Coping Decisions of Production Enterprises under Low-Carbon Economy

Abstract

It aims to study the production and emission reduction decisions of production enterprises under carbon constraints. In the case of carbon constraints in production, manufacturers have four strategic choices: production within the carbon quota, adopting emission reduction technologies, purchasing carbon emission rights, and using emission reduction technologies and purchasing carbon emission rights. Based on the income model of production enterprises under four different strategies, first, under the condition of maximizing the interests of production enterprises, the optimal profit, optimal production, optimal total carbon emission, and optimal emission reduction rate under different carbon constraints are determined, and summarize its laws. Afterward, in order to further optimize corporate profits, the impact of changes in the carbon reduction scale cost and consumers’ low-carbon preference was studied. Several important conclusions are shown as follows. First, the stricter the carbon constraint policy, the greater the optimal emission reduction rate of enterprises. Second, the adoption of emission reduction technology can effectively reduce the impact of carbon constraint on output. Third, the optimal strategy is to both reduce emissions and purchase carbon emission rights, which can realize environmental economic dividends. Fourth, the lower the cost factor of the carbon reduction scale and the higher the low-carbon preference of consumers, the easier it is for firms to achieve carbon sufficiency in their production.

1. Introduction

Since the goal of carbon peak and carbon neutral was proposed in 2020, the development of China’s green and low-carbon economy has been accelerating day by day. The development model of a low-carbon economy is the trend choice of most countries in the world today, and it is also the way to realize the transformation of China’s economic mode and sustainable economic development. The era of a low-carbon economy is both a challenge and an opportunity for business management. Responding actively and quickly to deal with this change will help companies stand in the context of a low-carbon economy and develop in the long run.

In the context of a low-carbon economy, production enterprises, as the main source of carbon emissions [1], are increasingly subject to carbon limit constraints in their production decisions, which has led to dramatic changes in their operation management. Therefore, it is of great practical importance to study how enterprises can balance their economic and environmental benefits and make optimal production and emission reduction decisions under the dual requirements of the government’s carbon limit constraint and increasing enterprise profits.

The adoption of carbon abatement technologies and the purchase of carbon allowances are the two main ways for production companies to alleviate carbon constraints, and there has been a great deal of research on the impact of these two ways on the production operations of supply chain members [2,3,4,5]. It has been suggested that the levers for firms to comply with the environment include investing in carbon abatement technologies and purchasing carbon allowances [6,7]. This suggests that companies need to make a trade-off between investing in carbon emission reduction and purchasing carbon emission rights, according to their own situation, so as to achieve emission reduction at the lowest cost. Meanwhile, with the government’s emphasis and implementation of carbon emission policies, consumers’ environmental awareness is increasing, and residents are more willing to purchase low-carbon products, making consumer low-carbon preferences an important influencing factor in business decisions [8,9,10].

In the above context, we put forward the following research questions:

(1) What are the trends of enterprises’ profits, production, total carbon emissions, and reduction rates with carbon allowances under different strategies?

(2) Which strategy should companies adopt to achieve environmental economic dividends under the carbon cap constraint?

(3) How do changes in parameters such as the proportional cost of the carbon reduction factor and consumers’ low-carbon preferences affect enterprises’ profits?

In order to solve the above problems, this paper establishes the revenue model of production enterprises under four different strategies from the perspective of production enterprises, considering carbon constraints and consumers’ low-carbon preferences. First, under the condition of maximizing the interests of production enterprises, the optimal profit, optimal production, optimal total carbon emission, and optimal emission reduction rate under different carbon constraints are determined, and the optimal emission reduction strategy is found. After that, in order to further optimize the profits of enterprises, the impact of changes in the carbon reduction scale cost and consumers’ low-carbon preference are studied.

The rest of our paper is arranged as follows. Section 2 introduces the literature review, Section 3 describes the problem and basic assumptions, Section 4 establishes and solves the return models under different strategies, Section 5 conducts a numerical analysis, and Section 6 summarizes some management implications for production enterprises and proposes future research directions.

2. Literature Review

The research area of this paper focuses on three aspects: the production decisions of enterprises under carbon constraints, emission reduction decisions of enterprises under carbon constraints, and low-carbon preferences of consumers, and therefore the related literature review starts from these three aspects.

2.1. Production Decisions of Enterprises under Carbon Constraints

Most of the carbon emissions of manufacturing companies come from their production activities, so the implementation of carbon policies can directly affect the production decisions of companies. Chang et al. [11] analyzed the impact of carbon cap-and-trade mechanisms and the related parameters on optimal production decisions for both independent and substitutable demand markets. The study shows that the CCT mechanism has a significant impact on firms’ manufacturing and remanufacturing production decisions. He et al. [12] explored the optimal production decisions of self-pricing manufacturers and the optimal carbon limit-setting decisions of regulators under a cap-and-trade mechanism. Xu et al. [13] analyzed the impact of the carbon trading price on the optimal production decision and optimal profit for an MTO (make-to-order) supply chain. Dobos [14] explores the impact of carbon trading on firms’ production decisions based on the dynamic Arrow–Karlin model and gives the optimal output of firms. Wang et al. [15] developed three optimization models under different capital conditions to study the production decision and financing decision of firms. The findings suggest that carbon emission restrictions can encourage capital-constrained manufacturers to produce more remanufactured products. Liu et al. [16] studied the impact of three policies on remanufacturing: the carbon cap, carbon tax, and carbon cap-and-trade mechanism with limited information on demand distribution. Miao et al. [17] studied remanufacturing production decisions considering trade-ins under a carbon tax and carbon cap-and-trade mechanism. However, this literature does not examine carbon constraints and carbon reduction inputs together.

2.2. Emission Reduction Decisions of Enterprises under Carbon Constraints

In the context of carbon limits, producers are under severe pressure to reduce emissions, and blindly adopting emission reduction measures may cause the profits of producers to fall, so it is crucial for companies to determine the optimal level of emission reduction. Benjaafar et al. [18] introduced carbon footprint parameters into various optimization models and analyzed how carbon emission reductions can be adjusted through operational decisions. Chen et al. [19] studied pricing and carbon reduction decisions considering the impact of social learning under a carbon cap-and-trade mechanism. Zou et al. [20] explored pricing and abatement decisions under four models: centralized decision making, decentralized decision making without consideration of equity concerns, decentralized decision making with consideration of manufacturers’ equity concerns, and decentralized decision making with consideration of retailers’ equity concerns. Wei et al. [21] studied emission reduction strategies and pricing strategies for low-carbon supply chains under two scenarios of information symmetry and information asymmetry. Cao et al. [22] studied manufacturers’ production and carbon emission reduction levels under carbon cap-and-trade and low-carbon subsidy policies. The results of the study showed that the level of carbon emission reduction increased with the price of carbon trading, independent of the unit low-carbon subsidy. Bai et al. [23] studied carbon reduction and coordination strategies in a make-to-order supply chain. Xue et al. [24] investigated the impact of government subsidies based on energy efficiency levels on the optimal decision of the supply chain when manufacturers independently conduct R&D on emission reduction technologies, considering both centralized and decentralized decision making. However, the above literature does not consider the impact of consumers’ low-carbon preferences on product demand.

