Research on Government Green Subsidy Strategy of Heterogeneous Agricultural Product Supply Chain

. To promote dual carbon targets, this study builds a heterogeneous agricultural product supply chain comprised of a green agricultural product producer, an ordinary agricultural products producer, and a retailer. Tis study constructs four strategic government green subsidy models, wherein the nongovernment subsidy, government subsidy for green agricultural producers, subsidy for green consumers, and dual subsidies. Based on Stackelberg’s game theory, this study analyzes the impact of differentiated green subsidies on heterogeneous agricultural product supply chain decision-making. Te results show that the government will provide green subsidies only when the input cost coefcient of green quality is within a certain threshold. Consumers’ preference for green agricultural products will improve the level of green quality. Te dual subsidy strategy of the government has the highest social welfare and can stimulate the motivation of green production. When considering the subsidy expenditure, the green consumer subsidy strategy can achieve a higher level of green quality and promote the consumption of green agricultural products. Te government should combine diferent subsidy target efects and subsidy efects to provide appropriate green subsidy strategies. According to the corresponding subsidy strategy, the producers and retailers of heterogeneous agricultural products carry out green investment, reasonable pricing, and precision marketing.


Introduction
As China's consumer upgrading trend becomes more pronounced, Chinese consumers' demand for agricultural products is changing from satisfying subsistence to healthy, high-quality, green, and organic agricultural products [1][2][3].In order to meet the diverse needs of consumers, the market will sell heterogeneous agricultural products with diferent qualities and values [4].Te government has continuously introduced policies to improve the quality of agricultural products and the development level of green agriculture.In 2023, the government issued the "No. 1 Central Document," proposing to accelerate the application of green agricultural technology, build a national agricultural green development pilot zone, and improve the monitoring system for agricultural ecological environment protection.Agriculture has become greener and more high quality.
However, green production requires advanced technology and higher production and R&D costs [5], which reduces the motivation of producers to produce green agricultural products [6], resulting in a smaller production scale and more difcult green quality improvement.To alleviate the fnancial pressure of green production, the Chinese government has formulated a green policy system oriented to green ecology and proposed comprehensive agricultural subsidies, direct subsidies for grain farmers, and subsidies for good crop seeds.In the face of various agricultural operators and green production technologies, the government has introduced supportive policies.However, the government has been providing green subsidies for the agricultural production process and business entities, but the subsidy link and object are relatively single.
Green agricultural products are more expensive, and Chinese consumers are less willing to pay a premium for green agricultural products [7], while it has become common for consumers in developed countries to pay a green premium for green produce [8,9].For instance, about 80% of German consumers are willing to buy green agricultural products at a higher premium [10].It has somewhat hindered the expansion of the consumer market for green agricultural products in China.Among green policies, the provision of subsidies plays a crucial role in the green development process [11,12].For instance, the government provides green subsidies to producers and consumers in the production and sales process of green products such as new energy vehicles and green appliances [13,14], which help green product producers while stimulating green consumers to purchase green products [15,16].When the market simultaneously sells heterogeneous agricultural products of ordinary quality and green quality, the government can adopt the subsidy methods of no green subsidies, subsidies for green agricultural products producers, subsidies for green consumers, and double subsidies.According to different subsidy methods, the green quality level and price of heterogeneous agricultural products are also afected by diferent decisions.Te government takes the maximum social welfare, the improvement of green quality, the promotion of green agricultural consumption, and the consideration of subsidy expenditure as subsidy objectives, which will produce diferent optimal subsidy strategies.
Based on this, in response to the diferent needs and preferences of consumers regarding the quality of agricultural products, this study mainly investigates the following questions: (1) Is it necessary for the government to provide green subsidies for consumers?(2) What is the impact of green subsidies on the optimal decision of heterogeneous agricultural product supply chain?(3) What is the optimal green subsidy strategy under diferent subsidy objectives?
Te remainder of the paper is organized as follows.Section 2 provides a review of relevant literature.Section 3 describes the problem description.Section 4 presents the construction and solution of the green subsidy strategy model of the government of heterogeneous agricultural products.Section 5 performs the analysis of model equilibrium results.Section 6 performs numerical analysis based on actual data.Section 7 summarizes the conclusions and management insights and discusses limitations and future research.

