The Impact of Differentiated Carbon Taxes on New Enterprises’ Strategies When Entering Original Markets with Different Degrees of Market Competition

Abstract

We view the development of industries with various market competition levels as a dynamic process and investigate the game between a new entrant and the original market with variable market competition degrees under the premise of considering the entry and exit of companies in the industry. Based on this, we explore the prerequisites for the new firm to enter the initial market and construct a recursive formula for the optimal output of individual firms entering the market one at a time, as well as the conditions for the new firm to enter the market in the three scenarios of the original market being mixed strategy, low-carbon type, and traditional type, respectively, and the optimal decision-making behavior once entering the market. We create diversified carbon tax rates for various cost bands of low-carbon production patterns in order to modify the original traditional market and allow the new enterprise to enter the market using a low-carbon production strategy. We anticipate that our study will serve as a theoretical guide for accomplishing a low-carbon shift in production patterns.

1. Introduction

With rapid social and economic development, carbon emissions have risen every year, environmental pollution issues brought on by excessive carbon dioxide emissions have progressively exacerbated global climate change, and the massive increase in carbon dioxide emissions has led to global warming, which poses serious threats to human survival and development [1,2]. The Third Assessment Report of the United Nations Intergovernmental Panel on Climate Change (IPCC) concluded that climate change has adverse effects on human health, and the Paris Agreement established a target for limiting the increase in global temperature to 2 °C and aiming to regulate it to 1.5 °C [3]. Complex interactions between social, political, economic, and technological systems result in carbon emissions [4], and implementing climate measures to lower carbon emissions and air pollution is beneficial for human health [5]. Therefore, it is obvious that cutting carbon emissions is crucial for human production and life.

Low-carbon production patterns are gradually gaining attention from all spheres of society for their ability to reduce the emission of harmful gases without affecting economic development. As a foundation for reducing emissions and preserving energy, enterprises’ low-carbon production behavior plays a crucial role in reducing carbon emissions, and the low-carbon production of enterprises can assist the development of low-carbon pathways, improving the potential to slow down global warming [6]. Low-carbon problems are externalities for both producers and consumers [7], and because the market alone cannot address these issues, governments must create a variety of low-carbon regulations to control enterprises’ low-carbon production behavior and consumers’ purchasing decisions across different industries in order to maximize societal gains [8]. There are typically two types of initial responses to external diseconomies: one involves internalizing negative external costs [9], while the other is an emissions trading scheme derived from the Coase Theorem’s property rights definition principle [10,11], for a carbon emissions trading scheme in the carbon emissions market [12,13]. To limit and reduce CO₂ emissions, many countries and regions have introduced corresponding carbon emission reduction policies to promote the transformation of the entire society toward a green and low-carbon direction, including market-based emission reduction policies, e.g., carbon emissions trading, mandatory emission reduction policies, e.g., carbon allowances, and incentive-based emission reduction policies, e.g., carbon taxes, and subsidy policies [14]. Earlier, several academics investigated the issues with programs to minimize carbon emissions and made related suggestions [15,16,17,18].

In September 2020, Chinese President Xi Jinping solemnly declared at the 75th session of the United Nations General Assembly that China will strive to achieve peak carbon by 2030 and carbon neutrality by 2060. As a national strategic deployment, many theoretical and practical issues need to be addressed, among which how to compel enterprises to choose a low-carbon production mode is the focus of our attention. The so-called environmental protection, low-carbon production, and control of carbon emission problems in the production processes of enterprises in industry are dynamic processes. In a similar vein, the development of the entire industry is likewise a dynamic process, and a dynamic process inevitably involves the entry and exit of enterprises. Additionally, the low-carbon production mode replaces the original traditional production pattern as a dynamic process. That is, the original enterprise changes its production mode, which is understood as a traditional enterprise exiting the market and then entering the market as a new enterprise with a low-carbon production pattern. The cost of creating a unit of the product differs between these two production patterns as a result of the various production technologies and scales used by businesses in industry. Here, we contend that there is no difference in production technology between enterprises and that the expansion of production scale within a sector is not the result of enterprises’ expansion on an internal basis but rather is brought about by the entry of new enterprises. Additionally, the original market’s competition degree also affects whether new firms enter the market and with which production mode. The government’s environmental and low-carbon initiatives also have an impact on enterprises’ behavior and the primary market. Therefore, it is crucial to research how the government designs differentiated carbon tax policies that encourage new enterprises to adopt low-carbon production patterns and force existing enterprises with old production patterns to leave the market, to achieve the 2030 carbon peaking and 2060 carbon neutrality targets.

In recent years, researchers have continuously studied the government’s carbon tax policy as an exogenous affecting influencing firms’ decisions about their production strategies [19,20]. Zhang et al., (2021) used the Stackelberg game method with government carbon tax policy as an exogenous variable to investigate the effect of a carbon tax on whether two competitive manufacturing enterprises produce low-carbon products or not [21]. Wang et al., (2022) studied the production decisions of a single manufacturing firm using three models: producing just traditional items, producing only low-carbon products, and producing both traditional and low-carbon products, with the government’s carbon tax policy as an exogenous variable [22].

In addition, existing studies have typically examined the impact of carbon tax policies on the production behavior of manufacturing firms [23]. Although some scholars have studied the impact of government carbon tax policies on firms’ choices of low-carbon production strategies based on separate market structures, they studied the game between the government and monopolistic enterprises or two manufacturing firms [24,25,26,27,28,29,30], with little consideration for the degree of competition in continuous-type markets, or how government carbon tax policies can be set to enable firms with different degrees of market competition to shift their production patterns.

Using a three-stage game methodology, Qu and Sun (2022) investigated how the government, low-carbon manufacturing companies, and traditional manufacturing companies interacted in a carbon tax policy scenario [31]. Krass et al., (2013) considered the interaction between monopolistic manufacturing enterprises and the government, combined the study of government carbon tax regulation with the study of firms’ technology choices, and investigated the relationship between government carbon tax policy and firms’ low-carbon strategy choices in the context of a monopoly [32]. However, the majority of the existing studies are based on a single market competitive scenario. The impact of carbon tax policies on enterprises’ decisions to adopt low-carbon production strategies when the degrees of market competition are different is not considered. Hu et al., (2014) considered n firms producing homogeneous ordinary products and m firms producing homogeneous green products, where each company only produces one product and there is a certain difference between ordinary and green products, established a multi-oligopoly competition model and studied the enterprises’ optimal production decisions, and analyzed the impact of the government’s carbon tax policy and subsidy policy on social welfare through numerical simulations [33]. Although the degree of competition among enterprises is also highlighted to some extent, it is obvious from the existing research that there is no unified framework to study the game between new firms and the original market in industries with different degrees of market competition. On the other hand, Zhu et al., (2019) analyzed a product differentiation model for conventional gasoline vehicles and new energy vehicles under the carbon credit quota regulation and investigated the effectiveness of carbon emission allowances in the case of manufacturers focusing on primary consumer segments, and optimal pricing and production strategies were derived. Then, they investigated the strategic decision of whether a firm enters an alternative market with limited total carbon emissions [34].

There are studies on the game between the government and enterprises, the game between enterprises themselves, and the game between new entrants and a particular incumbent in the existing research, but there is no study on the game between new entrants and the original market by taking the entire industry as a market entity, i.e., the current research does not consider the dynamic association when entering the market one by one. We believe that an industry’s development is a dynamic process that will inevitably lead to new companies entering the industry. Therefore, it is necessary to analyze the game between new enterprises and the original industry as a whole, as well as the prerequisites for new enterprises to enter the industry. Hence, under the scenario that the government implements a differentiated carbon tax policy, we investigate the game behavior between a new enterprise and the original market and explore how the government should set differentiated carbon tax rates that can transform the original traditional market and allow new enterprises to enter the market with a low-carbon production strategy. This will assist in achieving the national “carbon peaking and carbon neutrality” target as well as the green and low-carbon transition of various industries.

Unlike previous studies, this study considers the degrees of market competition in various industries. More crucially, the optimal carbon tax rate in the existing research is typically an isolated point under a specific cost or a particular market structure rather than a feasible range of carbon tax possibilities. We investigate how the government can create corresponding varied carbon tax mechanisms in accordance with various cost intervals based on the game between new enterprises and the original market and acquire feasible intervals for carbon tax options, so that new enterprises can enter the market with low-carbon production strategies and enterprises in the original conventional market can exit the market.

The rest of the paper is organized as follows: Section 2 introduces the model details, Section 3 proposes the concrete models and presents the corresponding results, and Section 4 provides the numerical analysis. Section 5 concludes the paper. Some proofs are provided in Appendix A.

2. Model Description

Suppose that the original market is anthropomorphized into a decision-maker, representing industries with different degrees of market competition, and the decision-maker is engaging in a game with the new entrant. Assuming that the original market and the new entrant have symmetrical information and that both the new company and the original market are completely rational, each makes decisions simultaneously to maximize utility function. The new enterprise’s behavior is to determine whether to enter the market and its optimal output $Q_e$ after entering the market; correspondingly, the new entrant’s strategy set is $Q_e\in [0,\infty )$. The original market’s behavior is to determine its optimal output $Q_m$, and the strategy set is $Q_m\in [0,\infty )$. In the subscript of the symbols, $e$ represents the new entrant and $m$ means the original market. The enterprises’ production costs in the original market differ from each other between $C_M$ and $(C_M+△C_M)$, i.e., for firms in the market, the cost of producing a unit of product is $c_m\in [C_M,C_M+△C_M]$, where $c_m$ can be interpreted as the average cost for companies in the original market. The cost per unit of product produced by a new entrant is $c_e$.

