1 Introduction

The issue of global warming has attracted more and more attention from the international community. All countries attach great importance to it and begin to take measures to reduce greenhouse gas emissions. Carbon emission reduction has become an important international issue. The European Union (EU), Japan, and South Korea, as well as more than 110 other countries, have committed to achieving carbon neutrality by 2050 (Li et al., 2022). Carbon peaking and carbon neutrality goals reflect China's initiative to shoulder the responsibility of dealing with global climate change, and it also means that China will face arduous tasks and severe challenges in carbon emission reduction. With the rapid development of economy and the rapid expansion of production activities, China has become the country with the largest total carbon emissions and the second largest per capita carbon emissions in the world. High carbon dioxide emissions and rising total energy consumption are the realistic problems facing China. Therefore, further improving carbon regulatory policies and promoting enterprises to carry out low-carbon technology innovation are the internal requirements and fundamental ways to achieve carbon peaking and carbon neutrality goals, as well as an important link to realize the sustainable development of environment.

The practice of China's carbon regulation policy began with the carbon emissions trading mechanism, referred to as “carbon trading”. The development of carbon trading in China started from local pilot projects, and after ten years of continuous exploration and innovation, it was officially launched in the national carbon trading market. After more than two years of online operation, the national carbon market has established a basic framework system, and initially played the role of carbon price discovery mechanism (Ministry of Ecology & Environment of the People's Republic of China, 2023). Since the implementation of carbon trading, some achievements have been made in carbon emission reduction, but many problems have also been exposed. Firstly, carbon trading can only cover key emission units, and the scope of regulation is relatively narrow. It is estimated that the carbon emissions controlled by the national carbon market only account for about 70% of the total national carbon emissions (Zhang, 2022). Secondly, the incentive effect of carbon trading policy on low-carbon innovation of enterprises also has some limitations (Shen et al., 2017). Thus, a single carbon trading policy cannot effectively reduce carbon emissions in the long run. Wu et al. (2014) also showed that the carbon emissions trading mechanism is more suitable for China at this stage. With the enhancement of emission reduction in China, the introduction of carbon tax can be considered to make up for the shortcomings of existing carbon emission policies.

The carbon tax policy taxes the emitters according to carbon dioxide emissions, which will cover all sources of carbon emissions. As a policy tool, carbon tax can rely on the existing tax system with low implementation cost. Moreover, carbon tax is flexible and can adjust the tax system and tax rate according to different levels of economic development. In terms of low-carbon development effect, scholars have found that carbon tax can reduce carbon emissions and promote the development of low-carbon economy (Xu & Mao, 2019). The NDRC and the Ministry of Finance have also been actively studying the introduction of carbon tax policies. Since China put forward carbon peaking and carbon neutrality goals, it has been mentioned in many documents to promote the landing of the carbon tax system. At the beginning of 2022, the NDRC and other seven departments jointly issued the Implementation Plan for Promoting Green Consumption, which once again clarified the working idea of promoting green and low-carbon development through fiscal and taxation tools. The specific time of levying carbon tax in China depends on the actual domestic situation, but the pace is indeed accelerating.

At present, there is only the national carbon market in China's carbon pricing mechanism, and the carbon tax is still in the stage of research and formulation. Once the carbon tax is levied, it will directly increase the tax burden of enterprises, which is a big challenge for small and micro enterprises. Although relevant research showed that the synergy mechanism of carbon trading and carbon tax has a positive impact on promoting low-carbon emission reduction of enterprises (Ye, 2019), in fact, enterprises, as the subject of bounded rationality, may choose to emit the excess carbon directly rather than purchase the emissions through the carbon market when facing high carbon emission costs, such as carbon market transaction costs and carbon tax. That is to say, the government needs to fully consider the tolerance of enterprises to policies when formulating compound carbon policy, otherwise it will discourage the enthusiasm of enterprises to trade in the carbon market through innovative behavior, which will lead to enterprises choosing to violate the carbon regulation policy. Therefore, the implementation of carbon regulation policy also needs appropriate government supervision as a guarantee.

In the real world, the strategy equilibrium between the carbon emission enterprises and governments is not the result of a game. Affected by information asymmetry and other external conditions, they will continue to play a dynamic game and gradually reach a strategy equilibrium state. In short, local governments and carbon emission enterprises are the actors of bounded rationality. Evolutionary game theory is a theory that studies the dynamic adjustment process of group strategies from the perspective of bounded rationality in order to find a stable equilibrium strategy. Many scholars have successfully applied it in the study of various social practical problems (Gu et al., 2022; Guttman, 2000; Kosfeld, 2002; Liu et al., 2022). Thus, the use of evolutionary game to analyze the low-carbon innovation between local governments and enterprises in carbon regulations is very suitable for the characteristics of game subjects in real life.

Therefore, various factors are comprehensively considered to optimize the existing carbon policy. Based on the evolutionary game theory, we introduce carbon tax policy into China's existing carbon regulations in this article, construct an evolutionary game model between local governments and carbon emission enterprises, and explore the synergy impact mechanism of carbon trading and carbon tax on low-carbon innovation of enterprises, with a view to designing a compound carbon policy that can effectively promote low-carbon innovation of enterprises and inhibit carbon emissions. The main contributions of this article are as follows.

  1. (1)

    Based on the assumption of bounded rationality, we take the government as the supervision subject into the game model, and explore the synergistic incentive mechanism of the compound carbon policy with carbon trading and carbon tax on enterprise innovation under government supervision.

  2. (2)

    In the design of the compound carbon policy, this article fully considers the tolerance of enterprises to the policy, and introduces the preferential income tax policy in carbon regulation.

  3. (3)

    The compound policy design is in line with the actual situation of China's carbon regulation, and fully reflects the punitive supplementary role of carbon tax on carbon trading policy.

The remainder of this article is structured as follows. Section 2 reviews the relevant literature. Section 3 constructs an evolutionary game model between local governments and carbon emission enterprises under the compound carbon policy. Section 4 analyzes the stability of governments and enterprises, and the evolutionary game system. Section 5 carries out numerical simulation and policy effect simulation. Section 6 gives the main conclusions and policy suggestions, and future research directions.

2 Literature review

2.1 Comparative analysis and policy optimization of carbon tax and carbon trading policy

In the process of implementing carbon emission reduction goal, the Chinese government has constantly updated low-carbon development measures (Lv et al., 2019), to guide enterprises to conduct low-carbon production and develop low-carbon economy. Among these policy tools, carbon tax and carbon trading system based on market mechanism are the two most important carbon emission reduction measures proposed by the United Nations Framework Convention on Climate Change (Yuan, 2022). The Carbon trading policy reduces emissions by trading air as a resource in the market, while carbon tax curbs carbon emissions by command and control. Theoretical research (Tian et al., 2024) and policies practice (Wissema & Dellink, 2006; Yu et al., 2022) showed that a single carbon tax or carbon trading system has a certain positive impact on promoting low-carbon innovation of enterprises and curbing carbon emissions.

Carbon trading system and carbon tax policy have their own advantages and disadvantages. Keohane (2009) and He et al. (2023) believed that carbon trading policy has more advantages. However, Pope and Owen (2009) compared the two from the aspects of management cost and contract performance cost, and found that carbon trading has no universal advantage in management cost. Meanwhile, some scholars also believe that carbon tax can regulate carbon emissions by controlling the level of tax rate, which is better than carbon trading system in reducing emissions (Avi-Yonah & Uhlmann, 2009) and has more practical significance (Zeng et al., 2014). Other scholars believe that the two policies have their own advantages: Carbon tax has low operating cost and can cover all polluting enterprises, and it is fairer and conducive to enterprises to control the cost of emission reduction; while carbon trading policy has more advantages in carbon emission control, emission reduction incentives, promoting technological innovation of enterprises and resource allocation efficiency (Deng et al., 2014; Guan et al., 2021).

