A model of sustainable development in economic, social, and environmental aspects: the role of social capital in China

This paper takes China as an example to quantitatively study the weight relationship between economic growth, social welfare, and environmental quality, and explores the role of social capital in promoting sustainable development. This paper constructs the Hamilton optimization model, the ‘sustainable’ element is introduced on the basis of the traditional utility function, and given China’s economic characteristics of numerical simulation analysis was carried out on the model parameters, to reveal the connotation of Chinese sustainable development from the mathematical, to explore the influence mechanism of social capital in promoting sustainable development. Research has found that sustainable development seeks coordinated economic, social, and environmental development. Setting the weight ratio of economic growth, social welfare, and environmental quality at 1:1.5:1.5 can better explain China’s concept of sustainable development. When social capital accelerates its accumulation, it can effectively control the development and utilization of natural resources, and the ecosystem environment can maintain the stable state of the ecosystem through its own regeneration and restoration ability. Social capital has become a new driving force for sustainable development, achieving a steady and rising trend of China’s economic development level. The research conclusions are helpful to promote the sustainable development, and to enrich and expand the relevant research on social capital.


Introduction
As the negative impact of human activities on the environment continues to intensify, the trend of global warming continues, with a sharp increase in sudden and extreme climate disasters.Factors such as climate change, loss of biodiversity, and excessive resource consumption will exacerbate the deterioration of the ecosystem environment.Earth's environment is on the brink of collapse.With the acceleration of industrialization and urbanization, China's long-term extensive economic growth mode has led to excessive consumption and shortage of resources.The resulting series of environmental problems have seriously hindered economic development and the improvement of people's quality of life, especially the environmental pollution problem, led by smog, has had a great impact on the health, production and life of Chinese residents (Liu andLin 2019, Huang et al 2021).So the traditional development models of simply pursuing rapid economic growth and 'pollution first, treatment later' have no longer adapted to the current and future development requirements of China.At the same time, the impact and restriction of natural resources and ecosystem environment on human economic and social development are more obvious than ever before.The contradiction between economic society and resource environment is also increasingly prominent.How to make the economic, social and environmental coordinated development has become a hot issue of academic research.Different from the traditional extensive economic growth model which is only oriented by economic growth, the sustainable development model can promote the balanced development of economy, society and environment.China has relatively insufficient per capita resources and weak ecosystem environment foundation.Seeking and implementing a sustainable development model is a strategy that must be implemented in China's modernization drive.
Environmental protection is an important aspect of sustainable development, but the core of sustainable development is development, which aims to achieve economic development while improving quality of life, improving social welfare, and protecting the environment and sustainable utilization of resources (Taylor et al 2017, Fan et al 2022).The Sustainable Development Goals aimed at thoroughly addressing development issues in the three dimensions of economy, society, and environment in a comprehensive manner, and promoting the transition of economies around the world towards sustainable development.Although the interpretation of sustainability varies in different fields, the coordinated development of economy, society and ecosystem is the definition of sustainable development generally accepted by international organizations (OECD 2001).Early quantitative studies on the coordination between economy, society and environment mainly focused on the constraint of resources and environment on economic growth (Panayotou 1997), and payed more attention on the interaction between economy and environment.This research method ignored the social level theoretical analysis (Zugravu et al 2020), especially the interrelationship between society and environment is considered to be the most lacking field in sustainable development research.
The essential feature of sustainable development is multidimensional, emphasizing that economic, social and environmental factors should be considered simultaneously in the process of human development, and ensuring that any one factor will not pose a threat to others (Bond et al 2001, Grzebyk andStec 2015).Therefore, sustainable development should focus on the analysis of trade-offs and synergies among economy, society and environment (Cook et al 2019).Its basic requirement is that economic growth should adapt to social welfare and ecosystem environment conditions.However, because each of the three has its own characteristics and logic, it is likely that their interests are in conflict with each other.For example, improving material welfare often damages natural ecosystems (Lehtonen 2004).The complexity of sustainable development goals lies in how to prioritize one dimension over another to ensure the realization of contributions to human well-being (Costanza et al 2016).Moreover, the importance levels of the three 'dimensions' of economy-society-environment in sustainable development are not equal, and the priority right of different dimensions is also different among countries.
With the continuous deepening of research on the United Nations Sustainable Development Goals, there have been many literature on measuring sustainable development indicators or quantifying sustainable development goals.However, there are relatively few mathematical models in existing literature that study the trade-offs and synergistic effects between economy, society, and environment.Only a few literatures explicitly model population as a function of well-being and as a feedback on resource use.(Costanza et al 2016, Collste et al 2017, Kirsten and Michel 2023) Therefore, it is necessary to construct a theoretical framework that can integrate the three dimensions of economy, society, and environment (Hukkinen 2003), and social capital can provide a new tool for solving sustainable development problems, allowing the exploration of the three variables of economy, society, and environment that have different goals but are interrelated within the overall framework (Chiesura and Groot 2003).The World Bank (1995) announced for the first time using the 'expanded wealth' index as a new measure of global or regional development, including four elements: natural capital, material capital, human capital and social capital.After successively emphasizing natural capital, material capital and human capital, social capital can make up for the deficiency of traditional factors, obtain greater output with the same or even less input, and become a new driving force for economic development.Therefore, in the sustainable development phase, social capital plays an active role in promoting economic, social and environmental sustainable development through traditional elements.
Therefore, the marginal contribution of this article is mainly reflected in three aspects.First, a theoretical framework that can integrate the three dimensions of economy, society and environment has been constructed.The Hamilton optimization model is used to introduce 'sustainable' elements on the basis of traditional utility functions to depict economic growth, social welfare, and environmental quality, thus clearly refining the core connotation of sustainable development in China.Second, based on the value added attribute and informal institutional characteristics of social capital, social capital is included in the production function and accumulation equation, and social capital, human capital, technological capital, and natural capital are internalized.Through mathematical models, the internal mechanism of social capital promoting economic, social, and environmental sustainable development is analyzed, which provides a research paradigm for demonstrating social capital as a new driving force for sustainable development.Thirdly, numerical simulation analysis should be conducted based on the actual situation in China, in order to set the weights for economic, social, and ecological dimensions.Sustainable development requires the coordination of the three dimensions of economy, society, and ecology.This paper uses numerical simulation to set the weights of indicators for each dimension.The simulation process can ensure that an increase in the weight coefficient of any dimension will not have a negative impact on other dimensions, thereby reflecting the consistency of the overall goal.