2.3. Consumers’ Low-Carbon Preference

As consumers become more environmentally conscious, more and more of them are concerned about the environmental characteristics of products and are willing to pay more [25]. Yang et al. [26] studied the pricing and abatement decisions under three different channel structures based on carbon cap-and-trade mechanisms and consumer low-carbon preferences. Kim and Sim [27] found that consumer preferences for low-carbon play a key role in environmental protection and that companies are more willing to produce low-carbon products to attract consumers in order to increase profits. Zhang et al. [28] studied the impact of consumer environmental awareness on order quantity and channel coordination in single manufacturer and single retailer supply chains. Wang et al. [29] explored the optimal pricing and optimal profits of dual-channel supply chain members under carbon cap-and-trade policies and consumer low-carbon preferences. Du et al. [30] examined the relationship between consumers’ low-carbon preferences and supply chain performance. The findings suggest that consumer low-carbon preferences increase both channel profits and emission reductions. Ji et al. [31] investigated the emission reduction behavior of supply chain members in the retail channel and dual-channel scenarios under carbon cap-and-trade and consumer low-carbon preferences, respectively. Wang et al. [32] investigated the effect of supply chain members’ risk aversion on supply chain performance under consumers’ low-carbon preferences.

Compared with the existing studies, the contributions of this paper are as follows: first, the strategy choice problem of carbon emission reduction and the carbon allowance purchase of production enterprises is studied in order to balance the economic and environmental benefits. Most of the existing literature studied enterprises’ decision making under a specific strategy, while little literature studies the problem of enterprises’ strategy choice. Second, various influencing factors, including consumer low-carbon preferences, are fully considered in the model construction. Consumer low-carbon preferences are one of the directions to focus on for low-carbon supply chain operation, while less literature focuses on consumers’ low-carbon preferences in the context of carbon constraints. Thirdly, the findings of this paper are combined to provide theoretical references for the emission reduction and output decisions of production enterprises subject to carbon constraints in a low-carbon environment.

3. Problem Description and Basic Assumptions

Widely distributed small- and medium-sized production enterprises are an important source of greenhouse gas emissions, and most of the carbon emissions come from their production links. In this paper, we stand in the perspective of carbon emission-dependent production enterprises and make production and emission reduction decisions in the context of carbon constraint in order to balance economic and environmental benefits.

It is assumed that in the initial stage, a producer can obtain a certain amount of carbon allowances allocated by the government for free, and if this amount is not enough to meet the target production requirements, the firm can either adopt emission reduction techniques or buy carbon allowances. Carbon allowances are a limited resource, and enterprises may not be able to buy enough carbon allowances. The strategic choices faced by the enterprises are shown in

Table 1. Strategic choices faced by enterprises.

Initial Carbon Allowances	Strategies Adopted by Enterprises	Carbon Allowances after Adopting Strategy
Adequate (no carbon constraints)	Neither reduce emissions nor buy carbon allowances	Adequate (NCC)
Inadequate (presence of carbon constraints)	Neither reduce emissions nor buy carbon allowances	Inadequate (NN)
	Reduce emissions without purchasing carbon allowances	Adequate (YN1)
		Inadequate (YN2)
	Buying carbon allowances without reducing emissions	Adequate (NY1)
		Inadequate (NY2)
	Both reduce emissions and purchase carbon allowances	Adequate (YY1)
		Inadequate (YY2)

. Therefore, enterprises are faced with the problems of how to respond to the government’s carbon constraint, how to make emission reduction strategy choices, how to carry out production, and how to invest in emission reduction.

Drawing on the research assumptions of the existing literature, this paper presents the following assumptions:

(1) The cost of carbon emission reduction is fully borne by the production enterprise, and the investment in emission reduction will not affect the production cost of the product. The cost of zero carbon emissions at the current level of technology is infinite, so companies cannot achieve zero carbon emissions, i.e.,$\lambda \in [1,0)$.

(2) Market demand $D$ is a linear function of product price and abatement rate. $D=a-bp+k\lambda$, where $a$ denotes the potential market size, $b$ denotes the elasticity coefficient of market demand with respect to product price, $k$ denotes the elasticity coefficient of market demand with respect to abatement rate, and $b>0$, $k>0$, $a>bp$. When the firm does not adopt abatement technology $k=0$. In addition, the information in the product market is completely shared, and the market can be completely cleared, that is, $q=D=a-bp+k\lambda$.

(3) The carbon emission reduction input $C\left(\lambda \right)$ of enterprises is a one-time input, and it is in line with the law of diminishing marginal effect. $C\left(\lambda \right)=\frac{1}{2}h{\lambda }^2$, $h$ is the carbon reduction scale cost factor of production enterprises, and the value is generally larger, $C^{\prime }\left(\lambda \right)>0$, $C^{″}\left(\lambda \right)>0$. After the introduction of emission reduction technology, the carbon emission per unit product is $e\left(1-\lambda \right)$.

(4) Both sides of the game are risk-neutral and completely rational, and they make decisions based on the principle of maximizing their own interests.

The main symbols and descriptions used in this paper are presented in

Table 2. Main symbols and descriptions.

Symbol	Description
$e$	The initial carbon emissions per unit product when carbon emission reduction technologies are not used
$p$	Unit product price
$\lambda$	The emission reduction rate per unit of product when carbon emission reduction technologies are adopted, $\lambda =\Delta e/e$ and $0\le \lambda \le 1$
$c_m$	Production cost per unit of product
${\omega }_s$	Price per unit of carbon emission right
$c_s$	Cost per unit of carbon emission right
$q$	Product output
$E$	Carbon allowances allocated for free by the government
$E_0$	Maximum purchase of carbon allowances
$E^x$	Total carbon emissions of production companies under $x$ strategy, $x=\mathrm{NN}, \mathrm{YN}1, \mathrm{YN}2, \mathrm{NY}1, \mathrm{NY}2, \mathrm{YY}1, \mathrm{YY}2$.
${\pi }_m$	Profits of manufacturing enterprise
${\pi }_s$	Profits of carbon trading center

.

4. Build and Solve Revenue Model

4.1. No Carbon Constraints

There are two kinds of production situations without carbon constraints: one is the traditional production mode without carbon constraint policies; the other is the situation in which there is a carbon constraint mechanism, but the enterprise has sufficient carbon quota to produce without carbon constraint.

When there is no carbon quota constraint, the production decisions of production enterprises are not affected by carbon emissions. The first scenario can be used as a control sample for carbon-constrained production-planning studies, and this paper focuses on the second scenario. Because this paper only studies the impact of carbon allowances in the primary market on production enterprises, and does not consider the secondary trading allocation system, this paper assumes that companies do not sell excess carbon allowances. Thus, in this case, the enterprise will not reduce carbon emissions. At this time, the profit function of the production enterprise is

$$\underset{q}{\max }{\pi }_m=\left(p-c_m\right)q$$

(1)

Theorem 1.

