Two-Way Carbon Options Game Model of Construction Supply Chain with Cap-And-Trade

Abstract

As one of the main sources of global greenhouse gas emissions, the low-carbon transformation and emission reduction in the construction industry are inevitable requirements for addressing climate change. Under cap-and-trade regulations, Carbon emission rights have become a key production factor. However, the price of carbon emission rights is highly random. Taking the EU carbon market in 2024 as an example, the carbon price fluctuated by more than 35%, soaring from 65 euros per ton to 80 euros per ton and then falling back. Such sharp fluctuations not only increase the cost uncertainty of enterprises but also complicate the investment decisions for emission reduction. Therefore, enterprises can enhance the flexibility of carbon emission rights trading decisions through option strategies, helping them hedge against the risks of carbon price fluctuations, and at the same time improve market liquidity and risk management capabilities. Against this background, based on the carbon cap-and-trade policy, this paper introduces the two-way option strategy into the construction supply chain game model composed of general contractors and subcontractors, and studies to obtain the optimal carbon reduction volume, carbon option purchase volume, maximum expected profit of general contractors, subcontractors and profit distribution ratio. This study shows that two-way options play a crucial role in optimizing supply decision-making and emission reduction strategies. Under the decentralized model, emission reduction responsibilities are often shifted to subcontractors by the general contractor, resulting in a decline in overall mitigation effectiveness. Furthermore, appropriately lowering the carbon emission benchmark can strengthen enterprises’ incentives for emission reduction and significantly enhance the profitability of the supply chain. The study further suggests that general contractors should enhance their competitiveness by developing environmentally friendly technologies and improving their ability to reduce emissions on their own. Meanwhile, subcontractors need to actively participate in the collaborative efforts through revenue-sharing contracts. This study reveals the strategic value of two-way carbon options in construction supply chain carbon trading and provides theoretical support for the formulation of carbon market policies, contributing to the low-carbon transition of the construction supply chain.

1. Introduction

The intensification of global warming and its cascading environmental consequences have galvanized international climate governance efforts, with carbon dioxide emission reduction emerging as a critical target for intervention [1]. Following the landmark 2015 Paris Agreement signed by 196 nations, the European Union established strategic leadership through its 2018 Climate Neutrality Roadmap, targeting net-zero emissions by 2050. This vision has materialized through comprehensive legislative frameworks, including the “Fit for 55 Package”, “Green Deal Industrial Plan”, and “Net-Zero Industry Act”, which recalibrate carbon market mechanisms and renewable energy benchmarks across strategic sectors such as transportation and manufacturing. The EU further reinforced this trajectory in April 2024 by unveiling its inaugural Projects of Common Interest and Projects of Mutual Interest to accelerate the energy transition. In this context, China has institutionalized carbon governance through key regulatory instruments, notably the Interim Measures for the Administration of Carbon Emissions Trading and its implementation rules, which together form the legal foundation of the national carbon market. The construction sector, responsible for 39% of global operational emissions and nearly half of China’s total emissions via direct and indirect sources, plays a critical role. Its decarbonization is essential to achieving China’s dual carbon goals—peaking emissions by 2030 and reaching neutrality by 2060—and advancing the broader sustainability agenda.

In response to global climate challenges, governments worldwide have intensified efforts to establish carbon emission control mechanisms, with carbon emissions trading emerging as a pivotal policy instrument. The European Union pioneered this initiative through its Emissions Trading System (EU ETS) in 2005, which is currently the world’s largest carbon market, encompassing power generation, industry, aviation, and is scheduled to include maritime transport by 2027. This cap-and-trade system employs market stabilization mechanisms and progressive reduction in free allowances to maintain carbon price stability. Parallel developments are evident in China, where the national carbon market launched in July 2021 has expanded to cover power generation, steel, chemicals, and cement industries, achieving a cumulative trading volume of 465 million tons worth ¥27 billion by July 2024, with carbon prices stabilizing around ¥90/ton as reported by the Ministry of Ecology and Environment.

The financialization of carbon markets has led to the development of sophisticated trading instruments, particularly carbon options, which serve as essential risk-hedging tools. Functioning as contractual rights rather than obligations, these derivatives enable enterprises to purchase call options to secure future acquisition costs for emission quotas or put options to guarantee selling prices for surplus allowances. This flexibility proves crucial in mitigating price volatility risks inherent in carbon markets while enhancing transactional adaptability. Market participants can strategically employ option combinations to optimize emission management decisions, thereby improving market liquidity, a critical factor in achieving dual carbon objectives. Current academic research increasingly incorporates option pricing models into carbon market analyses, recognizing their capacity to quantify risk exposure and inform corporate emission strategies within evolving regulatory frameworks.

The effectiveness of such market-based mechanisms fundamentally relies on robust top-tier policy frameworks. Currently, the predominant global carbon emission policies comprise three major systems: Cap-and-Trade, carbon taxation, and emission caps. Among these, the cap-and-trade has been widely recognized by both governments and enterprises for its market-oriented flexibility and has become one of the most important global tools for carbon reduction [2].

Against this backdrop, the construction industry faces an urgent need to implement systematic carbon management strategies to address the challenges posed by carbon constraints. The construction supply chain has emerged as a transformative organizational paradigm, demonstrating distinct advantages over traditional fragmented contracting models through its systematic integration of stakeholder resources, information flows, and risk allocation mechanisms. This collaborative framework not only strengthens project execution through enhanced quality control and schedule compliance but also optimizes cost efficiency and sustainability outcomes [3]. Among prevalent implementation models, the Engineering Procurement and Construction (EPC) stands out as the dominant form due to its centralized decision-making, high efficiency, and strong cost management capability. Based on this, this paper develops a construction supply chain decision-making model involving a general contractor and subcontractors.

The key contributions of the present research work are as follows:

(1)

Although previous studies have explored the role of option contracts in supply chain risk hedging, they have mainly focused on manufacturing or retail scenarios. In the construction industry, previous research has focused on the development of green building materials and the optimization of supply chain processes, and carbon financial tools have not yet been incorporated into the construction supply chain management framework. This study introduces two-way carbon options into the construction supply chain environment for the first time to manage the price fluctuations of carbon quotas. Through the above innovations, this study bridges the key gap between carbon financing and construction supply chain management and broadens the research scope of option contract theory.