Connotation of social capital
The concept of social capital originates from the social network theory.The French sociologist Bourdieu believes that the social capital is the sum of the actual and potential resources shared by the members of the social networks.Resources are embedded in the social networks and can be acquired or invoked by the members through taking certain actions (Lin 2001).Social network can convey sufficient information, and individual economic behavior will be affected by social network.Social capital is an exclusive resource that can bring benefits to group members (Bourdieu 1986).Coleman believes that social capital exists in the social relationships between groups, and that good social relationships not only promote mutual trust, but also enhance mutual cooperation (Fukuyama 1995).Social norms are binding only when group members trust each other.Social capital is a responsibility and an expectation that helps to bind individuals to fulfill common codes of conduct and to observe commitments (Coleman 1990).Fukuyama extended Coleman's definition of social capital by focusing on the correlation between social capital and the economy and society (Fukuyama 1995).Fukuyama believes that social capital is a set of values or moral norms that can promote the cooperation among group members, and social norms such as reciprocity and trust are the important basis of social capital.The evolution of social capital began with families and extended to social groups and institutions through culture, traditions, religion, and customs, etc (Fukuyama 1995).The economic prosperity of a society depends on the general trust of the society, and cultural traditions and religious customs are important drivers of economic development.The most recognized definition of social capital was proposed by Putnam. Putnam et al (1993) found that there was a strong atmosphere of trust and cooperation in the central and northern regions of Italy in his research.This rich social capital could coordinate people's actions, improved the investment returns of material capital and human capital, and promoted regional economic development.Therefore, Putnam (1995) believes that social capital is a characteristic of social organizations, which includes trust, association, common norms, norms and networks.Social capital is formed through the long-term accumulation of 'reciprocity' behaviors among people, which can improve the economic efficiency of organizations by promoting coordinated action.The modern concept of social capital mainly came from three authors, Bourdieu, Coleman and Putnam, many current definitions of social capital are related to those just described (Claridge 2004).Broadly, definitions of social capital include reference to social networks, trust, and norms, which can promote joint action and achieve economic goals (Claridge 2004).For example, Social capital refers to the social relationships between individuals that can generate economic effects (Szreter 2000), and the Oxford dictionary defines social capital as' networks of relationships among people' (Hellerstein and Neumark 2020).

Social capital and economic growth
Social capital has a positive impact on economic growth.Studies by Putnam et al show that regions with higher social capital have higher levels of economic growth compared to regions with lower social capital.The evidence provided by Knack and Keefer (1997) showed that high trust, high social participation rate was significantly positively associated with economic growth after controlling for other economic growth factors.Based on the characteristics of production factors of social capital, Ostrom (1999) pointed out that the contribution of social capital to economic growth lied in the creation of economic opportunities through the enhancement of mutually beneficial cooperation between individuals or organizations.Social capital could make up for the deficiency of traditional capital elements, reduce transaction costs, bring different economic benefits to the corresponding subject, and then affect the economic output level, showing a certain geographical gap (Li et al 2015).Therefore, social capital could be used as a factor of production to explain why there were some differences in economic growth performance and income levels in countries and regions with similar levels of traditional factors (material capital, human capital, technical level, natural resources, etc) (Rupasingha et al 2006).

Social capital and social welfare
Social capital can help vulnerable groups avoid risks and reduce the vulnerability of individuals and groups to adverse circumstances, that is, increase their resilience to external and internal changes and engage in more profitable economic activities (Paudel et al 2017).This will improve the employability and opportunities of workers, increase income returns, and reduce the incidence of poverty.Therefore, some scholars believe that social capital is the 'capital of the poor' (Grootaert 1999), Lin et al (2001) found that social capital, with trust and human relations mechanism, can help individuals to obtain sufficient market information, reduce transaction costs and promote cooperation, and prioritize business opportunities and jobs, and then obtain higher economic income.Antoni and Matthew (2004) found that social networks are the source of employment information and opportunities from the perspective of social networks and information.(Islam and Alam 2018, Khosla and Jena 2020).Therefore, there is a significant positive correlation between social capital and social welfare.Social capital can help families obtain long-term assets, smooth consumption, enjoy better public services, improve employment ability and opportunities, participate in mutual assistance and cooperation behaviors, and obtain more financial credit and community resources (Islam and Alarm 2018).

Social capital and the ecosystem environment
Social capital has a direct impact on the environmental behaviors and attitudes of individuals or groups and reduces pollution emissions (Farrow et al 2017, Marbuah et al 2021;), social capital with social networks, social norms and social trust as the core elements has a certain degree of influence on sustainable resource utilization and environmental protection (Arachchi and Managi 2022).Social trust will lead to mutually beneficial coordination and cooperation.Individual environmental behavior will be affected by other people's behavior or ideas.If people believe that other people will reduce resource consumption and environmental protection, then individuals will also take actions conducive to the environment (Abrahamse and Steg 2013).Social norms constrain most people to obey environmental rules and perceive cooperation, thus affecting the acceptance and reliability of information related to environmental policies (Jones et al 2009).Social network is the bridge of information dissemination and communication, and dense social networks are of great significance for enhancing public participation and awareness of environmental issues.Social network helps group members understand the implementation of environmental policies.For example, if they share information about clean energy policies or low-carbon technology projects within groups, group members will support the adoption of these behaviors to a greater extent (Karen and Anne 2002).
Based on the existing literature, it can be seen that social capital has a positive relationship with economy, society and environment, but few literature have constructed a unified theoretical framework to systematically analyze the comprehensive relationship between social capital and economic growth, social welfare and ecosystem environment.Mathematical model research is still in the exploratory stage, and it is urgent to further clarify the role of social capital in economic-social-environmental sustainable development mechanism, reduce the 'black box' nature of social capital as a new endogenous driving force for sustainable development.