Pricing and Quality Decisions of Heterogeneous Products.
Pricing and quality decisions of heterogeneous products have received much scholarly attention in recent years.Liu and Zhang [17] investigated the dynamic pricing problem of two enterprises producing vertically diferentiated products by considering strategic consumers and concluded that strategic consumers incur higher losses for low-quality enterprises than for high-quality enterprises.Li and Chen [18] investigated the product pricing problem for two manufacturers and one seller producing heterogeneous goods; when quality is an exogenous variable, price competition between the two manufacturers diminishes; when quality is an endogenous variable, manufacturers and retailers achieve a win-win situation, but price and quality competition between manufacturers increases.Rational decisions on pricing and quality of heterogeneous products increase the profts of supply chain members [19][20][21] and afect consumers' heterogeneous product demand and purchasing behavior [22].To improve product competitiveness and mitigate channel conficts, many scholars studied heterogeneous products from the dual-channel supply chain perspective.Liu et al. [23] studied the pricing of heterogeneity in a dual-channel supply chain by considering consumer acceptance of heterogeneous products.Hou et al. [24], on the other hand, considered nonprice characteristics, i.e., the impact of the "online to store" channel and product quality levels on product pricing and proftability.It has been found that the presence of both price competition and quality competition in producing heterogeneous products had impacts on the pricing and quality decisions of heterogeneous products.
Heterogeneous agricultural products are those with imperfect substitutability, which have two attributes, the performance of use to meet the basic needs of consumers (basic value function) and to meet higher quality needs and services of consumers (beyond value function) [4].In order to simplify the description, this paper divides heterogeneous agricultural products into two categories: one is ordinary agricultural products with basic value functions to meet the quality and safety of agricultural products, and the other is green agricultural products with both attributes to meet quality and safety while satisfying consumers' pursuit of green, organic, and nutritious health.Many scholars have also discussed the pricing and quality decision of heterogeneous agricultural products.Considering consumer equilibrium, Liu et al. [5] have explored behavior-based pricing between organic and ordinary food frms, with the pricing strategies of the two enterprises being opposite as market share increases.Pu et al. [25] found that the market entry mode of organic agricultural products afects the pricing and demand of organic and ordinary agricultural products.Perlman et al. [26] considered a dual-channel supply chain consisting of organic and ordinary agricultural products to analyze the pricing of heterogeneous agricultural products under dual channels.Green agricultural products require higher costs to improve green quality levels, but producers lack sufcient incentives to improve green quality levels alone.Terefore, government subsidies play an important role as a powerful tool to increase producers' profts and the green quality level of agricultural products.Based on the heterogeneity of consumers' evaluation of service and product quality, farmers and retailers will make strategic decisions of product competition or cooperation [27].Unlike previous studies, this paper considers the impact of diferentiated government green subsidies on the pricing and green quality level decisions of heterogeneous agricultural products.

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Discrete Dynamics in Nature and Society