(2)

A novel Holistic Stackelberg game-based model is proposed for a construction supply chain operating under cap-and-trade. The model jointly optimizes the profit-sharing ratio, carbon emission reduction levels, and the amount of carbon allowances purchased through carbon option contracts within a unified decision framework. This integrated approach captures the hierarchical leader–follower dynamics in the supply chain, thereby transcending the fragmented approaches of prior studies, which treat emission mitigation, contract design, and carbon credit procurement in isolation. The model advances the theoretical discourse at the intersection of green construction supply chain management and option contract theory, while offering practical insights for policymakers and industry stakeholders seeking integrated strategies in the construction sector.

(3)

From the perspective of practical value, the construction supply chain’s emission reduction is confronted with three major predicaments: high emission reduction costs, significant risks of carbon price fluctuations and difficulty in coordinating the interests of the main parties. Most existing research remains at the theoretical deduction stage and lacks an adaptive design for practical operation scenarios in the construction supply chain. This study verified the role of option tools in mitigating carbon price fluctuation risks and optimizing emission reduction decisions by constructing a building supply chain game model with two-way carbon options. Through numerical analysis, it revealed the correlation rules between key parameters such as the carbon price benchmark and the emission reduction cost coefficient and the optimal strategy. It should be noted that the model of this study is based on a series of simplified assumptions, such as complete information symmetry and complete risk transfer. These assumptions aim to strip away complex interfering factors, clearly depict the interaction mechanism between two-way options and supply chain emission reduction decisions, and make the model results unable to be directly used as precise quantitative tools in the actual operation of enterprises. Despite this, the research conclusions can still provide theoretical references and decision-making ideas for construction enterprises and offer a theoretical analysis framework for understanding the strategic interactions in the low-carbon transformation of the construction supply chain. This is in sharp contrast to existing research focusing on manufacturing.

In conclusion, this study not only expands the application boundaries of carbon option theory in the construction field but also provides a new analytical paradigm for supply chain collaborative emission reduction through an integrated model.

This paper has the following structure: Following the abstract and introduction, in Section 2 literature review introduces the research background and supporting literature of the carbon option game model of decision optimization of supply chain under carbon cap-and-trade policy, decision optimization of supply chain under option contracts and construction supply chain management in detail, in Section 3 it describes the model and the key assumptions, in Section 4 it introduces the General contractor’s two-way option strategy model for purchasing carbon emission rights, in Section 5 it shows the sensitivity analysis of decision variables, Section 6 presents the numerical analysis, including comparative scenarios. Finally, Section 7 provides the conclusion that summarizes key findings and offers managerial insights, and also discusses the limitations and directions for future work.

2. Literature Review

Literature closely related to this study can be divided into three perspectives: decision optimization of supply chains under carbon cap and trading, decision optimization of supply chains with option contracts, and decision optimization of construction supply chains.

2.1. Decision Optimization of Supply Chain Under Carbon Cap-And-Trade Policy

To achieve the strategic vision of “carbon peak” and “carbon neutrality”, building an institutional framework to strengthen green and low-carbon development is an inherent requirement for China’s economy to transform to a stage of high-quality development. The characteristics of carbon cap and trading policies are easy to implement and highly operational. Therefore, they have been adopted by many regional and national governments and widely used in practice. Different scholars have also conducted much research on carbon cap and trading policies in various scenarios. Ref. [4] Studied the impact of carbon cap-and-trade policy on decision-making within emission-dependent supply chains and the issue of distributive fairness in social welfare under the carbon cap-and-trade policy environment. Ref. [5] Investigated which policy, the carbon cap-and-trade policy or the low-carbon subsidy policy, is more beneficial to society. Ref. [6] The role and impact of inventory decisions on remanufacturing and systems operating under carbon emission restrictions were explored. Ref. [7] The optimal decision for warehouse management and technology investment under a carbon cap-and-trade policy was analyzed in research, and the impact of initial carbon emission quotas and the cost of unit carbon emission trading with external markets on the economic and environmental performance of warehousing operations was discussed. Ref. [8] The optimal strategy problem of a capital-constrained low-carbon supply chain under uncertain returns, incorporating green finance and a carbon cap-and-trade mechanism was studied. Ref. [9] In the context of the carbon cap-and-trade mechanism, an evolutionary game model between the government and manufacturers was established, and the impact of government policies on manufacturers’ decisions and the dynamic trend of the carbon cap-and-trade market were analyzed. Ref. [10] The carbon cap-and-trade policy was combined with consumers’ low-carbon preferences under the differential game model, and the carbon emission reduction efforts of supply chain members and the optimal profits obtained under the above scenarios were compared.

The carbon quota and carbon trading system not only have a profound impact on the daily operations of enterprises, but also profoundly affects many core decisions within the supply chain, including production processes, product pricing and order quantity selection strategies, as well as market channel selection. For example, in the production process, to improve energy efficiency and reduce carbon emissions, Ref. [11] studied the production and carbon emission reduction decision-making problems in a two-stage supply chain consisting of a manufacturer and a retailer. By establishing single and joint emission reduction modes, it was found that the joint emission reduction mode is the optimal emission option for the supply chain. Ref. [12] Used a two-stage Stackelberg game to study the production decision-making problems in a supply chain consisting of manufacturers and retailers, and the government’s cap setting problem under wholesale prices and revenue-sharing contracts under cap-and-trade regulations.

To further balance production costs and market demand, enterprises need to formulate reasonable product pricing and select the optimal order quantity based on their own carbon emission quotas, production costs, market demand and other factors under the constraints of carbon limits to achieve the optimal operation decision. For example, Ref. [13] explored the optimal decision of sales price and carbon footprint in a two-stage supply chain consisting of a single manufacturer and a single retailer, under the background of a carbon cap and trading policy. Ref. [14] studied the production and product pricing issues in an upstream manufacturer and a downstream retailer that produces two products based on MTO. Ref. [15] studied the pricing and carbon reduction decision issues in vertical and horizontal cooperative supply chains. The results showed that vertical cooperation leads to higher carbon reduction rates and lower retail prices. In contrast horizontal cooperation among manufacturers harms the profits of retailers and the welfare of consumers. Ref. [12] used a two-stage Stackelberg game to study the government quota setting problem under wholesale prices and revenue-sharing contracts under carbon cap-and-trade policies in a supply chain composed of manufacturers and retailers. The results showed that excessive government allocation of carbon credits may damage the profits of manufacturers and wholesale prices or revenue-sharing contracts, thereby increasing the difficulty of implementing cap-and-trade supervision. Ref. [16] studied the optimal ordering and transportation mode selection decisions of retailers under different carbon emission reduction policies. Ref. [17] by analyzing the optimal operating decisions of suppliers and manufacturers in a decentralized scenario, the performance of the supply chain was examined by comparing profits and carbon emissions with those in a centralized scenario.