Considering the initial market as an entirety, the products in the original market are fully substituted, and the new entrant’s products have certain differences from the original market’s products, The new competitor will have an impact on product pricing and output in the initial market; the inverse demand function between the original market and the new competitor’s products is $P_e=a-b{Q}_e-d{Q}_m$ and $P_m=a-b{Q}_m-d{Q}_e$, where $P_e$ represents the new entrant’s unit product price, $P_m$ indicates the original market’s unit product price, $a$ is the product’s total market volume, $b$ inverse portrays the price sensitivity of the demand for this product, and $d$ inverse portrays the price sensitivity of the demand for substitutes to the price of this product. $d/b$ indicates the substitution degree between the low-carbon product and the conventional product, and $d\in [0,b]$ demonstrates that demand for the product has a greater impact on price than demand for its substitute. The closer the value of $d$ is to $b$, the greater the substitutability and the less distinct the two products are from each other.

Since the basic model is focused on analyzing the dynamic association of new companies entering the original market one after another, we only use the aforementioned uniformized notation in the fundamental model and take into account the original market and the new firm’s type of strategy when talking about the critical cost for the new firm to enter the market.

In contrast to the traditional production strategy $C$ with excessive pollution and high emissions, the term “low-carbon production strategy $L$” refers to a production pattern that uses renewable energy, improved manufacturing technology, or both to limit the amount of carbon emissions produced during production. In line with this, the products produced by the new entrant and the original market using either the traditional production strategy or the low-carbon production strategy are both traditional and low-carbon products. The costs of producing a unit of traditional products and a unit of low-carbon products are $c_0$ and $(c_0+c)$, respectively, where $c$ is the extra cost associated with producing a unit of low-carbon products. The product produced by the new entrant and the product in the original market are completely substituted when the new entrant has the same production strategy as the original market, at which point $d=b$. Accordingly, the carbon emissions generated by producing a unit of conventional product and a unit of low-carbon product are $e_0$, $(e_0-\alpha c)$ respectively, and $\alpha$ is the abatement cost coefficient, which shows a positive relationship between the additional cost and the decrease in carbon emissions from producing a unit of low-carbon product, but the carbon emissions from manufacturing a unit of a low-carbon product will not be completely eliminated, i.e., $(e_0-\alpha c)>0$. The government applies a carbon tax at a rate $t_l$ on enterprises that adopt a low-carbon strategy and a carbon tax at a rate $t_c$ on companies that use a traditional strategy in accordance with the quantitative taxation concept. The parameters involved in the model are all positive.

Other relevant assumptions in the model are as follows.

Assumption 1: 

We measure the original market’s degrees of competition in various industries by introducing a parameter$\lambda$in the market return function. As a result, the original market’s degrees of competition in different industries are measured by$\lambda$times consumer surplus, where the value$\lambda$ranges from 0 to 1. When$\lambda$takes 0, it indicates a perfectly monopolistic market; when$\lambda$takes 1, it indicates a perfectly competitive market; and when$\lambda$takes a value between 0 and 1, it indicates other competitive markets. Therefore, the$\lambda$times consumer surplus in the original market revenue function can be used to explain all levels of market competition.

Assumption 2: 

The original market exists, indicating that companies in the original market can produce units of product profitably, i.e.,$a>C_m$.

In this work, we do not consider other market entry barriers besides cost.

3. Model Building and Solving

3.1. The Basic Game Model between the New Entrant and the Original Market

Combining the profit function in microeconomics, which is the revenue from the sale of a product minus the cost of producing the product, we obtain the new entrant’s utility function as

$$U_e(Q_e,Q_m)=(P_e-c_e)Q_e$$

(1)

i.e.,

$$U_e(Q_e,Q_m)=(a-b{Q}_e-d{Q}_m-c_e)Q_e$$

(2)

Combining the market revenue function in microeconomics, which is the firm’s revenue from selling the product minus the cost of producing the product, and $\lambda$ times consumer surplus function, and the consumer surplus formula, we obtain the revenue function for the original market with different degrees of market competition as

$$U_m(Q_e,Q_m)=\pi +\lambda CS$$

(3)

The original market’s utility function consists of the profit from producing and selling the product in the original market $\pi =(a-b{Q}_m-d{Q}_e-c_m)Q_m$ and the $\lambda$ times consumer surplus in the original market. Where the consumer surplus in the original market is $CS=b{Q}_m{}^2/2$, then, we have

$$U_m(Q_e,Q_m)=(a-b{Q}_m-d{Q}_e-c_m)Q_m+\lambda (\frac{b{Q}_m{}^2}{2})$$

(4)

Proposition 1:

The new entrant’s and original entire market’s optimal output are, respectively

$$Q_e{}^{\ast }=\frac{(b(2-\lambda )-d)a-b(2-\lambda )c_e+d{c}_m}{2b^2(2-\lambda )-d^2}$$

(5)

$$Q_m{}^{\ast }=\frac{(2b-d)a-2b{c}_m+d{c}_e}{2b^2(2-\lambda )-d^2}$$

(6)

In this instance, $Q_m{}^{\ast }$ is the total output of all companies in the original market, which is the game result of the companies in the original market.

Proposition 2:

The new entrant’s optimal utility, the original entire market’s optimal utility, and the enterprises’ profit in the original market are, respectively

$$U_e{}^{\ast }=\frac{b{((b(2-\lambda )-d)a-b(2-\lambda )c_e+d{c}_m)}^2}{{(2b^2(2-\lambda )-d^2)}^2}$$

(7)

$$U_m{}^{\ast }=\frac{b(2-\lambda ){((2b-d)a-2b{c}_m+d{c}_e)}^2}{2{(2b^2(2-\lambda )-d^2)}^2}$$

(8)

$${\pi }_m{}^{\ast }=\frac{b(1-\lambda ){((2b-d)a-2b{c}_m+d{c}_e)}^2}{{(2b^2(2-\lambda )-d^2)}^2}$$

(9)

Proof. 

Combining optimization theory, we first solve for the first-order derivatives of the respective revenue functions of the new entrant and the original market with respect to their respective yields. Then, we take the second-order derivatives of the respective revenue functions of the new entrant and the original market with respect to their respective output.

The first-order derivative of the new entrant’s utility function with respect to the output $Q_e$ is

$$\frac{\partial U_e}{\partial Q_e}=a-c_e-2b{Q}_e-d{Q}_m$$

(10)

The second-order derivative of the new entrant’s utility function with respect to the output $Q_e$ is

$$\frac{{\partial }^2{U}_e}{\partial Q_e{}^2}=-2b<0$$

(11)

Combining optimization theory, we know that the new entrant’s utility function is a concave function of the output $Q_e$ since the second-order derivative is less than 0; thus, the new entrant’s utility reaches its maximum value when the first-order derivative of the new entrant’s utility function with respect to the output $Q_e$ is equal to 0.

Similarly, the first-order derivative of the original market’s utility function with respect to the output $Q_m$ is

$$\frac{\partial U_m}{\partial Q_m}=a-c_m-b(2-\lambda )Q_m-d{Q}_e$$

(12)

Additionally, the second-order derivative of the original market’s utility function with respect to the output $Q_m$ is

$$\frac{{\partial }^2{U}_m}{\partial Q_m{}^2}=-b(2-\lambda )<0$$

(13)

Combining optimization theory, the original market’s utility function is a concave function of the output $Q_m$ since the second-order derivative is smaller than 0, and the original market’s utility reaches its maximum when the first-order derivative of the original market’s utility function with respect to the output $Q_m$ is equal to 0. Therefore, let

$$\frac{\partial U_e}{\partial Q_e}=a-c_e-2b{Q}_e-d{Q}_m=0$$

(14)

$$\frac{\partial U_m}{\partial Q_m}=a-c_m-b(2-\lambda )Q_m-d{Q}_e=0$$

(15)

Combining Equations (14) and (15), we can obtain the optimal output with Equations (5) and (6) for the new firm and the original market in Proposition 1. Substituting the equations $Q_e{}^{\ast }$ and $Q_m{}^{\ast }$ into the utility functions $U_e(Q_e,Q_m)$ and $U_m(Q_e,Q_m)$, we obtain the new entrant’s and the original market’s optimal utility in Proposition 2. □

The above proposition reveals the new entrant’s optimal behavior after entering the market and the equilibrium result of the game between the new firm and the original market.

Corollary 1:

When no new competitors enter the market, let$d=0$and$c_e=0$in$Q_m{}^{\ast }=((2b-d)a-2b{c}_m+d{c}_e)/(2b^2(2-\lambda )-d^2)$. Subsequently, the enterprises’ total output in the original market is$Q_m{}^{\ast }=(a-c_m)/b(2-\lambda )$.

Combining traditional economic theory, when the original market’s overall production is $Q_m{}^{\ast }=(a-c_m)/b$, it demonstrates that the original market is a perfectly competitive market, which corresponds to $\lambda =1$. When the entire output in the initial market is $Q_m{}^{\ast }=(a-c_m)/2b$, it proves that the original market is a perfect monopoly market, which corresponds to $\lambda =0$. When the original market’s total output $Q_m{}^{\ast }$ is between $[(a-c_m)/2b,(a-c_m)/b]$, it indicates that the original market’s market structure is between that of a perfect monopoly market and a perfect competition market, and the associated $\lambda$ is between 0 and 1. This confirms the accuracy of using $\lambda \in [0,1]$ to categorize the different degrees of market competition in the original market.