Green Supply Chain Government Subsidy Strategy and
Decision.In the green supply chain, government subsidy targets include product producers and green consumers.Productively, in order to encourage manufacturers to produce more green products, the government implements subsidy policies to bring more profts to manufacturers [28].
When governments subsidize manufacturers' costs of lowcarbon production innovation, retailers can adopt revenuesharing or cost-sharing strategies to enhance supply chain cooperation [29].Li et al. [30] studied three government subsidy strategies: a green nonsubsidy, a green product subsidy, and a green innovation subsidy and found that the green product subsidy strategy is optimal when the cost of green innovation is sufciently high and green innovation reduces the variable production cost of the green product signifcantly, while the green innovation subsidy strategy is optimal when subsidy efciency is maximum.Te government's direct subsidy strategy to farmers will have an important impact on the greening of agricultural products [31].
When it comes to subsidizing green consumers, Cohen et al. [32] argued that direct government subsidies to green consumers afect green product production and pricing decisions.Both governments and manufacturers will beneft if consumers are properly sensitive to green and low-carbon subsidies [33].Considering consumers' preferences for the environment, Wang et al. [34] constructed a three-stage Stackelberg model composed of the government, manufacturers, and retailers to analyze the incentive efect of consumers' green subsidy strategy.Zhao et al. [35] studied the problem of joint decision-making on remanufacturer pricing and subsidy shares between remanufacturers and consumers and suggested that remanufacturers can obtain higher returns by sharing subsidies with consumers.
In response to the existence of heterogeneous products in the consumer market that meets diferent consumer needs, scholars discussed the decision-making problem of the green subsidy strategy in a competitive market.Government subsidy strategies that take into account heterogeneous efects can bring about win-win-win results for farmers, raw material producers, and social welfare [36].In both cases without and with government subsidies, Meng et al. [37] analyzed the supply chain decisions of two competing enterprises and found that government subsidies reduce the price of green products; in contrast to this fnding, Barman et al. [38] studied the pricing decision problem of a bioligopolistic green supply chain consisting of two competing manufacturers and one retailer and found that government subsidies increase the price of green products.In agricultural supply chain, government subsidies involve a variety of subsidy strategies, including price subsidies [39], yield subsidies and innovation subsidies [40], crop insurance and premium subsidies [41], production subsidies with fxed or coefcient subsidies [42], the price loss coverage program, and the agriculture risk coverage program [43] for agricultural producers.Tese pieces of literature have only studied the situations of government subsidies for agricultural producers, and the target of subsidies is relatively single.
In summary, based on existing studies, this paper establishes a Stackelberg game model to analyze the diferentiated green subsidy strategy and the optimal pricing and green quality level decision of heterogeneous agricultural product supply chain.Te diference between this paper and other scholars' research is that frst, existing studies do not consider the impact of diferentiated green subsidies on ordinary agricultural producers.In real life, it is common to sell homogenous agricultural products in the consumer market, so it is important to analyze the impact of the government's green subsidy strategy on green agricultural producers and ordinary agricultural producers.Second, this paper proposes the dual subsidy strategy that provides subsidies of green agricultural producers and green consumers and compares the efects of diferent green subsidy strategies on heterogeneous agricultural product supply chain decisions.Finally, combining the results of comparative and arithmetic analysis, the optimal green subsidy strategy of the government is discussed for diferent subsidy targets.

Problem Description
We consider a heterogeneous agricultural supply chain model consisting of a green agricultural producer (hereafter referred to as producer 1), an ordinary agricultural producer (hereafter referred to as producer 2), and a retailer [26,38].Te Stackelberg game process is shown in Figure 1.According to previous studies, a linear demand function was used to represent the green agricultural product market demand d g and ordinary agricultural product market demand d n , respectively [38,44], with the expressions d g � ρα -p g + bp n + βθ and d n � (1 -ρ) α -p n + bp g .
Among them, α(α > 0) represents the total market demand capacity, ρ represents the preference coefcient of consumers of green agricultural products, and 1 -ρ represents the preference coefcient of consumers of ordinary agricultural products.Also, b(0 < b < 1) represents the coefcient of cross-price elasticity between green agricultural products and ordinary agricultural products and the value of b, and the larger the value of b, the more intense the market competition between the two agricultural products.β is the coefcient of consumer sensitivity to green quality levels, and θ is the green quality level of green agricultural products.Here, the subscript g represents green agricultural products, and the subscript n represents ordinary agricultural products.Since green agricultural products need to invest more production costs, assuming c g > c n , without loss of generality, assuming c n � 0, we use c to represent the unit production cost of green agricultural products [25].
In this paper, we consider green and ordinary agricultural products to be sold in the market at the same time.Diferent green subsidy strategies are considered to be provided by the government, including nongovernment subsidy (N), producer 1 subsidy strategy (G), green consumer subsidy strategy (C), and dual subsidy strategy (G + C).Te parameters used in this paper are shown in Table 1.
Discrete Dynamics in Nature and Society

Nongovernment Subsidy Strategy Model.
When the government does not provide green subsidies, the sequence of decision-making of supply chain members is as follows: producer 1 decides the green quality level of green agricultural products θ and the wholesale price w N g , while producer 2 decides the wholesale price w N n of ordinary agricultural products, and then, the retailer decides the sales price p N g and p N n of agricultural products.Based on demand functions and research assumptions, proft functions of the agricultural product producer and the retailer can be expressed as follows: Lemma 1.When the government does not provide green subsidies, π N S is a concave function of p g and p n , and the heterogeneous agricultural products' sales price has an optimal value; when

Both the green quality level of green agricultural products and the wholesale price of heterogeneous agricultural products have the optimal value. Te above optimal values are as follows (see the Proof Lemma A.1. in Appendix A for full workings):
Te green quality input cost coefcient is greater than a certain threshold, and the green quality level of green agricultural products and the wholesale price and the sales price of heterogeneous agricultural products have the optimal value.Since the government does not provide any green subsidies, it can only rely on green quality and preferential prices to attract customers to purchase green agricultural products.Terefore, the coefcient of cross-price elasticity and the coefcient of consumer sensitivity to the green quality level directly afect the decisions and earnings of producer 1, producer 2, and the retailer.