Construction and solution of the theoretical model
Due to the limitations and deficiencies of the neoclassical growth models, Lucas (1988) introduced resource constraints into the endogenous growth model to optimize the research on sustainable economic growth.Later, with the development of endogenous economic growth theory, more and more scholars study sustainable development based on endogenous growth model.These researches aimed to achieve the balance between economy, society and environment through rational resource allocation among departments, and ensured that resource consumption and environmental degradation can be controlled while economic and social development.

Theoretical model construction
The Hamilton model is a commonly used dynamic optimization model in the field of economics, whose core idea is to transform economic problems into a sequence of decisions made at different time points based on the principle of dynamic programming.The Hamilton model considers the choices faced by decision-makers at each point in time and their impact on the future.This model provides researchers with a powerful tool for analyzing economic problems and provides in-depth understanding and insight in resource allocation, investment decision-making, consumption choices, and economic growth.(Li and Zhong 2020, Ospanov and Kondybayeva 2022, Tarasyev et al 2023) The Hamilton optimization model is constructed to investigate the economic system of five sectors: final product, human capital, social capital, technology innovation, and environmental management.Social capital has the attribute of value appreciation, which is constantly accumulating and internalizing itself, and can be incorporated into the production function of the final product department as a new 'product element'.And social capital has informal institutional characteristics, which can be regarded as a valuable 'new resource' introduced into the dynamic accumulation equation of human capital, technological innovation and environmental management departments.

Final product department
Assuming that only one final product is produced, the scalable C-D production function form is adopted according to the model developed by Palma et al (2010) and Wan and Zhang 2023: Among them, Y is the production volume of final products, x is the productivity level parameter of the exogenous final product department, t Y is the proportion of human capital used for the production of final products, K H represents human capital, K P represents material capital, N Y represents the consumption of natural resources, K S represents social capital, K A represents knowledge capital, ( )  c = i 1, ,5 i represents the output elasticity of each production factor, and meets c c c c c The C-D production function with constant return to scale is adopted in the model, mainly considering that it can be assumed in the later numerical simulation that c , 1 c , 2 c 5 remains unchanged.When the output elasticity c 4 of social capital as a factor of production rises, the output elasticity of natural resources will inevitably decline.If the economic growth rate still shows an upward trend at this time, it can be verified that social capital has the value-added function and can break through the bottleneck constraints of traditional resource factors, achieving greater economic output with less resource input and promote sustainable economic development.
The net increase in the stock of material capital in the economic system is equal to the total output Y minus the total consumption C, z representing the depreciation rate,  K P is the increment of material capital.According to the identity of national income accounting, the accumulation equation of material capital is:

Human capital department
Social capital is a network that connects social relations.Higher social capital helps people get more educational opportunities (Dinda 2014), and creates the necessary fields for the generation and promotion of human capital.
On the one hand, human capital accumulation is related to its own stock.On the other hand, social capital can be transformed into human capital, and human capital accumulation can be accelerated through micro communication.According to Palma et al (2010) on the setting of human capital, the accumulation equation is set as: Among them,  K H is the increment of human capital, a 1 is the efficiency of human capital production, t H is the proportion of human capital used in human capital departments, and a 2 represents the coefficient of the transformation from social capital stock to human capital.Li and Huang (2009) believed that education level and health status are the two key factors determining the ownership of human capital.Human capital should be defined more broadly, with education and health as two types of capital.The specific forms of generalized human capital should be constructed according to Cobb-Douglas: Among them, K T is education capital, K M is health capital, b 2 represent the contribution rate of education and health to human capital respectively.

Social capital department
The accumulation equation of social capital is a dynamic development process of self growth and strengthening.
The improvement of education level is a key factor in the accumulation of social capital, which can help individuals better understand social trust, social communication and support mutually beneficial cooperation.Higher human capital is conducive to the establishment of social networks.The equation for social capital accumulation is set up as follows: Among them,  K S is the increment of social capital, f 1 is the production efficiency of human capital invested in social capital departments, t S represents the proportion of human capital allocated to social capital departments, and f 2 represents the self transformation coefficient of social capital stock.Akcomak and Weel (2009) believes that social capital makes the communication network between different subjects more convenient, and significantly promotes knowledge production and technological innovation.In view of this, social capital is introduced as an innovation-driven factor into the knowledge production function, and the knowledge capital equation is set as follows:

Research and development department
Among them,  K A is the increment of knowledge capital, d is the efficiency of R&D, t A is the proportion of human capital allocated to the research and development department, and ( ) represents the output elasticity of each element respectively.

Environmental management department
Environmental quality is measured by the rate of change of pollution stock.Social capital can enhance people's awareness of resource conservation, urge people to abide by the relevant rules of ecological protection, and improve the performance of environmental governance.The motion equation of environmental quality is constructed as follows: Among them,  E is the increment of pollution stock, q 1 represents the self regeneration parameter of the environment, q 2 represents the input parameter of social capital, q 3 represents the impact parameter of natural resource consumption on environmental pollution, q 4 represents the input parameter of green technological innovation, and t E represents the proportion of human capital allocated to environmental management departments.