Under the carbon cap-and-trade policy, as the carbon trading market continues to develop and improve, trading methods may also become more diverse and complex, so the choice of reasonable channels is particularly important. Among them, single-channel transactions are concentrated and have low costs, but their applicable scenarios are relatively limited. For example, Ref. [18] studied the channel selection and emission reduction decisions of manufacturers under carbon cap-and-trade policies when considering carbon emission constraints. The results showed that when a few consumers choose online channels, setting larger carbon quotas and encouraging manufacturers to develop dual-channel models would be more effective. Therefore, dual channels are more widely studied due to their flexible transaction forms and decentralized transaction risks. For example, Ref. [19] studied the decision-making and coordination issues in dual-channel supply chains with low-carbon preferences and channel substitution under carbon cap-and-trade policies. The results showed that carbon cap-and-trade policies can effectively reduce carbon emissions throughout the supply chain and facilitate the coordinated development of the economy and the environment. Ref. [20] introduced carbon cap-and-trade policies into dual-channel supply chain management and studied the pricing and emission reduction decisions of supply chain members under a low-carbon environment.

2.2. Decision Optimization of Supply Chain Under Option Contracts

Existing research on option contracts mainly allows supply chain companies to use option contracts as a financial tool in the ordering process to avoid risks caused by market price fluctuations, demand uncertainty or instability in production and supply and coordinate the supply chain. Therefore, many scholars have studied the role of option contracts in operations management. For example, Ref. [21] explored how retailers with capital constraints can order products through call option contracts to achieve supply chain coordination. Ref. [22] studied the optimal decision of option contracts for manufacturers in the flexible cap ETS system and used the newspaper vendor model to solve the joint emission quota ordering and production pricing problems under demand uncertainty. Ref. [23] moreover, others studied an option contract for coordinating the supply chain, finding the coordination conditions of the supply chain system under both decentralized and centralized decision-making. This approach optimizes the profit of the supply chain system and enables the profit of supply chain members to reach Pareto optimality. Ref. [24] investigated the fairness of channel members in a two-tier supply chain consisting of a single supplier and a single retailer. The results showed that under certain pricing parameters, the supply chain under the fair concerns of channel members can be coordinated through optional contracts. Ref. [25] studied the news supplier problem in the joint ordering and pricing setting under the option contract based on demand uncertainty. By comparing single ordering and mixed ordering, it was found that optimal pricing and ordering strategies exist and are unique for both cases. Ref. [26] developed a news supplier model to examine the impact of customer returns on the company’s pricing and ordering decisions. The study found that option contracts can be a tool to mitigate the negative impact of customer returns.

With the gradual improvement of the carbon emission trading system and the continuous development of the carbon financial market, scholars have turned their attention to this emerging field. However, there are still relatively few studies examining the role of carbon options in operations management. For example, Ref. [27] considered the government’s restrictions on the total carbon emissions of manufacturers and the manufacturers’ implementation of carbon emission reduction strategies, analyzed the optimal emission procurement and product pricing strategies of manufacturers under carbon option contracts, and analyzed the value of adopting carbon option contracts to enterprises. In carbon option contracts, one-way options are primarily categorized into call options and put options, which offer relatively low risks and returns, making them suitable for investors who have clear expectations for carbon asset prices. Ref. [28] studied the optimal operating decisions of suppliers and retailers with call option contracts under one-way options in the supply chain. The results show that call option contracts can benefit both retailers and suppliers, improve the performance of the entire supply chain and reduce ineffective carbon emissions. Bidirectional options have relatively high risks and return and can provide greater flexibility. For example, Ref. [29] proposed that one-way option (UO) contracts and two-way option (BO) contracts can effectively increase corporate profits and reduce carbon emissions. It was found that UO contracts are more beneficial to manufacturers, while BO contracts are more beneficial to retailers, the entire supply chain and the environment.

In recent years, Ref. [30] have introduced carbon emission options to hedge risks in the case of demand uncertainty. The maximum expected profits of enterprises under pure wholesale price contracts, pure carbon option contracts and portfolio contracts were studied. Ref. [31] investigated the changes in the optimal decisions of each entity under the two-way option trading model with and without carbon emission rights. Ref. [32] compared decentralized models with and without carbon option contracts. Studies have shown that carbon option contracts benefit both suppliers and manufacturers under certain conditions. Ref. [33] under emission limits, carbon option contracts and storage contracts are utilized to coordinate the supply chain, considering customers’ environmental awareness. When discussing carbon option games, the literature has all conducted in-depth research on the optimization of operational decisions in the supply chain based on continuous production scenarios in the manufacturing industry. However, they ignored the project-based characteristics of the construction supply chain. They failed to consider the importance of carbon choice games from the perspective of more specific construction project supply chains. This provides a clear theoretical gap for this study to focus on the application of two-way carbon options in the construction supply chain.

2.3. Construction Supply Chain Management

Ref. [3] first introduced the concept of supply chain management in the manufacturing industry into the construction industry, forming a preliminary framework for modern supply chain management ideas in the construction industry. Construction supply chain management introduces an innovative approach for all stages of construction projects, from planning, and engineering design to construction and operation management, aiming to effectively reduce construction costs and improve the reliability of construction projects. Ref. [34] pointed out that the construction supply chain is completely different from the common unilateral, long-term transaction relationship in the manufacturing supply chain. It is complex, diverse and decentralized, and it contains a complex binary relationship network involving many stakeholders.