In the absence of carbon constraints, production enterprises neither need to purchase carbon emission rights nor reduce carbon emissions. There is a unique optimal output $q^{*}$for the enterprise, which maximizes the enterprise’s profit.

Proof of Theorem 1.

From $q=a-bp$, inversely solve $p=\frac{a-q}{b}$, and bring it into Formula (1), we can obtain

$$\underset{q}{\max }{\pi }_m=\frac{a-q}{b}q-c_{m}q$$

(2)

where $\frac{{\partial }^2{\pi }_m}{\partial q^2}=-\frac{2}{b}<0$, taking the first derivative of (2) and making it equal to zero, the optimal output of the production enterprise can be obtained

$$q^{*}=\frac{a-c_{m}b}{2}$$

(3)

Bringing Equation (3) into Equation (2), we can obtain the optimal profit of the production enterprise

$${\pi }_m^{*}=\frac{a^2-2ab{c}_m+b^2{c}_m^2}{4b}$$

At this time, the carbon trading center has no income, and the total carbon emission of the production enterprises is $e{q}^{*}$. □

4.2. NN Strategy

Under the NN model, production enterprises respond negatively to the carbon cap policy by neither adopting emission reduction technologies nor purchasing carbon emission rights. At this time, the income of production enterprises only includes sales revenue, and the cost only includes production costs. Therefore, the profit function of the manufacturer is

$$\underset{q^{NN}}{\max }{\pi }_m^{NN}=\left(p-c_m\right)q^{NN}$$

(4)

Theorem 2.

There are carbon constraints, but manufacturers neither adopt carbon emission reduction technologies nor purchase carbon emission rights. There is a unique optimal output $q^{NN*}$for the enterprise, which maximizes the profit of the enterprise.

Proof of Theorem 2.

Under this strategy, the manufacturer neither reduces emissions nor purchases carbon emission rights, and its available carbon quota is $E$, so it can obtain the optimal output

$$q^{NN*}=\frac{E}{e}$$

(5)

Putting $p=\frac{a-q^{NN}}{b}$ and Formula (5) into Formula (4), the optimal profit of the enterprise can be obtained

$${\pi }_m^{NN*}=\frac{\left(ae-E-be{c}_m\right)E}{b{e}^2}$$

At this time, the carbon trading center has no income, and the total carbon emission of the production enterprises is $E^{NN}=E$. □

4.3. YN Strategy

Under the existing technical conditions, the cost for enterprises to achieve zero carbon emissions is infinite, that is to say, the current level of carbon emission reduction technology can only alleviate carbon constraints but cannot achieve zero carbon emissions. Therefore, when companies only adopt carbon reduction technologies, they may or may not achieve carbon sufficiency.

4.3.1. YN1 Strategy

Under the YN1 strategy, the production enterprise only adopts carbon emission reduction technologies without purchasing carbon emission rights, and finally achieves carbon sufficiency. At this time, the income of the production enterprise includes sales revenue, and the cost includes production cost and emission reduction cost. Therefore, the profit function of the manufacturer is

$$\underset{q^{YN1},{\lambda }^{YN1}}{\max }{\pi }_m^{YN1}=\left(p-c_m\right)q^{YN1}-\frac{1}{2}h{\lambda }^{YN{1}^2}$$

$$\mathrm{s}.\mathrm{t}. q^{YN1}\left(1-{\lambda }^{YN1}\right)e\le E$$

(6)

Theorem 3.

In the presence of carbon quota constraints, and the carbon quota is sufficient after the enterprise adopts the emission reduction technology, when the carbon quota satisfies $q^{YN1*}\left(1-{\lambda }^{YN1*}\right)e\le E$, there is unique optimal $q^{YN1*}$and ${\lambda }^{YN1*}$, which maximize the profit of the enterprise.

Proof of Theorem 3.

Bring $p=\frac{a+k{\lambda }^{YN1}-q^{YN1}}{b}$ into the profit function of the production enterprise, we can obtain

$$\underset{q^{YN1},{\lambda }^{YN1}}{\max }{\pi }_m^{YN1}=\left(\frac{a+k{\lambda }^{YN1}-q^{YN1}}{b}-c_m\right)q^{YN1}-\frac{1}{2}h{\lambda }^{YN{1}^2}$$

(7)

To find the second-order partial derivatives of the above formula with respect to $q^{YN1}$ and ${\lambda }^{YN1}$, respectively, we have $\frac{{\partial }^2{\pi }_m^{YN1}}{\partial q^{YN{1}^2}}=-\frac{2}{b}$,$\frac{{\partial }^2{\pi }_m^{YN1}}{\partial q^{YN1}\partial {\lambda }^{YN1}}=\frac{k}{b}$,$\frac{{\partial }^2{\pi }_m^{YN1}}{\partial {\lambda }^{YN1}\partial q^{YN1}}=\frac{k}{b}$,$\frac{{\partial }^2{\pi }_m^{YN1}}{\partial {\lambda }^{YN{1}^2}}=-h$. From this, the Hessian matrix of the profit function of the production enterprise about the decision variables $q^{YN1}$ and ${\lambda }^{YN1}$ can be obtained

$$H^{YN1}=\left|\begin{array}{cc}\frac{{\partial }^2{\pi }_m^{YN1}}{\partial q^{YN{1}^2}}& \frac{{\partial }^2{\pi }_m^{YN1}}{\partial q^{YN1}\partial {\lambda }^{YN1}}\\ \frac{{\partial }^2{\pi }_m^{YN1}}{\partial {\lambda }^{YN1}\partial q^{YN1}}& \frac{{\partial }^2{\pi }_m^{YN1}}{\partial {\lambda }^{YN{1}^2}}\end{array}\right|=\left|\begin{array}{cc}-\frac{2}{b}& \frac{k}{b}\\ \frac{k}{b}& -h\end{array}\right|=\frac{2bh-k^2}{b^2}$$

When the carbon reduction scale cost factor satisfies $h>\frac{k^2}{2b}$, it can be known that $-\frac{2}{b}<0$, $\frac{2bh-k^2}{b^2}>0$, Hessian matrix is negative definite, that is, there is an optimal output and emission reduction rate to maximize the profit of the manufacturer.