In terms of policy optimization, global researchers have also carried out a lot of research. For the carbon trading policy, Ye and Linghu (2015) proposed a differentiated carbon quota allocation strategy for high-emission enterprises and low-emission enterprises under the duopoly competition environment. Wang et al. (2016) proposed an optimization model based on DEA to calculate China's carbon quota. Based on empirical data research, Zhu (2023) found that the policy effect of carbon trading is not stable and sustainable. We should learn from other countries to strengthen the top-level design mechanisms in this sphere, legislation, information disclosure, trading rules, supervision and other systems all to be strengthened and promoted so as to facilitate the rapid development of carbon trading markets. For carbon tax, Timilsina et al. (2011) found that the coordinated implementation of carbon tax and subsidy policy can alleviate the problem of revenue reduction caused by excessive investment in innovation, and improve the enthusiasm of enterprises to reduce emissions. Drawing on the experience of the failed implementation of the carbon tax policy in Australia, Baranzini and Carattini (2016) pointed out that in the design of carbon tax policy, the acceptability of the public should be taken into account, and the income of the carbon tax should be dedicated to the environment, which may gain more support.

The above literatures have analyzed the respective policy effectiveness of carbon tax and carbon trading system, and put forward optimization suggestions based on the policy itself, but they have not aware of the correlation between the two emission reduction policies.

2.2 Discussion on the compatibility of carbon tax policy and carbon trading system

A single carbon trading or carbon tax policy may be restricted by realistic factors, and it is difficult to fully play the theoretical emission reduction effect (Guan et al., 2021). The practice of EU also showed that the implementation of a single carbon policy is often difficult to achieve the desired results (Bruvoll & Larsen, 2004). Therefore, some countries have begun to try to take multiple measures to reduce carbon emissions. Scholars have also gradually innovated research ideas and begun to explore whether carbon tax and carbon trading system can be implemented compatibly to achieve emission reduction goals.

In the early 1990s, Finland, Denmark, Norway and other European countries took the lead in introducing carbon tax to protect the environment. At the beginning of the implementation, certain emission reduction effect was achieved (Wissema & Dellink, 2006), but Norway's policy practice showed that the implementation of a single carbon tax policy is not entirely successful (Bruvoll & Larsen, 2004). Based on this, scholars proposed that the carbon emission trading mechanism could be used to make up for the lack of carbon tax. Lee et al. (2007) found that the effect of carbon tax on emission reduction among different industries is significantly different, and suggested that carbon tax should be implemented in combination with other emission reduction policies such as emissions trading to better achieve emission reduction goals. According to related research (Li & Shi, 2022), carbon tax is an important part of environmental regulations, and the strategic combination of carbon tax mechanism with subsidy mechanism, penalty mechanism and public supervision mechanism can effectively encourage enterprises to carry out green technology innovation.

At present, there is only carbon trading mechanism in China's carbon regulation. Based on the current background of China's low-carbon economy and the urgency of carbon emission reduction, scholars have also conducted extensive discussions on the compatibility of carbon tax and carbon trading, and actively provided policy suggestions for the implementation path (Xiao et al., 2023) and system design of China's carbon tax (Xu, 2021; Yang, 2024), calling on the government to introduce carbon tax policy as soon as possible. Wei (2015) analyzed the relationship between carbon tax and absolute emission reduction target and relative emission reduction target respectively, and found that the compatibility between carbon trading and carbon tax is related to the type of emission reduction targets. With the deepening of research, Piao and Yang (2016) proposed that, in the environmental economic market, carbon trading and carbon tax should be used together to achieve complementary advantages, so as to effectively avoid the failure of market mechanism. Moreover, the method of adopting carbon tax mechanism in the short term and developing carbon trading in the long term would be more suitable for China's future development (Li & Qian, 2018).

2.3 Establishment and implementation of compound carbon policy

In order to achieve the greenhouse gas emission reduction target proposed in 2000, the United Kingdom implemented the carbon emission trading system on the basis of carbon tax policy the next year (Dresner et al., 2006). As the first country in the world to implement two policies at the same time, the UK has fully considered link and harmony between the two policies, which provides a good reference for the design and implementation of compound carbon policy in the EU and other countries in the world.

Scholars who affirm the compound carbon policy believe that as a highly flexible policy combination, it can generate higher emission reduction efficiency (Pope & Owen, 2009; Tamura & Kimura, 2008), and also help to promote low-carbon innovation of enterprises (Sun et al., 2021). Ye (2023) suggested that we should learn from the experience of some European countries in coordinating carbon trading and carbon tax policies, and establish a carbon tax system that is conducive to achievingthe dual carbon goals. The practice of OECD countries also showed that the compound policy of carbon tax and carbon trading is better than the single policy in controlling greenhouse gas emissions (Oueslati, 2014). Based on the two-level decision-making mechanism of government-enterprise, Zhao and Yin (2016) found that enterprises enjoy higher flexibility in emission reduction decision-making under the compound policy system. At present, the implementation of a compound carbon tax and carbon trading policy is not only an objective need for emissions reduction, but also an inevitable result of achieving high-quality environmental governance (Wu et al., 2020).

2.4 Impact of carbon regulations on enterprise strategy choice

The development of low-carbon technologies is an important approach to realize urban carbon neutrality goals and sustainable urban development (Shang & Lv, 2022). R&D activities have proven crucial not only in enhancing the competitiveness of enterprises but also in sustaining a healthy growth of industries (Bai et al., 2022). Therefore, exploring the promotion mechanism of carbon regulatory policies on enterprises’ choice of creative strategy has become a major topic in recent years. Effective carbon regulation policy should be able to motivate enterprises to choose low-carbon emission reduction strategies and promote enterprises to carry out low-carbon technology transformation. By analyzing the strategic interaction mechanism between the government and enterprises with serious pollution, Lu et al. (2022) found that the government's dynamic carbon regulation strategy has a positive impact on the choice of enterprises’ emission reduction strategies.

Developing low-carbon technology innovation is an important means for enterprises to achieve emission reduction targets and reduce emission reduction costs. Meltzer (2014) found that the carbon tax policy has a positive incentive effect on the green technology development of enterprises, and has a “double dividend” of improving the economic performance of enterprises and protecting the ecological environment. Chen and Hu (2018) concluded that the carbon taxes levied by governments are proved more effective to encourage low-carbon manufacturing than governments subsidize the low-carbon technology. However, some scholars argue that carbon trading, as a market incentive tool, has a stronger impact on low-carbon innovation than command control tools such as carbon tax (Bergek & Berggren, 2014) Based on the empirical analysis. Liao et al. (2020) also found that carbon trading could effectively promote enterprises to increase innovation investment and improve the efficiency of green economic growth.

It is not difficult to find that the academic community has reached a consensus. In the context of global climate change, the implementation of compound policy of carbon trading and carbon tax is better than the implementation of a single policy. However, scholars mostly focused on demonstrating the compatibility of carbon tax and carbon trading, as well as the impact of carbon policy on the choice of emission reduction strategies of enterprises, and paid less attention to the issue of low-carbon technology innovation of enterprises under the compound carbon policy. Therefore, based on the fully consideration of enterprises’ tolerance to carbon policy, in this article we put forward the idea of establishing a compound policy design with carbon trading as the main and carbon tax as the supplement, to explore the perfect path of compound carbon policy to promote enterprises' low-carbon innovation.