Furthermore, under the influence of global climate change, the supply chain is facing stability risks. To address the challenges brought about by climate change, construction enterprises must enhance their equipment and business models to mitigate carbon emission issues and comply with corresponding carbon emission policies, thereby reducing the pressure of carbon emissions. For example, Ref. [35] emphasized the importance of green building project management. Ref. [36] studied green purchasing behavior in the context of real estate development. Ref. [37] comprehensively analyzed the progress of current low-carbon building development and inspected and optimized the application of carbon emission accounting methods in buildings. The existing research on green buildings mainly focuses on the development of low-carbon building materials and construction technologies. For example, Ref. [38] explored the adsorption performance of aerogel for CO₂ and discussed the application prospects of low-carbon building materials. Promote the development of low-carbon building materials. Ref. [39] research has shown that incorporating suitable low-carbon recycled materials to replace primary building materials can effectively reduce carbon emissions and help achieve the zero-carbon goal. Notably, the latest research has begun to focus on carbon synergy in the construction supply chain, such as Ref. [40], combined with prospect theory. The influence of perception parameters on the decision-making process of stakeholders in low-carbon buildings was explored.

However, studies exploring carbon emissions and profit distribution from the perspective of the green supply chain still mainly focus on traditional manufacturing industries. Most existing studies on the construction supply chain focus on improving the operational efficiency, such as [41] combining construction supply chain management with building information modeling in construction to utilize synergy effects and enhance the construction process. Based on the Sustainable Development Goals in an uncertain environment, Ref. [42] studied the incentives and coordination issues of the construction supply chain comprises owners, general contractors, subcontractors, and suppliers. Ref. [43] studied the production-distribution-construction integrated system composed of the construction department and material suppliers in a random environment, proposed a new two-level multi-stage planning method for multi-objective optimization to examine the interaction among decision-makers and conducted a quantitative analysis of the construction supply chain. Ref. [44] evaluated the sustainability level of sustainable supply chain management in the Turkish construction industry by using the analytical network process method and proposed a sustainable supply chain management model. Ref. [45] by constructing a system dynamics model, the dynamic process of construction demolition waste management was explored, and cross-organizational incentive strategies were proposed to achieve a coordinated balance between the construction period and quality goals. Ref. [46] provided a distributed coordination method to improve the interaction and collaboration between agents and personnel. Ref. [47] proposed a hybrid meta-heuristic algorithm to address multi-objective optimization problems in construction project planning and financing.

To sum up, although the above-mentioned literature on the construction supply chain has deeply explored the optimization issues of construction project supply chain management from aspects such as cost control, schedule management, quality control, supplier selection and cooperation, it has not yet incorporated financial tools such as carbon options into the supply chain coordination framework. It has ignored the issue of carbon option contract sequencing and coordination under the background of carbon quota and trading policies. Therefore, this study will focus on establishing a carbon selection game mechanism between general contractors and subcontractors in the construction supply chain to promote the overall sustainable development and collaborative optimization of the supply chain.

Through a literature review of supply chain decision-making under carbon cap-and-trade, the application of option contracts, and construction supply chain management, we can see that although existing research on supply chain decision-making, option contract application, and construction supply chain management under carbon cap-and-trade has accumulated to a certain extent in their respective fields, there are still obvious deficiencies in their integration with the characteristics of the construction industry. The unique demands of the construction supply chain have not been fully responded to. Specifically, most of the existing research is based on the continuous production mode of the manufacturing industry, ignoring the essential feature of the construction supply chain that is project-centered, that is, its carbon emissions have the characteristics of temporary and phased fluctuations, and the risk allocation is closely related to the contract design. Traditional carbon trading tools and single option mechanisms are difficult to adapt to the complex carbon risks under this project-oriented environment.

Meanwhile, in the research on the coordination of the construction supply chain, the focus is mostly on process optimization or technology application, lacking the integration of carbon financial tools, and failing to solve the problem of risk hedging and interest synergy between general contractors and subcontractors under the fluctuation of carbon prices. In response to these gaps, this study focuses on the construction supply chain based on EPC projects, introducing a two-way option mechanism into the supply chain decision-making model. By capturing the dual risks of quota surplus and shortage that may occur in the project, it provides flexible risk hedging tools for subcontractors. At the same time, combined with the Stackelberg game framework, it constructs a decision-making interaction model between general contractors and subcontractors. Combining the use of two-way options with the profit distribution mechanism not only makes up for the deficiency of existing research in paying insufficient attention to the particularity of carbon risks in construction projects, but also fills the gap in the application of carbon financial tools in the coordination of the construction supply chain, thereby forming a low-carbon collaborative decision-making framework that suits the characteristics of the construction industry.

3. Model Assumptions and Description

This paper considers a two-stage supply chain consisting of a general contractor and a subcontractor under the Engineering Procurement, and Construction (EPC). Under cap-and-trade, the general contractor and the subcontractor not only need to complete the project construction goals but also need to weigh the additional costs and carbon trading risks brought by carbon emission reduction. To incentivize the general contractor to meet emission reduction targets, the project owner introduces a performance-based emission reduction mechanism. Additional rewards are granted based on the contractor’s emission reduction performance, thereby adopting a form of contract that combines a lump-sum contract with an emission reduction incentive contract. Similarly, to incentivize the subcontractor to engage in emission reduction efforts, the general contractor allocates a portion of the incentive accordingly. It is essential for the general contractor to determine an appropriate allocation ratio that effectively motivates subcontractors to contribute to the achievement of emission reduction targets. In addition, the subcontract between the general contractor and the subcontractor also incorporates both a lump-sum contract and a performance-based emission reduction incentive mechanism. Considering the uncertainty in carbon market prices as well as the surplus or shortage of carbon allowances, this paper introduces bidirectional carbon options to enhance the flexibility and risk management capability of enterprises in carbon trading decisions. On this basis, it develops the General Contractor’s Two-Way Option Strategy Model for Purchasing Carbon Emission Rights and systematically analyzes the impact of different strategic arrangements on emission reduction performance and profit distribution under decentralized decision-making scenario.

Some used in this article, and their descriptions are listed in

Table 1. Notations and descriptions.