Corollary 2:

When the new enterprise enters the market, due to$\lambda \in [0,1]$and the new entrant’s output$Q_e{}^{\ast }=((b(2-\lambda )-d)a-b(2-\lambda )c_e+d{c}_m)/(2b^2(2-\lambda )-d^2)$, combining traditional economics theories, we know that when the new company’s output is$Q_e{}^{\ast }=((b-d)a-b{c}_e+d{c}_m)/(2b^2-d^2)$, it implies that the initial market is completely competitive, which corresponds to$\lambda =1$, demonstrating that when the original market is perfectly competitive, the new entrant’s output is equal to half of the original market’s capacity, which is consistent with Zhang and Tang’s (2002) conclusions [35]. When the new entrant’s output is$Q_e{}^{\ast }=((3b-2d)a-3b{c}_e+2d{c}_m)/2(3b^2-d^2)$, this is the outcome when a new entrant joins the existing market to create three oligopolistic markets, each with a different cost, indicating that the original market is a duopoly market, which corresponds to$\lambda =1/2$. When the new entrant’s output is$Q_e{}^{\ast }=((2b-d)a-2b{c}_e+d{c}_m)/(4b^2-d^2)$, which is equivalent to the duopoly market’s production, and the new entrant and the original market create a duopoly market, it demonstrates that the original market is a perfect monopoly market, corresponding to$\lambda =0$. When the new entrant’s output$Q_e{}^{\ast }$is between$[((2b-d)a-2b{c}_e+d{c}_m)/(4b^2-d^2),((b-d)a-b{c}_e+d{c}_m)/(2b^2-d^2)]$, it indicates that, at this time, the original market’s market structure is between that of a perfectly monopolistic market and a perfectly competitive market, and the$\lambda$is between 0 and 1.

It can be seen that, based on the new entrant’s optimal output equation, we can deduce the dynamic connection between each new entrant’s optimal output when each firm with different costs enters the market. At this point, the new entrant’s optimal output equation is also the optimal output’s recursive equation when each company continuously enters the market. This result is consistent with the results of studying the optimal output of firms with various costs [36].

Specifically, when $d=b$, it is the unique instance of product complete substitution; then $Q_e{}^{\ast }=((1-\lambda )a-(2-\lambda )c_e+c_m)/(3-2\lambda )b$ and $Q_m{}^{\ast }=(a-2c_m+c_e)/(3-2\lambda )b$. Furthermore, when $\lambda =0$, indicating that the original market is a perfect monopolistic market, the new entrant and the original market eventually create a duopoly market. In this instance, $Q_e{}^{\ast }=(a-2c_e+c_m)/3b$ and $Q_m{}^{\ast }=(a-2c_m+c_e)/3b$, which is consistent with existing research that investigated the output of duopoly enterprises with different costs [37]. When $\lambda =1/2$, it indicates that after the new company enters the market, the original duopoly market will combine with the new entrant to establish a triple oligopoly market. At this moment, $Q_e{}^{\ast }=(a-3c_e+2c_m)/4b$ and $Q_m{}^{\ast }=(a-2c_m+c_e)/2b$, and the result is consistent with existing research that explored triple oligopoly firms with various costs [38]. The previous investigations pertain to particular circumstances in this study, but there is no research considering companies’ various costs, product variability, and the dynamic relationships between enterprises as they successively enter the market.

Proposition 3:

The critical cost of enabling the new enterprise to enter the market is

$$c_e{}^{\ast }=\frac{((2-\lambda )b-d)a+d{c}_m}{b(2-\lambda )}$$

(16)

Proof 

: When a new entrant’s production quantity is positive, the participation constraint for that entrant is satisfied.

Since, $b>d$, then $2b^2(2-\lambda )-d^2>0$ in the equation $Q_e{}^{\ast }$. Let $Q_e{}^{\ast }=0$; we obtain the critical cost $c_e$ per unit of a product, i.e., $c_e{}^{\ast }=(((2-\lambda )b-d)a+d{c}_m)/b(2-\lambda )$. Further analysis reveals that when $c_e\le c_e{}^{\ast }$, then $Q_e{}^{\ast }\ge 0$, indicating that if the participation constraint for new firms entering the market is satisfied, then the new firms enter the market.

Since the critical cost for the new firm to enter the market is $c_e{}^{\ast }=(((2-\lambda )b-d)a+d{c}_m)/b(2-\lambda )$, based on this, we analyze the critical cost of a new firm entering the market in four scenarios, that is, when the original market is a traditional market $C$ or a low-carbon market $L$, the new company enters the market with a traditional production strategy $C$ or a low-carbon production strategy $L$, respectively. □

Corollary 3:

When the government does not impose a carbon tax policy, if the new firm enters the market with a traditional production strategy, currently, the cost per unit of a traditional product is$c_e=c_0$. Consequently, when the new entrant’s traditional product’s cost per unit satisfies the critical condition$c_e=c_0\le c_e{}^{\ast }$, the market entry barrier for the new firm is removed, i.e., if the profit is greater than 0, then the new firm enters the market. Further analysis shows that when the original market is a traditional market$C$, the average cost per unit of product in the original market is$c_m=c_0$; in line with this, the precondition for the new firm to enter the market with a conventional production strategy is$c_0\le (((2-\lambda )b-d)a+d{c}_0)/b(2-\lambda )$, that is,$a-c_0\ge 0$. When the original market is a low-carbon market$L$, the average cost per unit of product in the original market is$c_m=c_0+c$; accordingly, the prerequisite for the new company to enter the market with a traditional production strategy is$c_0\le (((2-\lambda )b-d)a+d(c_0+c))/(b(2-\lambda ))$, i.e.,$b(2-\lambda )(a-c_0)-d(a-c_0-c)\ge 0$.

Corollary 4:

When the government does not impose a carbon tax policy, if the new firm enters the market with a low-carbon production strategy, the new company’s low-carbon product cost per unit is$(c_0+c)$. Further analysis shows that when the original market is a conventional type$C$, the average cost of producing a unit of product in the original market is$c_m=c_0$. Since the cost of a low-carbon product is higher than that of a traditional product, for a new firm to enter the market with a low-carbon production strategy, the additional cost per unit of a low-carbon product$c$must satisfy the critical condition$c\le (((2-\lambda )b-d)a+d{c}_0)/(b(2-\lambda ))-c_0$, i.e.,$b(2-\lambda )(a-c_0-c)-d(a-c_0)\ge 0$. Additionally, it demonstrates that when the type of original market is$C$, it will be beneficial for a separate company to alter its original production mode and enter the market with a low-carbon production pattern. When the additional cost of producing a unit of low-carbon product is in the range of$(c_e{}^{\ast }-c_0)<c<(a-c_0)$, the new company will not enter the market with a low-carbon production strategy; this means that no company alters its conventional production strategy when the initial market is of the type$C$. In this case, the government can impose a carbon tax to regulate the costs of low-carbon products and traditional products, so that the new firm can enter the$C$type market with a low-carbon production strategy.

When the original market is of type $L$, the average cost of producing a unit of product in the original market is $c_m=c_0+c$; when the new company enters the market with a low-carbon production strategy, the additional cost of a low-carbon product needs to satisfy $(c_0+c)\le (((2-\lambda )b-d)a+d(c_0+c))/(b(2-\lambda ))$, i.e., $a-c_0-c\ge 0$, which demonstrates that despite the higher cost per unit of low-carbon products compared to traditional products, the new company may not necessarily enter the market with the traditional production strategy due to the different prices of the two products.

The above corollary essentially clarifies the critical cost for a new firm entering the market. Additional sensitivity testing reveals that $\partial Q_e{}^{\ast }/\partial c_e=-(2-\lambda )b/(2b^2(2-\lambda )-d^2)<0$,$\partial Q_e{}^{\ast }/\partial c_m=d/(2b^2(2-\lambda )-d^2)>0$, $\partial Q_m{}^{\ast }/\partial c_e=d/(2b^2(2-\lambda )-d^2)>0$ and $\partial Q_m{}^{\ast }/\partial c_m=-2b/(2b^2(2-\lambda )-d^2)<0$, demonstrating that the new entrant will produce less as its unit production costs rise. The new entrant is more competitive and produces more when the firm’s unit cost of production in the original market is higher. The original market’s enterprises are more competitive and produce more overall when the new entrant’s unit production cost is higher.

3.2. The Conditions for the New Company to Enter the Market and the Subsequent Behavior When the Original Market Is of a Different Type

In the fundamental game model between the new entrant and the original market, we analyze the critical cost for the new firm to enter the market without government intervention. Based on this, we further investigate the prerequisite cost conditions for the new firm to enter the market with two production modes under the government’s differentiated carbon tax policy, then we analyze the new entrant’s behavior when the original market is mixed, traditional, or low-carbon type, respectively.

Assume that both the new company and the enterprises in the original market have low-carbon $L$ and traditional $C$ production strategies. Therefore, there are four strategy portfolios, $(L,L)$, $(C,L)$, $(L,C)$, $(C,C)$. The first letter indicates the type of strategy to adopt when the new company enters the market, and the second letter indicates the type of the original market, indicating that the new company enters the market with a low-carbon strategy $L$ or traditional strategy $C$ when the original market is type $L$ or type $C$, respectively.

The new entrant’s behavior is to maximize its utility function by determining whether to select a low-carbon production strategy or a traditional production strategy to enter the original market, and the corresponding optimal output is $Q_e$ and the new entrant’s strategy set is $\{Q_{eLL},Q_{eLC},Q_{eCL},Q_{eCC}\}$, which denotes the new entrant’s output under diverse strategy portfolios, respectively. The new firm’s utility function includes the cost of producing the product, the revenue from selling the product, and the carbon tax paid.

Suppose that the original market is anthropomorphized into a decision-maker, representing industries with different degrees of market competition. Under the original market’s different types of scenarios, the original market’s behavior is to maximize its utility by deciding its optimal output $Q_m$ when a new firm enters the market. The original market’s strategy set is $\{Q_{mLL},Q_{mLC},Q_{mCL},Q_{mCC}\}$, which denotes the original market’s output under different strategy portfolios, respectively. The original market’s utility function contains the entire industry’s revenue from selling the product, manufacturing costs, carbon tax revenue, and $\lambda$ times the consumer surplus.

3.2.1. The New Entrant’s Behavior When the Original Market Is a Hybrid of Low-Carbon and Traditional

In the original market as a mixed strategy scenario, we analyze the prerequisites for new firms to enter the original market when it is evolutionarily stable, the circumstances in which the low-carbon strategy is the original market’s evolutionary stable strategy, and the new entrant’s corresponding decision behavior. We continue to investigate how to develop a differentiated carbon tax policy that will make the original market as well as new entrants low-carbon.