Government Subsidy to Producer 1 Strategy Model.
When the government provides green subsidies to producer 1, the sequence of decision-making of supply chain members is as follows: producer 1 decides the green quality level of green agricultural products θ and the wholesale price w G g , while producer 2 decides the wholesale price w G n of ordinary agricultural products, and then, the retailer decides the sales price p G g and p G n of agricultural products.Te government provides the s 1 ratio subsidy for the green quality input cost of producer 1. Considering that the government provides green subsidies to producer 1, the proft functions for the producer and the retailer can be obtained based on demand functions and research assumptions that

Lemma 2. When the government provides green subsidies to producers 1, π G S is a concave function of p g and p n , and the heterogeneous agricultural products' sales price has an optimal value; when
Mg is a concave function about θ and w g and π G Mn is a concave function about w n .Both the green quality level of green agricultural products and the wholesale price of heterogeneous agricultural products have the optimal value.Te above optimal values are as follows (see the Proof Lemma A.2. in Appendix A for full workings): If the green quality input cost coefcient is greater than a certain threshold, the government will provide green subsidies for producer 1. Due to the large capital demand for improving the level of green quality, green technology is difcult to obtain, and the government will enhance the green production motivation of producer 1 through green subsidies.Producer 1 will invest more money to improve the level of green quality.Te percentage of government subsidies for green quality input costs infuences the decisions and returns of producers 1, 2, and retailers.

Government Subsidy to Green Consumer Strategy Model.
When the government provides green subsidies to green consumers, the sequence of decision-making of supply chain members is as follows: producer 1 decides the green quality level of green agricultural products θ and the wholesale price w C g , while producer 2 decides the wholesale price w C n of ordinary agricultural products, and then, the retailer decides the sales price p C g and p C n of agricultural products.Te government provides the unit subsidy s 2 to green consumers according to the green quality level θ.Considering that the government provides green subsidies to green consumers, and at this instant, the demand functions are Te proft functions for the producer and the retailer can be obtained as follows: Discrete Dynamics in Nature and Society Lemma 3. When the government provides green subsidies to green consumers, π C S is a concave function of p g and p n , and the heterogeneous agricultural products' sales price has an optimal value; when k Mg is a concave function about θ and w g and π C Mn is a concave function about w n .Both the green quality level of green agricultural products and the wholesale price of heterogeneous agricultural products have the optimal value.Te above optimal values are as follows (see the Proof Lemma A.3. in Appendix A for full workings): where When the green quality input cost coefcient is greater than a certain threshold, the government will provide green subsidies for consumers.When the green quality level of green agricultural products is diferent, the sensitivity of consumers is also afected to diferent degrees.Te government provides green consumption subsidies to increase green consumption willingness.Consumers tend to purchase higher quality green produce.Te amount of government subsidy per unit for green consumers afects the decisions and earnings of producers 1, 2, and retailers.

Government Dual Subsidy Strategy Model.
When the government provides green subsidies to producer 1 and green consumers, the sequence of decision-making of supply chain members is as follows: producer 1 decides the green quality level of green agricultural products θ and the wholesale price w G+C g , while producer 2 decides the wholesale price w G+C n of ordinary agricultural products, and then, the retailer decides the sales price p G+C g and p G+C n of agricultural products.Te government not only provides the s 1 ratio subsidy for the green quality input cost of producer 1 but also provides the unit subsidy s 2 to green consumers according to the green quality level θ.Considering that the government provides dual subsidies to producer 1 and green consumers, and at this instant, the demand functions are Te proft functions for the producer and the retailer can be obtained as follows: Lemma 4. When the government provides dual subsidies, π G+C S is a concave function of p g and p n , and the heterogeneous agricultural products' sales price has an optimal value; when Mg is a concave function about θ and w g and π G+C Mn is a concave function about w n .Both the green quality level of green agricultural products and the wholesale price of heterogeneous agricultural products have the optimal value.Te above optimal values are as follows (see the Proof Lemma A. 4. in Appendix A for full workings): Te green quality input cost coefcient is greater than a certain threshold, and the government provides green subsidies for producers and consumers.Considering improving green production capacity and green consumption willingness of green agricultural products, the government adopts a double green subsidy strategy.Te percentage of government subsidies for green quality input costs and the amount of government subsidy per unit for green consumers also afect the decision and income of producer 1, producer 2, and the retailer.

