Differential Game Analysis of Emission Reduction and Preservation in the Tertiary Food Supply Chain under Different Government Subsidy Models

Abstract

Climate change has brought about huge environmental problems and economic losses worldwide, and low carbon has become a hot topic of research in the context of the climate crisis. The article outlines a three-tier food supply chain consisting of suppliers, manufacturers and retailers, based on a cooperative model in which manufacturers share the cost of emission reduction and retailers share the cost of freshness. The study finds that when the government grants the maximum amount of subsidies to the manufacturer, the government’s decision to reduce emissions, the government’s regulations to preserve freshness, and the government’s profits are not affected. The study found that when the government awarded the largest subsidy coefficient to manufacturers and the smallest subsidy coefficient to retailers, the non-specific one-off government subsidy model could lead to higher profits for suppliers, manufacturers, and retailers, increasing the amount of emission reduction while improving the preservation quality of food. When the government subsidy factor for manufacturers is minimal, the reduction in emissions is greater under the government’s dedicated abatement subsidy model but does not increase the freshness quality of the food; food supply chain players consistently produce higher levels of preservation effort and preservation quality under the government’s non-dedicated one-off subsidy model.

1. Introduction

Climate change is causing huge environmental problems and economic losses worldwide, and there is a consensus on the climate crisis. With the 2015 Paris Agreement’s temperature control targets, countries have launched emission reduction plans one after another. More than 130 countries and regions have already proposed “zero carbon” or “carbon neutral” climate targets. China will also increase its national contribution and aim to hit peak CO₂ emissions by 2030 and work towards achieving carbon neutrality by 2060.

Carbon emissions from the food supply chain are widespread and significant. According to a study, globally, approximately 13.7 billion metric tonnes of carbon dioxide equivalent (CO₂) are emitted through the food supply chain each year, across 119 countries and 38,000 commercial farms. This shows that the impact of carbon emissions from the food supply chain on climate change cannot be ignored. In order to reduce carbon emissions, suppliers need to reduce their carbon footprint in the production of raw materials; manufacturers need to invest in low-carbon research and development in the production and processing of food products and reduce carbon emissions from energy during transportation; retailers need to guide consumers’ low-carbon purchasing behaviour through promotional tools (e.g., low-carbon labels, low-carbon consumer rebates and price discounts) in the sales process [1]. These supply chain enterprises reduce carbon emissions, but in the process increase corporate costs; in order to increase the incentive of enterprises to reduce emissions, the government, as the advocate of emission reduction, often takes a series of measures to reduce the cost of emission reduction of enterprises in order to reduce carbon emissions.

In order to combat climate change and reduce carbon emissions, the implementation of national fiscal policies has become a key initiative. According to the International Monetary Fund (IMF), fiscal policy can play a central role in addressing climate change by promoting investment in mitigation and adaptation and driving quality economic development. Fiscal policies to address climate change are currently being developed from both expenditure and revenue perspectives to promote a low-carbon transition and enhance resilience to climate change. For fiscal revenue policies, countries have mainly adopted measures such as setting up carbon taxes, issuing sovereign green bonds, and establishing carbon market revenue funds to address climate change. In terms of fiscal expenditure policies, the main measures to mitigate and adapt to climate change are integrating climate expenditure into existing fiscal budgets, proposing special low-carbon expenditure plans, low-carbon fiscal subsidies, setting climate budgets, and promoting the formation of climate-friendly government procurement [2].

At present, the domestic government in China mainly uses fiscal spending to encourage local enterprises to reduce carbon emissions. In Anhui Wuhu Fanchang District, a subsidy of up to RMB 8 million was given to high energy-consuming and high-emission enterprises that took the initiative to close down in March 2021. Shanghai Xuhui District also issued a special fund management scheme for energy conservation, emission reduction, and carbon reduction in March 2021 to encourage enterprises to save energy, reduce emissions, and reduce carbon. Jiangsu Province implemented a fiscal policy linked to the effectiveness of pollution reduction and carbon reduction in February 2021, returning a certain percentage of the coordinated funds to cities and counties that met the target. Yunnan, Beijing, Hangzhou, Shenzhen, Guangzhou and other places have also released relevant policies. From the above policies, domestic policies are mainly in the form of incentives and subsidies, with enterprises being the main beneficiaries. Government incentives and subsidies for enterprises can be divided into two forms: one is a special government subsidy for emissions reduction, and the other is a non-specific one-off subsidy. Emission reduction subsidies refer to special production subsidies for suppliers and manufacturers to invest in research and development of emission reduction technologies and new energy devices, and special consumption subsidies for retailers of low-carbon products, thereby enhancing the competitiveness of low-carbon projects and achieving a reduction in carbon emissions. Non-specific one-off subsidies refer to the government’s provision of a certain percentage and share of subsidies directly to supply chain enterprises through incentives or subsidies, for example, a one-off subsidy for a company that has been certified as a “carbon neutral company” or a one-off subsidy or incentive for a company that has achieved a reduction in carbon emissions. It is important to explore the impact of the two different government subsidies on the decision making of food supply chain companies, on carbon emissions and food freshness, as well as on profits, and to provide a basis for government decisions.

In the context of the climate crisis and the global effort to reduce carbon emissions, the food supply chain, with its wide range of carbon emissions and large volume of emissions, has a practical significance in analysing the impact of carbon emission reduction policies on food supply chain enterprises to achieve the goal of carbon peaking and carbon neutrality. At the same time, exploring the impact of two different government subsidies on the decision making of food supply chain enterprises, on carbon emissions and food freshness, as well as on their profits is not only of theoretical significance but also provides a basis for the government to adopt different subsidy policies under different circumstances to achieve different policy objectives.