Representative household
This paper introduces material consumption C, generalized human capital K H and pollution stock E into the welfare function of representative families, and endows them with the concept of sustainable economic, social and environmental development.The economic goal of sustainable development is to ensure a high quality of living standards and meet people's needs for a better life.The introduction of material consumption C highlights people's internal demands for a high-quality life, reflecting the quality of economic growth.People are the center of sustainable development, and the free and comprehensive development of individual abilities is the real social development.The introduction of generalized human capital K H highlights that people pay more attention to the overall improvement of individual sustainable development (education and health), which helps all members of society have the ability to obtain equal opportunities to participate in economic activities and share economic achievements, and depicts the level of social welfare (Sen 2001).Ecosystem environment is a prominent shortcoming of sustainable development.China is still facing serious environmental pollution.Sustainable development must maintain a certain stock of natural capital, so that development can be controlled within the carrying capacity of ecosystem environment.The introduction of pollution stocks E reflects people's pursuit of sustainable development of ecosystem environment, and stands for environmental quality.The utility function of the representative household in the infinite time domain is as follows: Among them, ρ is the time discount rate, and the integral additive utility function is as follows: respectively corresponding to the weights of consumption, generalized human capital and pollution stock in the utility function.
The essence of social sustainable development is in order to improve the well-being of people's livelihood, by strengthening individual education, health and other self-generating capabilities (improving the level of human capital in a broad sense), it will help them to obtain more equitable and reasonable opportunities for self sustainable development, narrow the generalized human capital gap between different classes to obtain economic income, and achieve fair income distribution (Sen 2001).The higher the level of generalized human capital, the higher the development opportunities and the equality of national income distribution enjoyed by all members of society, and the more equitable the society will be.Therefore, the introduction of educational capital K T and health capital K M into the utility function can best reflect the high-quality connotation of capacity development, opportunity equality, social welfare and people's livelihood. 1 Based on this, this paper substitutes equation (4) into equation (9) to obtain utility function equation (10), that is, educational capital and health capital are introduced into the utility function, so as to pay attention to the balanced development of economy, social (education and health) ,environment and other fields.
, ln ln ln ln 10 respectively corresponding to the weights of material consumption (economy), education and health care (society), and pollution stock (environment) in the utility function.
1 By strengthening individual education, health and other self-sustaining capabilities (improving the level of human capital in a broad sense), it will help them to obtain more equitable and reasonable opportunities for self-development, achieve social equity and improve people's well-being.Therefore, the connotation of human capital promotion, human capacity development, education and medical (health) capital, equality of opportunity, social equity, social welfare and people's well-being in the broad sense are consistent, and are used as synonyms in the text.
The goal of social planners is to maximize the intertemporal utility of representative families in the infinite time domain, and build a Hamilton optimization model of family present value as follows: , 11

H P P H H S S
A A E

Model solving
It is assumed that homogeneous human capital can flow freely between departments to obtain the same wage income, so it is set in the model Y are decision variables and l l l l l l l , , , , , , M are shadow prices of covariant variables.
For each decision variable, the first-order condition of utility maximization is as follows: C Take the derivative of : 13 For the covariant variables respectively, the Euler equation is obtained as follows: The condition of transversality: Making decisions based on the overall position is the social optimal equilibrium model.Solving simultaneous systems of equations with 21 unknown variables, respectively growth rate variables g , H g ,

P P H H S S A A E T T M M
Economic growth rate g Y can be derived from simultaneous equations: c n q q a t a r a r q c a n t a q n t q q n r a t a a Due to the difficulty of solving the equation, Matlab software is used to conduct numerical simulation.This method can analyze the relationship between parameter factors and related variables, and intuitively demonstrate how social capital promotes sustainable development.

Parameter setting
Sustainable development aims to pursue the coordinated development of economy, society, environment and other aspects.Therefore, the weight of social welfare and environmental quality in the utility function should be equal or even more important than that of economic growth.In view of this, the weight values of the three in the utility function are initially set as and the adjusted values u u , 2 3 will be compared and analyzed in the subsequent simulation.The setting of relevant parameters should fully consider the economic characteristics of sustainable development.China is in the new normal of optimizing the input of factors and adjusting the economic structure, from extensive growth mainly relying on increasing the consumption of material resources to intensive growth mainly relying on technological progress, improving management and improving the quality of workers.Based on this, referring to some parameter settings in relevant literature (Palma et

Robustness and consistency analysis of simulation results
In this paper, the model is initially assigned based on the setting of the above related parameters, so as to obtain a 'benchmark model'.And the numerical value is adjusted within the reasonable value range of the related parameters for simulation analysis, so as to ensure that the numerical simulation results of the model can better reflect the internal relationship between variables.Based on the 'benchmark model' of social capital as the output elasticity c 4 of factors of production, and as a resource acceleration factor accumulation of parameters a , 2 f , 2 g , 3 q 2 repeated simulation test, found that the five parameter values have a critical point, respectively is 0.08, 0.03, 0.03, 0.19, 0.19.When the numerical simulation exceeds the critical value, the simulation results will become extremely unstable, so the theoretical model should be simulated in a reasonable range of parameters.The following simulation results fully prove that when the parameters of the 'benchmark model' are taken within a reasonable range, the simulation results have good robustness and consistency, and achieve optimal equilibrium.
(1) The simulation results of the optimal equilibrium solution of consumption output ratio

C Y
are basically kept at around 73%, which is approximately close to the 78% of global final consumption in GDP reported by the World Bank.
(2) The simulation results of capital output ratio K Y P are always within the range of [3.38,3.58].
(3) The simulated growth rate of natural resource consumption g N Y always keeps negative growth.If natural resources are consumed at a positive growth rate, resource depletion will inevitably accelerate, and it is difficult to achieve sustainable development.The consumption of resources should not only meet the needs of contemporary people, but also consider the needs of future generations.The sustainable use of resources is the only way to achieve sustainable development (Ekins 2003).
(4) The simulated value of pollution stock increase rate g E is always −3.5%, and the absolute value of this value is the self regeneration and repair coefficient of the environment.That is, the ecosystem environment maintains the stability of the system with its own regeneration and repair capacity, which is consistent with the views of Pelenc and Ballet (2015).Sustainable development emphasizes that development is limited and cannot exceed the carrying capacity of the environment.Whether to maintain the stable balance of the natural ecosystem environment system is a question related to the sustainable survival of human beings.Therefore, under the influence of social capital, the development and utilization of natural resources by human beings, as well as the environmental pollution caused by production activities, can be kept within the limits of the self repair and renewal capacity of the ecosystem.