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Discrete Dynamics in Nature and Society

Analysis of Equilibrium Results under Nongovernment Subsidy Strategy
Proposition 5. zθ N * /zρ > 0; zw N * g /zρ > 0; zw N * n /zρ > 0, Proposition 5 reveals that under the nongovernment subsidy strategy, the optimal decision of the green quality level, the wholesale price, and the sales price of green agricultural products increase as the preference coefcient of consumers of green agricultural products increases.Te wholesale price and the sales price of ordinary agricultural products are related to the coefcient of green quality input cost, the coefcient of consumer sensitivity to green quality levels, and the coefcient of cross-price elasticity of heterogeneous agricultural products.Because, in order to cater to the green preferences of consumers, producer 1 strives to carry out green production inputs, and the green quality level of agricultural products is improved accordingly.Due to the rising production cost, producer 1 will correspondingly increase the wholesale price, and sellers will increase the price of green agricultural products, thereby increasing their own profts.Terefore, with the increase of consumers' green preference, the level of green quality and the price of green agricultural products will increase even if producer 1 does not receive green subsidies from the government.When consumers' green preference changes, the pricing of ordinary agricultural products is afected by green quality input cost and other factors.

Analysis of Equilibrium Results under Producer 1 Subsidy Strategy
Proposition 6 has essentially the same conclusion as Proposition 5 and will not be repeated here.When the government provides green subsidies to producer 1, the wholesale price and sales price of ordinary agricultural products are not only afected by k, β, and b but also related to the percentage of government subsidies for green quality input costs.Because producer 1 has received green subsidies from the government, the enthusiasm for green production has increased.With the increase of consumers' green preference, the green quality level and pricing of green agricultural products have been further improved.Te subsidy funds received by producer 1 afect the pricing of green agricultural products and thus afect the pricing of ordinary agricultural products.

Corollary 7. zθ
Corollary 7 reveals that under the producer 1 subsidy strategy, the optimal decisions on the green quality level of green agricultural products and the wholesale and sales prices of heterogeneous agricultural products increase as the percentage of government subsidies for green quality input costs increases.Producer 1 will increase green quality levels of agricultural products due to the increase in the subsidy percentage, which in turn will increase the pricing of their products, while ordinary agricultural products will increase their product pricing accordingly to increase their profts.

Analysis of Equilibrium Results under Green Consumer Subsidy Strategy
When the government provides green subsidies to green consumers, as the preference coefcient of consumers of green agricultural products increases, the wholesale and sales price of ordinary agricultural products will only increase subsequently when k is satisfed within a certain threshold.As consumers receive green subsidies from the government, they are more willing to consume green.With the increase of consumers' green preference, the green quality level and pricing of green agricultural products have been further improved.Unit subsidies received by consumers according to the level of green quality afect the pricing of green agricultural products and thus afect the pricing of ordinary agricultural products.
Corollary 9 reveals that under the green consumer subsidy strategy, the green quality level and pricing of green agricultural products and the sales price of ordinary agricultural products increase as the amount of government subsidy per unit for green consumers increases.Te wholesale price of traditional agricultural products is affected by the range of the green quality input cost coefcient.Tis strategy positively infuences the decision of producer 1 and the retailer, and the decision-making of producer 2 is related to the coefcient of green quality input cost.

Discrete Dynamics in Nature and Society
Similarly, Proposition 10 illustrates that the optimal decision of green quality levels and the wholesale price and sales price of green agricultural products increase as the preference coefcient of consumers of green agricultural products increases.Also, the wholesale price and sales price of ordinary agricultural products are infuenced by the coefcient of green quality input cost.Terefore, increasing consumer preference for green agricultural products can promote the level of green quality.However, with the increase of the coefcient of green quality input cost, the impact of consumer preference for green agricultural products on the pricing of ordinary agricultural products shows a trend of increasing and then decreasing.Because the input cost of green quality is too high, proft can hardly ofset the high cost, production and consumption enthusiasm gradually decline, and the infuence of consumers' green preference on the pricing of agricultural products will be weakened.

Corollary 11. (1)
Corollary 11 reveals that under the dual subsidy strategy, with the increase of s 1 and s 2 , the optimal decision of producer 1 and the retailer increases and promotes the green quality level, pricing, and sales price of ordinary agricultural products.However, with the increase of s 2 , afected by the coefcient of green quality input cost, the wholesale prices of ordinary agriculture products show a trend of frst increasing and then decreasing.