Symbol Description
Parameters
$P_i$	Fixed contract price for the enterprise ($i=1,2$, represents general contractor and subcontractor respectively)
$s$	Total construction area of the project
$w_0$	The option price per unit of carbon quota in a carbon option contract
$w_b$	The strike price per unit of carbon quota in a carbon option contract
$\phi$	The incentive factor given by the owner to the general contractor for each unit of emission reduction
$C_i$	Fixed costs of the enterprise ($i=1,2$, represents general contractor and subcontractor)
$R_i$	Emission reduction input cost of enterprise ($i=1,2$, represents general contractor and subcontractor)
${\epsilon }_i$	Emission reduction cost coefficient of the enterprise ($i=1,2$, represents general contractor and subcontractor)
$e_0$	Initial carbon emissions per unit building area of the project
$k$	Carbon emission benchmark per unit building area of engineering projects (advanced carbon emission level). The carbon emission benchmark $k$ refers to the “General Specification for Building Energy Efficiency and Renewable Energy Utilization” (GB 55015-2021) [48]. For residential buildings, $k$ is taken as 0.6 tCO2/m2, and for public buildings, $k$ is taken as 0.8 tCO2/m2. The numerical analysis in this paper adopts the weighted average value of the location of the construction unit.
${\pi }_i$	Expected profit of the enterprise ($i=1,2,$ represents general contractor and subcontractor)
$\pi$	Expected profits from construction supply chain
$t$	The carbon price of the instant purchase of carbon emission rights has a distribution function and density function of $G\left(t\right)$ and $g\left(t\right)$ respectively, the market carbon price adopted in this paper follows a uniform distribution $U\left(m,n\right)$, a mean of $\mu$, a minimum carbon price of $m$, a maximum carbon price of $n$
Decision variables
$q_0$	The amount of carbon allowances purchased through carbon option contracts
$\lambda$	Profit-sharing ratio given by the general contractor to the subcontractor
$e_i$	Carbon emission reduction per unit building area of the enterprise ($i=1,2$, represents general contractor and subcontractor)

.

In this paper, the following assumptions are made:

(1)

Under the Engineering Procurement and Construction (EPC), the general contractor, as the primary party contracted by the project owner, is responsible for overall project coordination and resource allocation, thereby holding a dominant position. The subcontractor, by contrast, performs specific tasks under the general contractor’s direction. Therefore, a Stackelberg game is established between the two parties, with the general contractor as the leader and the subcontractor as the follower.

(2)

The general contractor and the subcontractor must comply with government regulations, and the carbon market is relatively active. The third party in the market can fulfill carbon option contracts and meet the general contractor’s carbon option needs.

(3)

It is assumed that the information between the general contractor and the subcontractor is completely symmetrical, both parties are completely rational, and both aim to maximize their expected profits. Given the prevalence of information asymmetry in the construction supply chain, this study temporarily adopts the assumption of complete information symmetry to simplify the model. This is mainly based on the reality that the general contractor and subcontractors in the EPC model of the construction supply chain achieve information sharing through standardized contracts, and the general contractor and subcontractors achieve carbon emission data symmetry through BIM technology. However, the general contractor reserves the right to make decisions. Future research can explore the impact of information asymmetry by combining signal games or contract design.

(4)

Under emission reduction incentives, both the general contractor and the subcontractor will increase their carbon reduction efforts by investing more in technological and managerial innovations. As the level of investment increases, their emission reduction costs will also rise accordingly. The emission reduction investment cost of firm $i$ is $R_i\left(e_i\right)={\displaystyle \frac{1}{2}}{\epsilon }_i{e_i}^2$.

(5)

Assuming that the project’s initial carbon emissions per unit building area exceed the benchmark level (advanced carbon emissions level), that is, $e_0>k$. This assumption reflects the common high-emission status of construction projects in the absence of emission reduction measures. Excess emissions will result in additional carbon costs.

(6)

Assume that the general contractor and subcontractor engage in a single short-term cooperation without any long-term cooperation constraints, and that the risks and costs of emission reduction are completely transferred to the subcontractor. Under such restrictive assumptions, the equilibrium result that the general contractor’s optimal emission reduction effort is zero $e_1=0$ emerges as a theoretical artifact rather than a universal practical rule.

4. General Contractor’s Two-Way Option Model for Purchasing Carbon Emission Rights

In Section 4.1, the models corresponding to four distinct scenarios are formulated, and subsequently, Section 4.2 presents the solution procedures for these scenarios.

4.1. The Model

The amount of carbon emission rights two-way options purchased by the general contractor in the carbon trading market is $q_0$. The execution of the two-way option by the general contractor depends on the real-time price of the carbon trading market and the carbon quota surplus. In this paper, the superscript $p$ is used to indicate the execution of a put option, and $c$ is used to indicate the execution of a call option.

When $w_b\ge t$, that is, the execution price of the two-way option is higher than the instant price, the general contractor determines the number of two-way options to be executed, the insufficient carbon emission rights are purchased at the instant price, and the surplus carbon emission rights are sold to third parties in the market. The expected profit function of the construction supply chain enterprise that the general contractor executes the put option is

$$\begin{array}{ll}{\mathit{\pi }}_{\mathbf{1}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\mathit{P}}_{\mathbf{1}}& - {\mathit{C}}_{\mathbf{1}}-{\mathit{P}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}+(\mathbf{1}-\mathit{\lambda })\left[\mathit{\phi }\mathit{s}\right({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}})-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}\\ & +{\mathit{w}}_{\mathit{b}}\mathit{min}\left(\mathit{max}\left(\mathit{s}\right.\right.\left(\mathit{k}+{\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}-{\mathit{e}}_{\mathbf{0}}\right),\mathbf{0}),{\mathit{q}}_{\mathbf{0}})\\ & +\mathit{t}\mathit{max}\left(\mathit{s}\right.\left(\mathit{k}+{\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}-{\mathit{e}}_{\mathbf{0}}\right)-{\mathit{q}}_{\mathbf{0}},\mathbf{0})\\ & -\mathit{t}\mathit{max}(\mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right),\mathbf{0}\left)\right]\end{array}$$

$$\begin{array}{ll}{\mathit{\pi }}_{\mathbf{2}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}& - {\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }\left[\mathit{\phi }\mathit{s}\right({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}})-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}\\ & +{\mathit{w}}_{\mathit{b}}\mathit{min}\left(\mathit{max}\left(\mathit{s}\right.\right.\left(\mathit{k}+{\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}-{\mathit{e}}_{\mathbf{0}}\right),\mathbf{0}),{\mathit{q}}_{\mathbf{0}})\\ & +\mathit{t}\mathit{max}\left(\mathit{s}\right.\left(\mathit{k}+{\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}-{\mathit{e}}_{\mathbf{0}}\right)-{\mathit{q}}_{\mathbf{0}},\mathbf{0})\\ & -\mathit{t}\mathit{max}(\mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right),\mathbf{0}\left)\right]\end{array}$$

$$\begin{array}{ll}{\mathit{\pi }}^{\mathit{p}}={\mathit{\pi }}_{\mathbf{1}}^{\mathit{p}}+{\mathit{\pi }}_{\mathbf{2}}^{\mathit{p}}=& {\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}})-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}\\ & +{\mathit{w}}_{\mathit{b}}\mathit{min}\left(\mathit{max}\left(\mathit{s}\right.\right.\left(\mathit{k}+{\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}-{\mathit{e}}_{\mathbf{0}}\right),\mathbf{0}),{\mathit{q}}_{\mathbf{0}})\\ & +\mathit{t}\mathit{max}\left(\mathit{s}\right.\left(\mathit{k}+{\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}-{\mathit{e}}_{\mathbf{0}}\right)-{\mathit{q}}_{\mathbf{0}},\mathbf{0})\\ & -\mathit{t}\mathit{max}(\mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right),\mathbf{0})\end{array}$$

