }^{NRP}>{\chi }^{NP}>{\chi }^{NR}>{\chi }^N$. Further comparing ${\chi }^{NPR}$ and ${\chi }^{NRP}$, we have ${\chi }^{NPR}-{\chi }^{NRP}={\displaystyle \frac{1-e^{-\theta t}}{\theta }}({\mathrm{B}}^{NPR}-{\mathrm{B}}^{NRP})$, which leads to ${\mathrm{B}}^{NPR}-{\mathrm{B}}^{NRP}={\displaystyle \frac{\theta {\mathrm{\beta }}_F^2({\pi }_R-{\pi }_P)}{k_F(\theta +\rho )}}$, when ${\pi }_R<{\pi }_P$, ${\chi }^{NPR}<{\chi }^{NRP}$, and vice versa. Based on the above results regarding carbon emission reductions and the preservation quality of fresh agricultural products, we derive Proposition 7. Proof Completed. □

Compared to the Nash equilibrium scenario, the producer’s cost sharing with the TPL provider and retailer increases the incentives for emission reductions, thereby enhancing the overall carbon reductions in the fresh agricultural product supply chain and lowering carbon emissions. The same holds for the retailer sharing emission reduction costs. Similarly, the sharing of preservation costs by the producer and retailer enhances the TPL provider’s incentive to invest in preservation technology research, thereby promoting the improvement of preservation quality. However, adding the sharing of preservation costs on top of the carbon reduction cost sharing does not affect the reduction in carbon costs, meaning that the sharing of preservation costs cannot further lower carbon emission levels. Similarly, the sharing of carbon reduction costs does not impact preservation costs and thus cannot improve preservation quality.

In Proposition 7, it is concluded that when the producer’s marginal profit is lower than that of the retailer, the retailer’s sharing of carbon reduction costs results in greater carbon reductions, and the retailer’s sharing of preservation costs leads to a higher level of preservation quality. In the cooperative game model discussed in this section, the retailer can only share either carbon reduction costs or preservation costs, so in this case, they must choose between higher carbon reductions and better preservation quality. Is there a cooperative model that can both increase carbon reductions and improve preservation quality?

Through calculation, it is found that when $1<{\displaystyle \frac{{\pi }_P}{{\pi }_R}}<{\displaystyle \frac{{2\eta \beta }_T^2{k}_P+2{\eta \beta }_P^2{k}_T}{{2\eta \beta }_T^2{k}_P+{\eta \beta }_P^2{k}_T}}$, the conditions ${\mathrm{\tau }}^{NPR}<{\mathrm{\tau }}^{NRP},{{\chi }^{NPR}<\chi }^{NRP}$ are satisfied. Thus, Proposition 8 can be concluded.

Proposition 8.

When the marginal profits of producer and retailer meet a certain level, a cooperative contract that includes both carbon reduction cost sharing and preservation cost sharing can simultaneously enhance the carbon reduction level and preservation quality of the fresh agricultural product supply chain.

Proposition 8 indicates that compared to a single cost-sharing strategy, the cooperative contract for sharing both carbon reductions and preservation costs can enhance both the carbon reduction level and preservation quality level of the supply chain without the need to make a trade-off between the two. It only requires the relevant quantity conditions to be met.

Based on the calculations related to carbon reductions and preservation quality levels: $J_P^{NRP\mathrm{*}}>J_P^{NR\mathrm{*}}>J_P^{N\mathrm{*}}$, $J_P^{NPR\mathrm{*}}>J_P^{NP\mathrm{*}}>J_P^{N\mathrm{*}}$, $J_T^{NRP\mathrm{*}}>J_T^{NR\mathrm{*}}>J_T^{N\mathrm{*}}$, $J_T^{NPR\mathrm{*}}>J_T^{NP\mathrm{*}}>J_T^{N\mathrm{*}}$, $J_R^{NRP\mathrm{*}}>J_R^{NR\mathrm{*}}>J_R^{N\mathrm{*}}$, $J_R^{NPR\mathrm{*}}>J_R^{NP\mathrm{*}}>J_R^{N\mathrm{*}}$, Proposition 9 can be concluded.