Build Lagrangian functions to solve

$$L\left(q^{YN1},{\lambda }^{YN1}\right)=-\left[\left(\frac{a+k{\lambda }^{YN1}-q^{YN1}}{b}-c_m\right)q^{YN1}-\frac{1}{2}h{\lambda }^{YN{1}^2}\right]+{\beta }^{YN1}\left[q^{YN1}\left(1-{\lambda }^{YN1}\right)e-E\right]$$

where ${\beta }^{YN1}$ is a Lagrange multiplier not less than 0, the optimal solution to the above problem should satisfy the following KKT conditions:

$$\{\begin{array}{l}\frac{\partial L\left(q^{YN1},{\lambda }^{YN1}\right)}{\partial q^{YN1}}=\frac{2q^{YN1}-a-{k\lambda }^{YN1}}{b}+c_m+e{\beta }^{YN1}\left(1-{\lambda }^{YN1}\right)=0\\ \frac{\partial L\left(q^{YN1},{\lambda }^{YN1}\right)}{\partial {\lambda }^{YN1}}=h{\lambda }^{YN1}-\frac{k{q}^{YN1}}{b}-e{\beta }^{YN1}{q}^{YN1}=0\\ {\beta }^{YN1}\left[q^{YN1}\left(1-{\lambda }^{YN1}\right)e-E\right]=0\\ q^{YN1}\left(1-{\lambda }^{YN1}\right)e\le E\end{array}$$

Discuss:

① ${\beta }^{YN1}=0$, under the YN1 strategy, the optimal output of the production enterprise is

$$q^{YN1*}=\frac{bh\left(a-b{c}_m\right)}{2bh-k^2}$$

(8)

the optimal emission reduction rate is

$${\lambda }^{YN1*}=\frac{k\left(a-b{c}_m\right)}{2bh-k^2}$$

(9)

Bring Equations (8) and (9) into Equation (7) to obtain the optimal profit of the production enterprise

$${\pi }_m^{YN1*}=\frac{h{\left(a-b{c}_m\right)}^2}{2\left(2bh-k^2\right)}$$

At this time, the carbon trading center has no income, and the total carbon emission of the production enterprises is

$$E^{YN1}=e{q}^{YN1*}\left(1-{\lambda }^{YN1*}\right)=\frac{ebh\left(a-b{c}_m\right)\left[2bh-k^2-k\left(a-b{c}_m\right)\right]}{{\left(2bh-k^2\right)}^2}$$

② ${\beta }^{YN1}>0$, $q^{YN1}\left(1-{\lambda }^{YN1}\right)e-E=0$, at this time, the carbon quota allocated by the government exactly to meet the production of the production enterprises. Because the profit function of the production enterprise is continuous in the real number range, this unit point may not be considered as a special case. □

Corollary 1.

When$E=E1=\frac{bheA\left(B-kA\right)}{B^2}$, the total carbon emission of the production enterprise after investing in carbon emission reduction technology just reaches the initial carbon emission quota freely allocated by the government.

Proof of Corollary 1.

Let $A=a-b{c}_m$, $B=2bh-k^2$, then $q^{YN1*}=\frac{bhA}{B}$, ${\lambda }^{YN1*}=\frac{kA}{B}$, and then obtain $E=E1=e{q}^{YN1*}\left(1-{\lambda }^{YN1*}\right)=\frac{bheA\left(B-kA\right)}{B^2}$. □

4.3.2. YN2 Strategy

Under the YN2 strategy, the production enterprise only adopts carbon emission reduction technology without purchasing carbon emission rights and has not yet achieved carbon sufficiency. At this time, the production enterprise’s income includes sales revenue, and the cost includes production cost and emission reduction cost. The carbon emission reduction rate meets the ${\lambda }^{YN2}\in [0,1-\frac{E}{e{q}^{YN2}})$. Therefore, the profit function of the manufacturer is

$$\underset{q^{YN2},{\lambda }^{YN2}}{\max }{\pi }_m^{YN2}=\left(p-c_m\right)q^{YN2}-\frac{1}{2}h{\lambda }^{YN{2}^2}$$

$$\mathrm{s}.\mathrm{t}. q^{YN2}\left(1-{\lambda }^{YN2}\right)e>E$$

(10)

Theorem 4.

There are carbon quota constraints, and companies only use carbon emission reduction technologies without purchasing carbon emission rights, and they have not yet achieved carbon sufficiency. When the carbon quota satisfies $q^{YN2*}\left(1-{\lambda }^{YN2*}\right)e>E$, there is only one $q^{YN2*}$and ${\lambda }^{YN2*}$, which maximizes the profit of the enterprise.

Proof of Theorem 4.

Bring $p=\frac{a+k{\lambda }^{YN2}-q^{YN2}}{b}$ into the profit function of the production enterprise, we can obtain

$$\underset{q^{YN2},{\lambda }^{YN2}}{\max }{\pi }_m^{YN2}=\left(\frac{a+k{\lambda }^{YN2}-q^{YN2}}{b}-c_m\right)q^{YN2}-\frac{1}{2}h{\lambda }^{YN{2}^2}$$

(11)

Under this strategy, the available carbon quota of the manufacturer is $E$, so the optimal output of the manufacturer is

$$q^{YN2*}=\frac{E}{e\left(1-{\lambda }^{YN2}\right)}$$

Putting the above Equation into Equation (11), we can obtain

$$\underset{q^{YN2}}{\max }{\pi }_m^{YN2}=\left(\frac{e\left(a+k{\lambda }^{YN2}\right)\left(1-{\lambda }^{YN2}\right)-E}{be\left(1-{\lambda }^{YN2}\right)}-c_m\right)\frac{E}{e\left(1-{\lambda }^{YN2}\right)}-\frac{1}{2}h{\lambda }^{YN{2}^2}$$

According to the extreme value theorem, there must be an optimal carbon emission reduction rate ${\lambda }^{YN2*}$ to maximize the profit of the production enterprise. Therefore, the optimal emission reduction rate of production enterprises

$${\lambda }^{YN2*}=arg\underset{{\lambda }^{YN2}}{\max }{\pi }_m^{YN2}\left({\lambda }^{YN2}\right)$$

At this time, the carbon trading center has no income, and the total carbon emission of the production enterprises is

$$E^{YN2}=E$$

□

4.4. NY Strategy

Carbon allowances are limited resources, so when the carbon allowances freely allocated by the government are not enough to enable enterprises to achieve optimal production, companies may not purchase enough carbon allowances for production. Therefore, when companies only purchase carbon emission rights, they may or may not achieve carbon sufficiency. The maximum purchase amount of carbon allowances in this article is $E_0$.

4.4.1. NY1 Strategy

Under the NY1 strategy, production enterprises do not use carbon emission reduction technologies but only purchase carbon emission rights to achieve carbon sufficiency. At this time, the income of production enterprises includes sales revenue, and the cost includes production costs and carbon rights purchase costs. The profit function of production enterprises and carbon trading centers is

$$\underset{q^{NY1}}{\max }{\pi }_m^{NY1}=\left(p-c_m\right)q^{NY1}-\left(e{q}^{NY1}-E\right){\omega }_s^{NY1}$$

$$\underset{{\omega }_s^{NY1}}{\max }{\pi }_s^{NY1}=\left({\omega }_s^{NY1}-c_s\right)\left(e{q}^{NY1}-E\right)$$

$$\mathrm{s}.\mathrm{t}. E<e{q}^{NY1}<E+E_0$$

(12)

Theorem 5.

There are carbon constraints, companies do not use carbon emission reduction technologies but only purchase carbon emission rights and ultimately achieve carbon sufficiency. When the carbon quota satisfies $E<e{q}^{NY1*}<E+E_0$, there is only one $q^{NY1*}$and ${\omega }_s^{NY1*}$, which maximizes the profit of the enterprise.

Proof of Theorem 5.