Numerical simulation analysis of the core connotation of sustainable development
Based on the 'benchmark model' constructed above, this part intends to conduct a multi-group numerical simulation analysis on the weights of utility functions, in order to reveal the core connotation of sustainable development.

Simulation analysis of economic growth weight u 1 in utility function
According to Formula (10), u , 1 u , 2 u 3 respectively represent the weights of economic growth (material consumption), social welfare (education and health care), and environmental quality (pollution stock) in the utility function.In the benchmark model, the initial state of utility weight is set as three cases:①u As shown in table 1(1), the numerical simulation results of u u show that when the u u , 2 3 value remains unchanged and the material consumption weight u 1 gradually increases from the initial value 1, people are more willing to increase consumption expenditure and enjoy life, so the consumption output ratio shows an upward trend, from 0.73 to 0.8, but the economic growth rate g Y gradually decreases and shows a negative growth, from 2.48% to −2.81%.The reason is that if the consumption-output ratio rises, the investment rate will decline, which will inevitably lead to insufficient investment in the whole society, which cannot meet the high consumption and restrain economic growth.Therefore, only by maintaining a reasonable consumption-output ratio, not pursuing excessive consumption, and reducing unnecessary resource consumption, can we promote sustainable economic growth.Through comparative analysis of tables 1(1) (2), the simulation results show that when the Table 1.Numerical simulation results of economic growth weight u .
(1)u initial value of the u u , 2 3 unchanged, and value u 1 more than the initial value 1, system simulation will gradually become unstable, economic growth rate g Y all showed a downward trend and showed negative growth, so the numerical simulation of the follow-up to economic growth (material consumption) weights in the utility function of the value is set to 1.
It is worth noting that the third simulation case, that is, the weight of economic growth is u = 1, 1 while the weight of social welfare and environmental quality is u u = = 0.5, 2 3 shown in table 1(3).The simulation results of economic growth rate g Y are completely negative, and the simulation values of g E and K Y P no longer remain stable, which is obviously not in line with the actual situation.But this fully illustrate that the one-sided pursuit of economic growth and the development concept of 'GDP first', while ignoring people's urgent needs in social welfare, ecosystem environment and other fields, will lead to a series of social problems such as industrial structure imbalance, income inequality, lack of public resources such as education and health care, and serious environmental pollution, which will inevitably fail to ensure sustainable economic development, The numerical simulation results are consistent with the views of some scholars(Coscieme et al 2020) Therefore, the value of u 2 and u 3 in the utility function should not be lower than the value of u .it can be seen from table 2 that as the social welfare weight u 2 in the utility function rises from 1 to 1.5, it means that people will continuously improve their education and culture level and health status, the growth rate of human capital rises g K H from 3.07% to 3.26%, and the economic growth rate g H rises from 2.83% to 3.03% accordingly.This result strongly proves that sustainable development can balance the economy and society, and the two are mutually unified.Constantly improve the social welfare (u 2 up), means that people can enjoy better education, such as health care, public resources, thereby reducing access to economic income between different classes of education health capital gap (g K T and g K M up), it will mobilize people initiative, creativity, innovation for the development of production, improve labor productivity, promote economic growth.
However, it can be found from table 2 that when the social welfare u 2 reaches 1.5, far exceeding the economic growth value u 1 of 1, the economic growth efficiency g Y has an an inflection point, the growth rate drops sharply, and the simulated values of g , E g , N Y K Y P no longer maintain a steady state.The simulation results shows that economic growth is still the basic principles of sustainable development.People's aspirations for a better life should be commensurate with the current level of economic development.Therefore, the weight of social welfare u 2 in the utility function is set as 1.5 in the subsequent simulation.

Simulation analysis of environmental quality weights u 3 in utility functions
The initial state of utility weight is set as: According to the adjusted parameter values u , 3 as shown in table 3, when the value of environmental quality weight u 3 is lower than u 1 and u , 2 between the interval [0.5 and 1.5), the system simulation results become extremely unstable and the economy shows negative growth.It shows that the improvement of ecosystem environment must not be ignored in the stage of sustainable development, and the extensive economic growth at the expense of environment cannot bring about sustainable economic growth.Environmental quality is the basis of economic growth and social welfare, and ecological protection is the development of productive forces.The quality of ecosystem environment construction directly restricts the degree of economic development and is the source of sustainable development of human society.However, when the weight value u 3 exceeds u , 2 from 1.5 to 2, as shown in This just shows that it is not advisable to overemphasize environmental quality while inhibiting economic and social development.Sustainable development should adhere to the concept of harmonious coexistence and all-round development between people, between people and society, between people and nature, and social welfare and environmental quality are equally important, regardless of priority.Therefore, the weight of environmental quality u 3 in the utility function is set as 1.5 in the subsequent simulation.