A number of scholars have studied the quality of food preservation in and around the low-carbon food supply chain. In terms of low-carbon food supply chains, Zhang Zongming [3] considered the carbon emissions in the transportation process of products, and by constructing a three-tier green supply chain consisting of manufacturers, third-party logistics suppliers, and retailers, he investigated the coordination mechanism of the green cost-sharing contract and the low-carbon cost-sharing contract when greenness and carbon emissions have an impact on product demand at the same time, so as to study the optimal decision of the supply chain under the two scenarios. By studying the supply chain of cold chain logistics services in the context of carbon cap-and-trade, Zhang Fang [4] investigated the impact of suppliers’ risk-averse tendency on the reduction of emissions and supply chain coordination in the supply chain of cold chain logistics services. Chen Jing [5] explored the carbon emissions of the meat food supply chain by establishing a system dynamics model of the fresh meat supply chain and found that the transportation link had the highest contribution to the carbon emissions of the whole supply chain. Zhang Tianxia [6] considered transportation speed and carbon emissions in the perishable food supply chain and concluded that carbon emissions and transportation speed are related; he also derived the optimal transportation speed when the supply chain reaches coordination. In terms of food preservation quality, Wu Qing [7] studied a secondary supply chain composed of distributors and third-party logistics providers, incorporated freshness and price into the demand function, and designed benefit-sharing, cost-sharing and price discount contracts. Cao Yu [8] constructed a secondary supply chain composed of suppliers and retailers, set demand as a linear function of price and freshness effort, and analysed the freshness decisions among supply chain subjects. Liu Molin [9] constructed a second-tier supply chain consisting of fresh food suppliers and e-merchants, set the demand function as a linear function of price, preservation effort, and value-added service level, analysed preservation decisions as well as value-added service level decisions, and designed a contractual incentive mechanism based on benefit sharing and two-way cost sharing. Ma Xueli [10] constructed a three-tier supply chain consisting of farmers and third-party logistics providers. Wang Lei [11] used the Stackelberg game method to construct the revenue function of the fresh produce supply chain, compare the optimal pricing and optimal preservation effort under decentralized and centralized decision making, and design a revenue sharing contract. Wang Shuyun [12] introduced freshness preservation effort and quality loss into the model to study the optimal ordering quantity of supply chain members when the profit of the supply chain system is maximized. Cao Xiaoning [13] et al. constructed a dual-channel supply chain, considered the influence of supplier freshness preservation effort on freshness decay, and studied the decision making and coordination of supply chain subjects. The above scholars studied the decisions related to food supply chain agents when freshness preservation effort was considered. Singh et al. [14] analysed the barriers to growth in fresh produce supply chains (FPSC) and found that it was possible to improve the quality of food products by improving cold chain facilities. Lavelli [15] studied circular supply chains in food systems and found that new loops in food supply chains may also present food safety risks. Barman et al. [16] discussed the impact of COVID-19 on food supply chains and agribusiness. Most of the above studies analysed firms’ reduction or preservation decisions in terms of games between firms within the supply chain and did not include government in the supply chain to consider the role it plays in the decisions of supply chain firms.

Some scholars have further studied the role of government subsidies on enterprises’ emission reduction and carbon reduction. Wang Daoping [1] studied the dynamic optimisation of long-term cooperative emission reduction and promotion in supply chains where the government provides subsidies to manufacturers for emission reduction costs and to retailers for promotion costs. Zhang Fu’an and Li Na et al. [17] considered consumer low-carbon preferences and government subsidy policies and analysed the impact of different subsidy policies and consumer low-carbon preferences on emission reduction decisions. Zhang Lingrong and Peng Bo et al. [18] investigated the optimal strategy of low-carbon subsidies under four scenarios, including government subsidies for emission reduction technology inputs and subsidies for low-carbon product output. Cao et al. [19] studied the effects of carbon cap-and-trade policies and low-carbon subsidy policies on products and manufacturers’ emission reduction levels and explored which policy is more beneficial to society. Meng Fansheng and Han Bing [20] analysed the impact of three environmental regulatory instruments, including carbon emission reduction innovation subsidies, carbon taxes, and carbon trading, on firms’ carbon emission reduction behaviour by building an evolutionary game model between government and firms.

Most of the above studies only consider manufacturers and retailers in the emission reduction supply chain and do not consider suppliers in the low-carbon supply chain. In the literature related to government emission reduction subsidies, there are also fewer studies analysing the situation where the government provides subsidies to suppliers, manufacturers, and retailers at the same time. In reality, suppliers are also involved in the process of carbon emissions, and government subsidies to suppliers exist. Compared with existing studies, the innovations of this paper are as follows: (1) The research on carbon emission reduction in the supply chain focuses on the food supply chain; in fact, there is a more serious carbon emission problem in the food supply chain, and therefore it is more relevant to achieve the goal of carbon peaking and carbon neutrality by analysing the impact of carbon emission reduction policies on food supply chain enterprises in order to achieve emission reduction. (2) At present, most scholars only consider the secondary food supply chain formed by two of the suppliers, manufacturers and retailers, but rarely analyse the tertiary or more than tertiary food supply chain formed by suppliers, manufacturers, and retailers. This thesis constructs a differential game model for the three-tier food supply chain, which is more appropriate to the actual operation of the supply chain and more difficult to analyse, laying the foundation for future analysis of supply chains above the three-tier level or even the whole supply chain system. (3) In the current research on government low-carbon subsidies, the impact of government subsidies on emission reduction and food quality in the low-carbon food supply chain is also less explored in the literature. By adding government factors to the three-tier food supply chain and analysing the impact of different government subsidy policies on emission reduction, food preservation quality, and profits of food supply chain enterprises, this not only broadens the number of main actors in the food supply chain, but also provides a basis for enterprises to make decisions and adjustments according to different government subsidy policies.

In this paper, we consider a three-tier food supply chain system consisting of suppliers, manufacturers, and retailers in the model. The suppliers of raw food materials, the manufacturers of food products, and the retailers of food produce carbon emissions during the production, processing, transportation, and distribution of food and thus incur corresponding abatement costs. In addition, suppliers, manufacturers, and retailers all bear a certain amount of preservation costs in order to ensure the freshness of the food. In order to encourage companies to respond positively to the national emissions reduction policy, the government subsidises the costs of food supply chain companies. It is assumed that there are two forms of government subsidies: firstly, the government designates a specific purpose for the funds, stipulating that the subsidies can only be used for energy saving, green branding, energy management, new and renewable energy construction, green activities, and other special projects related to emission reduction; secondly, the government provides one-off financial subsidies directly to enterprises in the form of incentives and subsidies but does not specify the purpose of the subsidies, in which case the subsidies can be used for both emission reduction and other costs. In this paper, the government subsidies are incorporated into the Stackelberg game model based on a cooperative model in which manufacturers share the cost of emission reduction and retailers share the cost of freshness, and the government subsidies are compared with two different types of government subsidies, namely government-specific emission reduction subsidies and non-specific one-off subsidies. The impact of different government subsidies on the reduction decisions, freshness decisions, and profitability of food supply chain enterprises is under the same cost subsidy factor.