Bring $p=\frac{a-q^{NY1}}{b}$ into the profit function of the production enterprise, we can obtain

$$\underset{q^{NY1}}{\max }{\pi }_m^{NY1}=\left(\frac{a-q^{NY1}}{b}-c_m\right)q^{NY1}-\left(e{q}^{NY1}-E\right){\omega }_s^{NY1}$$

(13)

Build Lagrangian functions to solve

$$L\left(q^{NY1}\right)=-\left[\left(\frac{a-q^{NY1}}{b}-c_m\right)q^{NY1}-\left(e{q}^{NY1}-E\right){\omega }_s^{NY1}\right]+{\beta }^{NY1}\left(E-e{q}^{NY1}\right)$$

where ${\beta }^{NY1}$ is a Lagrange multiplier not less than 0, the optimal solution to the above problem should satisfy the following KKT conditions:

$$\{\begin{array}{l}\frac{\partial L\left(q^{NY1}\right)}{\partial q^{NY1}}=\frac{2q^{NY1}-a}{b}+c_m+e{\omega }_s^{NY1}-e{\beta }^{NY1}=0\\ {\beta }^{NY1}\left(E-e{q}^{NY1}\right)=0\\ e{q}^{NY1}>E\end{array}$$

There must be ${\beta }^{NY1}=0$, then

$$q^{NY1}=\frac{a-b{c}_m-be{\omega }_s^{NY1}}{2}$$

(14)

Bring the above formula into the profit function of the carbon trading center, and take the second derivative of ${\omega }_s^{NY1}$ to obtain $\frac{{\partial }^2{\pi }_s^{NY1}}{\partial {\omega }_s^{NY{1}^2}}=-b{e}^2<0$, and set $\frac{\partial {\pi }_s^{NY1}}{\partial {\omega }_s^{NY1}}=0$ to obtain the optimal carbon right price

$${\omega }_s^{NY1*}=\frac{b{e}^2{c}_s+e\left(a-b{c}_m\right)-2E}{2b{e}^2}$$

(15)

Bring the above formula into Formula (14) to obtain the optimal output of the production enterprise

$$q^{NY1*}=\frac{e\left(a-b{c}_m\right)-b{e}^2{c}_s+2E}{4e}$$

(16)

Bring Formulas (15) and (16) into Formula (12) to obtain the optimal profit of the production enterprise

$${\pi }_m^{NY1*}=\frac{\left[3e\left(a-b{c}_m\right)+b{e}^2{c}_s-2E\right]\left[e\left(a-b{c}_m\right)-b{e}^2{c}_s+2E\right]-2\left[e\left(a-b{c}_m\right)-b{e}^2{c}_s-2E\right]\left[e\left(a-b{c}_m\right)+b{e}^2{c}_s-2E\right]}{16b{e}^2}$$

The optimal profit of carbon trading center

$${\pi }_s^{NY1*}=\frac{{\left[e\left(a-b{c}_m\right)-b{e}^2{c}_s-2E\right]}^2}{8b{e}^2}$$

At this time, the total carbon emissions of the production enterprises

$$E^{NY1}=\frac{e\left(a-b{c}_m\right)-b{e}^2{c}_s+2E}{4}$$

□

Corollary 2.

When$E=E2=\frac{eA-b{e}^2{c}_s}{2}-2E_0$, the producer can just buy enough carbon allowances to meet its production.

Proof of Corollary 2.

Let $A=a-b{c}_m$, then $q^{NY1*}=\frac{eA-b{e}^2{c}_s+2E}{4e}$, $E=e{q}^{NY1*}-E_0=\frac{eA-b{e}^2{c}_s+2E}{4}-E_0$, and simplify to $E=E2=\frac{eA-b{e}^2{c}_s}{2}-2E_0$. □

4.4.2. NY2 Strategy

Under the NY2 strategy, production enterprises do not use carbon emission reduction technologies but only purchase carbon emission rights, which are not sufficient to achieve carbon sufficiency. At this time, the income of production enterprises includes sales revenue, and the cost includes production costs and carbon rights purchase costs. Expected profits of production companies and carbon trading centers

$$\underset{q^{NY2}}{\max }{\pi }_m^{NY2}=\left(p-c_m\right)q^{NY2}-E_0{\omega }_s^{NY2}$$

$$\underset{{\omega }_s^{NY2}}{\max }{\pi }_s^{NY2}=\left({\omega }_s^{NY2}-c_s\right)E_0$$

$$\mathrm{s}.\mathrm{t}. e{q}^{NY2}>E+E_0$$

(17)

Theorem 6.

There are carbon constraints, companies do not adopt carbon emission reduction technologies, only purchase carbon emission rights, but they have not yet achieved carbon sufficiency. When the carbon quota satisfies $e{q}^{NY2*}>E+E_0$, there is only one $q^{NY2*}$and ${\omega }_s^{NY2*}$, which maximizes the profit of the enterprise.

Proof of Theorem 6.

Under this strategy, the available carbon quota of the manufacturer is $E+E_0$, so the optimal output of the manufacturer is

$$q^{NY2*}=\frac{E+E_0}{e}$$

Bring the above formula into Formula (17) to obtain the optimal profit of the production enterprise

$${\pi }_m^{NY2*}=\frac{\left(E+E_0\right)\left[e\left(a-b{c}_m\right)-\left(E+E_0\right)\right]}{b{e}^2}-{\omega }_s^{NY2*}$$

(18)

The optimal profit of carbon trading center

$${\pi }_s^{NY2*}=({\omega }_s^{NY2*}-c_s)E_0$$

(19)

From Equations (18) and (19), it can be seen that the profit of an enterprise is negatively correlated with the carbon price, and the profit of the carbon trading center is positively correlated with the carbon price (in the following numerical analysis, it is assumed that ${\omega }_s^{NY2*}$ takes the minimum value, that is, ${\omega }_s^{NY2*}=4$). At this point, the manufacturer’s total carbon emissions

$$E^{NY2}=E+E_0$$

□

4.5. YY Strategy

From Section 4.3 and Section 4.4 above, it can be seen that under the two strategies of only using emission reduction technologies and purchasing carbon emission rights, there are two situations of carbon sufficiency and carbon deficiency. Therefore, when companies adopt carbon emission reduction technologies and purchase carbon emission rights, there may also be both carbon-sufficient and carbon-deficient scenarios.

4.5.1. YY1 Strategy

Under the YY1 strategy, production enterprises adopt carbon emission reduction technologies and purchase carbon emission rights to achieve carbon sufficiency. At this time, the income of production enterprises includes sales revenue, and the cost includes production cost, emission reduction cost and carbon rights purchase cost. Profit function of production enterprises and carbon trading centers

$$\underset{q^{YY1},{\lambda }^{YY1}}{\max }{\pi }_m^{YY1}=\left(p-c_m\right)q^{YY1}-\frac{1}{2}h{\lambda }^{YY{1}^2}-\left[e\left(1-{\lambda }^{YY1}\right)q^{YY1}-E\right]{\omega }_s^{YY1}$$

$$\underset{{\omega }_s^{YY1}}{\max }{\pi }_s^{YY1}=\left({\omega }_s^{YY1}-c_s\right)\left[e\left(1-{\lambda }^{YY1}\right)q^{YY1}-E\right]$$

$$\mathrm{s}.\mathrm{t}. E<e\left(1-{\lambda }^{YY1}\right)q^{YY1}<E+E_0$$

(20)

Theorem 7.