The weight relationship between economic growth, social welfare, and environmental quality in sustainable development
The essential characteristic of sustainable development is its multidimensionality.The key analysis should focus on the balancing relationship among the economic, social, and environmental dimensions.The fundamental requirement is that economic growth should be in harmony with social welfare and environmental quality.This paper takes the reality of China as an example for numerical analysis.Through the simulated results mentioned above, it is evident that there exists a mutual influence and coordination among the dimensions of economic growth, social welfare, and environmental quality.The weight ratio u u u , , 1 2 3 among them is set as 1:1.5:1.5, effectively interpreting the core essence of sustainable developme.nt in China.This is manifested in the following three aspects: (1) Economic growth forms the foundational premise of sustainable development, but the weight of economic growth should not surpass the weights assigned to social welfare and environmental quality.Simulation results indicate that solely pursuing economic growth speed while neglecting the urgent needs of the populace in areas such as education, healthcare, and ecosystem environment would fail to ensure sustained and healthy economic development.The results of economic growth lack sharing, and the economy cannot maintain high-speed and sustained growth.Sustainable development emphasizes the 'participatory' process of economic growth and the 'sharing' of growth outcomes, creating a large number of employment and entrepreneurial opportunities through sustainable economic growth methods, especially providing more opportunities for the poor to participate to the maximum extent possible.It tends to achieve 'pro poverty' economic growth through increasing productive employment opportunities rather than direct income distribution methods.
(2) The weight assigned to social welfare should take precedence over the weight allocated to economic growth.
Simulation results indicate that the core concept of sustainable development is the equalization of social welfare, with primary attention given to the comprehensive development of individuals.By increasing investments in fields such as education and healthcare, enhancing the degree of equalization in public services, and improving the basic qualities and self-development capabilities of individuals, particularly the impoverished population, the aim is to promote economic and social coordinated development through equal opportunities and shared outcomes.The importance of equal opportunity (equal access to basic public services such as education and healthcare) surpasses that of equal outcomes (income distribution equality).Inequality in opportunities arising from individual backgrounds in education and health is a waste of human talents and a fundamental issue of social injustice, exacerbating the widening income gap and affecting the progress of sustainable development.However, it is worth noting that when the weight of social welfare reaches 1.5, far exceeding the weight of economic growth at 1, the economic growth rate declines linearly.This indicates that economic growth remains a fundamental principle of sustainable development, and the level of social welfare should be adapted to the current stage of economic development.Otherwise, there is a risk of falling into a 'high welfare trap,' where economic growth stagnates.
(3) The value assigned to the weight of environmental quality should not be lower than the weights assigned to economic growth and social welfare; otherwise, negative economic growth may occur.Simulation results indicate that green growth is a crucial avenue for achieving sustainable economic development.Sustainable development must not overlook improvements in the ecosystem environment, as environmental quality forms the foundation of both economic growth and social welfare.Sustainable development entails environmentally friendly economic growth, and economic development decisions must fully consider environmental carrying capacity, improve resource utilization efficiency, and reduce pollution emissions.Neglecting environmental issues during economic development inevitably leads to the depletion of natural resources and continuous degradation of the ecosystem environment.This will bring greater uncertainty to the production and livelihoods in impoverished areas.Moreover, compared to other dimensions of poverty, ecological poverty tends to be more persistent.However, it is noteworthy that an increase in the value of the social welfare weight from 1.5 to 2 does not necessarily result in a continued increase in the economic growth rate.This indicates that excessively emphasizing environmental quality at the expense of suppressing economic development is also undesirable.
4.5.Analysis of China's sustainable development model 4.5.1.Construction of the sustainable development indicator system Sustainable development represents a new economic development paradigm that emphasizes the comprehensive development of the economy, society, and environment.These three dimensions form an interconnected and interdependent whole, where sacrificing the interests of one aspect can impede the realization of others.China's sustainable development model considers economic growth as indispensable but places a strong emphasis on social welfare and environmental quality.Fair social welfare and a well-preserved ecosystem environment are deemed essential prerequisites for achieving sustainable economic growth.By promoting human feasibility and maintaining the regenerative ability of the ecosystem environment, efficient economic growth and maximizing social welfare and ecological welfare can be achieved, So as to make the achievements of economic development more equitable and benefit all the people.In order to analyze the status of sustainable development in China more intuitively, based on the theoretical definition and weight setting of the concept of sustainable development mentioned earlier, a scientific and reasonable sustainable development indicator system is constructed to quantitatively measure and evaluate the level of sustainable development in China from 2001 to 2021.The Sustainable Development Index mainly selects indicators from three dimensions: economy, society, and environment.Economic growth is measured from four dimensions: economic development level, industrial structure, economic growth momentum, and highquality consumption.Six dimensions are selected as social welfare indicators: high-quality education, public healthcare, employment opportunities, social security, income growth and distribution, and social assistance, To measure environmental quality from two dimensions: pollution reduction and green environmental protection, as shown in table 4. The weights for the three primary indicators of economy, society and environment are based on the simulated results of their weight factors, which are set at 1:1.5:1.5.The weights for secondary and tertiary indicators are calculated using the entropy weight method.Ultimately, the Sustainable Development Index is derived, where the total index equals the economic growth index multiplied by 1, added to the social welfare index multiplied by 1.5, and added to the environmental quality index multiplied by 1.5.