Proposition 9.

The carbon reduction cost-sharing contract can increase the profit of the fresh agricultural product supply chain enterprises based on the Nash equilibrium, while the preservation cost-sharing contract can further increase the profit obtained by enterprises during the carbon reduction process, on top of the carbon reduction cost-sharing contract.

Proposition 9 shows that the profits of the producer, TPL provider, and retailer without cost sharing are all lower than their profits under cost sharing, indicating that supply chain participants reduce profit losses caused by double marginal effects through cost-sharing contracts, resulting in higher profit levels than in the absence of cost sharing.

Adding preservation cost sharing on top of carbon reduction cost sharing is equivalent to further adding the effect of cost-sharing contracts. The roles of preservation cost sharing and carbon reduction cost sharing in increasing profits are similar.

8. Example Analysis

This section verifies the conclusions obtained in the previous text through numerical simulation and performs a sensitivity analysis on several important parameters. The following values are assumed: ${\beta }_P=2$, ${\beta }_F=3,{\beta }_T=2.8$, ${\beta }_R=2$, ${\pi }_P=8$, ${\pi }_T=1.2$, ${\pi }_R=3$, $\eta =2$, $k_P=20$, $k_T=12$, $k_F=16$, $k_R=19$, $\theta =1$, $\lambda =1$, $\rho =0.8$, $\delta =3.2$, $D_0=20$, ${\tau }_0=3$, $\xi =2$.

(1) The Impact of Marginal Profit on Emission Reduction Effort and Freshness Effort

Figure 2, Figure 3 and Figure 4 show the effect of marginal profit on emission reduction efforts and investment in freshness technology development. Figure 2 shows that when the marginal profit of the producer is greater than that of the retailer, the TPL provider will exert more effort in emission reductions when the producer shares the emission reduction costs and will invest more in freshness technology development when the producer shares the freshness costs. Conversely, when the marginal profit of the producer is less than that of the retailer, the opposite occurs. This indicates that, for the TPL provider, when the entity with a larger marginal profit shares the costs, the TPL provider will exert greater effort.

Figure 3a shows that regardless of the relationship between the marginal profits of the producer and the retailer, the producer always exerts a greater reduction effort when the retailer shares the emission reduction costs. Figure 3b shows that the retailer always exerts greater low-carbon promotion reduction effort when the producer shares the costs of low-carbon promotion. The changes in emission reduction efforts of the producer and retailer indicate that when one party bears the corresponding emission reduction costs, it increases the reduction efforts of the other party, thereby validating Proposition 6.

(2) The impact of marginal profit on emission reduction volume and preservation quality level

Figure 4a shows the change in product emission reductions over time when the producer’s marginal profit is less than the retailer’s. When the producer or retailer shares the emission reduction costs, the reduction volume is greater than that without cost sharing. This indicates that cost sharing promotes the increase in emission reductions. Furthermore, when the producer’s marginal profit is less than the retailer’s, the emission reduction volume when the retailer shares the emission reduction costs is greater than when the producer shares the costs. This suggests that in this case, the retailer sharing the emission reduction costs better enhances the reduction effort, leading to higher emission reductions.

Figure 5a shows the change in agricultural product preservation quality over time when the producer’s marginal profit is less than the retailer’s. The preservation quality under cost sharing is higher than that without cost sharing. This indicates that sharing preservation costs enhances preservation efforts, thereby improving preservation quality. Furthermore, when the producer’s marginal profit is less than the retailer’s, the retailer sharing the preservation costs leads to better preservation quality. This suggests that in this case, the retailer sharing preservation costs better enhances preservation efforts and improves agricultural product freshness. This validates Proposition 7.