2. Model Establishment and Solution

2.1. Assumption

The following conditions are assumed in this paper:

(1)

The abatement costs of suppliers and manufacturers are an increasing function with respect to the level of abatement effort, and as the level of abatement effort continues to increase, the cost of abatement increases at an accelerating rate as the level of abatement effort increases, i.e., satisfying ${C_S}^{\prime }\left(E_S\left(t\right)\right)>0$ ${C_S}^{″}\left(E_S\left(t\right)\right)>0$ ${C_M}^{\prime }\left(E_M\left(t\right)\right)>0$ ${C_M}^{″}\left(E_M\left(t\right)\right)>0$. Similarly, the cost of promotion is an increasing function of the degree of low-carbon effort. Therefore, abatement costs and promotion costs are convex functions of abatement effort and promotion effort [21], respectively, so that the abatement costs of suppliers and manufacturers are

$$\begin{gathered} C_S\left(E_S\left(t\right)\right)=\frac{1}{2}{k}_S{E}_S^2\left(\mathrm{t}\right) \\ {C}_M\left(E_M\left(t\right)\right)=\frac{1}{2}{k}_M{E}_M^2\left(\mathrm{t}\right) \end{gathered}$$

The cost of low-carbon promotions for retailers is

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

where $E_S,E_M,E_R$ represents the abatement efforts of suppliers, manufacturers, and retailers, and $k_S,k_M,k_R$ represents the abatement cost factors of suppliers, manufacturers, and retailers.

(2)

The same as abatement efforts and abatement costs, the cost of preservation for suppliers, manufacturers, and retailers is a convex function of preservation effort. The higher the level of preservation effort, the higher the cost of further increasing the level of preservation effort. Therefore, the expression for the cost of preservation is

$$\begin{gathered} C_S\left({\tau }_S\left(t\right)\right)=\frac{1}{2}{\alpha }_S{\tau }_S^2\left(\mathrm{t}\right) \\ {C}_M\left({\tau }_M\left(t\right)\right)=\frac{1}{2}{\alpha }_M{\tau }_M^2\left(\mathrm{t}\right) \\ {C}_R\left({\tau }_R\left(t\right)\right)=\frac{1}{2}{\alpha }_R{\tau }_R^2\left(\mathrm{t}\right) \end{gathered}$$

where ${\tau }_S,{\tau }_M,{\tau }_R$ represents the preservation effort of the supplier, manufacturer, and retailer, and ${\alpha }_s, {\alpha }_M, {\alpha }_R$ represents the preservation cost factor of the supplier, manufacturer, and retailer.

(3)

The amount of emission reduction per unit of product depends on the level of emission reduction efforts of suppliers and manufacturers [22] and has a natural tendency to decline over time; its change process is dynamic. The low-carbon promotional behaviour of retailers in the sales process reduces carbon emissions by increasing consumer demand for low-carbon food, leading to low-carbon food consumption, but does not affect the emission reduction per unit of product. Therefore, the differential equation for emission reduction per unit of product is

$$\stackrel{•}{X}\left(t\right)={\beta }_S{E}_S+{\beta }_M{E}_M-\lambda X\left(t\right)$$

At the initial moment, the product emission reduction is $X\left(0\right)=X_0\ge 0$, $\stackrel{•}{X}\left(t\right)=\frac{\partial X\left(t\right)}{\partial t}$. ${\beta }_s, {\beta }_M$ denote the coefficient of influence of the supplier’s and manufacturer’s emission reduction efforts on the product emission reduction (both greater than 0), respectively, λ denotes the natural rate of decline in the supply chain product emission reduction, and $X\left(t\right)$ denotes the unit product emission reduction at moment t.

(4)

Drawing on studies by Wang et al. [23], Yu Rong et al. [24], Jorgensen et al. [25], and Tang Run et al. [26] on the dynamics of food quality in green and fresh foods, the quality of freshness ultimately purchased by consumers is the result of the preservation efforts of suppliers, manufacturers, and retailers. Therefore, the differential equation for food preservation quality is

$$\stackrel{•}{Q}\left(t\right)={\eta }_S{\tau }_S+{\eta }_M{\tau }_M+{\eta }_R{\tau }_R-\gamma Q\left(t\right)$$

At the initial moment, the preservation quality is $Q\left(0\right)=Q_0, Q\left(T\right)=0$, $\stackrel{•}{Q}\left(t\right)=\frac{\partial Q\left(t\right)}{\partial t}$. ${\eta }_S, {\eta }_M, {\eta }_R$ denote the coefficient of influence of the level of preservation effort of the supplier, manufacturer, and retailer on the preservation quality (all greater than 0), γ denotes the quality loss rate of the food product, and $Q\left(t\right)$ denotes the preservation quality of the food product at moment t.

(5)

Market demand for the product is assumed to be a linear combination of the amount of emission reduction, the level of retailer effort to promote low-carbon food, and the quality of freshness. The amount of product emission reduction, low-carbon promotion, and freshness quality all positively affect market demand. To control for variables and focus on key factors such as emission reduction effort and preservation effort, this paper assumes that the retail price of the food product remains constant and does not consider the effect of price on market demand [27]. Therefore the market demand is

$$D\left(X\left(t\right),Q\left(t\right),E_R\left(t\right)\right)=D_0+mX+nQ+\omega E_R$$

where $D_0$ represents the potential demand for the product at the initial moment, and $m, \omega , n$ represent the coefficient of influence of product reduction, retailer low-carbon promotion, and freshness quality on market demand (all greater than 0).

(6)

The abatement cost factors shared by the manufacturer for the supplier and the retailer are $l_S$ and $l_R$, respectively; the freshness cost factors shared by the retailer for the supplier and the manufacturer are ${\epsilon }_S$ and ${\epsilon }_M$. The cost-sharing factors are all real numbers between 0 and 1.

(7)

The government subsidy for abatement costs for suppliers, manufacturers, and retailers is ${\Phi }_S, {\Phi }_M, {\Phi }_R$ in the case of dedicated abatement subsidies, and the government subsidy factor is ${\psi }_S, {\psi }_M, {\psi }_R$ in the case of non-dedicated one-off government subsidies. In this paper, the superscript X indicates the case of government-specific abatement subsidies, and no superscript indicates the case of non-specific one-off government subsidies.

(8)

Taking into account the time value of money, the discount rate for suppliers, manufacturers, and retailers is $\rho >0$.

(9)

The marginal profit per unit of product for the supplier, manufacturer, and retailer are ${\pi }_S$, ${\pi }_M$, and ${\pi }_R$, respectively.

(10)

The values of the relevant parameters meet the basic conditions for the existence of an optimal solution to the decision.