Numerical Analysis
Tis study discusses the supply chain optimal decision under diferentiated government green subsidies by constructing a heterogeneous agricultural supply chain model.Te impact of other key parameters on the optimal decision of the supply chain and the supply chain members' strategies under the green subsidies are further studied through numerical analysis.Tis study takes the Beijing Daxing watermelon industry as an example, then combines the survey data to assign reasonable values to the relevant parameters in this model, and tries to ensure the existence of a practical basis for the research parameter settings.In order to facilitate the operation, the relevant research parameter settings are appropriately reduced, and the specifc data are assumed as follows: α � 100, ρ � 0. In addition, assume that the government social welfare SW is expressed as SW � π Mg + π Mn + π S -GS + CS + EI [45], where the government subsidy expenditure is GS, where GS G � s 1 kθ 2 /2, GS C � s 2 kθd C g , and GS G+C � s 1 kθ 2 /2 + s 2 kθd G+C g .Consumer surplus is expressed as CS � (d 2 g + d 2 n )/2, and environmental improvement is expressed as EI � θd g .From Lemmas 1-4, the condition that the case of diferent government green subsidies holds simultaneously is k > 286.06.

Comparative Analysis of Government Social Welfare under Diferent Green Subsidy Strategies.
As shown in Figure 2, we set k � 400, β � 25 and analyze the efect of s 1 and s 2 on the diference in social welfare functions under diferent green subsidy strategies.Here, ∆SW 1 � SW G − SW C , ∆SW 2 � SW C − SW G+C , and ∆SW 3 � SW G − SW G+C .
Figure 2 shows that ∆SW 1 > 0, ∆SW 2 < 0, ∆SW 3 < 0 and |∆SW 2 | > |∆SW 3 |.It can be inferred that no matter the size of subsidy ratio s 1 and unit subsidy amount s 2 , government social welfare is always represented as SW G+C > SW G > SW C .It shows that when the government conducts green subsidies with the goal of improving social welfare, the optimal green subsidy strategy is the dual subsidy strategy.s 1 and s 2 only afect the size of social welfare but do not afect the choice of the government's green subsidy strategy.

Infuence of the Coefcient of Green Quality Input Cost on
Supply Chain Decisions.By setting s 1 � 0.3 and s 2 � 1.5, this paper compares and analyzes the infuence of the green quality input cost coefcient on the optimal decision of heterogeneous agricultural product supply chain under four subsidy strategies.
Table 2 describes the infuence of the green quality input cost coefcient on the optimal green quality level and pricing of heterogeneous agricultural products under diferent government green subsidies.As shown in Table 2, increasing the coefcient of green quality input cost, the green quality level of agricultural products, and the pricing of heterogeneous agricultural products showed a decreasing trend and eventually levelled of.Te results show that the coefcient of green quality input cost has the same efect on the decision of heterogeneous agricultural product supply chain members.Te green quality level is expressed as θ G + C * > θ G * > θ C * > θ N * .Terefore, compared with the nongovernment subsidy, government green subsidies can improve the green quality level of agricultural products, and the green quality level under the dual subsidy strategy is the highest.
Table 3 describes the infuence of the coefcient of green quality input cost on demands for heterogeneous agricultural products and the profts of supply chain members under diferent government green subsidies.Table 3 shows that increasing the coefcient of green quality input cost will reduce the demands for heterogeneous agricultural products and the profts of supply chain members, and the reduction will be slower and slower.Te lower the green quality input cost coefcient within a certain threshold, the higher the demand for heterogeneous agricultural products and the profts of supply chain members can be maintained, so as to achieve a win-win situation in the supply chain.When the coefcient of green quality input cost is a fxed value, the demands for heterogeneous agricultural products and profts of supply chain members are the highest under the dual subsidy strategy, followed by the producer 1 subsidy strategy, then the green consumer subsidy strategy, and fnally the nongovernment subsidy strategy.
Tat is, the government provides dual subsidies as the optimal green subsidy strategy expected by supply chain members, and the demands and profts of agricultural products under each green subsidy are higher than those under the nongovernment subsidy strategy.Terefore, the optimal government green subsidy strategy expected by supply chain members is dual subsidies, independent of the size of the coefcient of green quality input cost.In summary, the government's optimal green subsidy strategy is dual subsidies when the government's subsidy objective is to increase the green quality level and demands for green agricultural products.