When $w_b<t$, that is, the execution price of the two-way option is lower than the spot price, the general contractor determines the number of two-way options to be executed, the surplus carbon emission rights are sold at the spot price, and the insufficient carbon emission rights are purchased from the third party in the market. The expected profit function of the construction supply chain enterprise that the general contractor executes the call option is

$$\begin{array}{ll}{\pi }_1^c\left(e_1,q_0,\lambda \right)=P_1& - C_1-P_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2+(1-\lambda )\left[\phi s\right(e_1+e_2)-w_0{q}_0\\ & -w_b\mathit{min}\left(\mathit{max}\left(s\right.\right.\left(e_0-e_1-e_2-k\right),0),q_0)-t\mathit{max}\left(s\right.\left(e_0-e_1-e_2-k\right)-q_0,0)\\ & +t\mathit{max}(s\left(k+e_1+e_2-e_0\right),0\left)\right]\end{array}$$

$$\begin{array}{ll}{\mathit{\pi }}_{\mathbf{2}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}-{\mathit{C}}_{\mathbf{2}}& - {\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }\left[\mathit{\phi }\mathit{s}\right({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}})-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-{\mathit{w}}_{\mathit{b}}\mathit{min}\left(\mathit{max}\left(\mathit{s}\right.\right.\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right),\mathbf{0}),{\mathit{q}}_{\mathbf{0}})\\ & -\mathit{t}\mathit{max}\left(\mathit{s}\right.\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)-{\mathit{q}}_{\mathbf{0}},\mathbf{0})+\mathit{t}\mathit{max}(\mathit{s}\left(\mathit{k}+{\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}-{\mathit{e}}_{\mathbf{0}}\right),\mathbf{0})]\end{array}$$

$$\begin{array}{ll}{\mathit{\pi }}^{\mathit{c}}={\mathit{\pi }}_{\mathbf{1}}^{\mathit{c}}+{\mathit{\pi }}_{\mathbf{2}}^{\mathit{c}}=& {\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}})-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}\\ & -{\mathit{w}}_{\mathit{b}}\mathit{min}\left(\mathit{max}\left(\mathit{s}\right.\right.\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right),\mathbf{0}),{\mathit{q}}_{\mathbf{0}})-\mathit{t}\mathit{max}\left(\mathit{s}\right.\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)-{\mathit{q}}_{\mathbf{0}},\mathbf{0})\\ & +\mathit{t}\mathit{max}(\mathit{s}\left(\mathit{k}+{\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}-{\mathit{e}}_{\mathbf{0}}\right),\mathbf{0})\end{array}$$

In summary, the expected profit objective function of the general contractor, subcontractor and construction supply chain is

$${\mathit{\pi }}_{\mathbf{1}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\int }_{\mathit{m}}^{{\mathit{w}}_{\mathit{b}}}{\mathit{\pi }}_{\mathbf{1}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)\mathit{g}\left(\mathit{t}\right)\mathit{d}\mathit{t}+{\int }_{{\mathit{w}}_{\mathit{b}}}^{\mathit{\infty }}{\mathit{\pi }}_{\mathbf{1}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)\mathit{g}\left(\mathit{t}\right)\mathit{d}\mathit{t}$$

$${\mathit{\pi }}_{\mathbf{2}}\left({\mathit{e}}_{\mathbf{2}}\right)={\int }_{\mathit{m}}^{{\mathit{w}}_{\mathit{b}}}{\mathit{\pi }}_{\mathbf{2}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{2}}\right)\mathit{g}\left(\mathit{t}\right)\mathit{d}\mathit{t}+{\int }_{{\mathit{w}}_{\mathit{b}}}^{\mathit{\infty }}{\mathit{\pi }}_{\mathbf{2}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{2}}\right)\mathit{g}\left(\mathit{t}\right)\mathit{d}\mathit{t}$$

$$\mathit{\pi }={\int }_{\mathit{m}}^{{\mathit{w}}_{\mathit{b}}}{\mathit{\pi }}^{\mathit{p}}\mathit{g}\left(\mathit{t}\right)\mathit{d}\mathit{t}+{\int }_{{\mathit{w}}_{\mathit{b}}}^{\mathit{\infty }}{\mathit{\pi }}^{\mathit{c}}\mathit{g}\left(\mathit{t}\right)\mathit{d}\mathit{t}$$

Under the two-way option model, the expected profits of enterprises under four option execution conditions are obtained according to the carbon emission reductions in the general contractor and subcontractor and the fluctuation of the market carbon price (indicated by superscripts $\mathrm{I},\mathrm{I}\mathrm{I},\mathrm{I}\mathrm{I}\mathrm{I}$ and $\mathrm{I}\mathrm{V}$), which are partially executed put, fully executed put, fully executed call, and partially executed call. To better illustrate the four possible option execution scenarios under the two-way carbon option model, a diagram is provided below. Figure 1 visually explains the conditions and outcomes of each execution case—partially executed put, fully executed put, fully executed call, and partially executed call.

Then, the profits of the general contractor, subcontractor and construction supply chain under the cap-and-trade policy are expressed as follows (

Table 2. The actual meaning of option execution conditions.