Analysis of the measurement of sustainable development levels in China
Based on the accessibility of statistical data, this paper calculates the China Sustainable Development Index and three sub indices from 2001 to 2021, including the Economic Growth Index, Social Welfare Index, and Environmental Quality Index.The data for each indicator primarily come from the 'China Statistical Yearbook,' 'China Environmental Statistical Yearbook,' and the China Economic and Social Development Statistics Database.
From the perspective of China's Sustainable Development Index, as shown in figure 1, the index gradually increased from 1.0 in 2001 to 2.05 in 2021, indicating significant progress in China's sustainable development process in recent years and a significant overall improvement.In the process of economic growth, the Chinese government adheres to the principle of putting people first, optimizing industrial structure, strengthening infrastructure construction, focusing on ensuring and solving people's livelihood issues such as education, healthcare, and employment, continuously improving the social security system, and vigorously promoting the construction of ecological civilization, accelerating the creation of a 'beautiful China', allowing more people to enjoy the fruits of economic development more fairly and reasonably, and achieving coordinated development of the economy, society, and environment.
From the composition of the sustainable development index, as shown in figure 2, the environmental quality index accounts for the largest proportion, followed by the social welfare index, and the economic growth index accounts for the smallest proportion.This indicates that China's sustainable development model fully focuses on ecological and environmental issues, and focuses on ensuring people's livelihoods.China has a vast territory and abundant natural resources, with obvious advantages in ecological and economic development.Under the strategic goal of building an ecological powerhouse, the Chinese government adheres to ecological priority and green development, effectively promoting significant improvement in environmental quality, and vigorously developing ecological industries, ecological agriculture, and ecological tourism.It transforms green ecological advantages into economic advantages, constructs a green, low-carbon, and circular economic system, and the ecosystem environment becomes an important force in promoting sustainable development in China.

Mathematical analysis
It can be seen from Formula (29) that, based on the position of social planners and driven by social capital, there is a determined social optimal balanced growth path BGP (Balanced-growth path, BGP) .Along this path, economic variables {Y, K P , C} grow at the same balanced growth rate, and c The model solution results reveal the internal mechanism of social capital promoting sustainable development: the elasticity of social capital output c , 4 the coefficient of social capital stock transforming to human capital a , 2 the elasticity of social capital output to technological innovation g , 3 and the partial derivative of social capital input parameters q 2 in environmental governance to economic growth rate are all positive, indicating that social capital is included in the production function of the final product sector as a 'new element' based on the value added attribute.It indicates that social capital, based on the value-added attribute of value, is incorporated into the production function of the final product sector as a 'new element', as well as introduced into the production or accumulation equation of human resources, technological innovation, and environmental management departments as a 'new resource' based on informal institutional characteristics.It can directly or indirectly promote sustainable development, and social capital has become an endogenous driving force for sustainable development.It can be seen from table 5 that when the output elasticity of social capital c 4 as a new factor into the production function increases from the benchmark value of 0.02 to the critical point of 0.08, the output elasticity of natural resources a 3 decreases from 0.14 to 0.08, but the economic growth rate g Y steadily increases from the initial simulation value of 2.69% to 3.03%; In addition, social capital has always accelerated its growth at a rate greater than economic growth, that is 4 so as to offset the negative benefits of economic growth caused by slow resource input, and obtain greater economic output with less resource input.This shows that social capital can break through the bottleneck constraints of traditional resource elements and become a new engine and endogenous power for sustainable economic growth.The output elasticity of natural resources c 3 has subsequently decreased from 0.14 to 0.08, which means that the rapid economic growth is no longer dependent on a large increase in the consumption of material resources.Instead, social capital is used to effectively control the total use and development speed of natural resources and improve the efficiency of the use of various economic resources.Accelerated accumulation of social capital can enhance the environmental awareness and social responsibility of enterprises and residents, promote the innovation and application of clean technology, optimize the allocation of resources with social networks as the carrier, and improve the utilization efficiency of natural capital.Furthermore, it can effectively reduce the pressure of obtaining economic output by relying on massive input of material resources, promote the transformation of China's economic growth from extensive to intensive, and achieve more sustainable economic growth.
The numerical simulation further found that when the parameters of social capital a 2 in the human capital accumulation equation increased from 0.01 to the critical point 0.03, the parameters of social capital f 2 in the self accumulation equation increased from 0.01 to the critical point 0.03, the parameters of social capital g 3 in the technological innovation equation increased from 0.1 to the critical point 0.19, and the parameters of social capital q 2 in the environmental quality movement equation increased from 0.1 to the critical point 0.19, the economic growth rate g Y rose steadily to 3.4%, It is basically consistent with the prediction of China's potential economic growth rate by Institute of Economics, Chinese Academy of Social Sciences.China is entering a stage of sustainable development in which economic growth is slowing down gradually.With the continuous improvement of development level, China's economic growth rate is expected to continue to decline in the future(Long and Ji 2019).The reason is that sustainable development is not simply pursuing high-speed economic growth, but seeking coordinated development of economy, society and environment, focusing on social welfare and environmental quality.The transformation of development mode and economic structure will certainly promote the transformation of China's economic development from 'post industrialization' with high-speed economic growth to 'urbanization' with high-level social welfare.As the industrial structure is adjusted from the secondary industry with high emissions and high productivity growth to the tertiary industry with low emissions and relatively low growth rate, economic growth is bound to show 'structural deceleration'.The simulated value of economic growth in this paper is based on the numerical simulation results under the premise that the weight ratio of economic growth, social welfare and environmental quality in the utility function is set as 1:1.5:1.5, which is in line with the sustainable development trend in the future.The direct mechanism through which social capital propels sustainable development is as follows: Social capital consistently accelerates at a rate greater than economic growth.Simultaneously, the contribution of the accumulation rate of social capital to the economic growth rate consistently exceeds its own output elasticity.Under the influence of social capital, human exploitation of natural resources and the environmental pollution resulting from production activities can be maintained within the limits of the ecosystem's self-repair and renewal capacity.Consequently, the accelerated accumulation of social capital, particularly the portion exceeding output elasticity due to economic growth, is sufficient to offset the negative effects on economic growth caused by a slowdown in resource inputs.This facilitates the transition of China's economic growth from extensive to sustainable development, ensuring a stable and progressively advancing trend.Social capital has thus become a new engine and endogenous force for sustainable development in China.
The indirect mechanism of social capital in promoting sustainable development lies in: (1) Social capital uses social networks as carriers, embedding knowledge and technology into mutually beneficial social relationship When social capital is introduced into the environmental management department, the input parameter of social capital in environmental governance is q 2 Social capital can reduce energy consumption, ensure environmental quality, and indirectly promote sustainable development q 2 rises from the benchmark value of 0.1 to the critical value of 0.19, and the growth rate of environmental pollution stock g E remains at a steady state value of −3.5%.Its absolute value is the self regeneration and repair coefficient of the environment, which is 0.035.In addition, the growth rate of resource consumption g R is negative, and g Y steadily increa- ses to 3.05% ( ) networks, promoting knowledge sharing and technology diffusion, enhancing collaborative cooperation between R&D personnel and organizations, avoiding duplication of R&D activities, stimulating R&D personnel's innovative thinking and inspiration, improving their independent innovation ability, injecting a continuous stream of innovative vitality into economic development.The prosperous development of the economy and society will inevitably create more employment opportunities and new job positions for society, thereby helping people increase income through productive employment, improve their quality of life, share the fruits of economic development, and achieve sustainable economic growth.Sustainable development is an efficient development driven by technological innovation and progress, and social capital drives the transformation of economic growth towards an innovation driven sustainable development model.
(2) Social capital is a network that connects social relationships.Through interaction and exchange between individuals, it can promote the dissemination and sharing of knowledge, which is beneficial for individuals to absorb and master advanced knowledge and experience, thereby creating a favorable social environment for the improvement of human capital level and accelerating the accumulation of generalized human capital.Social capital can help individuals expand and enrich their access to educational opportunities and medical information through the construction of interpersonal networks and information network platforms.It helps people better enjoy high-quality public resources such as education, culture, healthcare, leisure and entertainment, improve the equalization of education and medical services, and narrow the broad human capital gap in obtaining economic income between different social classes.The more social capital is occupied, the higher the level of generalized human capital, and the higher the development opportunities and equal distribution of national income enjoyed by all members of society, which helps to achieve the maximization of social welfare.
(3) Social capital, relying on social trust and reciprocity norms, can promote residents to comply with relevant environmental protection regulations, continuously enhance public environmental awareness, and force enterprises to innovate green.It can effectively control the total utilization of natural resources, improve resource utilization efficiency, supervise enterprises to prioritize clean technologies and renewable energy, thereby reducing the negative impact of pollution emissions on environmental stock.Social capital promotes collaborative cooperation, breaks the 'prisoner's dilemma' in environmental protection, alleviates conflicts of interest in environmental governance, improves the performance of environmental governance, enables the ecosystem environment to self repair and maintain stability with regenerative capacity, and promotes the transformation of ecological advantages into economic development advantages.A good ecosystem environment is the basic right of all members of society, the most inclusive livelihood, and the long-term driving force for sustainable economic and social development.