When the original market is a mixed strategy, we assume that the probability that the original market is a low-carbon $L$ type strategy is $x$, and the probability that the original market is a traditional $C$ type is $(1-x)$. Currently, the new company enters the market at the mixed strategy’s Nash equilibrium in the original market, and the new firm also selects a low-carbon production strategy with probability $x$ and the traditional production strategy with probability $(1-x)$.

Under the scenario that the government implements differentiated carbon taxation, we first analyze the game matrix’s components of the new company and the original market, and it is worth noting that when the new entrant and the original market have a consistent strategy, the new entrant’s and the original market’s products are completely interchangeable; accordingly, $d=b$. In addition, when $t_c=t_l$, it represents the special scenario of the uniform carbon tax policy. When $t_c$ and $t_l$ take 0, it indicates the specific case where there is no carbon tax imposed by the government.

Proposition 4:

If the original market is of the type$L$and the new firm enters the market with the strategy$L$, i.e., the strategy profile$(L,L)$scenario, the new entrant’s and the original market’s optimal output are, respectively

$$Q_{eLL}{}^{\ast }=\frac{(1-\lambda )(a-c_0-c-(e_0-\alpha c)t_l)}{(3-2\lambda )b}$$

(17)

$$Q_{mLL}{}^{\ast }=\frac{a-c_0-c-(e_0-\alpha c)t_l}{(3-2\lambda )b}$$

(18)

Proposition 5:

If the original market is of the type$L$and the new firm enters the market with the strategy$L$, i.e., the strategy profile$(L,L)$scenario, the new entrant’s and the original market’s optimal utility functions and the enterprises’ profit in the original market are, respectively

$$U_{eLL}{}^{\ast }=\frac{b{((1-\lambda )(a-c_0-c-(e_0-\alpha c)t_l))}^2}{{((3-2\lambda )b)}^2}$$

(19)

$$U_{mLL}{}^{\ast }=\frac{b(2-\lambda ){(a-c_0-c-(e_0-\alpha c)t_l)}^2}{2{((3-2\lambda )b)}^2}$$

(20)

$${\pi }_{mLL}{}^{\ast }=\frac{b(1-\lambda ){(a-c_0-c-(e_0-\alpha c)t_l)}^2}{{((3-2\lambda )b)}^2}$$

(21)

Proof. 

When the original market is of the type $L$ and the new firm enters the market with the strategy $L$, the cost of producing a unit of low-carbon product for both the firm in the original market and the new entrant is $(c_0+c+(e_0-\alpha c)t_l)$, and the products produced by the new entrant and the original market are completely substituted, i.e., $b=d$. The new entrant’s and the original market’s optimal output and maximum revenue corresponding to this scenario can be obtained from Equations (5) and (6). □

Proposition 6:

If the original market is of the type$L$and the new firm enters the market with the strategy$C$, i.e., the strategy profile$(C,L)$scenario, the new entrant’s and the original market’s optimal output are, respectively

$$Q_{eCL}{}^{\ast }=\frac{(b(2-\lambda )-d)a-b(2-\lambda )(c_0+e_0{t}_c)+d(c_0+c+(e_0-\alpha c)t_l)}{2b^2(2-\lambda )-d^2}$$

(22)

$$Q_{mCL}{}^{\ast }=\frac{(2b-d)a-2b(c_0+c+(e_0-\alpha c)t_l)+d(c_0+e_0{t}_c)}{2b^2(2-\lambda )-d^2}$$

(23)

Proposition 7:

If the original market is of the type$L$and the new firm enters the market with the strategy$C$, i.e., the strategy profile$(C,L)$scenario, the new entrant’s, the original market’s optimal utility functions and the enterprises’ profit in the original market are, respectively

$$U_{eCL}{}^{\ast }=\frac{b{((b(2-\lambda )-d)a-b(2-\lambda )(c_0+e_0{t}_c)+d(c_0+c+(e_0-\alpha c)t_l))}^2}{{(2b^2(2-\lambda )-d^2)}^2}$$

(24)

$$U_{mCL}{}^{\ast }=\frac{b(2-\lambda ){((2b-d)a-2b(c_0+c+(e_0-\alpha c)t_l)+d(c_0+e_0{t}_c))}^2}{2{(2b^2(2-\lambda )-d^2)}^2}$$

(25)

$${\pi }_{mCL}{}^{\ast }=\frac{b(1-\lambda ){((2b-d)a-2b(c_0+c+(e_0-\alpha c)t_l)+d(c_0+e_0{t}_c))}^2}{{(2b^2(2-\lambda )-d^2)}^2}$$

(26)

Proof. 

When the original market is of the type $L$ and the new company enters the market with the strategy $C$, the cost of producing a unit of low-carbon product for the firm in the original market is $(c_0+c+(e_0-\alpha c)t_l)$, while the traditional product’s cost for the new entrant is $(c_0+e_0{t}_c)$. The new entrant’s and the original market’s optimal output and maximum revenue corresponding to this scenario can be obtained from Equations (5) and (6). □

Proposition 8:

If the original market is of the type$C$and the new firm enters the market with the strategy$L$, i.e., the strategy profile$(L,C)$scenario, the new entrant’s and the original market’s optimal output are, respectively

$$Q_{eLC}{}^{\ast }=\frac{(b(2-\lambda )-d)a-b(2-\lambda )(c_0+c+(e_0-\alpha c)t_l)+d(c_0+e_0{t}_c)}{2b^2(2-\lambda )-d^2}$$

(27)

$$Q_{mLC}{}^{\ast }=\frac{(2b-d)a-2b(c_0+e_0{t}_c)+d(c_0+c+(e_0-\alpha c)t_l)}{2b^2(2-\lambda )-d^2}$$

(28)

Proposition 9:

If the original market is of the type$C$and the new firm enters the market with the strategy$L$, i.e., the strategy profile$(L,C)$scenario, the new entrant’s and original market’s optimal utility functions and the enterprises’ profit in the original market are, respectively

$$U_{eLC}{}^{\ast }=\frac{b{((b(2-\lambda )-d)a-b(2-\lambda )(c_0+c+(e_0-\alpha c)t_l)+d(c_0+e_0{t}_c))}^2}{{(2b^2(2-\lambda )-d^2)}^2}$$

(29)

$$U_{mLC}{}^{\ast }=\frac{b(2-\lambda ){((2b-d)a-2b(c_0+e_0{t}_c)+d(c_0+c+(e_0-\alpha c)t_l))}^2}{2{(2b^2(2-\lambda )-d^2)}^2}$$

(30)

$${\pi }_{mLC}{}^{\ast }=\frac{b(1-\lambda ){((2b-d)a-2b(c_0+e_0{t}_c)+d(c_0+c+(e_0-\alpha c)t_l))}^2}{{(2b^2(2-\lambda )-d^2)}^2}$$

(31)

Proof. 

When the original market is of the type $C$ and the new company enters the market with the strategy $L$, the cost of producing a unit of traditional product for the firm in the original market is $(c_0+e_0{t}_c)$, and the cost of producing a unit of low-carbon product for the new entrant is $(c_0+c+(e_0-\alpha c)t_l)$. The new entrant’s and the original market’s optimal output and maximum revenue corresponding to this scenario can be obtained from Equations (5) and (6), respectively. □

Combined with Proposition 3, it is evident that the more intense the degree of market competition, the smaller the additional critical cost per unit of product for the new firm to enter the market with a low-carbon production strategy under the strategy profile $(L,C)$, which indicates that when the original market is a traditional market, the higher the degree of competition among firms within the market, and the less easy for the new firm to enter the market with a low-carbon production strategy.

Proposition 10:

If the original market is of the type$C$and the new firm enters the market with the strategy$C$, i.e., the strategy profile$(C,C)$scenario, the new entrant’s and the original market’s optimal output are, respectively

$$Q_{eCC}{}^{\ast }=\frac{(1-\lambda )(a-c_0-e_0{t}_c)}{(3-2\lambda )b}$$

(32)

$$Q_{mCC}{}^{\ast }=\frac{a-c_0-e_0{t}_c}{(3-2\lambda )b}$$

(33)

Proposition 11:

If the original market is of the type$C$and the new firm enters the market with the strategy$C$, i.e., the strategy profile$(C,C)$scenario, the new entrant’s and the original market’s optimal utility functions and the enterprises’ profit in the original market are, respectively

$$U_{eCC}{}^{\ast }=\frac{b{((1-\lambda )(a-c_0-e_0{t}_c))}^2}{{((3-2\lambda )b)}^2}$$

(34)

$$U_{mCC}{}^{\ast }=\frac{b(2-\lambda ){(a-c_0-e_0{t}_c)}^2}{2{((3-2\lambda )b)}^2}$$

(35)

$${\pi }_{mCC}{}^{\ast }=\frac{b(1-\lambda ){(a-c_0-e_0{t}_c)}^2}{{((3-2\lambda )b)}^2}$$

(36)

Proof. 

When the original market is of the type $C$ and the new firm enters the market with the strategy $C$, the cost of producing a unit of traditional product for both the firm in the original market and the new entrant is $(c_0+e_0{t}_c)$. The new entrant’s and the original market’s optimal output and maximum revenue corresponding to this scenario can be obtained from Equations (5) and (6), respectively. □

The participation constraint of a new firm entering the market is satisfied when the profit of a new entrant is greater than 0. Therefore, we can obtain the following corollary by combining the utility functions of a new firm entering the market under the above four types of strategy profiles.

Corollary 5:

When the original market is of the type$C$, the condition for the new company to enter the market with a traditional production strategy is$a-c_0-e_0{t}_c\ge 0$; while the condition for the new company to enter the market with a low-carbon production strategy is$b(2-\lambda )(a-c_0-c-(e_0-\alpha c)t_l)-d(a-c_0-e_0{t}_c)\ge 0$.