Case Number	Carbon Quota Surplus or Deficit	Actual Meaning
Case 1	$0\le \mathrm{s}\left({\mathrm{e}}_1+{\mathrm{e}}_2+\mathrm{k}-{\mathrm{e}}_0\right)\le {\mathrm{q}}_0$	Low surplus: The surplus amount of carbon credits generated by emissions reductions ≤ the amount of bidirectional options purchased.
Case 2	${\mathrm{q}}_0\le \mathrm{s}\left({\mathrm{e}}_1+{\mathrm{e}}_2+\mathrm{k}-{\mathrm{e}}_0\right)$	High surplus: The surplus amount of carbon allowances generated by emissions reductions > the amount of bidirectional options purchased.
Case 3	${\mathrm{q}}_0\le \mathrm{s}\left({\mathrm{e}}_0-{\mathrm{e}}_1-{\mathrm{e}}_2-\mathrm{k}\right)$	High deficit: Carbon allowance deficit after emissions reduction still exceeds the amount of two-way options purchased.
Case 4	$0\le \mathrm{s}\left({\mathrm{e}}_0-{\mathrm{e}}_1-{\mathrm{e}}_2-\mathrm{k}\right)\le {\mathrm{q}}_0$	Low deficit: The carbon quota deficit remaining after emissions reductions is ≤ the amount of purchased two-way options.

):

(1)

Case 1: $\mathbf{0}\le \mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)\le {\mathit{q}}_{\mathbf{0}}$

If the general contractor and subcontractor achieve a surplus of carbon quotas through carbon emission reduction, and the amount does not exceed the carbon emission rights option purchase quantity of the two-way option, the general contractor will compare the exercise price of the two-way option with the immediate purchase price and choose whether to execute the two-way option.

When $w_b\ge t$, the partial bidirectional option is executed, and the expected profit function of the general contractor, subcontractor and construction supply chain is:

$${\mathit{\pi }}_{\mathbf{1}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{P}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}+\left(\mathbf{1}-\mathit{\lambda }\right)\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+{\mathit{w}}_{\mathit{b}}\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)\right]$$

$${\mathit{\pi }}_{\mathbf{2}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+{\mathit{w}}_{\mathit{b}}\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)\right]$$

$${\mathit{\pi }}^{\mathit{p}}={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+{\mathit{w}}_{\mathit{b}}\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)$$

When $w_b<t$, the two-way option is not executed, and the excess carbon emission rights are sold at the instant price. The expected profit function of the general contractor, subcontractor and construction supply chain is

$${\mathit{\pi }}_{\mathbf{1}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{P}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}+\left(\mathbf{1}-\mathit{\lambda }\right)\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+\mathit{t}\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)\right]$$

$${\mathit{\pi }}_{\mathbf{2}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+\mathit{t}\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)\right]$$

$${\mathit{\pi }}^{\mathit{c}}={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+\mathit{t}\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)$$

In summary, the expected profits of the general contractor, subcontractors and construction supply chain are as follows:

$$\begin{array}{ll}{\pi }_1^I\left(e_1,q_0,\lambda \right)=& P_1-C_1-P_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2\\ & +\left(1-\lambda \right)\left[\phi s\left(e_1+e_2\right)-w_0{q}_0+\mu s\left(e_1+e_2+k-e_0\right)+s\left(e_1+e_2+k-e_0\right){\int }_m^{w_b}G\left(t\right)dt\right]\end{array}$$

(1)

$${\pi }_2^I\left(e_2\right)=P_2-C_2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\lambda \left[\phi s\left(e_1+e_2\right)-w_0{q}_0+\mu s\left(e_1+e_2+k-e_0\right)+s\left(e_1+e_2+k-e_0\right){\int }_m^{w_b}G\left(t\right)dt\right]$$

(2)

$$\begin{array}{ll}{\pi }^I=P_1-C_1-C_2& - {\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\phi s\left(e_1+e_2\right)-w_0{q}_0+\mu s\left(e_1+e_2+k-e_0\right)\\ & +s\left(e_1+e_2+k-e_0\right){\int }_m^{w_b}G\left(t\right)dt\end{array}$$

(3)

(2)

Case 2: ${\mathit{q}}_{\mathbf{0}}\le \mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)$

If the general contractor and subcontractor achieve a carbon quota surplus through carbon emission reduction, and it is not less than the carbon emission rights option purchase quantity of the two-way option, the general contractor will compare the exercise price of the two-way option with the immediate purchase price and choose whether to execute the two-way option.

When $w_b\ge t$, all bidirectional options are executed, and the excess quotas are sold at the instant price. The expected profit function of the general contractor, subcontractor and construction supply chain is

$${\mathit{\pi }}_{\mathbf{1}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{P}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}+\left(\mathbf{1}-\mathit{\lambda }\right)\{\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+{\mathit{w}}_{\mathit{b}}{\mathit{q}}_{\mathbf{0}}+\mathit{t}\left[\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)-{\mathit{q}}_{\mathbf{0}}\right]\}$$

$${\mathit{\pi }}_{\mathbf{2}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }\{\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+{\mathit{w}}_{\mathit{b}}{\mathit{q}}_{\mathbf{0}}+\mathit{t}\left[\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)-{\mathit{q}}_{\mathbf{0}}\right]\}$$

$${\mathit{\pi }}^{\mathit{p}}={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+{\mathit{w}}_{\mathit{b}}{\mathit{q}}_{\mathbf{0}}+\mathit{t}\left[\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)-{\mathit{q}}_{\mathbf{0}}\right]$$

When $w_b<t$, the two-way option is not exercised, and the excess carbon emission rights are sold at the instant price. The expected profit function of the general contractor, subcontractor and construction supply chain is

$${\mathit{\pi }}_{\mathbf{1}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{P}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}+\left(\mathbf{1}-\mathit{\lambda }\right)\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+\mathit{t}\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)\right]$$

$${\pi }_2^c\left(e_2\right)=P_2-C_2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\lambda \left[\phi s\left(e_1+e_2\right)-w_0{q}_0+ts\left(e_1+e_2+k-e_0\right)\right]$$

$${\mathit{\pi }}^{\mathit{c}}={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}+\mathit{t}\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)$$

In summary, the expected profits of the general contractor, subcontractors and construction supply chain are as follows:

$${\pi }_1^{II}\left(e_1,q_0,\lambda \right)=P_1-C_1-P_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2+\left(1-\lambda \right)\left[\phi s\left(e_1+e_2\right)-w_0{q}_0+\mu s\left(e_1+e_2+k-e_0\right)+q_0{\int }_m^{w_b}G\left(t\right)\mathrm{d}t\right]$$