With carbon constraints, companies adopt carbon emission reduction technologies and purchase carbon emission rights to achieve carbon sufficiency. When the carbon quota satisfies $e{q}^{YY1*}>E$, there is only one $q^{YY1*}$and ${\lambda }^{YY1*}$, which maximizes the profit of the enterprise.

Proof of Theorem 7.

Bring $p=\frac{a+k{\lambda }^{YY1}-q^{YY1}}{b}$ into the profit function of the production enterprise, we can obtain

$$\underset{q^{YY1},{\lambda }^{YY1}}{\max }{\pi }_m^{YY1}=\left(\frac{a+k{\lambda }^{YY1}-q^{YY1}-b{c}_m}{b}\right)q^{YY1}-\frac{1}{2}h{\lambda }^{YY{1}^2}-\left[e\left(1-{\lambda }^{YY1}\right)q^{YY1}-E\right]{\omega }_s^{YY1}$$

(21)

Because $\frac{{\partial }^2{\pi }_m^{YY1}}{\partial q^{YY{1}^2}}=-\frac{2}{b}<0$, $\frac{{\partial }^2{\pi }_m^{YY1}}{\partial \lambda { }^{YY{1}^2}}=-h<0$, let $\frac{\partial {\pi }_m^{YY1}}{\partial q^{YY1}}=0$, $\frac{\partial {\pi }_m^{YY1}}{\partial {\lambda }^{YY1}}=0,$ we can obtain

$$q^{YY1}=\frac{a+k{\lambda }^{YY1}-b{c}_m-be\left(1-{\lambda }^{YY1}\right){\omega }_s^{YY1}}{2}$$

(22)

$${\lambda }^{YY1}=\frac{q^{YY1}\left(k+be{\omega }_s^{YY1}\right)}{bh}$$

(23)

Simultaneous Equations (22) and (23) solve the optimal output of the production enterprise

$$q^{YY1*}=\frac{bh\left(a-b{c}_m-be{\omega }_s^{YY1}\right)}{2bh-{\left(k+be{\omega }_s^{YY1}\right)}^2}$$

(24)

optimal emission reduction rate

$${\lambda }^{YY1*}=\frac{\left(a-b{c}_m-be{\omega }_s^{YY1}\right)\left(k+be{\omega }_s^{YY1}\right)}{2bh-{\left(k+be{\omega }_s^{YY1}\right)}^2}$$

(25)

In order to satisfy $E3\ge 0$, after calculation (substituting into the parameters in Table 2), ${\omega }_s^{YY1}\le 4.859$ can be obtained, and because ${\omega }_s\ge c_s=4$, so $4\le {\omega }_s^{YY1}\le 4.859$. According to the extreme value theorem, there must be an optimal carbon price ${\omega }_s^{YY1*}$ to maximize the profit of the carbon trading center.

$${\omega }_s^{YY1*}=arg\underset{{\omega }_s^{YY1}}{\max }{\pi }_s^{YY1}\left({\omega }_s^{YY1}\right)$$

(26)

Bring Formulas (24)–(26) into Formula (20) to obtain the optimal profit of the production enterprise

$$\max {\pi }_m^{YY1*}=\left(\frac{a+k{\lambda }^{YY1*}-q^{YY1*}}{b}-c_m\right)q^{YY1*}-\frac{1}{2}h{\lambda }^{YY1{*}^2}-\left[e\left(1-{\lambda }^{YY1*}\right)q^{YY1*}-E\right]{\omega }_s^{YY1*}$$

The optimal profit of carbon trading center

$${\pi }_s^{YY1*}=\left({\omega }_s^{YY1*}-c_s\right)\left[e\left(1-{\lambda }^{YY1*}\right)q^{YY1*}-E\right]$$

The optimal expected profit of production enterprises and carbon trading centers is a function of the price of carbon rights. Due to the complicated formula, it will be further analyzed in the case analysis. At this time, the total carbon emissions of the production enterprises

$$E^{YY1}=e\left(1-{\lambda }^{YY1*}\right)q^{YY1*}$$

□

Corollary 3.

When$E=E3=e\left(1-{\lambda }^{YY1*}\right)q^{YY1*}-E_0$, after adopting the carbon emission reduction technology, the production enterprise can just purchase enough carbon allowances from the carbon trading center to produce products.

4.5.2. YY2 Strategy

Under the YY2 strategy, the production enterprise adopts carbon emission reduction technology and purchases carbon emission rights but has not yet achieved carbon sufficiency. At this time, the production enterprise’s income includes sales revenue, and the cost includes production cost, emission reduction cost and carbon right purchase cost. The discharge rate satisfies ${\lambda }^{YY2}\in [0,1-\frac{E+E_0}{e{q}^{YY2}})$. Profit function of production enterprises and carbon trading centers is

$$\underset{q^{YY2},{\lambda }^{YY2}}{\max }{\pi }_m^{YY2}=\left(p-c_m\right)q^{YY2}-\frac{1}{2}h{\lambda }^{YY{2}^2}-E_0{\omega }_s^{YY2}$$

$$\underset{{\omega }_s^{YY2}}{\max }{\pi }_s^{YY2}=\left({\omega }_s^{YY2}-c_s\right)E_0$$

$$\mathrm{s}.\mathrm{t}. e\left(1-{\lambda }^{YY2}\right)q^{YY2}>E+E_0$$

(27)

Theorem 8.

There are carbon constraints, and companies adopt carbon emission reduction technologies and purchase carbon emission rights, but they have not yet achieved carbon sufficiency. When the carbon quota satisfies $e\left(1-{\lambda }^{YY2*}\right)q^{YY2*}>E+E_0$, there is only one $q^{YY2*}$and ${\lambda }^{YY2*}$, which maximizes the profit of the enterprise.

Proof of Theorem 8.

Bring $p=\frac{a+k{\lambda }^{YY2}-q^{YY2}}{b}$ into the profit function of the production enterprise, we can obtain

$$\underset{q^{YY2},{\lambda }^{YY2}}{\max }{\pi }_m^{YY2}=\left(\frac{a+k{\lambda }^{YY2}-q^{YY2}}{b}-c_m\right)q^{YY2}-\frac{1}{2}h{\lambda }^{YY{2}^2}-E_0{\omega }_s^{YY2}$$

(28)

Under this strategy, the available carbon allowance of the manufacturer is $E+E_0$, so the optimal output of the manufacturer is

$$q^{YY2*}=\frac{E+E_0}{e\left(1-{\lambda }^{YY2*}\right)}$$

Bring the above formula into Formula (28) to obtain the optimal profit of the production enterprise

$$\underset{{\lambda }^{YY2}}{\max }{\pi }_m^{YY2}=\left(\frac{e\left(a+k{\lambda }^{YY2}\right)\left(1-{\lambda }^{YY2}\right)-\left(E+E_0\right)}{be\left(1-{\lambda }^{YY2}\right)}-c_m\right)\frac{E+E_0}{e\left(1-{\lambda }^{YY2}\right)}-\frac{1}{2}h{\lambda }^{YY{2}^2}-E_0{\omega }_s^{YY2}$$