3 Problem description and model construction

Currently, China's national carbon market is still in the stage of free allocation. Inspired by some research (Tamura & Kimura, 2008), this article introduces carbon tax on the basis of carbon trading policy: Enterprises without carbon quotas are taxed according to their actual emissions, and enterprises with carbon quotas are taxed according to their excess carbon tax. Since this article aims to explore the impact mechanism of the compound policy of carbon trading and carbon tax on enterprise innovation behavior, the research object is set as enterprises with carbon quotas.

Considering the tolerance of enterprises to the compound policy and other issues, the policy environment of this article is set to implement a carbon tax policy with punitive supplementary effect on the basis of carbon trading, that is, when the government authorities supervise the illegal carbon emission behavior of enterprises, they will levy carbon tax on the part exceeding the quota as a punishment.

In fact, both the local governments and enterprises are subjects with bounded rationality and pursuing maximum benefits, and they are in a state of incomplete information symmetry. In the course of the long lasting relationship, both of them constantly adjust and improve their strategies through repeated games to achieve equilibrium. Unlike traditional game theory that emphasizes static equilibrium, evolutionary game theory emphasizes more on the development and change process of equilibrium strategies. It believes that subjects with bounded rationality cannot find their optimal equilibrium point in each game, but achieve game equilibrium through a long-term constant trying by mistake and imitational learning of advantageous strategies. We can conclude that the game characteristics of the subjects in our study are in line with the main ideas of evolutionary game theory. Therefore, we will use evolutionary game theory to onstruct an evolutionary game model between local governments and carbon emission enterprises, and explore the optimal path of compound carbon policy to encourage enterprise low-carbon innovation under government supervision.

According to the above description, to more intuitively demonstrate the game characteristics of the cooperative evolution between the government and enterprise, the following assumptions are given. The definitions of each parameter involved in the above assumptions can be summarized in Table 1.

Table 1 The definitions of parameters in assumptions
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Hypothesis

1 As the regulator of policy implementation, the local government is one side of the game, its strategy set is {strict supervision, relaxed supervision}. As the object of government supervision, the production enterprises of carbon emission are the other side of the game, and enterprises will choose from the strategy set {low-carbon innovation, traditional production}. To analyze the evolutionary process of strategy selection for the both, it is assumed that at time \(t\), the probability of local governments choosing “strict supervision” strategy is, and the probability of enterprises choosing “low-carbon innovation” strategy is \(x\), both \(x\) and \(y\) are functions of time \(t\).

Hypothesis

2 The construction of carbon trading market and carbon tax system by the government needs certain human and material costs. In the process of policy implementation, local governments supervise the behavior of enterprises and invest supervision cost, which is a monotonic increasing function of supervision intensity \(\alpha\). When \(\alpha { = }1\), local governments choose the “strict supervision” strategy, and its supervision cost is \(C_{1}\). The government departments allocate carbon emission quota \(E\) to enterprises free of charge, and the carbon tax rate is \(t_{3}\).

Hypothesis

3 When enterprises choose the “low-carbon innovation” strategy, they need to invest in equipment improvement, technology research and development and other innovation costs \(C_{2}\) to obtain basic income \(R\). After technology innovation, the carbon emission \(E_{1}\) of the enterprise will fall below the carbon quota, and there is no need to pay carbon tax. At the same time, enterprises can also sell the surplus carbon emission \(\Delta E_{1}\) in the carbon trading market at a price \(p\), and obtain profits, where \(\Delta E_{1} = E - E_{1}\). The low-carbon production process of enterprises will bring about environmental improvement, and will also bring economic benefits \(R_{e}\) for the government, such as investment attraction (Yang & Xu, 2021). In order to encourage enterprises innovation, preferential tax policies are implemented (Guan & Yam, 2015), and innovative enterprises can pay income tax at preferential tax rate \(t_{1}\).

Hypothesis

4 If enterprises do not innovate and continues to carry out production with traditional technology, they will get additional income \(r\), but they cannot enjoy preferential tax policies. In this case, enterprise can only pay income tax according to the normal tax rate \(t_{2}\), and their carbon emissions \(E_{2}\) will exceed the carbon quota. If enterprises consciously abide by the carbon policy, they need to purchase emission amount of \(\Delta E_{2}\) in the carbon market at the price of \(p\), where \(\Delta E_{2} = E_{2} - E\); however, in order to save costs, the enterprise may also choose illegal emissions behavior with probability \(\beta\). When enterprise’s violation is supervised by the government department, it shall pay a fine, which is the carbon tax on the part exceeding the quota.

It is assumed that the above elements are common knowledge between local governments and carbon emission enterprises, that is, the revenue function of each player is known among all players, where \(0 \le x \le 1\), \(0 \le y \le 1\), \(0 \le \alpha \le 1\), \(0 \le \beta \le 1\). When the government levies income tax on enterprises, it implements a preferential tax rate for innovative enterprises, so the preferential tax rate \(t_{1}\) is less than the normal tax rate \(t_{2}\). According to the corresponding relationship between the carbon emissions of innovative enterprises and traditional enterprises, it can be known that \(E_{1} < E < E_{2}\). Based on the above assumptions, the payment matrix of the game is shown in Table 2.

Table 2 The payment matrix of the evolutionary game model
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4 Evolutionary stability analysis

The expected returns of carbon emission enterprises with the two strategies (low-carbon innovation or traditional production) and the average expected returns are as follows:

$$ \left\{ \begin{gathered} U_{x} = (R + p\Delta E_{1} )(1 - t_{1} ) - C_{2} \hfill \\ U_{{\overline{x}}} = (R + r)(1 - t_{2} ) - [(1 - \beta )p + \alpha \beta t_{3} ]\Delta E_{2} - (1 - \alpha )\beta \Delta E_{2} t_{3} y \hfill \\ \overline{U}_{x} = xU_{x} + (1 - x)U_{{\overline{x}}} \hfill \\ \end{gathered} \right. $$
(1)

According to the Malthusian dynamic equation theorem (Oueslati, 2014), the replicated dynamic equation \(H_{1} (x)\) of enterprises is:

$$ H_{1} (x) = \frac{dx}{{dt}} = x(U_{x} - \overline{U}_{x} ) $$
(2)

Combined with Formula (1), Formula (2) can be expressed as:

$$ H_{1} (x) = x(1 - x)\{ (R + p\Delta E_{1} )(1 - t_{1} ) - (R + r)(1 - t_{2} ) + [(1 - \beta )p + \alpha \beta t_{3} ]\Delta E_{2} - C_{2} + (1 - \alpha )\beta \Delta E_{2} t_{3} y\} $$
(3)

Similarly, the expected returns of local governments with the two strategies (strict supervision or relaxed supervision) and the average expected returns are as follows:

$$ \left\{ \begin{gathered} U_{y} = (R + r)t_{2} + \beta \Delta E_{2} t_{3} - C - C_{1} + [(R + p\Delta E_{1} )t_{1} + R_{e} - (R + r)t_{2} - \beta \Delta E_{2} t_{3} ]x \hfill \\ U_{{\overline{y}}} = (R + r)t_{2} + \alpha \beta \Delta E_{2} t_{3} - C - \beta C_{1} + [(R + p\Delta E_{1} )t_{1} + R_{e} - (R + r)t_{2} - \alpha \beta \Delta E_{2} t_{3} ]x \hfill \\ \overline{U}_{y} = yU_{y} + (1 - y)U_{{\overline{y}}} \hfill \\ \end{gathered} \right. $$
(4)