2.2. Cooperation Game in the Case of Government-Specific Emission Reduction Subsidies

• The cooperative game model is based on the sharing of abatement costs by manufacturers for suppliers and retailers and by retailers for suppliers and manufacturers, with the addition of a role for the government, which guarantees a certain amount of abatement by issuing a special subsidy to food supply chain companies, stipulating that the subsidy can only be used for abatement. The objective function in this case is

  $$\begin{gathered} J_S^X\left(x,Q,t\right)={\displaystyle {\int }_0^{\infty }{e}^{-\rho t}}\left[{\pi }_s\left(D_0+mX+nQ+\omega E_R\right)-\frac{1}{2}(1-l_S-{\Phi }_S)k_s{E}_s^2-\frac{1}{2}(1-{\epsilon }_s){\alpha }_s{\tau }_s^2\right]dt \\ {J}_R^X\left(x,Q,t\right)={\displaystyle {\int }_0^{\infty }{e}^{-\rho t}}\left[{\pi }_R\left(D_0+mX+nQ+\omega E_R\right)-\frac{1}{2}(1-l_R-{\Phi }_R)k_R{E}_R^2-\frac{1}{2}{\alpha }_R{\tau }_R^2-\frac{1}{2}{\epsilon }_S{\alpha }_S{\tau }_S^2-\frac{1}{2}{\epsilon }_M{\alpha }_M{\tau }_M^2\right]dt \\ {J}_M^X\left(x,Q,t\right)={\displaystyle {\int }_0^{\infty }{e}^{-\rho t}}\left[{\pi }_M\left(D_0+mX+nQ+\omega E_R\right)-\frac{1}{2}(1-{\Phi }_M)k_M{E}_M^2-\frac{1}{2}{l}_s{k}_s{E}_s^2-\frac{1}{2}{l}_R{k}_R{E}_R^2-\frac{1}{2}(1-{\epsilon }_M){\alpha }_M{\tau }_M^2\right]dt \end{gathered}$$

Theorem 1.

The equilibrium results under the government-specific abatement subsidy scenario are

(1) The optimal equilibrium decision of the food supply chain agents is

$$\begin{array}{l}{E}_S^{X*}=\frac{m{\beta }_S(2{\pi }_M+{\pi }_S)}{2k_S(\rho +\lambda )(1-{\Phi }_S)}\\ E_M^{X*}=\frac{m{\beta }_M{\pi }_M}{k_M(\rho +\lambda )(1-{\Phi }_M)}\\ E_R^{X*}=\frac{\omega ({\pi }_R+2{\pi }_M)}{2k_R(1-{\Phi }_R)}\end{array}\begin{array}{l}{\tau }_S^{X*}=\frac{n{\eta }_S(2{\pi }_R+{\pi }_S)}{2{\alpha }_S(\rho +\gamma )}\\ {\tau }_M^{X*}=\frac{n{\eta }_M(2{\pi }_R+{\pi }_M)}{2{\alpha }_M(\rho +\gamma )}\\ {\tau }_R^{X*}=\frac{n{\eta }_R{\pi }_R}{{\alpha }_R(\rho +\gamma )}\end{array}\begin{array}{l}{l}_S^X=\frac{(2{\pi }_M-{\pi }_S)(1-{\Phi }_S)}{(2{\pi }_M+{\pi }_S)}\\ l_R^X=\frac{(2{\pi }_M-{\pi }_R)(1-{\Phi }_R)}{(2{\pi }_M+{\pi }_R)}\end{array}\begin{array}{l}{\epsilon }_S^X=\frac{2{\pi }_R-{\pi }_S}{(2{\pi }_R+{\pi }_S)}\\ {\epsilon }_M^X=\frac{2{\pi }_R-{\pi }_M}{(2{\pi }_R+{\pi }_M)}\end{array}$$

(2) The optimal trajectories for product reduction and freshness retention are

$$\begin{array}{l}X=(x_0-x_{\infty }^X)e^{-\lambda t}+x_{\infty }^X\\ Q^X=(Q_0-\frac{B^X}{\gamma })e^{-\gamma t}+\frac{B^X}{\gamma }\end{array}$$

Among the formulas, $x_{\infty }=\frac{A^X}{\lambda }, A^X=\frac{m{\beta }_S^2(2{\pi }_M+{\pi }_S)}{2k_S(\rho +\lambda )(1-{\Phi }_S)}+\frac{m{\beta }_M^2{\pi }_M}{k_M(\rho +\lambda )(1-{\Phi }_M)}$

$B^X=\frac{n{\eta }_S^2(2{\pi }_R+{\pi }_S)}{2{\alpha }_S(\rho +\gamma )}+\frac{n{\eta }_M^2(2{\pi }_R+{\pi }_M)}{2{\alpha }_M(\rho +\gamma )}+\frac{n{\pi }_R{\eta }_R^2}{{\alpha }_R(\rho +\gamma )}$

(3) The optimal profit value function for suppliers, manufacturers, and retailers is

$$\begin{array}{l}{J}_S^X\left(x,Q,t\right)=e^{-\rho t}\cdot \left(a_1^{\mathrm{X}}X+a_2^{X}Q+a_3^X\right)\\ J_M^X\left(x,Q,t\right)=e^{-\rho t}\cdot \left(b_1^{X}X+b_2^{X}Q+b_3^X\right)\\ J_R^X\left(x,Q,t\right)=e^{-\rho t}\cdot \left(c_1^{X}X+c_2^{X}Q+c_3^X\right)\end{array}$$