Corollary 6:

When the original market is of the type$L$, the condition for the new firm to enter the market with a traditional production strategy is$b(2-\lambda )(a-c_0-e_0{t}_c)-d(a-c_0-c-(e_0-\alpha c)t_l)\ge 0$, while the condition for the new firm to enter the market with a low-carbon production strategy is$a-c_0-c-(e_0-\alpha c)t_l\ge 0$.

In conclusion, when the original market is a mixed strategy, the game matrix between the new entrant and the original market is as follows

Table 1. Game matrix of the new entrant and the original market.

		Original Market
		L(x)	C(1 − x)
New Entrant	$L$$(x)$	$(U_{eLL}{}^{\ast },U_{mLL}{}^{\ast })$	$(U_{eLC}{}^{\ast },U_{mLC}{}^{\ast })$
	$C$$(1-x)$	$(U_{eCL}{}^{\ast },U_{mCL}{}^{\ast })$	$(U_{eCC}{}^{\ast },U_{mCC}{}^{\ast })$

.

Lemma 1:

The conditions of the differential carbon tax rate for the new firm entering the market when the original market is a hybrid strategy are$t_c{}^{\prime }<t_c<\frac{a-c_0}{e_0}$and$t_l{}^{Left}<t_l<t_l{}^{Right}$; or$t_c{}^{Left}<t_c<t_c{}^{\prime }$and$0<t_l<t_l{}^{Right}$, where$t_c{}^{\prime }=\frac{b(2-\lambda )(a-c_0)-d(a-c_0-c)}{b(2-\lambda )e_0}$,$t_c{}^{Left}=\frac{d(a-c_0)-b(2-\lambda )(a-c_0-c)}{d{e}_0}$,$t_l{}^{Left}=\frac{d(a-c_0-c)-b(2-\lambda )(a-c_0-e_0{t}_c)}{d(e_0-\alpha c)}$,$t_l{}^{Right}=\frac{b(2-\lambda )(a-c_0-c)-d(a-c_0-e_0{t}_c)}{b(2-\lambda )(e_0-\alpha c)}$. (See Appendix A.1 for the Proof of Lemma 1).

Proposition 12:

When the original market is a mixed strategy, the replication dynamic equation of the new entrant has three stable points,$x_1$,$x_2$and$x_3$, where$x_1=0$,$x_2=1$and$x_3=(-(U_{eLC}{}^{\ast }-U_{eCC}{}^{\ast }))/((U_{eLL}{}^{\ast }-U_{eCL}{}^{\ast })-(U_{eLC}{}^{\ast }-U_{eCC}{}^{\ast }))$.

Proof. 

When the original market is a mixed strategy, the utility functions of the new entrant when it selects a low-carbon or traditional strategy are, respectively

$$U(L,x)=x{U}_{eLL}{}^{\ast }+(1-x)U_{eLC}{}^{\ast }$$

(37)

$$U(C,x)=x{U}_{eCL}{}^{\ast }+(1-x)U_{eCC}{}^{\ast }$$

(38)

For the new entrant, the undifferentiated utility function when choosing different strategies is $U(L)=U(C)$, i.e., $x{U}_{eLL}{}^{\ast }+(1-x)U_{eLC}{}^{\ast }=x{U}_{eCL}{}^{\ast }+(1-x)U_{eCC}{}^{\ast }$. Based on this, we can obtain the original market’s mixed-strategy Nash equilibrium as $x_1$ $x_2$, $x_3$. At this point, the replication dynamics equation for the new firm is

$$F(x)=\frac{dx}{dt}=x(1-x)(U(L,x)-U(C,x))$$

(39)

$$F(x)=x(1-x)(x(U_{eLL}{}^{\ast }-U_{eCL}{}^{\ast })+(1-x)(U_{eLC}{}^{\ast }-U_{eCC}{}^{\ast }))$$

(40)

Let $F(x)=\frac{dx}{dt}=0$; we obtain the $x_1$, $x_2$, and $x_3$, where, $x_1=0$, $x_2=1$ and $x_3=\frac{-(U_{eLC}{}^{\ast }-U_{eCC}{}^{\ast })}{(U_{eLL}{}^{\ast }-U_{eCL}{}^{\ast })-(U_{eLC}{}^{\ast }-U_{eCC}{}^{\ast })}$. □

Corollary 7:

It is more attractive for the new enterprise to enter the market with a low-carbon production strategy when the original market is of the type$C$than when the original market is of the type$L$.

The analysis shows that there is a constant $((U_{eLL}{}^{\ast }-U_{eCL}{}^{\ast })-(U_{eLC}{}^{\ast }-U_{eCC}{}^{\ast }))<0$ in the range of $\lambda \in [0,1]$ and $d\in [0,b]$ (see Appendix A.2 for the Proof of Corollary 7), indicating that corollary 7 is proved.

Since the original market is a mixed strategy, the new firm enters the market at the original market’s mixed-strategy Nash equilibrium. When the stationary solution $x_1$, $x_2$ and $x_3$ is evolutionarily stable, it is also the original market’s mixed-strategy Nash equilibrium. Therefore, it is necessary to analyze the new entrant’s behavior after entering the market when the stationary solution $x_1$, $x_2$ and $x_3$ is evolutionarily stable.

Theorem 1:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are $0<c<c^{\ast }$,$0<t_c<(a-c_0)/e_0$ and  $t_l{}^{\ast }<t_l<t_l{}^{Right}$, respectively, and  $U_{eLC}{}^{\ast }<U_{eCC}{}^{\ast }$ and  $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, then  $x_1=0$ is the evolutionarily stable strategy $ESS$.

Theorem 2:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are $c^{\ast }<c<a-c_0$,$t_c{}^{\ast }<t_c<(a-c_0)/e_0$ and  $t_l{}^{\ast }<t_l<t_l{}^{Right}$, respectively, and  $U_{eLC}{}^{\ast }<U_{eCC}{}^{\ast }$ and  $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, then  $x_1=0$ is the  $ESS$.

Theorem 3:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are  $c^{\ast }<c<c^{Left}$, $0<t_c<t_c{}^{\ast }$ and $0<t_l<t_l{}^{Right}$, respectively, and $U_{eLC}{}^{\ast }<U_{eCC}{}^{\ast }$ and $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, then $x_1=0$ is the $ESS$.

Theorem 4:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are  $c^{Left}<c<a-c_0$,$t_c{}^{Left}<t_c<t_c{}^{\ast }$ and  $0<t_l<t_l{}^{Right}$, respectively, and  $U_{eLC}{}^{\ast }<U_{eCC}{}^{\ast }$ and  $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, then  $x_1=0$ is the  $ESS$.

where $c^{\ast }=(a-c_0)-(((2b^2(2-\lambda )-d^2)(1-\lambda )+bd(3-2\lambda ))(a-c_0))/b^2(3-2\lambda )(2-\lambda )$, $c^{Left}=(b(2-\lambda )-d)(a-c_0)/b(2-\lambda )$, $c^{\prime }=(d-b(2-\lambda ))(a-c_0)/d$ and $c^{\ast \ast }=(a-c_0)-b^2(3-2\lambda )(2-\lambda )(a-c_0)/((1-\lambda )(2b^2(2-\lambda )-d^2)+(3-2\lambda )bd)$ are different additional costs per unit of low-carbon product and there are constant $c^{Left}>c^{\ast }>0>c^{\ast \ast }>c^{\prime }$, $t_c{}^{Left}<t_c{}^{\ast }<t_c{}^{\ast \ast }<t_c{}^{\prime }$ and $0<t_c{}^{\ast \ast }<t_c{}^{\prime }$ in the range of $\lambda \in [0,1]$ and $d\in [0,b]$.

In Theorems 1–4, $x_1=0$ is the $ESS$, which demonstrates that selecting a traditional strategy is preferable to selecting a low-carbon strategy for a new firm entering the market, regardless of whether the original market is of the type $L$ or $C$.

The government sets separate carbon tax rates when the additional cost of producing a unit of a low-carbon product is at different intervals, but all strive to enable the new firm to enter the market by regulating the cost of producing a unit of low-carbon product, which alters the profit, and the new company enters the market with the traditional production strategy, which is the dominating strategy for the new firm. However, the government’s objective is to achieve a green and low-carbon transition for the entire industry, so policymakers should avoid developing the carbon tax rate in the interval of Theorems 1–4 when the original market is a mixed strategy and the additional cost of producing a unit of low-carbon product is in different intervals, respectively.

If the original market is of the type $C$, when $(c_0+c+(e_0-\alpha c)t_l)>(c_0+e_0{t}_c)$, the total cost of producing a unit of a low-carbon product is higher than the total cost of producing a unit of a traditional product. Since the price per unit of a traditional product differs from the price per unit of low-carbon product, the new entrant also may not select the traditional production strategy, but must also fulfill $(a-c_0-c-(e_0-\alpha c)t_l)b^2(3-2\lambda )(2-\lambda )<((2b^2(2-\lambda )-d^2)(1-\lambda )+bd(3-2\lambda ))(a-c_0-e_0{t}_c)$. The new company decides to enter the market with a traditional production strategy. If the original market is of the type $L$, when $(c_0+c+(e_0-\alpha c)t_l)>(c_0+e_0{t}_c)$, surely there is $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, proving that when a differentiated carbon tax policy is implemented by the government, the new company will choose to enter the market with a traditional production strategy if the total cost of producing a unit of a low-carbon product is higher than the total cost of producing a unit of a traditional product. Therefore, under the combined consequences of the additional cost $c$ per unit of a low-carbon product, the carbon tax rate $t_c$ for traditional strategy and the carbon tax rate $t_l$ for low-carbon strategy, when the additional cost of producing a unit of a low-carbon product is in diverse intervals, respectively, if the government’s two tax rates are satisfied $(a-c_0-c-(e_0-\alpha c)t_l)b^2(3-2\lambda )(2-\lambda )<((2b^2(2-\lambda )-d^2)(1-\lambda )+bd(3-2\lambda ))(a-c_0-e_0{t}_c)$, the new enterprise will opt for the traditional production strategy to enter the market when the original market is a mixed strategy.