The replicated dynamic equation \(H_{2} (y)\) of local governments is:

$$ H_{2} (y) = \frac{dy}{{dt}} = y(U_{y} - \overline{U}_{y} ) $$
(5)

Combined with Formula (4), Formula (5) can be expressed as:

$$ H_{2} (y) = y(1 - y)(1 - \alpha )[(\beta \Delta E_{2} t_{3} - C_{1} ) - \beta \Delta E_{2} t_{3} x] $$
(6)

4.1 Evolutionary stable strategy of carbon emission enterprises

In the replicated dynamic equation \(H_{1} (x)\) of carbon emission enterprises, let \(y_{0} = \frac{{C_{2} - (R + p\Delta E_{1} )(1 - t_{1} ) + (R + r)(1 - t_{2} ) - [(1 - \beta )p - \alpha \beta t_{3} ]\Delta E_{2} }}{{(1 - \alpha )\beta \Delta E_{2} t_{3} }}\). According to the stability theorem for differential equations and the properties of evolutionary stability strategies, when \(x^{*}\) satisfies \(F(x^{*} ) = 0\) or \(F^{\prime}(x^{*} ) < 0\), \(x^{*}\) is an evolutionarily stable strategy (ESS) (Weibull, 1995). The evolutionary stable strategies of carbon emission enterprises under different conditions will be analyzed as follows.

When \(y = y_{0}\), there is always \(H_{1} (x) = 0\), then any value of \(x\) in the interval range of \([0,1]\) is an evolutionary stable state.

When \(y \ne y_{0}\), let \(H_{1} (x) = 0\), two stable states of \(x^{*} = 0\) and \(x^{*} = 1\) can be obtained: If \(y < y_{0}\), then \(H_{1}^{\prime } (0) < 0\), \(H_{1}^{\prime } (1) > 0\), so \(x^{*} = 0\) is an evolutionary stable strategy; if \(y > y_{0}\), then \(H_{1}^{\prime } (0) > 0\), \(H_{1}^{\prime } (1) < 0\), so \(x^{*} = 1\) is an evolutionary stable strategy.

Therefore, the evolutionary stable state of carbon emission enterprises is related to the probability of local governments’ initial strategy choices, as shown in Fig. 1.

Fig. 1
figure 1

Replicated dynamic phase diagram of local governments and carbon emission enterprises

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4.2 Evolutionary stable strategy of local governments

In the replicated dynamic equation \(H_{2} (y)\) of local governments, let \(x_{0} = \frac{{\beta \Delta E_{2} t_{3} - C_{1} }}{{\beta \Delta E_{2} t_{3} }}\). Similarly, the evolutionary stable strategies of local governments under different conditions will be analyzed as follows.

When \(x = x_{0}\), there is always \(H_{2} (y) = 0\), then any value of \(y\) in the interval range of \([0,1]\) is an evolutionary stable state.

When \(x \ne x_{0}\), let \(H_{2} (y) = 0\), two stable states of \(y^{*} = 0\) and \(y^{*} = 1\) can be obtained: If \(x > x_{0}\), then \(H_{2}^{\prime } (0) < 0\), \(H_{2}^{\prime } (1) > 0\), so \(y^{*} = 0\) is an evolutionary stable strategy; if \(x < x_{0}\), then \(H_{2}^{\prime } (0) > 0\), \(H_{2}^{\prime } (1) < 0\), so \(y^{*} = 1\) is an evolutionary stable strategy.

Therefore, the evolutionary stable state of local governments is related to the probability of carbon emission enterprises’ initial strategy choices, as shown in Fig. 1.

4.3 Stability analysis of the evolutionary game system

The Formula (7) composed of the replicated dynamic equations of carbon emission enterprises and local governments, is the replicated dynamic system of the evolutionary game, and describes the dynamic process of the strategy evolution of the two game players.

$$ \left\{ \begin{gathered} H_{1} (x) = x(1 - x)\{ (R + p\Delta E_{1} )(1 - t_{1} ) - (R + r)(1 - t_{2} ) + [(1 - \beta )p + \alpha \beta t_{3} ]\Delta E_{2} - C_{2} + (1 - \alpha )\beta \Delta E_{2} t_{3} y\} \hfill \\ H_{2} (y) = y(1 - y)(1 - \alpha )[(\beta \Delta E_{2} t_{3} - C_{1} ) - \beta \Delta E_{2} t_{3} x] \hfill \\ \end{gathered} \right. $$
(7)

Let \(H_{1} (x) = 0\), \(H_{2} (y) = 0\), the equilibrium point \((x^{*} ,y^{*} )\) of the evolutionary system can be obtained, namely \((0,0)\), \((0,1)\), \((1,0)\), \((1,1)\) and \((x_{0} ,y_{0} )\), where \(0 \le x_{0} ,y_{0} \le 1\). However, these five equilibrium points are not necessarily the ESS of the system, we need to analyze according to the method proposed by Friedman (1991): For a dynamic system described by a differential equation, the stability of its equilibrium point can be obtained from the local stability analysis of the Jacobian matrix of the system. If and only if the value of Jacobian determinant is greater than zero and the trace of Jacobian matrix is less than zero, the strategy corresponding to the equilibrium point is the evolutionary stable strategy of the system.

In Formula (7), the Jacobian matrix \(A\) of the system can be obtained by taking the first-order partial derivative with respect to \(x\) and \(y\).

$$ A = \left[ {\begin{array}{*{20}c} {{{\partial H_{1} } \mathord{\left/ {\vphantom {{\partial H_{1} } {\partial x}}} \right. \kern-0pt} {\partial x}}} & {{{\partial H_{1} } \mathord{\left/ {\vphantom {{\partial H_{1} } {\partial y}}} \right. \kern-0pt} {\partial y}}} \\ {{{\partial H_{2} } \mathord{\left/ {\vphantom {{\partial H_{2} } {\partial x}}} \right. \kern-0pt} {\partial x}}} & {{{\partial H_{2} } \mathord{\left/ {\vphantom {{\partial H_{2} } {\partial y}}} \right. \kern-0pt} {\partial y}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right], $$
(8)

where

$$ a_{11} = (1 - 2x)\{ (R + p\Delta E_{1} )(1 - t_{1} ) - (R + r)(1 - t_{2} ) + [(1 - \beta )p + \alpha \beta t_{3} ]\Delta E_{2} - C_{2} + (1 - \alpha )\beta \Delta E_{2} t_{3} y\} , $$
(9)
$$ a_{12} = x(1 - x)(1 - \alpha )\beta \Delta E_{2} t_{3} , $$
(10)
$$ a_{21} = - y(1 - y)(1 - \alpha )\beta \Delta E_{2} t_{3} , $$
(11)
$$ a_{22} = (1 - 2y)(1 - \alpha )[(\beta \Delta E_{2} t_{3} - C_{1} ) - \beta \Delta E_{2} t_{3} x]. $$
(12)

\(Det(A)\) represents the value of determinant \(A\), and \({\text{tr}}(A)\) represents the trace of matrix \(A\). If the equilibrium point meets the conditions in Formula (13), the strategy corresponding to this point is the ESS of the system.