Among the formulas,

$$\begin{array}{l}\begin{array}{l}{a}_1^X=\frac{m{\pi }_s}{\rho +\lambda };a_2^X=\frac{n{\pi }_s}{\rho +\gamma }\\ a_3^X=\frac{1}{\rho }\left({\pi }_s{D}_0+\frac{{\beta }_S^2{m}^2{\pi }_s({\pi }_s+2{\pi }_M)}{4k_s{(\rho +\lambda )}^2(1-{\Phi }_S)}+\frac{{\beta }_M^2{m}^2{\pi }_s{\pi }_M}{k_M{(\rho +\lambda )}^2(1-{\Phi }_M)}+\frac{{\omega }^2{\pi }_s({\pi }_R+2{\pi }_M)}{2k_R(1-{\Phi }_R)}+\frac{{\eta }_S^2{n}^2{\pi }_s({\pi }_s+2{\pi }_R)}{4{\alpha }_S{(\rho +\gamma )}^2}+\frac{{\eta }_M^2{n}^2{\pi }_s(2{\pi }_R+{\pi }_M)}{2{\alpha }_M{(\rho +\gamma )}^2}+\frac{{\eta }_R^2{n}^2{\pi }_s{\pi }_R}{{\alpha }_R{(\rho +\gamma )}^2}\right)\end{array}\\ \begin{array}{l}{b}_1^X=\frac{m{\pi }_M}{\rho +\lambda };b_2^X=\frac{n{\pi }_M}{\rho +\gamma }\\ b_3^X=\frac{1}{\rho }\left({\pi }_M{D}_0+\frac{{\beta }_S^2{m}^2{({\pi }_s+2{\pi }_M)}^2}{8k_s{(\rho +\lambda )}^2(1-{\Phi }_S)}+\frac{{\beta }_M^2{m}^2{\pi }_M^2}{2k_M{(\rho +\lambda )}^2(1-{\Phi }_M)}+\frac{{\omega }^2{({\pi }_R+2{\pi }_M)}^2}{8k_R(1-{\Phi }_R)}+\frac{{\eta }_S^2{n}^2{\pi }_M(2{\pi }_R+{\pi }_S)}{2{\alpha }_S{(\rho +\gamma )}^2}+\frac{{\eta }_M^2{n}^2{\pi }_M(2{\pi }_R+{\pi }_M)}{4{\alpha }_M{(\rho +\gamma )}^2}+\frac{{\eta }_R^2{n}^2{\pi }_R{\pi }_M}{{\alpha }_R{(\rho +\gamma )}^2}\right)\end{array}\\ \begin{array}{l}{c}_1^X=\frac{m{\pi }_R}{\rho +\lambda };c_2^X=\frac{n{\pi }_R}{\rho +\gamma }\\ c_3^X=\frac{1}{\rho }\left({\pi }_s{D}_0+\frac{{\beta }_S^2{m}^2{\pi }_R({\pi }_s+2{\pi }_M)}{2k_s{(\rho +\lambda )}^2(1-{\Phi }_S)}+\frac{{\beta }_M^2{m}^2{\pi }_R{\pi }_M}{k_M{(\rho +\lambda )}^2(1-{\Phi }_M)}+\frac{{\omega }^2{\pi }_R({\pi }_R+2{\pi }_M)}{4k_R(1-{\Phi }_R)}+\frac{{\eta }_S^2{n}^2{(2{\pi }_R+{\pi }_S)}^2}{8{\alpha }_S{(\rho +\gamma )}^2}+\frac{{\eta }_M^2{n}^2{(2{\pi }_R+{\pi }_M)}^2}{8{\alpha }_M{(\rho +\gamma )}^2}+\frac{{\eta }_R^2{n}^2{\pi }_R^2}{2{\alpha }_R{(\rho +\gamma )}^2}\right)\end{array}\end{array}$$

From Theorem 1, it follows that when government subsidies are used only for abatement, the government subsidy cost factor increases the abatement effort of food supply chain firms while increasing the abatement per unit of product, the extent of which is related to the government’s cost subsidy factor for the firm. On the other hand, government abatement subsidies reduce the manufacturer’s cost-sharing of abatement to suppliers and retailers, indicating that the manufacturer’s original cost-sharing role of abatement to suppliers and retailers is replaced to some extent by the government; it is noteworthy that government abatement cost subsidies do not have an impact on firms’ preservation decisions and preservation cost-sharing coefficients, and also fail to improve the preservation quality of food products; from the profit expressions, the government-specific abatement subsidies increase the profits of suppliers, manufacturers, and retailers under the cooperative game.

2.3. Cooperation Game in the Case of Non-Specific One-off Government Subsidies

The cooperative game model in Section 2.2, in which manufacturers share the cost of abatement and retailers share the cost of preservation, is still used as a basis, except that the government grants one-off subsidies in the form of incentives or subsidies that are not specified for a specific purpose, i.e., the government subsidies can be used for both abatement and preservation. In this case, the impact of the government’s one-off subsidy on the decision of the food supply chain firms is analysed. The objective function in this case is

$$\begin{gathered} J_S\left(x,Q,t\right)={\displaystyle {\int }_0^{\infty }{e}^{-\rho t}}\left\{{\pi }_s\left(D_0+mX+nQ+\omega E_R\right)-(1-{\psi }_s)\left[\frac{1}{2}(1-l_s)k_s{E}_s^2+\frac{1}{2}(1-{\epsilon }_s){\alpha }_s{\tau }_s^2\right]\right\}dt \\ {J}_M\left(x,Q,t\right)={\displaystyle {\int }_0^{\infty }{e}^{-\rho t}}\left\{{\pi }_M\left(D_0+mX+nQ+\omega E_R\right)-(1-{\psi }_M)\left[\frac{1}{2}{k}_M{E}_M^2+\frac{1}{2}{l}_s{k}_s{E}_s^2+\frac{1}{2}{l}_R{k}_R{E}_R^2+\frac{1}{2}(1-{\epsilon }_M){\alpha }_M{\tau }_M^2\right]\right\}dt \\ {J}_R\left(x,Q,t\right)={\displaystyle {\int }_0^{\infty }{e}^{-\rho t}}\left\{{\pi }_R\left(D_0+mX+nQ+\omega E_R\right)-\left(1-{\psi }_R\right)\left[\frac{1}{2}(1-l_R)k_R{E}_R^2+\frac{1}{2}{\alpha }_R{\tau }_R^2+\frac{1}{2}{\epsilon }_S{\alpha }_S{\tau }_S^2+\frac{1}{2}{\epsilon }_M{\alpha }_M{\tau }_M^2\right]\right\}dt \end{gathered}$$

Theorem 2.