Theorem 5:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are  $0<c<a-c_0$,$t_c{}^{\ast \ast }<t_c<t_c{}^{\prime }$ and  $0<t_l<t_l{}^{\ast \ast }$, respectively, and  $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$ and  $U_{eLL}{}^{\ast }>U_{eCL}{}^{\ast }$, then  $x_2=1$ is the  $ESS$.

Theorem 6:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are  $0<c<a-c_0$,  $t_c{}^{\prime }<t_c<(a-c_0)/e_0$ and  $t_l{}^{Left}<t_l<t_l{}^{\ast \ast }$, respectively, and  $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$ and  $U_{eLL}{}^{\ast }>U_{eCL}{}^{\ast }$, then  $x_2=1$ is the  $ESS$.

In Theorems 5 and 6, $x_2=1$ is the $ESS$, which demonstrates the total cost of producing a unit of a traditional product and a unit of a low-carbon product can be adjusted to ensure that the new firm can enter the market by regulating the corresponding carbon tax rate $t_c$ for the conventional strategy and the carbon tax rate $t_l$ for the low-carbon strategy, that is, the new entrant can be profitable regardless of whether it opts for the conventional production strategy or the low-carbon production strategy. For the new firm entering the market, the revenue of choosing the low-carbon strategy is larger than that of adopting the traditional strategy, regardless of whether the initial market is of the type $L$ or type $C$.

If the original market is of the type $C$, when $(c_0+c+(e_0-\alpha c)t_l)<(c_0+e_0{t}_c)$, there must be $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$, illustrating that the new company will select a low-carbon production strategy to enter the market if the total cost of producing a unit of low-carbon product is lower than that of manufacturing a unit of traditional product under the effects of the government’s differentiated carbon tax policy. If the original market is of the type $L$, when $(c_0+c+(e_0-\alpha c)t_l)<(c_0+e_0{t}_c)$, the new entrant will not always adopt the low-carbon production strategy, since the prices of the two types of products are currently different. Even though the overall cost of producing a unit of a low-carbon product is less than that of manufacturing a unit of a traditional product, it must also satisfy $(a-c_0-c-(e_0-\alpha c)t_l)((2b^2(2-\lambda )-d^2)(1-\lambda )+bd(3-2\lambda ))>b^2(3-2\lambda )(2-\lambda )(a-c_0-e_0{t}_c)$ to make the new firm choose the low-carbon mode to enter the market.

In light of the combined effects of the additional cost $c$ per unit of low-carbon product, the carbon tax rate $t_c$ for the conventional strategy and the carbon tax rate $t_l$ for the low-carbon strategy, when the additional cost of producing a unit of low-carbon product is in different intervals, respectively, and the original market is a mixed strategy type, if the government’s corresponding carbon tax rates $t_l$ and $t_c$ satisfy $(a-c_0-c-(e_0-\alpha c)t_l)((2b^2(2-\lambda )-d^2)(1-\lambda )+bd(3-2\lambda ))>b^2(3-2\lambda )(2-\lambda )(a-c_0-e_0{t}_c)$, the new company will choose a low-carbon production strategy to enter the market.

Theorem 7:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are  $0<c<c^{\ast }$,  $0<t_c<t_c{}^{\ast \ast }$ and  $0<t_l<t_l{}^{\ast }$, respectively, and  $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$ and  $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, then  $x_3$ is the  $ESS$.

Theorem 8:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are  $c^{\ast }<c<a-c_0$,  $t_c{}^{\ast }<t_c<t_c{}^{\ast \ast }$ and  $0<t_l<t_l{}^{\ast }$, respectively, and  $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$ and  $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, then  $x_3$ is the  $ESS$.

Theorem 9:

When the additional cost per unit of low-carbon product, the carbon tax rate for the traditional strategy, and the carbon tax rate for the low-carbon strategy are  $0<c<a-c_0$,  $t_c{}^{\ast \ast }<t_c<(a-c_0)/e_0$ and  $t_l{}^{\ast \ast }<t_l<t_l{}^{\ast }$, respectively, and  $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$ and  $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, then  $x_3$ is the  $ESS$.

In Theorems 7–9, $x_3$ is the $ESS$, which indicates that when the original market is a mixed strategy type and the additional cost of producing a unit of low-carbon product is in various intervals, respectively, if the policymaker creates the corresponding differential carbon tax rates between the intervals in Theorems 7–9, the new firm may enter the market with either a conventional production strategy or a low-carbon production strategy.

In conclusion, we may find the Theorems 5 and 6 are the ideal state since the government’s objective is to enable the new company to enter the original market with a low-carbon production strategy. Therefore, the policymaker can select the differentiated carbon tax rates corresponding to different additional cost intervals of producing a unit of low-carbon product in the range of Theorems 5 and 6 when the original market is a mixed strategy type, to enable $x_2=1$ to be the $ESS$ and ensure that the new firm can always enter the original market with a low-carbon production strategy (see Appendix A.3 for the Proof of Theorems 1–9).

It is worth noting that when $\lambda$ equals 1, it indicates that the original market is perfectly competitive and no firm enters the market at that point. Because the profit of a firm in a perfectly competitive market is 0, a new firm will not enter the market with the same type. However, new firms will enter the market in another way, concluding that the specific case of $\lambda =1$ in the above conclusion. This clarifies the need to deter new firms from entering the market in the traditional way by enacting a carbon tax if the original market is type L, that is, so that new firms do not change their production pattern; and the need to make new firms enter the market with the L strategy by enacting a carbon tax when the original market alone is type C.

3.2.2. The New Entrant’s Behavior When the Original Market Is a Traditional Type

In this subsection, we investigate the prerequisites for the new enterprise to enter the market, the production strategy it selects to enter the market, as well as its production behavior when the initial market is of the traditional type. Additionally, we explore the conditions for the original traditional market to transform its production patterns. The purpose is to ensure the new firm enters the market with a low-carbon production strategy via the governmental carbon tax policy, transforming the original traditional market. We further analyze the conditions under which the new entrant’s profit with a low-carbon production strategy is higher than that of a traditional production strategy, resulting in the original market converting into a low-carbon market.

Lemma 2.

When the original market is of the type$C$, the prerequisite for a new company to enter the market is$t_c{}^{Left}<t_c<(a-c_0)/e_0$and$0<t_l<t_l{}^{Right}$, where$t_c{}^{Left}$and$t_l{}^{Right}$are as above.

Theorem 10:

When the additional cost for each unit of a low-carbon product is between  $0<c<c^{\ast }$, and the differentiated carbon tax rates are  $0<t_c<(a-c_0)/e_0$ and  $0<t_l<t_l{}^{\ast }$, then  $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$, that is, the new company enters the market with a low-carbon production strategy when the original market is of the type  $C$.

Theorem 11:

When the additional cost per unit of low-carbon product is between  $c^{\ast }<c<a-c_0$, and the differentiated carbon tax rates are  $t_c{}^{\ast }<t_c<(a-c_0)/e_0$ and  $0<t_l<t_l{}^{\ast }$, then  $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$, that is, the new enterprise enters the market with a low-carbon production strategy when the original market is of the type  $C$.

Theorems 10 and 11 provide the insight that under the combined impacts of the additional cost $c$ per unit of low-carbon product, the carbon tax rate for the traditional strategy $t_c$ and the carbon tax rate for the low-carbon strategy $t_l$, when $(c_0+c+(e_0-\alpha c)t_l)<(c_0+e_0{t}_c)$, there must be $U_{eLC}{}^{\ast }>U_{eCC}{}^{\ast }$, that is, if the total cost of producing a unit of a low-carbon product is less than that of producing a unit of a traditional product, the new firm will choose a low-carbon production strategy to enter the market when the original market is of the type $C$, under the effect of the government’s differentiated carbon tax policy. Thus, the policymaker can only design the differentiated carbon tax rates within the range of the aforementioned Theorems 10 and 11 when the original market is of the type $C$, so that the new enterprise will enter the market with the $L$ strategy, and then the companies in the original traditional market will change their production patterns to enable the transfer of the entire industry to the low-carbon market.

When the original market is of the type $C$, combining Theorems 10 and 11 with Corollary 4, we can obtain that when $t_l=0$ and $t_c=0$, that is, when there is no carbon tax policy, even though the cost of producing a unit of low-carbon product is higher, the new company may nevertheless enter the market with a low-carbon strategy since the price of the low-carbon product is different from the traditional product’s price. Additionally, if a new company’s additional production costs for a unit of a low-carbon product are greater than the prerequisite in Corollary 4, the government can impose a differentiated carbon tax policy to regulate the total cost per unit of product and then achieve the purpose of forcing the new firm to enter the original traditional market with a low-carbon production strategy, which also reveals that the original traditional market may be transformed into a low-carbon market.

Theorem 12:

When the additional cost per unit of low-carbon product is between  $0<c<c^{\ast }$, and the differentiated carbon tax rates are  $0<t_c<(a-c_0)/e_0$ and  $t_l{}^{\ast }<t_l<t_l{}^{Right}$, then  $U_{eLC}{}^{\ast }<U_{eCC}{}^{\ast }$, that is, the new firm enters the market with a traditional production strategy when the original market is of the type  $C$.

Theorem 13:

When the additional cost per unit of low-carbon product is between  $c^{\ast }<c<a-c_0$, and the differentiated carbon tax rates are  $t_c{}^{\ast }<t_c<(a-c_0)/e_0$ and  $t_l{}^{\ast }<t_l<t_l{}^{Right}$, then  $U_{eLC}{}^{\ast }<U_{eCC}{}^{\ast }$, that is, the new company enters the market with a traditional production strategy when the original market is of the type  $C$.