$$ \left\{ \begin{gathered} Det(A) = a_{11} a_{22} - a_{12} a_{21} > 0 \hfill \\ {\text{tr}}(A) = a_{11} + a_{22} < 0 \hfill \\ \end{gathered} \right. $$
(13)

According to the above analysis method, the equilibrium points \((0,0)\), \((0,1)\), \((1,0)\), \((1,1)\) and \((x_{0} ,y_{0} )\) are substituted into \(A\) respectively, and the corresponding \(Det(A)\) and \({\text{tr}}(A)\) can be obtained. To simplify the statement, let.

$$ T_{1} = (R + p\Delta E_{1} )(1 - t_{1} ) - (R + r)(1 - t_{2} ) + [(1 - \beta )p + \alpha \beta t_{3} ]\Delta E_{2} - C_{2} , $$
(14)
$$ S_{1} = (1 - \alpha )(\beta \Delta E_{2} t_{3} - C_{1} ), $$
(15)
$$ T_{2} = (R + p\Delta E_{1} )(1 - t_{1} ) - (R + r)(1 - t_{2} ) + [(1 - \beta )p + \beta t_{3} ]\Delta E_{2} - C_{2} , $$
(16)
$$ S_{2} = - (1 - \alpha )C_{1} , $$
(17)

where \(T_{1} < T_{2}\), \(S_{1} > S_{2}\) and \(S_{2} < 0\).Then, the results of \(Det(A)\) and \({\text{tr}}(A)\) for the five equilibrium points are shown in Table 3.

Table 3 Results of determinant and trace corresponding to each equilibrium point
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In Table 3, \({\text{tr}}(A)\) of equilibrium point \((x_{0} ,y_{0} )\) is equal to 0, which does not meet the second condition in Formula (13), so this point must not be the evolutionary stable point of the system. Next, the local stability of the other four equilibrium points will be analyzed. The stability analysis results of equilibrium points \((0,0)\), \((0,1)\), \((1,0)\) and \((1,1)\) under different conditions are summarized in Table 4.

Table 4 Stability analysis results of four equilibrium points under different conditions
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According to Table 4, the system does not have an asymptotically stable point in Case 4, and the evolutionary trajectory of the system is a closed-loop orbit around the central point \((x_{0} ,y_{0} )\) (Taylor & Jonker, 1978). However, in the other five cases, the system has an asymptotically stable point:

In Case 1 and Case 2, \((1,0)\) is the asymptotically stable point of the system, representing that under the relaxed supervision mode of the government, carbon emission enterprises independently choose the “low-carbon innovation” strategy. This situation is the optimal ESS of government-enterprise cooperative game.

In Case 3 and Case 5, \((0,0)\) is the asymptotically stable point of the system, that is, the ESS of government-enterprise is (traditional production, relaxed supervision).

In case 6, \((0,1)\) is the asymptotically stable point of the system, representing that under the strict supervision mode of the government, carbon emission enterprises choose traditional production, namely non-innovation strategy. This situation is the worst evolutionary stability state in government-enterprise non-cooperative game, and a manifestation of policy failure, which will cause serious waste of policy costs invested by the government.

Through further analysis, it can be concluded that \(T_{1}\) represents the net innovation return of the carbon emission enterprise under the relaxed supervision mode of the government, \(T_{2}\) represents the net innovation return of the carbon emission enterprise under the strict supervision mode of the government, \(S_{1}\) represents the net return of local government’s strict supervision when carbon emission enterprises carry out traditional production, \(S_{2}\) represents the net return of local government’s strict supervision when carbon emission enterprises carry out low-carbon innovation. Combined with the preconditions of each ESS in Table 4, it is easy to find that government-enterprise strategy choices are related to the benefits brought by the strategy.

5 Numerical simulation analysis

This study is to discuss the carbon policy that encourages carbon emission enterprises to choose low-carbon innovation behavior under a certain degree of government supervision. Therefore, effective carbon policy should be able to promote the system to gradually stabilize in the equilibrium state (low-carbon innovation, relaxed supervision), and the state is called the Ideal Stable State (ISS); while avoiding its gradual stabilization in the equilibrium state (traditional production, strict supervision), and the state is called the Policy Failure State (PFS). Next, we will focus on the analysis of these two evolutionary equilibrium states. Next, with the help of MATLA 7.0, the numerical simulation experiments are carried out to simulate the evolutionary path of government-enterprise strategies and convergence results of the system, and the values of specified parameters are changed for sensitivity analysis and policy effect simulation. The simulation experiment environment is shown in Table 5.

Table 5 Simulation environment for evolutionary strategies of government-enterprise
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5.1 Numerical simulation experiment

5.1.1 The policy failure state (PFS) simulation

According to Table 4, when point \((0,1)\) is an evolutionary stable point, the asymptotically stable condition of the system must meet \(T_{2} < 0\) and \(S_{1} > 0\), that is, \((R + p\Delta E_{1} )(1 - t_{1} ) - (R + r)(1 - t_{2} ) + [(1 - \beta )p + \beta t_{3} ]\Delta E_{2} < C_{2}\) and \(\beta \Delta E_{2} t_{3} > C_{1}\). Further analysis shows that after the implementation of carbon trading policy and carbon tax policy, the total profit under the enterprise innovation strategy is less than the cost, which leads to the lack of innovation. As a result, the enterprise will eventually evolve into the stable “traditional production” strategy; the benefits brought by the government's strict supervision can compensate for their regulatory costs. Therefore, the government will gradually converge to the “strict supervision” strategy. Finally, the evolutionary system is stable in the ISS.

In this system state, the probability of enterprises’ illegal carbon emission behavior is at a high level, and the government has not implemented strict supervision, setting α < β. Considering the interests of carbon emission enterprises in China, the cost of low-carbon technology innovation in the short term is much higher than the benefit, and lower than the benefits of traditional production, setting R-C2 < 0 and R < r.According to current regulations, the income tax rate for general enterprises is 25%, and the preferential income tax policy is not implemented for low-carbon innovative enterprises, setting t1 = t2 = 0.25. Meanwhile, based on the asymptotically stable conditions and the simulation data setting rules of related research (Wang & Yao, 2019; Yang & Xu, 2021), the experimental values of parameters are set as follows: C1 = 160, C2 = 1800, R = 300, r = 800, ∆E1 = 150, ∆E2 = 180, p = 10, t1 = 0.25, t2 = 0.25, t3 = 5.5, α = 0.3, β = 0.8. In order to analyze the influence of the initial strategy selection state of government-enterprise (i.e. the initial values of \(x\) and \(y\)) on the system evolutionary results, five groups of values (0,1,0.2), (0.3,0.9), (0.5,0.5), (0.9,0.2) and (0.8,0.9) are set, which cover different initial states that may occur in the dynamic system. The simulation experiment aims to intuitively display the dynamic evolutionary process of the game system. The simulation results are shown in Fig. 2, the horizontal axis in the figure is the probability of carbon emission enterprises choosing the “low-carbon innovation” strategy, the vertical axis is the probability of local governments choosing the “strict supervision” strategy, the curve is the evolutionary path of government-enterprise strategies under different initial states of the system.