The equilibrium decision in the case of a one-off government subsidy is

(1) The optimal equilibrium decision of the food supply chain agents is

$$\begin{gathered} \begin{array}{l}{E}_S^{*}=\frac{m{\beta }_S\left[2{\pi }_M(1-{\psi }_S)+{\pi }_S(1-{\psi }_M)\right]}{2k_S(\rho +\lambda )(1-{\psi }_M)(1-{\psi }_S)}\\ E_M^{*}=\frac{m{\beta }_M{\pi }_M}{k_M(\rho +\lambda )(1-{\psi }_M)}\\ E_R^{*}=\frac{\omega \left[{\pi }_R(1-{\psi }_M)+2{\pi }_M(1-{\psi }_R)\right]}{2k_R(1-{\psi }_M)(1-{\psi }_R)}\end{array}\begin{array}{l}{\tau }_S^{*}=\frac{n{\eta }_S\left[2{\pi }_R(1-{\psi }_S)+{\pi }_S(1-{\psi }_R)\right]}{2{\alpha }_S(\rho +\gamma )(1-{\psi }_S)(1-{\psi }_R)}\\ {\tau }_M^{*}=\frac{n{\eta }_M\left[2{\pi }_R(1-{\psi }_M)+{\pi }_M(1-{\psi }_R)\right]}{2{\alpha }_M(\rho +\gamma )(1-{\psi }_M)(1-{\psi }_R)}\\ {\tau }_R^{*}=\frac{n{\eta }_R{\pi }_R}{{\alpha }_R(\rho +\gamma )(1-{\psi }_R)}\end{array} \\ \begin{array}{l}{l}_S=\frac{2{\pi }_M(1-{\psi }_S)-{\pi }_S(1-{\psi }_M)}{2{\pi }_M(1-{\psi }_S)+{\pi }_S(1-{\psi }_M)}\\ l_R=\frac{2{\pi }_M(1-{\psi }_R)-{\pi }_R(1-{\psi }_M)}{2{\pi }_M(1-{\psi }_R)+{\pi }_R(1-{\psi }_M)}\end{array}\begin{array}{l}{\epsilon }_S=\frac{2{\pi }_R(1-{\psi }_S)-{\pi }_S(1-{\psi }_R)}{2{\pi }_R(1-{\psi }_S)+{\pi }_S(1-{\psi }_R)}\\ {\epsilon }_M=\frac{2{\pi }_R(1-{\psi }_M)-{\pi }_M(1-{\psi }_R)}{2{\pi }_R(1-{\psi }_M)+{\pi }_M(1-{\psi }_R)}\end{array} \end{gathered}$$

(2) The optimal trajectories for product reduction and freshness retention are

$$\begin{array}{l}X=(x_0-x_{\infty })e^{-\lambda t}+x_{\infty }\\ Q=(Q_0-\frac{B}{\gamma })e^{-\gamma t}+\frac{B}{\gamma }\end{array}$$

Among the formulas, $x_{\infty }=\frac{A}{\lambda }, A=\frac{m{\beta }_S^2\left[2{\pi }_M(1-{\psi }_S)+{\pi }_S(1-{\psi }_M)\right]}{2k_S(\rho +\lambda )(1-{\psi }_M)(1-{\psi }_S)}+\frac{m{\beta }_M^2{\pi }_M}{k_M(\rho +\lambda )(1-{\psi }_M)}$

$B=\frac{n{\eta }_S^2\left[2{\pi }_R(1-{\psi }_S)+{\pi }_S(1-{\psi }_R)\right]}{2{\alpha }_S(\rho +\gamma )(1-{\psi }_S)(1-{\psi }_R)}+\frac{n{\eta }_M^2\left[2{\pi }_R(1-{\psi }_M)+{\pi }_M(1-{\psi }_R)\right]}{2{\alpha }_M(\rho +\gamma )(1-{\psi }_M)(1-{\psi }_R)}+\frac{n{\pi }_R{\eta }_R^2}{{\alpha }_R(\rho +\gamma )(1-{\psi }_R)}$

(3) The optimal profit value function for suppliers, manufacturers, and retailers is

$$\begin{array}{l}{J}_S\left(x,Q,t\right)=e^{-\rho t}\cdot \left(a_{1}X+a_{2}Q+a_3\right)\\ J_M\left(x,Q,t\right)=e^{-\rho t}\cdot \left(b_{1}X+b_{2}Q+b_3\right)\\ J_R\left(x,Q,t\right)=e^{-\rho t}\cdot \left(c_{1}X+c_{2}Q+c_3\right)\end{array}$$

Among the formulas,

$$\begin{array}{l}\begin{array}{l}{a}_1=\frac{m{\pi }_s}{\rho +\lambda };a_2=\frac{n{\pi }_s}{\rho +\gamma }\\ a_3=\frac{1}{\rho }\left(\begin{array}{l}{\pi }_s{D}_0+\frac{{\beta }_S^2{m}^2{\pi }_s\left[{\pi }_s(1-{\psi }_M)+2{\pi }_M(1-{\psi }_S)\right]}{4k_s{(\rho +\lambda )}^2(1-{\psi }_M)(1-{\psi }_S)}+\frac{{\beta }_M^2{m}^2{\pi }_s{\pi }_M}{k_M{(\rho +\lambda )}^2(1-{\psi }_M)}+\frac{{\omega }^2{\pi }_s\left[{\pi }_R(1-{\psi }_M)+2{\pi }_M(1-{\psi }_R)\right]}{2k_R(1-{\psi }_M)(1-{\psi }_R)}\\ +\frac{{\eta }_S^2{n}^2{\pi }_s\left[{\pi }_s(1-{\psi }_R)+2{\pi }_R(1-{\psi }_S)\right]}{4{\alpha }_S{(\rho +\gamma )}^2(1-{\psi }_S)(1-{\psi }_R)}+\frac{{\eta }_M^2{n}^2{\pi }_s\left[2{\pi }_R(1-{\psi }_M)+{\pi }_M(1-{\psi }_R)\right]}{2{\alpha }_M{(\rho +\gamma )}^2(1-{\psi }_M)(1-{\psi }_R)}+\frac{{\eta }_R^2{n}^2{\pi }_s{\pi }_R}{{\alpha }_R{(\rho +\gamma )}^2(1-{\psi }_R)}\end{array}\right)\end{array}\\ \begin{array}{l}{b}_1=\frac{m{\pi }_M}{\rho +\lambda };b_2=\frac{n{\pi }_M}{\rho +\gamma }\\ b_3=\frac{1}{\rho }\left(\begin{array}{l}{\pi }_M{D}_0+\frac{{\beta }_S^2{m}^2{\left[{\pi }_s(1-{\psi }_M)+2{\pi }_M(1-{\psi }_S)\right]}^2}{8k_s{(\rho +\lambda )}^2(1-{\psi }_M){(1-{\psi }_S)}^2}+\frac{{\beta }_M^2{m}^2{\pi }_M^2}{2k_M{(\rho +\lambda )}^2(1-{\psi }_M)}+\frac{{\omega }^2{\left[{\pi }_R(1-{\psi }_M)+2{\pi }_M(1-{\psi }_R)\right]}^2}{8k_R(1-{\psi }_M){(1-{\psi }_R)}^2}\\ +\frac{{\eta }_S^2{n}^2{\pi }_M\left[2{\pi }_R(1-{\psi }_R)+{\pi }_S(1-{\psi }_S)\right]}{2{\alpha }_S{(\rho +\gamma )}^2(1-{\psi }_S)(1-{\psi }_R)}+\frac{{\eta }_M^2{n}^2{\pi }_M\left[2{\pi }_R(1-{\psi }_M)+{\pi }_M(1-{\psi }_R)\right]}{4{\alpha }_M{(\rho +\gamma )}^2(1-{\psi }_M)(1-{\psi }_R)}+\frac{{\eta }_R^2{n}^2{\pi }_R{\pi }_M}{{\alpha }_R{(\rho +\gamma )}^2(1-{\psi }_R)}\end{array}\right)\end{array}\\ \begin{array}{l}{c}_1=\frac{m{\pi }_R}{\rho +\lambda };c_2=\frac{n{\pi }_R}{\rho +\gamma }\\ c_3=\frac{1}{\rho }\left(\begin{array}{l}{\pi }_R{D}_0+\frac{{\beta }_S^2{m}^2{\pi }_R\left[{\pi }_s(1-{\psi }_M)+2{\pi }_M(1-{\psi }_S)\right]}{2k_s{(\rho +\lambda )}^2(1-{\psi }_S)(1-{\psi }_M)}+\frac{{\beta }_M^2{m}^2{\pi }_R{\pi }_M}{k_M{(\rho +\lambda )}^2(1-{\psi }_M)}+\frac{{\omega }^2{\pi }_R\left[{\pi }_R(1-{\psi }_M)+2{\pi }_M(1-{\psi }_R)\right]}{4k_R(1-{\psi }_M)(1-{\psi }_R)}\\ +\frac{{\eta }_S^2{n}^2{\left[2{\pi }_R(1-{\psi }_S)+{\pi }_S(1-{\psi }_R)\right]}^2}{8{\alpha }_S{(\rho +\gamma )}^2(1-{\psi }_R){(1-{\psi }_S)}^2}+\frac{{\eta }_M^2{n}^2{\left[2{\pi }_R(1-{\psi }_M)+{\pi }_M(1-{\psi }_R)\right]}^2}{8{\alpha }_M{(\rho +\gamma )}^2{(1-{\psi }_M)}^2(1-{\psi }_R)}+\frac{{\eta }_R^2{n}^2{\pi }_R^2}{2{\alpha }_R{(\rho +\gamma )}^2(1-{\psi }_R)}\end{array}\right)\end{array}\end{array}$$