Infuence of the Government Subsidy Expenditure on the
Supply Chain 6.3.1.Infuence of the Government Subsidy Expenditure on the Green Quality Level. Figure 3 shows the impact of government subsidy expenditure on the green quality level of green agricultural products under diferent green subsidies.From Figure 2, it can be seen that the green quality level tends to increase with the increase in government subsidy expenditure.Moreover, the green quality level achieved by the green consumer subsidy strategy is the highest when the government subsidy expenditure is a fxed value.Also, all the green subsidies act directly on green consumers and motivate producer 1 to attract more consumers by improving the green quality level.Terefore, when the government considers subsidy expenditures and aims to improve the green quality level as the subsidy objective, the optimal subsidy strategy is to subsidize green consumers, and the green consumer subsidy strategy has the least subsidy expenditure at the same green quality level.

Infuence of the Government Subsidy Expenditure on
Demands for Heterogeneous Agricultural Products.As shown in Figure 4, with the increase of government subsidy expenditure, the demands for green agricultural products shows an upward trend, among which the promotion efect of the green consumer subsidy strategy is more signifcant, followed by the producer 1 subsidy strategy and fnally the dual subsidy strategy.Although the dual subsidy strategy is more efective in promoting the demands for green agricultural products at the same coefcient of green quality input cost, it is not optimal for the same subsidy expenditure.Terefore, when the government considers the subsidy    Discrete Dynamics in Nature and Society expenditure and promotes the demands for green agricultural products as subsidy objectives, the optimal government green subsidy strategy is to subsidize green consumers.As shown in Figure 5, the demands for ordinary agricultural products increase with the increase in government subsidy expenditure.When the government subsidizes green consumers and provides dual subsidies, green consumers can directly get concessions for purchasing green agricultural products, and the demands for ordinary agricultural products increase more slowly.From the perspective of producer 2, it expects the government to provide the green consumer subsidy strategy, which is the same as the optimal strategy under the government subsidy objective of promoting the demands for green agricultural products.

Infuence of the Government Subsidy Expenditure on
Supply Chain Members' Profts.Figure 5 shows the impact of government subsidy expenditure on the profts of supply chain members when the government provides diferent green subsidy strategies.As shown in Figure 6(a), with the increase in government subsidy expenditure, the profts of producer 1 show an upward trend, which means that the green subsidy promotes profts of producer 1, among which the promotion of the green consumer subsidy strategy is more signifcant.When considering the subsidy expenditure and promoting the green quality level and green agricultural product demands as subsidy objectives, the optimal strategy of the government is to subsidize green consumers, and the profts of producer 1 are also optimal.As shown in Figure 6(b), the impact of government subsidy expenditure on producer 2's profts is essentially the same as its impact on demands for ordinary agricultural products.Te profts of producer 2 are not necessarily optimal when the government provides the green consumer subsidy strategy.As shown in Figure 6(c), the retailer's profts increase as government subsidy expenditure increases, and the retailer's profts are higher under both the green consumer subsidy strategy than under the other two green subsidies.