Fig. 2
figure 2

Evolutionary trend chart of the system in the PFS

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From Fig. 2, it can be obtained that no matter what the initial probability of government-enterprise strategy selection is, the dynamic system is asymptotically stable to the ESS corresponding to point \((0,1)\): Carbon emission enterprises choose the “traditional production” strategy, and local governments choose the “strict supervision” strategy. However, the selection probability of the initial strategy will affect the evolutionary rate of government-enterprise to the stable point, and the initial probability and the evolution rate change in the same direction. In a word, the positive benefits of strict supervision have improved the enthusiasm of local governments for supervision, but governments’ high intensity supervision makes the space for enterprise innovation income smaller, so that the innovation costs invested cannot be offset. This shows that the high-intensity carbon regulation policy is not conducive to guiding enterprises to carry out technological innovation, which can easily to lead to policy failure.

5.1.2 The ideal stable state (ISS) simulation

According to Table 4, when point \((1,0)\) is an evolutionary stable point, the asymptotically stable conditions of the system must meet \(T_{1} > 0\), \(S_{1} < 0\) or \(T_{1} > 0\), \(S_{1} > 0\). That is to say, whether \(S_{1}\) is less than zero or greater than zero, as long as \(T_{1} = (R + p\Delta E_{1} )(1 - t_{1} ) - (R + r)(1 - t_{2} ) + [(1 - \beta )p + \alpha \beta t_{3} ]\Delta E_{2} - C_{2} > 0\), the system will converge to the ESS corresponding to point \((1,0)\), which means that the net innovation return of the carbon emission enterprise under the relaxed supervision mode of local government is the substantive condition affecting the strategy selection of government-enterprise.

In order to explore carbon trading and carbon tax policies that encourage low-carbon innovation of enterprises, the PFS is regarded as the benchmark case of policy intensity, and the change of system evolutionary state is simulated by changing the set values of carbon trading price \({\text{p}}\), innovative enterprise income tax rate \(t_{1}\) and carbon tax rate \(t_{3}\). Accordingly, the values of other parameters in the benchmark case are kept unchanged, and let p = 12, t1 = 0.15, t3 = 9.5. At this time, the dynamic evolution process of government-enterprise is shown in Fig. 3. In this figure, both local governments and carbon emission enterprises make the system in the ISS by learning and constantly adjusting their own strategies.

Fig. 3
figure 3

Evolutionary trend chart of the system in the ISS

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5.2 Policy effect simulation

5.2.1 Carbon trading policy simulation

The carbon trading policy mainly guides and encourages carbon emission enterprises to carry out low-carbon innovation to reduce greenhouse gas emissions through price signals. The basic principle is to trade carbon dioxide emission rights as a commodity. Enterprises that have successfully reduced their carbon emissions can sell excess carbon emission quotas and gain profits, while enterprises that have exceeded their carbon emissions quotas have to purchase quotas in the carbon market. In this study, the parameters related to carbon trading policy include carbon price \(p\), the marketable carbon quota \(\Delta E_{1}\) after enterprise innovation, and the carbon quota \(\Delta E_{2}\) that enterprises need to purchase when they do not innovate. Therefore, the control variable method can be used to simulate the impact of different carbon trading policies on the system evolutionary process by changing the parameter values. Taking the parameter values in the PFS as the control group, two experimental groups were set for sensitivity analysis: (1) increase the values of three parameters, and set p = 12, ∆E1 = 200, ∆E2 = 220; (2) decrease the values of three parameters, and set p = 8, ∆E1 = 100, ∆E2 = 140. Since the initial probability of government-enterprise strategy selection does not affect the evolution result of the system, the initial state of the system is randomly set as \((0.5,0.5)\) to simulate the policy effect, as shown in Fig. 4.

Fig. 4
figure 4

Simulation of system evolution under different carbon trading policies

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According to Fig. 4, the evolution result of experimental group 1 is the ideal stable point \((1,0)\), that is, when the carbon price, the marketable carbon quota after enterprise innovation and the carbon quota that enterprises need to purchase when they do not innovate are simultaneously raised, the expected control target of carbon trading policy can be achieved, carbon emission enterprises can be guided to carry out low-carbon technology innovation, and the system can be transformed from PFS to ISS. Further analysis shows that even if the government does not participate in supervision, carbon emission enterprises will also carry out independent innovation and converge to the ideal equilibrium state. On the contrary, the policy strength of experimental group 2 failed to make the system evolve to the ideal equilibrium state. The stability point is still \((0,1)\), that is, when the carbon price, the marketable carbon quota after innovation and the carbon quota that enterprises need to purchase when they do not innovate are reduced, the dilemma of carbon emission enterprises adopting traditional technology cannot be improved, and the incentive effect of carbon trading policy is very limited at this time.

Carbon trading policy is a regulatory means based on the market mechanism. The carbon price is mainly affected by the relationship between supply and demand in the carbon market, while the carbon trading volume is subject to the innovation ability of enterprises and product market competition, which are not regulated by the government. Therefore, when the innovation incentive mechanism of carbon trading policy fails, it needs to be remedied by other means. Through simulation analysis, it is found that under the carbon trading policy of experimental group 2, the system can evolve to the ISS by increasing the carbon tax rate, the probability of government supervision and reducing the income tax rate, but the policy cost is too high at this time. Chen (2014) believed that in the face of abnormal fluctuations or long-term low carbon prices, it is very necessary for the government to regulate carbon prices scientifically according to law. Inspired by this research idea, we analyze the sensitivity of carbon price \(p\). Only the value of carbon price is changed, and other parameter values in the benchmark case are kept unchanged to simulate the evolution trend of the system, and p = 7, p = 15 and p = 20 are the three experimental values, as shown in Fig. 5.

Fig. 5
figure 5

The evolutionary trend of government-enterprise strategy under different carbon pricing mechanisms

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According to Fig. 5, increasing the carbon price can effectively improve the evolution dilemma of the system, so that the system gradually converges to the stable point \((1,0)\), and the higher the carbon price is, the faster the system will evolve to the equilibrium point. Further analysis shows that when the carbon price \(p\) is lower than 10.8742, the system will evolve into the PFS; and when the carbon price \(p\) is higher than 10.8742, the system will evolve into the ISS. In other words, there is a carbon price threshold in the carbon trading policy. When the carbon price is lower than the threshold, the policy intensity is not enough to change the strategy choice of carbon emission enterprises, and the carbon trading policy fails to play its expected role; and when the carbon price is higher than the threshold, the carbon emission enterprises will actively implement carbon emission reduction, and tend to choose “low-carbon innovation” strategy.

In summary, carbon price has an important impact on the implementation effect of carbon trading policy, but it is regulated by market mechanism and is a non-governmental regulatory element. Therefore, the role of carbon trading policy is limited, and it cannot stably and effectively stimulate carbon emission enterprises to carry out low-carbon innovation. Although it is optional for the government to regulate carbon price, the national carbon market has only been open for more than a year, and the policy effect has not been fully reflected, so it is still early to discuss this issue.

5.2.2 Carbon tax policy simulation

The carbon tax policy studied in this article is a punitive carbon policy. Its mechanism is that local governments will levy carbon tax on the on the over-quota part as punishment after supervising the illegal carbon emission behavior of enterprises, thus increasing the carbon emission cost of enterprises and forcing them to carry out low-carbon technology innovation. The policy effect is mainly affected by the government supervision and carbon tax rate. Next, the simulation analysis is used to intuitively show the impact of relevant factors on the strategy selection of government-enterprise. The ideal policy effect should be to enable enterprises to choose the low-carbon innovation strategy. Therefore, the minimum carbon tax rate of 21.19 can be obtained, by substituting the parameter values in the benchmark case into asymptotically stable conditions that make the system reach the ISS, and the probability of government supervision \(\alpha\) is 0.3. Keep other parameter values unchanged, and change the value of \(\alpha\) to simulate the evolution trend of governments and enterprise, as shown in Fig. 6.