It follows from Theorem 2 that in the case of a non-specific one-off government subsidy, the government subsidy affects both the abatement decision and the preservation decision of the food supply chain firms. The degree of influence on both the abatement decision and the preservation decision depends on the relationship between the coefficients. In contrast to Theorem 1, some of the government subsidies in the case of one-off government subsidies are used to improve the freshness of the food, not just to increase the amount of emission reductions.

2.4. Comparative Analysis

For the purpose of comparative analysis, it is assumed that the government’s dedicated abatement subsidy coefficient for food supply chain enterprises is equal to the non-specific one-off subsidy coefficient that the government awards to enterprises, i.e., ${\psi }_S={\Phi }_S, {\psi }_M={\Phi }_M, {\psi }_R={\Phi }_R$. The government gives the same subsidy coefficient and compares the change in enterprises’ decision making in the two cases of government-specific subsidies and government non-specific one-off subsidies, and analyses which government subsidy method is more beneficial to supply chain enterprises’ decision making in different cases.

2.4.1. Comparison of Profit Optimal Values

Theorem 3.

When${\Phi }_S<{\Phi }_M$and${\Phi }_R<{\Phi }_M$, introduce$a_3^X<a_3, b_3^X<b_3$to get$J_S^X<J_S, J_M^X<J_M$; when${\Phi }_R<{\Phi }_S<{\Phi }_M$, introduce$c_3^X<c_3$to get$J_R^X<J_R$.

Theorem 3 shows that, for both suppliers and manufacturers, the government’s non-specific one-off subsidy is more profitable for both suppliers and manufacturers when the government gives the manufacturer the highest subsidy factor; for retailers, the government’s non-specific one-off subsidy is more profitable for retailers when the government’s subsidy factor for retailers is the smallest. The combination shows that government non-specific one-off subsidies increase the profits of food supply chain companies when the government gives the largest subsidy factor to manufacturers and the smallest subsidy factor to retailers. The government subsidy replaces the cost of emission reduction and freshness, thus regulating the profit distribution and game of supply chain members.

2.4.2. Comparison of Emission Reductions and Preservation Quality

Theorem 4.

For abatement effort: when${\Phi }_S>{\Phi }_M$, we get$E_S^X>E_S$; when${\Phi }_R>{\Phi }_M$, we get$E_R^X>E_R$;$E_M^X=E_M$holds regardless of the change in the subsidy factor. For preservation efforts:${\tau }_S^X<{\tau }_S, {\tau }_M^X<{\tau }_M, {\tau }_R^X<{\tau }_R$always holds regardless of the variation of the subsidy factor.

Since $E_S^X>E_S$ holds when ${\Phi }_S>{\Phi }_M$, and $E_M^X=E_M$ also holds, the abatement volume satisfies $X^X>X$; since ${\tau }_S^X<{\tau }_S,{\tau }_M^X<{\tau }_M,{\tau }_R^X<{\tau }_R$, the preservation mass satisfies $Q^X<Q$.

Theorem 4 shows that for both suppliers and retailers, the government-specific subsidy model leads to higher abatement efforts for both suppliers and retailers when the government subsidy factor for the manufacturer is minimal. At the same time, manufacturers’ abatement efforts are the same under both government subsidy models. Therefore, emission reductions are greater under the government-specific subsidy model.

On the other hand, for preservation efforts, the government non-specific one-off subsidy model is more conducive for companies to increase their preservation efforts. The subsidy that was originally used for abatement is partially transferred to the preservation cost. For companies, the preservation costs are shared to a certain extent, which in turn leads to an increase in preservation effort and preservation quality.

The above process of analysis shows that when the government subsidy factor for manufacturers is the highest, supply chain companies make more profit under the non-specific one-off subsidy model and are able to increase the amount of emission reduction while improving the freshness quality of the food. When the government subsidy factor for manufacturers was the lowest, the food supply chain made more emission reductions under the dedicated abatement subsidy model but did not improve the freshness quality of the food. (Note: The calculation process involved in the above is detailed in the supplementary material).