(29)

Similar to Section 4.3.2, it can be obtained from the extreme value theorem that there must be an optimal carbon emission reduction rate, which maximizes the profit of the production enterprise. Therefore, the optimal emission reduction rate of production enterprises

$${\lambda }^{YY2*}=arg\underset{{\lambda }^{YY2}}{\max }{\pi }_m^{YY2}({\lambda }^{YY2})$$

From observation Formula (29), it can be concluded that the optimal carbon price ${\omega }_s^{YY2}$ has no effect on the optimal carbon emission reduction rate, and the profits of enterprises are negatively correlated with carbon prices, and the profits of carbon trading centers are positively correlated with carbon prices. At this time, the total carbon emissions of the production enterprises

$$E^{YY2}=E+E_0$$

□

5. Numerical Analysis

This section conducts a numerical analysis to visualize the optimal abatement rate, total carbon emissions, optimal production, and optimal profit for the four strategies under different carbon allowances and further analyzes the impact of changes in the scale cost of carbon abatement and low-carbon consumer preferences. According to the research hypothesis and previous research settings, the basic parameters are set as shown in

Table 3. Basic parameters

$\mathit{a}$	$\mathit{b}$	$\mathit{k}$	${\mathit{c}}_{\mathit{m}}$	${\mathit{c}}_{\mathit{s}}$	$\mathit{h}$	$\mathit{e}$	${\mathit{E}}_0$
100	0.5	0.3	5	4	1500	5	50

.

5.1. The Optimal Decision of Production Enterprises

5.1.1. The Maximum Profits for Production Enterprises under Carbon Constraints

Under the operating goal of profit maximization, this section focuses on the relationship between maximum profit and emission reduction strategies and compares and analyzes the optimal profits of the four strategies under different carbon constraints (as shown in Figure 1). It can be found that:

(a) No matter what strategy the company adopts, the relaxation of the carbon cap policy will have a positive impact on corporate profits. This shows that the carbon cap has certain constraints on the income of enterprises.

(b) On the whole, the corporate profit relationship is YY strategy > YN strategy > NY strategy > NN strategy. The profit of the YY strategy is always the largest. This is because under the YY strategy, the company eases the carbon quota constraints through appropriate emission reduction investments, enabling the company to produce more products to meet the market demand, and at the same time, the products after carbon emission reduction are more in line with the consumers’ low-carbon demand, which will further expand the demand and enable enterprises to obtain more benefits. Under the YN strategy, corporate profits are better than those under the NY strategy, because under the YN strategy, although the enterprise has to pay the cost of emission reduction, it obtains additional carbon emission rights in disguise, which increases the output; under the NY strategy, the enterprise saves emission reduction costs, but it is overly dependent on the purchase of carbon emission rights and does not have an advantage in carbon rights price negotiations.

5.1.2. The Optimal Output of Producers under Carbon Constraints

The implementation of the carbon cap policy will have a direct impact on the output of enterprises. Enterprises need to consider how to adjust their output to maximize profits while meeting market demand as much as possible. In this section, we compare and analyze the variation law of optimal yield under the four strategies (as shown in Figure 2), and we can find that:

(a) Under the NN and NY strategies, the output of enterprises is sensitive to the initial quota and grows nearly linearly with the relaxation of the carbon constraint policy. Therefore, when an enterprise only purchases additional carbon rights or responds negatively to the carbon cap policy, it should gradually expand production with the relaxation of the carbon constraint policy and should not blindly increase production to avoid penalties for exceeding carbon emissions.

(b) The output of enterprises under the YN and YY strategies is weakly affected by the initial carbon quota and is stably at a high level. This is because, on the one hand, the carbon emission per unit of product produced by the enterprise is reduced, enabling the enterprise to produce more products under the carbon limit set by the government. On the other hand, consumers have low-carbon preferences, and consumers are more willing to buy this product after companies reduce emissions, which increases market demand. Therefore, companies can increase product output by adopting emission reduction technologies to meet market demand as much as possible. It is worth noting that, under the YY strategy, the company’s emission reduction efforts are larger, but its output is slightly lower than that of the YN strategy, which is inconsistent with the general intuition (higher carbon emission reduction levels lead to higher output).

(c) When the carbon cap policy is stricter, the optimal output of the enterprise is less than the optimal output without carbon constraints. This shows that the carbon quota has certain constraints on the production of enterprises.

5.1.3. The Optimal Total Carbon Emissions of Production Enterprises under Carbon Constraints

In the future, carbon emission restrictions will become increasingly strict. As an important source of carbon emissions, manufacturers should effectively manage carbon emissions in the production process of products. Then, what impact will enterprises adopting emission reduction technologies and purchasing carbon allowances have on the total carbon emissions? This section presents the changes in the total carbon emissions under the four strategies (as shown in Figure 3), and we can find that:

(a) The total carbon emissions of enterprises under the four strategies are all on the rise. This is because with the relaxation of the carbon cap policy, companies will not only expand production scale but also reduce emission reduction intensity to reduce emission reduction costs.

(b) On the whole, the relationship between the total carbon emissions of enterprises is YY strategy < YN strategy < NY strategy. Combined with Section 5.1.1 (b), it can be concluded that enterprises can achieve a win–win situation of “development” and “carbon reduction” by purchasing additional carbon allowances while researching and developing emission reduction technologies. When the carbon quota is small, the environmental benefit of the YY strategy is not the highest. This is slightly different from the view in [3] that “investment in abatement technologies can compensate for the limited control of carbon emissions by the cap-and-trade mechanism”. This is because when the carbon constraints are stricter, the enterprises under the YY strategy have less emission reduction efforts and purchased additional carbon allowances, so their carbon emissions are greater than those of the YN strategy. The total carbon emissions under the NY strategy are higher, which indicates that the commercialization of carbon rights will lead to an increase in corporate carbon emissions.

(c) The carbon emissions of enterprises under the YN and YY strategies can usually meet the standards, while those under the NY strategy usually fail to meet the carbon emission standards. Under the YN strategy, when the carbon allowance allocated by the government for free is $E1$, the enterprise can just use emission reduction technologies to meet production requirements; under the NY strategy, when the carbon allowance allocated for free by the government is $E2$, the enterprise can just buy enough carbon quota. Under the parameter setting of this paper, when the enterprise chooses the YY strategy, there is no situation where the enterprise not only reduces emissions but also purchases carbon rights and still has insufficient carbon allowances ($E3<0$). Inferences 1, 2, and 3 are verified.