Fig. 6
figure 6

The evolutionary path of government-enterprise strategy under different levels of government supervision

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Further analysis shows that there is a threshold of 0.2999 for the probability of government supervision \(\alpha\). When \(\alpha \in [0,0.2999)\), the system has no evolutionary stable state, and both local governments and carbon emission enterprises play a strategic game around the center point, showing a periodic oscillation trend. When \(\alpha \in [0.2999,1]\), the system will gradually stabilize in the ISS, and the enterprise will choose the low-carbon innovation strategy independently without government supervision. This shows that a certain degree of government supervision is a necessary prerequisite to ensure the effect of carbon tax policy.

Similarly, the carbon tax rate \(t_{3}\) also has a threshold. When \(t_{3}\) is greater than the threshold, the system will gradually converge to the ISS. On the contrary, the carbon tax policy cannot normally play the role in carbon emission reduction, and the system will gradually converge to the PFS.

The carbon tax paid by illegal carbon emission enterprises is a part of the government's supervision income, and the government's regulatory income will directly affect its supervision probability. Therefore, it is necessary for us to explore the relationship between carbon tax rate and the probability of government supervision. The values of parameters in the benchmark case remain unchanged, and different carbon tax rates \(t_{3}\) are set to simulate the minimum probability of government supervision \(\alpha\) that makes the system reach the “ideal steady state”, as shown in Table 6.

Table 6 The minimum probability of government supervision under different carbon tax rates to stabilize the system in the ISS
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According to Table 6, with the increase of carbon tax rate, the probability of government supervision shows a downward trend. This shows that when the carbon tax rate set by the government is high, carbon emission enterprises will implement carbon emission reduction in order to reduce the cost of illegal carbon emissions. At this time, the government does not need to implement high-intensity supervision, and the cost of supervision will be reduced accordingly, so as to achieve good policy effect. However, the excessive carbon tax rate will greatly increase the economic burden of enterprises, and occupy the innovative research and development costs, which is not conducive to encouraging innovation.

5.3 Analysis on the synergistic incentive mechanism of carbon trading policy and carbon tax policy

After the launch of the national carbon trading market, carbon emission enterprises included in the emission control list must purchase their excess carbon emission rights from the carbon market. Since the trading market was launched more than a year ago, the trading price has risen steadily, and the carbon price has fluctuated reasonably with the carbon emission management cycle. But near the end of the performance cycle, the carbon price skyrocketed. Taking an enterprise with an annual carbon emission of 10 million tons as an example, assuming that its free carbon quota only accounts for 90% of the annual emissions, and based on the carbon price of 50 yuan per ton, if it does not implement low carbon emission reduction, it will have to pay a carbon emission cost of 50 million yuan. The maximum fine imposed by government departments on violations such as concealing and falsely reporting carbon emission data is only 30,000 yuan (Lan, 2022), which often fails to serve as a warning for enterprises with high output value. Driven by interests, the probability (\(\beta\)) of non-innovative enterprises choosing illegal carbon emission behavior may remain at a high level, that is to say, the probability (\(1 - \beta\)) of choosing carbon trading is relatively small. The illegal carbon emission probability can reflect the strength of carbon trading policy from the side, and the carbon tax rate can directly reflect the strength of carbon tax policy. Therefore, numerical simulation can be used to analyze the synergistic impact mechanism of carbon trading policy and carbon tax policy on enterprise innovation incentives. The values of other parameters in the benchmark case remain unchanged, and different values of illegal carbon emission probability \(\beta\) are given to simulate the lowest carbon tax rate \(t_{3}\) that can promote enterprise innovation, as shown in Table 7.

Table 7 The lowest carbon tax rate under different probability of illegal carbon emission to stabilize the system in the ISS
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According to Table 7, with the increase of the probability of illegal carbon emission and the decrease of the probability of choosing carbon trading behavior, the lowest carbon tax rate for non-innovative enterprises to choose the low-carbon innovation strategy is increasing, but it does not affect the evolution of both sides to the ideal stable point (1, 0), as shown in Fig. 7.

Fig. 7
figure 7

Evolutionary path of government-enterprise under different probabilities of illegal carbon emission and carbon tax rates

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From the perspective of carbon policy effect on encouraging innovation, the increase in the probability of illegal carbon emission weakens the incentive strength of carbon trading policy. Because the above sensitivity analysis is based on the premise that carbon emission enterprises tend to choose the low-carbon innovation strategy, it is not difficult to conclude that the strengthening effect of carbon tax policy on encouraging enterprise innovation is greater than the weakening effect of carbon trading policy. That is to say, when the effect of carbon trading policy is limited, the government can still achieve the goal of encouraging enterprise innovation. Therefore, the compound carbon policy of carbon trading and carbon tax can effectively stimulate enterprises to carry out low-carbon innovation and help to achieve the goal of carbon peaking and carbon neutrality.

5.4 Simulation of incentive effect of preferential income tax policy

Developing technology innovation and promoting low-carbon technology transformation are important means for enterprises to achieve low-carbon emission reduction goals. The income tax rate of high-tech enterprises recognized by the government can be reduced to 15% (Sun & Zhang, 2021). Accordingly, in the assumptions of Sect. 3, we assume that the government sets up preferential income tax policy for low-carbon innovative enterprises, so as to reduce the tax burden of enterprises and reduce their net cost of low-carbon innovation. From the perspective of carbon regulation policy, this preferential policy increases the economic benefits of carbon trading policy to low-carbon innovative enterprises, and strengthens the positive incentive effect of carbon trading policy. The values of other parameters in the benchmark case remain unchanged, different income tax rates of innovative enterprises \(t_{1}\) to simulate the lowest carbon tax rate \(t_{3}\) that can promote enterprise innovation, as shown in Table 8.

Table 8 The lowest carbon tax rate under different preferential income tax rates to stabilize the system in the ideal equilibrium state
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According to Table 8, when the government sets preferential income tax policy for innovative enterprises, the lowest carbon tax rate that drives enterprises to choose the low-carbon innovation strategy will decrease, and it does not affect the evolution of both sides to the ideal stable point (1,0), as shown in Fig. 8. This shows that the positive incentive effect of preferential policy for income tax can replace the higher intensity of carbon tax punishment effect to a certain extent. The research and development cycle of enterprises is long, and the innovation often cannot bring profits quickly. However, the preferential tax is a direct preference given by the government to enterprises for innovation, with great strength and considerable benefits (Liu, 2021). From the perspective of policy effect, the preferential policy on income tax can stimulate the innovation vitality of enterprises, guide the rational allocation of resources, and realize the double dividend of enterprises and the environment by reducing risks, improving returns and other channels.

Fig. 8
figure 8

Evolutionary path of government-enterprise under different preferential income tax rates and carbon tax rates

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6 Conclusions and policy implications

Based on the assumptions of bounded rationality, this article constructs an evolutionary game model between local governments and carbon emission enterprises under the compound carbon policy, analyzes the evolutionarily stable states of both sides of the system under different conditions, On this basis, the PFS and the ISS of the system are simulated, and numerical simulation experiments are carried out to simulate the effects of carbon trading policy and carbon tax policy. Meanwhile, with the help of sensitivity analysis of various elements of the system, the incentive mechanism of carbon policy is analyzed in depth. The main conclusions and policy recommendations of our study are as follows.