Conclusion and Prospect
Based on a heterogeneous agricultural product supply chain consisting of producer 1, producer 2, and a retailer, this study compares the optimal decision of the supply chain under four strategies: nongovernment subsidy, the producer 1 subsidy, the green consumer subsidy, and explore the government's optimal green subsidy strategy under diferent subsidy targets, and the main conclusions are as follows: (1) Te government will provide green subsidies only if the coefcient of green quality input cost reaches a certain threshold.(2) As the coefcient of green quality input cost increases, the decision-making of supply chain members will decrease.Compared with nongovernment subsidies, green subsidies always increase the green quality level, pricing, and demands for heterogeneous agricultural products, thus increasing supply chain members' profts.(3) Te preference coefcient of consumers for green agricultural products has a positive impact on the green quality level and pricing of green agricultural products, and only under certain conditions can the pricing of ordinary agricultural products be reduced.(4) Te optimal green subsidy strategy of the government difers under diferent subsidy objectives.Te government tends to provide the dual subsidy strategy when the government has maximum social welfare as the subsidy objective.When the government only considers improving the level of green quality and promoting the consumption of green agricultural products, the government also tends to provide the dual subsidy strategy.Te government prefers to provide the green consumer subsidy strategy when it considers the minimum subsidy expenditure and improving the green quality level and the demands for green agricultural products.
Based on the above research conclusions, we can obtain some managerial insights.
First, green subsidies can efectively promote the green quality of agricultural products, and with the increase of subsidies, green agricultural producers are more active in production.Terefore, it is an efective policy tool for the government to stimulate green production behavior by providing green subsidies.Government subsidy objectives will infuence the choice of the green subsidy strategy, and the government should combine the green subsidy efect and subsidy objectives to clarify the green subsidy strategy and set a reasonable subsidy intensity.In addition to fnancial subsidies, the government can implement tax incentives for energy conservation and carbon reduction and launch agricultural carbon reduction support tools to open up green agricultural development prospects through institutional innovation.
Second, higher green quality input costs will reduce the enthusiasm for green production.Green agricultural product producers should reasonably make green inputs according to the market environment and government subsidies, control green production costs while improving green quality, and increase consumer trust and green preferences.Ordinary agricultural producers should reasonably adjust the pricing of agricultural products according to the government's green subsidy policy and ensure the quality of agricultural products, while actively transforming and upgrading to green agriculture.
Finally, retailers can always beneft from green subsidies.Tey can cooperate with online e-commerce platforms, ofine supermarkets, community group buying, etc., to achieve omnichannel sales of heterogeneous agricultural products.Also, they can also use big data to accurately position agricultural products and identify consumer groups to realize precise marketing of heterogeneous agricultural products to improve their revenue.
Tis study can be extended to several promising aspects.First, we only studied a heterogeneous agricultural supply chain consisting of two producers and one retailer, and future research could be extended to a network of agricultural supply chains with multiple producers and retailers.Second, the government's green subsidy decisions deserve further study.Tird, future research can explore the decision-making of heterogeneous agricultural supply chains under imperfectly competitive markets.Discrete Dynamics in Nature and Society (A.3) We fnd the Hessian matrix for θ N , w N g , and w N n : By fnding the second partial derivative, the Hessian matrix is obtained: We know that H(G) 1 is negative defnite, and there is an optimal solution for p G g and p G n .Simultaneous equations zπ G S /zp G g � 0 and zπ G S /zp G n � 0 yield When k > β 2 /(4 − b 2 )(1 − s 1 ) is satisfed, H(G) 2 is negative defnite, and there is an optimal solution for θ G , w G g , and w G n , Finally, the optimal decision value of θ 6, b � 0.5, c � 0.4, β � 25, s 1 � [0.1, 0.35], and s 2 � [0, 1.5].

Figure 2 :
Figure 2: Te infuence of s 1 and s 2 on the diference in social welfare function.

Figure 5 :Figure 6 :
Figure 5: Te relationship between GS and d n .

Table 1 :
a concave functionFigure 1: Te main body game model diagram of agricultural product supply chain under government green subsidy.Descriptions for the parameters.
ParametersDescriptions α Te total market demand capacity ρ Te preference coefcient of consumers of green agricultural products b Te coefcient of cross-price elasticity c Te unit production cost of green agricultural products β Te coefcient of consumer sensitivity to green quality levels k Te coefcient of green quality input cost s 1 Percentage of government subsidies for green quality input costs s 2 Amount of government subsidy per unit for green consumers θ S Te proft of the retailer 4 Discrete Dynamics in Nature and Society about θ and w g and π N Mn is a concave function about w n .

Table 2 :
Te infuence of k on the optimal green quality level and price of heterogeneous agricultural products.

Table 3 :
Te infuence of k on demands for heterogeneous agricultural products and profts of supply chain members.
4 − b 2 is satisfed, H(N) 2 is negative defnite.π N Mg is a concave function of θ N and w N g , and π N Mn is a concave function of w N n .Simultaneous equations are as follows: zπ N Mg /zθ N g � 0, zπ N Mg /zw N g � 0, and zπ N Mn /zw N n � 0. Te solution can be obtained: θ N * , w N * g , and w N * n , and substituting θ N * , w N * g , and w N * n into p N g and p N n yields p N * (7)nd p N * n .Lemma A.2.It is also solved by backward induction, and according to equation(7), we can obtain the following equationzπ When k > (β + s 2 )[(2 − b 2 )s 2 + 2β]/2(4 − b 2 ) is satisfed, H(C) 2 isnegative defnite, and there is an optimal solution for θ C * , w C * g , w C * n , p C * g , and p C * n .