Conclusions
This article uses the Hamilton optimization model and numerical simulation methods to analyze the core connotation of sustainable development in China, and explores the mechanism and path of social capital promoting sustainable development.The research results indicate that: (1) Sustainable development seeks comprehensive economic, social, and environmental development, with a weight ratio of 1:1.5:1.5 for economic growth, social welfare, and environmental quality, which can better interpret China's concept of sustainable development.
(2) Social capital can effectively control and slow down the development and utilization of natural resources, improve the efficiency of various economic resources, and economic growth is no longer dependent on a significant increase in resource consumption, thereby promoting the transformation of China's economic growth model towards sustainable development.(3) Social capital indirectly affects the sustainable development of the economy, society, and environment through three paths: accelerating the accumulation of broad human capital, promoting technological innovation and progress, and improving environmental governance performance.
The implications of the above conclusion are very clear.Firstly, in the stage of sustainable development, investment in people's livelihoods and environmental protection should be increased to promote equalization of basic public services.The government should accelerate the improvement of public service systems such as education, healthcare, social security, and employment, vigorously develop circular economy, and improve the level of environmental infrastructure construction.Secondly, expand social capital investment, promote a highly open model of social capital, and effectively optimize the allocation of resources by building a network platform for communication and exchange between the government, enterprises, and residents.This will promote the formation of low-carbon production and green consumption environmental awareness among the general public, and promote sustainable development.Thirdly, increase the participation of social capital in talent cultivation, technological innovation, and environmental governance.Accelerate the construction and application of social network platforms such as online education and remote healthcare, strengthen mutual cooperation among innovative entities, improve independent innovation capabilities, and vigorously cultivate the ecological moral awareness and behavioral norms of the whole population.

(
al 2010, Wan and Zhang 2023) , the parameters set in this paper are as follows: c = 0

14. 2 .
Simulation analysis of social welfare weights u 2 in utility functions The initial state of the utility weight is set as:

Figure 2 .
Figure 2. in China's growth index, social welfare index, and environmental quality index from 2001 to 2021.
LetC Y be the change rate of Y , that is, the economic growth rate 2 , others are interpreted in the same way.

Table 2 .
Numerical simulation results of social welfare weights u .

table 3 ,
the economic growth rate g Y cannot continue to rise and the value g Y remains unchanged.

Table 3 .
Numerical simulation results of environmental quality weights u .

Table 5 .
Numerical simulation results of the output elasticity of social capital c 4 as a production 'new factor'.Analysis of the mechanism of social capital's impact on sustainable development Based on the numerical simulation results of the effect of social capital on sustainable development, it can be seen that the impact of social capital on sustainable development mainly includes direct and indirect effects, as shown in table 6.Firstly, social capital has value added attributes and can be included as a new 'factor' in the production function of the final product sector, directly affecting sustainable development; The second is that social capital has informal institutional characteristics and can be seen as a new 'resource' introduced into the dynamic accumulation equations of various departments such as technological innovation, human resources, and environmental management, indirectly affecting the sustainable development of the economy, society, and environment.

Table 6 .
Numerical simulation results of the effect of social capital on sustainable development.