(3) Collaborative Model for Simultaneously Enhancing Emission Reduction and Preservation Quality

Figure 6 shows the changes in emission reductions and preservation quality when the marginal profits of the producer and retailer satisfy the conditions of Proposition 8. The thick line in the figure represents preservation quality, while the thin line represents emission reductions. When $1<{\displaystyle \frac{{\pi }_P}{{\pi }_R}}<{\displaystyle \frac{{2\eta \beta }_T^2{k}_P+2{\eta \beta }_P^2{k}_T}{{2\eta \beta }_T^2{k}_P+{\eta \beta }_P^2{k}_T}}$, the model where the retailer shares the emission reduction costs and the producer shares the preservation costs can also simultaneously enhance emission reductions and preservation quality. This indicates that as long as the emission reduction cost coefficient, preservation cost coefficient, the impact of reduction effort on emission reductions, the impact of preservation effort on preservation quality, and marginal profits satisfy certain conditions, both emission reductions and preservation quality can be enhanced in the cooperative model of cost sharing for emission reductions and preservation. This validates Proposition 8.

(4) The Profit Size Between Fresh Agricultural Product Supply Chain Enterprises

Figure 7 shows the profit sizes of enterprises under different models. The profit of supply chain entities with cost sharing is always greater than that without cost sharing. Furthermore, the profit under the joint cost-sharing model of two costs is always higher than the profit under a single cost-sharing model. Compared with the Nash equilibrium state without cost sharing, the emission reduction cost sharing increases the enterprises’ motivation for emission reductions, leading to higher emission reduction volumes, which in turn increases market demand for fresh agricultural products, resulting in higher profits for the enterprises. The preservation cost-sharing contract improves market demand by enhancing the preservation quality of fresh agricultural products, thereby gaining higher profits. Moreover, adding preservation cost sharing on top of emission reduction cost sharing acts like an additional layer of cost sharing, making the enterprise’s profit significantly higher than under the single cost-sharing model, further enhancing the enterprise’s profit level. This validates Proposition 9.

Then, we analyze the trajectory of carbon emission reductions and preservation quality levels of fresh agricultural products under decentralized decision-making. As shown in Figure 8, the carbon emission reductions and preservation quality levels of fresh agricultural products gradually increase over time, but their growth rate decreases and gradually tends towards a steady state. This indicates that in the initial stages, technological innovations led to significant improvements, but as the technology continued to mature, the growth rate gradually slowed, ultimately reaching a balance point in terms of both technology and management. External factors, such as changes in market demand, policy guidance, and climate change, may cause short-term disturbances to carbon reduction levels and preservation quality. However, the impact of the external environment is ultimately temporary.

9. Conclusions, Managerial Implications, and Future Research Directions

9.1. Findings and Managerial Implications

This paper studies the dynamic carbon reduction decision-making problem in a three-tier fresh agricultural product supply chain composed of producers, TPL providers, and retailers, considering factors such as consumers’ low-carbon preferences and preservation preferences. Fresh products have no fixed shelf life, and market demand is influenced by the combined effects of preservation and carbon reduction efforts, as well as low-carbon promotion efforts. A dynamic carbon reduction differential game model is constructed, and the equilibrium decisions under five different cost-sharing models are compared to analyze the impact of different cooperation models on carbon reductions, preservation quality, and enterprise profits, leading to the following conclusions:

(1) For producers, the level of carbon reductions and preservation efforts depends on the marginal profit levels of producers and retailers. When the party with the larger marginal profit bears the costs, TPL providers will make greater efforts in carbon reductions or preservation. For both retailers and producers, when one party bears the carbon reduction costs, it motivates the other party to exert greater efforts in carbon reductions.