Theorem 14:

When the additional cost per unit of low-carbon product is between  $c^{\ast }<c<c^{Left}$, and the differentiated carbon tax rates are  $0<t_c<t_c{}^{\ast }$ and  $0<t_l<t_l{}^{Right}$, then  $U_{eLC}{}^{\ast }<U_{eCC}{}^{\ast }$, that is, the new enterprise enters the market with a traditional production strategy when the original market is of the type  $C$.

Theorem 15:

When the additional cost per unit of low-carbon product is between  $c^{Left}<c<a-c_0$, and the differentiated carbon tax rates are  $t_c{}^{Left}<t_c<t_c{}^{\ast }$ and  $0<t_l<t_l{}^{Right}$, then  $U_{eLC}{}^{\ast }<U_{eCC}{}^{\ast }$, that is, the new firm enters the market with a traditional production strategy when the original market is of the type  $C$.

From Theorems 12–15, we can infer that under the combined effect of the additional cost $c$ per unit of low-carbon product, the carbon tax rate $t_c$ for the traditional strategy and the carbon tax rate $t_l$ for the low-carbon strategy, when $(c_0+c+(e_0-\alpha c)t_l)>(c_0+e_0{t}_c)$, i.e., the total cost of producing a unit of low-carbon product is higher than that of producing a unit of traditional product, since the price of the traditional product and that of the low-carbon product are different, the new entrant’s optimal output and profit are also affected by the original market’s competition structure and the substitution degree between traditional and low-carbon products, then the new entrant does not necessarily choose the traditional production strategy at this point. It is further found that when $(a-c_0-c-(e_0-\alpha c)t_l)b^2(3-2\lambda )(2-\lambda )<((2b^2(2-\lambda )-d^2)(1-\lambda )+bd(3-2\lambda ))(a-c_0-e_0{t}_c)$, the new entrant chooses the traditional production strategy when the original market is of the type $C$. However, since the government’s objective is to enable the new firm to enter the market with a low-carbon production strategy at this point, then the government’s differentiated carbon tax rates must not fall within the interval of the above Theorems 12–15.

Proof. 

Corollaries 5–6 demonstrate that when the original market is of the type $C$, the conditions for the new firm entering into the market are $(a-c_0-e_0{t}_c)>0$, $(b(2-\lambda )-d)a-b(2-\lambda )(c_0+c+(e_0-\alpha c)t_l)+d(c_0+e_0{t}_c)>0$ and $(2b-d)a-2b(c_0+e_0{t}_c)+d(c_0+c+(e_0-\alpha c)t_l)>0$, i.e., $0<t_c<(a-c_0)/e_0$, $0<t_l<(a-c_0-c)/(e_0-\alpha c)-d(a-c_0-e_0{t}_c)/b(2-\lambda )(e_0-\alpha c)$. That is, $0<t_c<(a-c_0)/e_0$, $0<t_l<t_l{}^{Right}$. To be realistic, it is necessary to satisfy $t_l{}^{Right}>0$, which leads to Lemma 2, similarly referring to the Proof of Theorems 1–9 when the original market is a mixed strategy.

Combining Corollaries 3–6, we can derive the following regulatory directives and policy implications: If the original market exists and the government does not impose a carbon tax policy, it must be possible for the new firm to enter the market with a conventional production strategy $C$; however, the ideal state of this scenario is to enable the new firm to enter the market with a low-carbon production strategy $L$. If the additional cost of producing a unit of low-carbon product and the cost of producing a unit of traditional product satisfy $b(2-\lambda )(a-c_0-c)-d(a-c_0)\ge 0$, the new enterprise enters the market with a low-carbon production strategy. In contrast, if the additional cost per unit of low-carbon products and the cost per unit of traditional products do not satisfy $b(2-\lambda )(a-c_0-c)-d(a-c_0)\ge 0$, even though the additional cost of the production unit of low-carbon product is too high to enable the new firm to enter the market with a low-carbon production strategy, the government can impose a differentiated carbon tax policy by setting differentiated carbon tax rates at $t_l$ and $t_c$, to enable the additional cost per unit of low-carbon product, the cost per unit of traditional product and the carbon tax rates to satisfy $b(2-\lambda )(a-c_0-c-(e_0-\alpha c)t_l)-d(a-c_0-e_0{t}_c)\ge 0$, regulating the total cost of both types of products to enable the new enterprise to enter the market with a low-carbon production strategy via the governmental carbon tax policy.

This also demonstrates that when the original market is of the traditional type $C$, if one of the enterprises transitions from the original, traditional production mode and produces with another profitable low-carbon production strategy, it will cause the original production pattern to be transformed into a low-carbon mode. The governmental policymakers can develop the corresponding differentiated carbon tax rate interval based on the specific degree of market competition in different industries in reality, which enables the profit of the new enterprise entering the market with a low-carbon production strategy to be higher than that of a traditional production strategy. At this time, the enterprises in the original market will also adopt a low-carbon production strategy, and eventually the entire industry will be transformed to the low-carbon type. □

3.2.3. The New Entrant’s Behavior When the Original Market Is a Low-Carbon Type

In this section, we investigate the prerequisites for a new firm to enter the original market and the circumstances in which the new entrant’s profit from a low-carbon production strategy exceeds that of a conventional production pattern when the original market is of the low-carbon type $L$. This further suggests that, when the original market is of the type $L$, no enterprises will alter their initial production patterns.

Lemma 3.

When the original market is of the type$L$, the prerequisites for the new enterprise to enter the market are$t_c{}^{\prime }<t_c<(a-c_0)/e_0$and$t_l{}^{Left}<t_l<(a-c_0-c)/(e_0-\alpha c)$, or$0<t_c<t_c{}^{\prime }$and$0<t_l<(a-c_0-c)/(e_0-\alpha c)$, where$t_c{}^{\prime }$and$t_l{}^{Left}$are as above.

Theorem 16:

When the carbon tax rate for the traditional strategy is $0<t_c<t_c{}^{\ast \ast }$, and the carbon tax rate for the low-carbon strategy is $0<t_l<(a-c_0-c)/(e_0-\alpha c)$, then $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, that is, the new company enters the market with a traditional production strategy when the original market is of the type  $L$.

Theorem 17:

When the carbon tax rate for the traditional strategy is  $t_c{}^{\ast \ast }<t_c<(a-c_0)/e_0$, and the carbon tax rate for the low-carbon strategy is  $t_l{}^{\ast \ast }<t_l<(a-c_0-c)/(e_0-\alpha c)$, then  $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, that is, the new firm enters the market with a traditional production strategy when the original market is of the type  $L$.

We conclude from Theorems 16 and 17 that, under the combined impact of both the carbon tax rate $t_c$ for the traditional production strategy and the carbon tax rate $t_l$ for the low-carbon production strategy, if $(c_0+c+(e_0-\alpha c)t_l)>(c_0+e_0{t}_c)$, there must be $U_{eLL}{}^{\ast }<U_{eCL}{}^{\ast }$, that is, when the original market is of the type $L$, the new firm will choose the traditional production strategy to enter the market if the total cost of producing a unit of low-carbon product is higher than that of a traditional product as a result of the government’s differentiated carbon tax policy. This also indicates that the new firm will enter the market with a traditional production strategy regardless of the degree of competition in the original market. Additionally, this undermines the stability of the initial market, causing enterprises that currently use low-carbon production modes to switch to more conventional patterns that have lower overall costs per unit of production. Therefore, when the original market is of the type $L$, the government’s differentiated carbon tax rates must not fall within the range of the aforementioned Theorems 16 and 17.

Theorem 18:

When the carbon tax rate for the traditional strategy is  $t_c{}^{\ast \ast }<t_c<t_c{}^{\prime }$ and the carbon tax rate for the low-carbon strategy is  $0<t_l<t_l{}^{\ast \ast }$, then  $U_{eLL}{}^{\ast }>U_{eCL}{}^{\ast }$, that is, the new company enters the market with a low-carbon production strategy when the original market is of the type  $L$.

Theorem 19:

When the carbon tax rate for the traditional strategy is  $t_c{}^{\prime }<t_c<(a-c_0)/e_0$ and the carbon tax rate for the low-carbon strategy is  $t_l{}^{Left}<t_l<t_l{}^{\ast \ast }$, then  $U_{eLL}{}^{\ast }>U_{eCL}{}^{\ast }$, that is, the new enterprise enters the market with a low-carbon production strategy when the original market is of the type  $L$.

Theorems 18 and 19 demonstrate that under the combined effects of both the carbon tax rate $t_c$ for the traditional production strategy and the carbon tax rate $t_l$ for the low-carbon production strategy, when $(c_0+c+(e_0-\alpha c)t_l)<(c_0+e_0{t}_c)$, i.e., the total cost of producing a unit of low-carbon product is less than that of producing a unit of traditional product, the new entrant does not always opt for a low-carbon production strategy because the price per unit of a traditional product differs from the price per unit of a low-carbon product, and the new entrant’s optimal output and profit are also affected by the original market’s competitive structure as well as the degree of substitution between the traditional product and the low-carbon product. It is further found that when $(a-c_0-c-(e_0-\alpha c)t_l)((2b^2(2-\lambda )-d^2)(1-\lambda )+bd(3-2\lambda ))>b^2(3-2\lambda )(2-\lambda )(a-c_0-e_0{t}_c)$, the new company selects a low-carbon production strategy to enter the market. Thus, the policymaker can only develop differentiated carbon tax rates within this range when the initial market is of the type $L$, preventing the enterprises in the initial low-carbon market from altering their production patterns. Theorems 18 and 19 also assume that the government ought to impose a carbon tax for the traditional strategy at a range of between $0<t_c{}^{\ast \ast }<t_c<(a-c_0)/e_0$ when the initial market is of the type $L$. At this point, the new firm can profitably enter the market with a low-carbon production strategy, and the new firm’s profits entering the market with a low-carbon production strategy are higher than those of a traditional production strategy regardless of the degree of competition in the original market. As a result, the new company chooses to enter the market with a low-carbon production strategy. This demonstrates that no firm will return to the conventional production pattern when the original market is a low-carbon market, meaning the original market’s enterprises will not change and the entire industry will continue to follow low-carbon patterns. The policymakers can develop a specific range of differentiated carbon tax rates for the corresponding industries according to the actual degree of market competition in the respective industries.