  1. (1)

    The study in Sect. 5.2.1 concludes that the carbon trading policy mainly uses the market mechanism to guide enterprises to carry out low-carbon innovation. The policy effect is mainly affected by factors such as carbon price and carbon trading volume (the amount of carbon quota sold or purchased), and the carbon trading volume will directly affect the carbon price. Therefore, the carbon price has a direct impact on the policy effect. To prevent the failure of the carbon price market mechanism, the government needs to establish a scientific carbon price regulation mechanism. Based on the dynamic change of market demand, the government can study and judge the trend of future market development, properly regulate and intervene the carbon price, so that the carbon price is higher than the carbon price threshold without being at a high level for a long time, so as to effectively play the incentive role of carbon trading policy.

  2. (2)

    From the simulation results of carbon tax policy, it is found that the effect of carbon tax policy is directly affected by the carbon tax rate, and is regulated by the probability of government supervision. The carbon tax rate is positively related to the probability of government supervision. The higher the tax rate, the greater the government benefits from supervision behavior (reducing supervision costs), and the higher the probability of supervision. Some scholars (Wang & Yao, 2019) suggested it was impossible to establish a third-party (media, public, etc.) supervisory mechanism for carbon emission reduction, so as to promote enterprises to establish carbon emission information disclosure system and release platform, thus effectively reducing the cost of government supervision.

  3. (3)

    When carbon emission enterprises fear the high transaction costs in the carbon market and have a strong motivation to steal emissions, the carbon trading policy will fail to motivate innovation. At this time, it is necessary to introduce moderate government supervision and the punitive carbon tax policy, to effectively encourage enterprises to choose the low-carbon innovation strategy.

  4. (4)

    When the government implements preferential income tax policy for low-carbon innovative enterprises, it reduces the net cost of enterprises’ low-carbon innovation, indirectly increases their innovation performance, thus further strengthening the positive incentive effect of carbon trading policy.

The above conclusions can give some enlightenment to managers to better improve and optimize existing policy regulations. The policy recommendations are as follows.

  1. (1)

    Optimize the carbon quota allocation mechanism and perfect carbon trading policy. The government can improve the quota allocation mechanism from the following aspects. Firstly, formulate the reasonable amount of carbon quotas. In the allocation of quotas, industrial affordability and competitiveness of enterprises should be taken into account (Peng & Zhong, 2021), so that the issued carbon quotas are more in line with the actual demand of enterprises, which is conducive to form a reasonable market-oriented carbon price. Secondly, establish a paid quota allocation mechanism. The current full free quota allocation mechanism makes the price discovery mechanism missing, and also weakens the cost awareness of enterprises’ carbon emissions (Lai & Yao, 2021). The quota allocation of seven pilot carbon markets in China is mainly free allocation, and at the same time, combined with their own actual situation, they have designed various paid allocation systems. Although there are differences in the proportion setting, frequency of implementation, minimum price setting, bidding ceiling and other specific rule designs of paid distribution in each pilot carbon market, the overall practical results show that the implementation of paid distribution has played a role in price discovery to a certain extent. This provides important practical experience for further improving the quota allocation scheme of the national carbon market and timely carrying out paid allocation. The government could learn from these experiences and establish a paid allocation mechanism as soon as possible, to improve the liquidity and pricing efficiency of the carbon trading market. Meanwhile, a scientific carbon price regulation mechanism can be tried to establish. It is worth noting that there are also some challenges in the paid allocation of quotas in the pilot carbon markets. At present, some pilot markets lack normative documents for paid distribution, in case of disputes, there is a lack of lack of institutional basis of credibility, which can easily lead to doubts from stakeholders about the relevant rules and effectiveness of paid distribution. Therefore, the task of promoting paid distribution mechanism in the national carbon market has a long way to go.

  2. (2)

    Timely introduce the carbon tax policy and formulate an optimal carbon tax rate. Some foreign scholars have found in empirical research that carbon tax rates have a significant impact on unemployment rates (Fakoya, 2014) and default rate of commercial loans (Aiello & Angelico, 2023). Domestic scholars' research on the optimal tax rate of carbon tax is still in its infancy, and the research results are relatively lacking. Yao & Liu (2010) estimated the optimal carbon tax rate in China for the first time with the help of DICE model. Based on the actual situation in China, Lu et al. (2022) simulated the optimal carbon tax rate under market equilibrium conditions was 30 yuan per ton. Therefore, when China implements a carbon tax policy, how to formulate a reasonable carbon tax rate is a question worthy of in-depth consideration. In practical work, the government can appropriately refer to the carbon tax rate of countries with the same level of economic development, determine the carbon tax rate above the threshold, and pay attention to the connection with energy tax and related emission reduction policies. Meanwhile, we should try to reduce the supervision cost of local governments to improve the enthusiasm of supervision.

  3. (3)

    Establish a compound carbon policy led by carbon trading policy and supplemented by carbon tax policy. The policy practice of EU countries has tested the positive effect of compound carbon policy on carbon emission reduction (Dresner et al., 2006). On the basis of Chinese situation, scholars have used the Computable General Equilibrium (CGE) model for comparative analysis, and found that the carbon trading carbon tax policy was more beneficial to China's macro-economy and people's welfare (Shen & Zhao, 2022). In specific implementation, the carbon tax policy should serve the carbon trading policy. For enterprises that actively participate in carbon trading, the government can exempt or reduce carbon tax to mobilize their innovation enthusiasm; on the contrary, for enterprises that emit carbon illegally and do not innovate, the government should impose a punitive carbon tax to force them to carry out low-carbon innovation.

  4. (4)

    Implement preferential income tax policy for low-carbon innovative enterprises. According to the deployment of China's “dual carbon” goals, carbon peaking and carbon neutrality are long-term strategic goals, and the return cycle of enterprise innovation investment is long. Therefore, long-term incentive policy tools are very necessary. The empirical results showed that the preferential income tax policy had a positive impact on technological innovation of environmental protection companies (Li & Liu, 2017). For example, some scholars found that tax preference had a significant role in promoting the R&D investment level of environmental protection companies (Fan & Lv, 2023). Based on this, the government should speed up the construction of low-carbon technology standards, set preferential income tax rates for enterprises that have passed the standard certification, so as to reduce the tax burden of enterprises and stimulate their internal innovation power. This measure will effectively reduce the difficulty of encouraging enterprise innovation and regulatory costs, and further strengthen the effect of carbon trading policy.

Despite the valuable insights obtained by our study, our research has several limitations. The low-carbon innovation behavior of enterprises is not only affected by carbon policies, but also interfered by the external dynamic environment. The low-carbon behavior choices of competitors in the same industry and the dynamic changes of consumers' demand for low-carbon products are also the driving factors influencing Chinese enterprises to implement low-carbon technology innovation (Shi, 2015). Future studies could consider expanding the game model from the perspective of stochastically stable equilibrium (Foster & Young, 1990), to better achieve carbon reduction and sustainable development goals.

There are many factors influencing low-carbon innovation of enterprises. Effective environmental regulations will promote enterprises to choose low-carbon innovation behavior, and the quality of innovative talents can directly affect the innovation ability and quality of enterprises. These factors interact with each other and jointly affect the low-carbon innovation activities and development direction of enterprises. While strengthening policy construction, we should also attach importance to the cultivation of innovative talents and establish a scientific and effective talent development mechanism. The concept of green and low-carbon education could be integrated into the entire cultivation process of practical and innovative talents through higher education (Liu et al., 2023). It not only helps to improve the practical quality of green and low-carbon innovation, but also helps to achieve the carbon peaking and carbon neutrality goals.