3. Numerical Experiments

Numerical simulations are used to verify the findings obtained earlier and to analyse the impact of different government subsidy models on the decisions and profits of food supply chain agents. The basic parameters are set as $k_S=24$ $k_M=20$ $k_R=14$ ${\alpha }_S=18$ ${\alpha }_M=20$ ${\alpha }_R=12$ ${\beta }_S=2$ ${\beta }_M=3$ $\lambda =0.1$ $x_0=0$ ${\eta }_S=5$ ${\eta }_M=6$ ${\eta }_R=4$ $\gamma =0.2$ $Q_0=100$ $D_0=500$ $m=2$ $n=3$ $\omega =1$ ${\pi }_S=4$ ${\pi }_M=6$ ${\pi }_R=5$ $\rho =0.9$ $t=1$ ${\Phi }_S=0.3$ ${\Phi }_M=0.4$ ${\Phi }_R=0.2$.

3.1. Changes in Profits of Enterprises under Different Government Subsidy Methods

Figure 1 shows a comparison of the profits of firms in the food supply chain under different government subsidies when the government subsidy coefficient for retailers is smaller than that for suppliers and the subsidy coefficient for suppliers is smaller than that for manufacturers. The graph shows that the government provides non-specific one-off subsidies that bring more profits to suppliers, manufacturers, and retailers in this case. This verifies Theorem 3.

3.2. Changes in Emission Reduction by Enterprises under Different Government Subsidy Methods

The first observation is the change in the emission reduction efforts of enterprises. Figure 2 shows that when the government’s subsidy coefficient for suppliers is greater than that for manufacturers, the government’s specific emission reduction subsidy model makes suppliers reduce emissions more diligently. In Figure 3, when the government subsidy factor for retailers is greater than that for manufacturers, the government-specific subsidy model makes retailers work harder to reduce emissions. This is because the government-specific abatement subsidies share more of the cost of abatement for both suppliers and retailers than the non-specific one-off government subsidies, which greatly increases their incentive to reduce emissions. Figure 4 indicates that manufacturers’ abatement efforts are the same under both government subsidy approaches and are only related to the government subsidy coefficient for manufacturers, so the graphs of abatement efforts under the two government subsidy approaches overlap. This illustrates that the government subsidy approach used does not affect manufacturers’ abatement efforts.

3.3. Changes in Corporate Emissions Reductions under Different Government Subsidy Approaches

Further analysis of the change in emission reductions by firms. Figure 5 and Figure 6 show the change in emission reductions when the size of the government subsidy factor for suppliers and manufacturers varies, respectively. The graphs show that emission reductions are greater in the case of government-specific abatement subsidies when the government provides a larger subsidy factor for the supplier and greater product emission reductions when the government provides a larger subsidy factor for the manufacturer, with a non-specific one-off government subsidy. Since the manufacturer’s abatement effort is independent of the government subsidy method, when the government gives the supplier a larger subsidy factor than the manufacturer, the supplier abatement effort is greater in the case of the government-specific abatement subsidy and, accordingly, the abatement is greater in that case. Similarly, when the government subsidy factor for the supplier is less than that of the manufacturer, the supplier makes more effort to reduce emissions in the case of a non-specific one-off government subsidy, and therefore the emission reductions in this case are greater.

3.4. Changes in the Quality of Corporate Preservation under Different Government Subsidy Approaches

Firstly, we observe the change in preservation effort. Figure 7 shows the effect of different government subsidies on the preservation effort of enterprises when the size of government subsidy coefficients to enterprises varies. Corresponding to Figure 7, Figure 8 and Figure 9 represent the effect of different government subsidy approaches on the freshness quality of food when the size of the government subsidy coefficient to the firm varies. The figures show that regardless of the size of the government subsidy coefficients, firms’ preservation efforts were consistently higher in the non-specific subsidy scenario, and correspondingly, the preservation quality of the food products was higher. This is because part of the subsidy provided by the government in the non-specific subsidy case is used to share the preservation costs of the firm, thus enhancing the preservation effort, and this pattern is independent of the relationship between the size of the government subsidy coefficients provided to the firm.

4. Conclusions

This paper draws the following conclusions by comparing and analysing the abatement decisions, preservation decisions, and profits of food supply chain subjects under two government subsidy models.

For the profits of food supply chain subjects: when the government gives the largest subsidy coefficient to manufacturers and the smallest subsidy coefficient to retailers, then the model of non-specific one-off government subsidies can bring higher profits to enterprises.

For the reduction decision of enterprises: when the government gives the smallest subsidy coefficient to manufacturers, the reduction of products is greater under the government’s dedicated reduction subsidy model; when the subsidy coefficient to manufacturers is the largest, the reduction of products is greater under the non-dedicated subsidy model.

For firms’ preservation decisions: firms consistently receive higher levels of preservation effort and preservation quality under the government’s non-specific one-off subsidy model.

The above findings yield the impact of government subsidy models on the profitability, abatement decisions, and preservation decisions of food supply chain enterprises under different subsidy coefficient sizes, which provides a basis for enterprises to make decision adjustments according to different government subsidy policies. At the same time, it also provides a basis for the government to make decisions on subsidy policies under different circumstances. Thus, the corresponding policy recommendations based on the above findings are as follows.

(1)

When the government’s subsidy coefficient for manufacturers is the highest, it is recommended that a non-specific one-off subsidy be given to enterprises to maximise their profitability and to increase the reduction in emissions in the food supply chain while also improving the freshness quality of the food products.

(2)

For food supply chains with stringent freshness requirements, a non-specific subsidy model is recommended. A dedicated abatement subsidy model will increase emissions reductions but has no impact on the quality of food preservation. When the government adopts a non-specific one-off subsidy model, it can both increase emission reductions and improve food preservation quality.

This paper only analysed the relationships between individual food supply chain agents and did not analyse the interlocking network of relationships between multiple food supply chain agents; further analysis of the relationships between suppliers, manufacturers, and retailers could be undertaken in the future. On the other hand, with the development of logistics, logistics enterprises also play a very important role in the food supply chain. In the future, logistics enterprises can be further included in the main body of analysis, and the decision making of the food supply chain including logistics enterprises can be analysed. Finally, with the continuous development of e-commerce, online markets are also a sales channel for food products and cannot be ignored; the analysis of food supply chain models for both online and offline channels is also a hot spot for research.