The essence of the Theorems is via the combined effect of the additional cost $c$ per unit of low-carbon product, the carbon tax rate $t_c$ for the traditional production strategy, and the carbon tax rate $t_l$ for the low-carbon production strategy, controlling the total cost per unit of a traditional product and a low-carbon product, which alters the profits of the two production strategies so that the profit of the new entrant selecting the low-carbon production strategy is higher than that of the traditional production strategy, so as to direct the new company to enter the market with a low-carbon production strategy.

Proof. 

According to Corollaries 5–6 when the original market is of the type $L$, the prerequisites for a new company to enter the market are $(a-c_0-c-(e_0-\alpha c)t_l)>0$,$(b(2-\lambda )-d)a-b(2-\lambda )(c_0+e_0{t}_c)+d(c_0+c+(e_0-\alpha c)t_l)>0$ and $(2b-d)a-2b(c_0+c+(e_0-\alpha c)t_l)+d(c_0+e_0{t}_c)>0$, i.e., $t_l{}^{Left}<t_l<(a-c_0-c)/(e_0-\alpha c)$. Consequently, the $t_l{}^{Left}<t_l<(a-c_0-c)/(e_0-\alpha c)$, when $t_c{}^{\prime }<t_c<(a-c_0)/e_0$; the $0<t_l<(a-c_0-c)/(e_0-\alpha c)$, when $0<t_c<t_c{}^{\prime }$, which brings us to Lemma 3. Similarly, refer to the proof of Theorems 1–9 when the original market is a mixed strategy.

We can determine the following regulatory directions and policy implications by combining Corollaries 3–6: The new company must be able to enter the market with the low-carbon strategy $L$ when the government does not implement a carbon tax and the original market is of the type $L$. When the low-carbon product’s additional cost and the traditional product’s cost do not satisfy $b(2-\lambda )(a-c_0)-d(a-c_0-c)\ge 0$, the new company will not enter the market with a traditional production strategy, that is, the enterprises in the original market will not alter the production strategy; at this point, the original low-carbon market can be stabilized without government-imposed policies; when the low-carbon product’s additional cost and the traditional product’s cost satisfy $b(2-\lambda )(a-c_0)-d(a-c_0-c)\ge 0$, the new enterprise will enter the market with a traditional production strategy $C$, that is, at this point, the initial low-carbon market will shift, and then the government can impose a differentiated carbon tax policy and set the differentiated carbon tax rates $t_l$ and $t_c$, to ensure the low-carbon product’s extra cost and carbon tax rates do not satisfy $b(2-\lambda )(a-c_0-e_0{t}_c)-d(a-c_0-c-(e_0-\alpha c)t_l)\ge 0$ under the effect of carbon tax policy, so the new enterprise will not enter the market with a traditional production strategy, that is, the original low-carbon type market will not change the low-carbon production strategy. □

3.2.4. Analysis of the Original Market’s Behavior

Corollary 8:

When the additional cost per unit of low-carbon product and the carbon tax rates lie in the range in Theorems 5 and 6, i.e.,$t_l<t_l{}^{\ast \ast }$, there must be$t_l<t_l{}^{\ast \ast }<t_L{}^{\ast \ast }$, which leads the original market’s utility$U_{mLL}{}^{\ast }>U_{mLC}{}^{\ast }$. This indicates that the enterprises in the original market are transformed into the low-carbon$L$type if the original market is of the type$C$. If the original market is of the type$L$, then the firms in the original market are not transformed. (See Appendix A.4 for the Proof of Corollary 8.)

4. Numerical Analysis

Combining the model’s assumptions, the parameters’ actual significance, and the actual product’s market volume and other parameters as positive, we assign the following values to the other parameters in the model except the decision variables and the market competition degree, and use Matlab R2020b software to simulate the effect of the market competition degree on the new entrant’s optimal output $Q_{eLL}{}^{\ast }$, $Q_{eCL}{}^{\ast }$, $Q_{eLC}{}^{\ast }$, $Q_{eCC}{}^{\ast }$, i.e., Equations (17), (22), (27) and (32) and the original market’s optimal output $Q_{mLL}{}^{\ast }$, $Q_{mCL}{}^{\ast }$, $Q_{mLC}{}^{\ast }$, $Q_{mCC}{}^{\ast }$, i.e., Equations (18), (23), (28) and (33) and the new entrant’s optimal profit $U_{eLL}{}^{\ast }$, $U_{eCL}{}^{\ast }$, $U_{eLC}{}^{\ast }$, $U_{eCC}{}^{\ast }$, i.e., Equations (19), (24), (29) and (34) and the original market’s optimal revenue $U_{mLL}{}^{\ast }$, $U_{mCL}{}^{\ast }$, $U_{mLC}{}^{\ast }$, $U_{mCC}{}^{\ast }$, i.e., Equations (20), (25), (30) and (35).

$a=100$, $b=25$, $d=12$, $c_0=5$, $c=1$, $e_0=6$, $\alpha =0.5$, $t_l=1.8$, $t_c=0.8$.

Figure 1a reveals that for both strategy portfolios, the new firm’s optimal production declines as the degree of competition in the original market rises. This suggests that the higher the degree of competition in the original market, the more difficult it is for the new company to enter the market. Figure 1b shows that the original market’s optimal total output $Q_m{}^{\ast }$ increases as the degree of competition $\lambda$ in the original market rises.

Figure 2a demonstrates that the profit when the new entrant chooses the traditional strategy is constantly greater than that of choosing the low-carbon strategy when the original market is of type L. The profit when the new entrant selects the low-carbon production strategy is higher than that of adopting the traditional strategy when the initial market is the traditional type L. Figure 2b illustrates that the original market’s expected return is at its maximum when the new firm enters the market with the traditional production strategy and the original market is a traditional type C. It is worth noting that the market revenue $U$ includes enterprises’ profit $\pi$ and $\lambda$ times the consumer surplus.

5. Conclusions

Based on the unified framework of the degree of competition in the original market, we investigated the game between a new entrant and the entire original market with various degrees of competition. We viewed the development of the original market as a dynamic process in which the new company enters the market and firms in the original market exit continuously. Firstly, we developed a fundamental game model between a new enterprise and the original market in the event that the government does not impose a carbon tax policy. We then explored the dynamic relationship between the optimal output upon entry of each firm and analyzed the critical production cost for the new firm to enter the original market, as well as the critical cost for the new firm to enter the market with a low-carbon strategy and a traditional strategy, respectively, where the original market is a low-carbon and traditional market. We discovered a recursive relationship between enterprises’ optimal entry-level behavior that is not covered in previous research. Second, we investigated the game behavior of the new firm and the original market when the original market was of different types under a government-imposed differentiated carbon tax policy, and investigated how the government should set the differentiated carbon tax rate to drive out traditional firms and allow low-carbon firms to enter the market. We specifically explored the circumstances that allow the low-carbon strategy to develop into a stable strategy in the original market and the new entrant’s behavior once they have entered the market when the original market is a mixed market of low-carbon and traditional. On this basis, we concentrated on how to develop a corresponding differentiated carbon tax rate band for different additional cost bands per unit of low-carbon product that can enable both the new entrant and the original market for low-carbon type. When the original market was a traditional market, we analyzed the conditions under which the new enterprise could enter the market and the circumstances in which the new entrant’s profit with a low-carbon production strategy would be higher than that of a traditional production strategy, that is, the conditions under which the original market is transformed into a low-carbon type market; We investigated the prerequisites for a new firm to enter a low-carbon market and the circumstances under which the new entrant’s profit from a low-carbon production strategy would exceed that of a conventional production strategy, i.e., the enterprises in the original market would not alter their production patterns. Finally, we analyze the differential carbon tax rates according to the low-carbon product’s diverse additional cost intervals, so that the enterprises in the traditional-type original market could be transformed to low-carbon modes, while the enterprises in the low-carbon type original market maintained their current production patterns.

When a new company enters a traditional market with a low-carbon strategy, and the entire conventional market is converted to a low-carbon market, it represents the learning and renewal process of that specific organization. When the government imposes a differentiated carbon tax policy, if the low-carbon product’s additional cost is outside the purview of the low-carbon strategy as the dominant strategy, then adjusting the differentiated carbon tax rates can enable the new enterprise to enter the market with a low-carbon strategy when the original market is a type C and no enterprise changes its original low-carbon production pattern when the original market is a type L, which emphasizes the objective of the government’s differentiated carbon tax policy. We discovered that market competition levels affect the values at both ends of the carbon tax range; consequently, based on the mechanism designed by the aforementioned theorems, the policymaker can develop the corresponding carbon tax rate intervals in accordance with the diverse additional cost intervals of the low-carbon product. According to the degree of market competition in each industry in reality, authorities can design diverse actual carbon tax rate ranges. The implementation of a carbon tax policy can facilitate the exit of traditional high-emission enterprises and the entry of new low-carbon enterprises so that the entire industry can transition to the low-carbon production pattern. This has major practical implications for both the greening and decarbonization of industry as well as the achievement of the national objective of “carbon peaking and carbon neutrality.”

Existing studies are usually based on single market competition levels, and the optimal carbon tax rate in existing studies is usually an isolated point rather than a feasible interval for the carbon tax rate. In contrast, our conclusions include the results of different intermediate market competition levels and were a continuous process, so we derived a feasible range for carbon tax rate selection. In addition, we further considered investigating the new entrant’s behavior when there are different cost structures among enterprises in the original market.

In the future, we can study the scenario when the new entrant and the original market are strategically aligned and the product is not completely substituted.