5.1.4. The Optimal Emission Reduction Rate of Production Enterprises under Carbon Constraints

In the face of carbon emission reduction requirements of different intensities, how enterprises should make emission reduction plans is the primary problem that enterprises must face in the carbon economy era. In this section, the optimal emission reduction rate is used to represent the company’s emission reduction efforts, and the two cases of purchasing carbon emission rights as a supplement and not purchasing carbon emission rights are considered to analyze the emission reduction countermeasures of the company (as shown in Figure 4). It can be found:

(a) Under the two strategies of YN and YY, the optimal emission reduction rate of enterprises decreases with the increase in carbon allowances, and then remains unchanged. This is because when carbon allowances are low, the benefit of increased production outweighs the increase in abatement costs by increasing the abatement rate, while the benefits are offset by the increase in abatement costs when carbon allowances are high. Therefore, the stricter the carbon cap constraint policy, the more companies should increase their emission reduction efforts to ease carbon constraints.

(b) The optimal emission reduction rate changes rapidly with carbon allowances under the YN strategy, and it changes slowly under the YY strategy. This is because under the YN strategy, companies can only expand production by adopting emission reduction technologies and are more sensitive to changes in carbon allowances. Under the YY strategy, companies can alleviate some of the pressure on emission reductions by purchasing carbon rights, so the emission reduction rate changes relatively slowly with carbon allowances.

(c) There is a critical value where, when the carbon allowance is less than this value, the emission reduction rate of the enterprise under the YN strategy is larger, or otherwise, the opposite. This is because when the carbon allowances allocated for free by the government are small, the production enterprises face strong carbon constraints. Under the YY strategy, companies can purchase additional carbon allowances in addition to emission reductions, while the YN strategy can only be alleviated by increasing the emission reduction rate. Therefore, when the carbon quota is small, the emission reduction in the YN strategy is larger. With the increase in carbon allowances, under the YY strategy, in order to further expand production and obtain higher profits, enterprises still maintain a high emission reduction rate while purchasing additional carbon allowances.

(d) On the whole, the YY strategy is better than the YN strategy in terms of emission reduction. This shows that, on the premise that companies decide to adopt emission reduction technologies, purchasing carbon allowances can promote companies to increase emission reduction rates to a certain extent.

5.2. Other Factors Influencing Production Decisions

5.2.1. The Impact of Carbon Reduction Scale Cost Coefficient

An efficiency of abatement investment is only involved when companies adopt carbon abatement technologies. Thus, in this section, on the premise of adopting an emission reduction technology, two cases of purchasing carbon emission permits and not purchasing carbon emission permits are considered, respectively, to analyze the impact of the change in carbon reduction scale cost factor (as shown in Figure 5 and Figure 6). It can be found:

(a) Corporate profits are negatively correlated with the carbon reduction scale cost factor. The larger the carbon reduction scale cost factor, the more difficult it is to reduce emissions, and more costs need to be invested under the same degree of emission reduction. Therefore, enterprises will reduce the cost risk by reducing the emission reduction rate when making emission reduction decisions. Therefore, enterprises can further improve their own profits by improving the efficiency of emission reduction investment.

(b) The higher the emission reduction efficiency of enterprises, the easier it is to achieve carbon sufficiency. Therefore, enterprises should actively carry out low-carbon transformation and vigorously promote the research and development of low-carbon technologies to improve the efficiency of emission reduction and reduce the dependence on carbon quotas.

5.2.2. The Impact of Consumers’ Low-Carbon Preference

Consumers’ low-carbon preferences are only involved when companies adopt carbon reduction technologies; therefore, in this section, on the premise of adopting emission reduction technologies, the two scenarios of purchasing and not purchasing carbon emission permits are considered, respectively, to analyze the impacts of changes in consumers’ low-carbon preferences (as shown in Figure 7 and Figure 8). It can be found that:

(a) Corporate profits are positively correlated with consumers’ low-carbon preference. This is consistent with the findings in the literature [31]. The higher the consumers’ low-carbon preference, the higher the incentive for production enterprises to reduce emissions, prompting enterprises to increase their emission reduction efforts to produce, thereby expanding production within a limited carbon quota and increasing profits. Therefore, enterprises can further improve their own profits by increasing consumers’ low-carbon preference.

(b) The higher the consumer’s low-carbon preference, the easier it is for enterprises to achieve carbon sufficiency in their production. Therefore, enterprises should vigorously publicize the low-carbon concept and carry out green marketing while producing low-carbon, so as to enhance the low-carbon awareness of the whole people and expand the demand.

6. Conclusions and Discussion

6.1. Findings and Managerial Implications

This paper establishes the revenue model of production enterprises under four different strategies from the perspective of production enterprises. Firstly, the optimal emission reduction rate, total carbon emission, optimal production, and optimal profit of production enterprises under different carbon constraints are derived with the maximum profit that production enterprises can obtain as the standard, and the laws are summarized. After that, in order to further improve the profitability of production enterprises, the impact of the scale cost of carbon emission reduction and the change in consumers’ low-carbon preference are studied. The main findings of the study are as follows: (1) The stricter the carbon constraint policy, the greater the optimal emission reduction rate of enterprises. (2) The adoption of emission reduction technology can effectively reduce the impact of carbon constraint on output. (3) The optimal strategy is to both reduce emissions and purchase carbon emission rights, which can realize environmental economic dividends. (4) The lower the cost factor of the carbon reduction scale (the higher the efficiency of emission reduction) and the higher the low-carbon preference of consumers, the easier it is for firms to achieve carbon sufficiency in their production.

Based on the above analysis, the following management insights are obtained. Facing the carbon constraint, production companies should do the following:

(1) Improve the sensitivity to carbon policy regulation. Faced with the carbon limit constraint policy, enterprises should actively improve their sensitivity to carbon policy and actively cooperate with carbon emission reduction, and the stronger the government’s carbon emission control, the more enterprises should increase emission reduction to ease the carbon constraint.

(2) The adoption of carbon emission reduction technology is the main strategy, supplemented by the purchase of carbon allowances. The emission reduction effect and profitability of enterprises adopting emission reduction technology are better than purchasing carbon emission rights, and the strategy of adopting carbon emission reduction technology and purchasing carbon emission rights at the same time is the optimal strategy for enterprises, which can realize environmental economic dividends.

(3) Further expand profit space by vigorously promoting low-carbon concepts and actively exploring low-carbon transformation. In order to further optimize corporate profits, enterprises should actively carry out low-carbon transformation, accelerate the application of low-carbon technology research and development, improve the efficiency of emission reduction, and at the same time, improve consumers’ low-carbon perception, vigorously promote low-carbon concepts, and carry out green marketing.

6.2. Limitations and Future Directions

There are limitations to the research in this paper. First, the model in this paper is only applicable to carbon markets where initial quotas are allocated by the government free of charge, but not to carbon markets where there is carbon leakage and initial quotas are obtained through auctions. Second, in this paper, the producers are self-producing and self-selling, and only the secondary supply chain consisting of a single producer and a single carbon trading center is considered, while in practice, the supply chain operation also involves retailers and suppliers, which is more complicated. Third, carbon quotas will be adjusted with the change in the production cycle, and the carbon quota at the beginning of the period will have an impact on the subsequent production, while this paper only considers the single-cycle scenario. Therefore, the next research direction will be to study the emission reduction strategy choices of producers in more complex supply chain structures and multi-cycle scenarios.