(2) Carbon reduction cost sharing helps to increase the reduction amount, while preservation cost sharing can improve preservation quality. However, preservation cost sharing does not affect the reduction amount, and carbon reduction cost sharing does not affect preservation quality. When the marginal profit of the producer is lower than that of the retailer, the retailer’s reductions or preservation quality under cost-sharing contracts for carbon reductions or preservation is higher. If specific relationships are met between the carbon reduction cost coefficient, preservation cost coefficient, the effect of carbon reduction efforts on the reduction amount, the effect of preservation efforts on preservation quality, and marginal profits, cost-sharing models for both carbon reductions and preservation can simultaneously increase both the reduction amount and preservation quality.

(3) Compared to the situation with no cost sharing, carbon reduction cost sharing can increase the profits of the participants in the fresh agricultural product supply chain. The addition of preservation cost sharing on top of carbon reduction cost sharing can further enhance profit levels. In the absence of cost sharing, all participants must independently bear their own costs, which may lead to inefficiency, unequal distribution of benefits, and a lack of effective cooperation incentives. However, the introduction of carbon reduction cost sharing can break this situation by distributing the carbon reduction costs within the supply chain, motivating all parties to adopt more active carbon reduction measures, thus improving overall efficiency.

(4) Over time, the carbon reduction amount of fresh products shows a slow and steady growth trend. This change is not merely a simple linear relationship; it reflects the increasing importance of agricultural production activities in reducing greenhouse gas emissions in the context of global warming. Although the rate of carbon reduction growth has slowed in this process, overall, the process continues to develop steadily. Nevertheless, fluctuations in the external environment still impact the carbon reduction amount of fresh products. For example, extreme weather events, crop pests, and changes in international trade policies may lead to temporary deviations or fluctuations in the reduction amount. However, with appropriate management and adaptation to these external factors, these fluctuations will eventually return to a more balanced state. This balanced state not only ensures the sustainability of carbon reductions but also ensures that the entire supply chain can effectively respond to potential future challenges. Therefore, despite uncertainty and changes, the efforts to reduce the carbon footprint of fresh products will continue and are expected to achieve long-term environmental benefits.

(5) The preservation quality level of fresh agricultural products first increases and then stabilizes. In the early stages, with the investment and improvement in cold chain technology, preservation quality has significantly improved. This may be due to the introduction of technological innovations, improved supply chain coordination, and the continuous optimization of equipment and facilities, which allow preservation quality to improve quickly. However, as technology matures and supply chain coordination mechanisms stabilize, the rate of improvement in preservation quality begins to slow and eventually stabilizes. Furthermore, although changes in the external environment may affect short-term preservation quality, in the long term, internal coordination within the supply chain and technological advancements will become key factors in ensuring product preservation quality. Therefore, supply chain managers should focus on long-term development, continuously invest in technological innovation, and optimize management processes to ensure the long-term stability and efficiency of the cold chain system.

(6) Consumers’ low-carbon preferences have a greater impact on the carbon reduction efforts of TPL providers than on the efforts of producers. In a centralized decision-making scenario, the carbon reduction efforts of producers, TPL providers, and preservation efforts are all higher than those in decentralized decision-making. As the discount rate increases, the carbon reduction efforts of producers and TPL providers continue to decline, while TPL providers’ preservation efforts increase. Through the implementation of popular science activities, society as a whole can pay more attention to environmental issues, and through enhanced education, people can learn more about environmental protection knowledge, thus cultivating more low-carbon behaviors. Enhancing consumers’ environmental awareness helps increase the carbon reduction efforts of fresh supply chain enterprises, promoting the improvement of the carbon reduction level in low-carbon fresh supply chains, especially improving TPL providers’ carbon reduction efforts.

9.2. Future Research Directions

This study still has some limitations that require further exploration and improvement. For instance, it only considers a single business entity within the fresh agricultural product supply chain and does not account for the impact of competitive relationships on dynamic carbon reductions in the supply chain. Additionally, future research could consider the effects of government subsidies and carbon trading policies on the dynamic carbon reduction mechanisms within the fresh agricultural product supply chain.