(4)

$${\pi }_2^{II}\left(e_2\right)=P_2-C_2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\lambda \left[\phi s\left(e_1+e_2\right)-w_0{q}_0+\mu s\left(e_1+e_2+k-e_0\right)+q_0{\int }_m^{w_b}G\left(t\right)\mathrm{d}t\right]$$

(5)

$${\pi }^{II}=P_1-C_1-C_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\phi s\left(e_1+e_2\right)-w_0{q}_0+\mu s\left(e_1+e_2+k-e_0\right)+q_0{\int }_m^{w_b}G\left(t\right)dt$$

(6)

(3)

Case 3: ${\mathit{q}}_{\mathbf{0}}\le \mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)$

If the general contractor and subcontractor still exceed emissions after carbon emission reduction, and the missing quota is not less than the number of carbon emission rights options purchased under the two-way option, the general contractor will compare the exercise price of the two-way option with the immediate purchase price and choose whether to execute the two-way option.

When $w_b\ge t$, the bidirectional option is not exercised, and the missing quota is purchased at the instant price. The expected profit function of the general contractor, subcontractor and construction supply chain is

$${\mathit{\pi }}_{\mathbf{1}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{P}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}+\left(\mathbf{1}-\mathit{\lambda }\right)\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}\mathit{s}({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k})\right]$$

$${\mathit{\pi }}_{\mathbf{2}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}\mathit{s}({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k})\right]$$

$${\mathit{\pi }}^{\mathit{p}}={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}\mathit{s}({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k})$$

When $w_b<t$, all bidirectional options are executed, and the remaining missing quotas are purchased at the instant price. The expected profit function of the general contractor, subcontractor and construction supply chain is

$${\mathit{\pi }}_{\mathbf{1}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{P}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}+\left(\mathbf{1}-\mathit{\lambda }\right)\{\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-{\mathit{w}}_{\mathit{b}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}[\mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)-{\mathit{q}}_{\mathbf{0}}\left]\right\}$$

$${\mathit{\pi }}_{\mathbf{2}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }\{\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-{\mathit{w}}_{\mathit{b}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}[\mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)-{\mathit{q}}_{\mathbf{0}}\left]\right\}$$

$${\mathit{\pi }}^{\mathit{c}}={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-{\mathit{w}}_{\mathit{b}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}[\mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)-{\mathit{q}}_{\mathbf{0}}]$$

In summary, the expected profits of the general contractor, subcontractors and construction supply chain are as follows:

$$\begin{array}{ll}{\pi }_1^{III}\left(e_1,q_0,\lambda \right)=& P_1-C_1-P_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2\\ & +\left(1-\lambda \right)\left\{\phi s\left(e_1+e_2\right)-w_0{q}_0-w_b{q}_0-\mu \left[s\left(e_0-e_1-e_2-k\right)-q_0\right]+q_0{\int }_m^{w_b}G\left(t\right)\mathrm{d}t\right\}\end{array}$$

(7)

$${\pi }_2^{III}\left(e_2\right)=P_2-C_2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\lambda \left\{\phi s\left(e_1+e_2\right)-w_0{q}_0-w_b{q}_0-\mu \left[s\left(e_0-e_1-e_2-k\right)-q_0\right]+q_0{\int }_m^{w_b}G\left(t\right)\mathrm{d}t\right\}$$

(8)

$$\begin{array}{ll}{\pi }^{III}=P_1-C_1& - C_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\phi s\left(e_1+e_2\right)-w_0{q}_0-w_b{q}_0-\mu [s\left(e_0-e_1-e_2-k\right)-q_0]\\ & + q_0{\int }_m^{w_b}G\left(t\right)\mathrm{d}t\end{array}$$

(9)

(4)

Case 4: $\mathbf{0}\le \mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)\le {\mathit{q}}_{\mathbf{0}}$

If the general contractor and subcontractor still exceed emissions after carbon emission reduction, and the missing quota does not exceed the number of carbon emission rights options purchased under the two-way option, the general contractor will compare the exercise price of the two-way option with the immediate purchase price and choose whether to execute the two-way option.

When $w_b\ge t$, the bidirectional option is not exercised, and the missing quota is purchased at the instant price. The expected profit function of the general contractor, subcontractor and construction supply chain is

$${\mathit{\pi }}_{\mathbf{1}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{1}},{\mathit{q}}_{\mathbf{0}},\mathit{\lambda }\right)={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{P}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}+\left(\mathbf{1}-\mathit{\lambda }\right)\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}\mathit{s}({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k})\right]$$

$${\mathit{\pi }}_{\mathbf{2}}^{\mathit{p}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }\left[\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}\mathit{s}({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k})\right]$$

$${\mathit{\pi }}^{\mathit{p}}={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-\mathit{t}\mathit{s}({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k})$$

When $w_b<t$, some bidirectional options are executed, and the expected profit function of the general contractor, subcontractor and construction supply chain is

$${\pi }_1^c\left(e_1,q_0,\lambda \right)=P_1-C_1-P_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2+\left(1-\lambda \right)[\phi s\left(e_1+e_2\right)-w_0{q}_0-w_{b}s\left(e_0-e_1-e_2-k\right)]$$

$${\mathit{\pi }}_{\mathbf{2}}^{\mathit{c}}\left({\mathit{e}}_{\mathbf{2}}\right)={\mathit{P}}_{\mathbf{2}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\lambda }[\mathit{\phi }\mathit{s}\left({\mathit{e}}_1+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-{\mathit{w}}_{\mathit{b}}\mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)]$$

$${\mathit{\pi }}^{\mathit{c}}={\mathit{P}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{1}}-{\mathit{C}}_{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}^{\mathbf{2}}-{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}{\mathit{\epsilon }}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}^{\mathbf{2}}+\mathit{\phi }\mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}\right)-{\mathit{w}}_{\mathbf{0}}{\mathit{q}}_{\mathbf{0}}-{\mathit{w}}_{\mathit{b}}\mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)$$

In summary, the expected profits of the general contractor, subcontractors and construction supply chain are as follows:

$$\begin{array}{ll}{\pi }_1^{IV}\left(e_1,q_0,\lambda \right)=& P_1-C_1-P_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2+\left(1-\lambda \right)[\phi s\left(e_1+e_2\right)-w_0{q}_0-w_{b}s\left(e_0-e_1-e_2-k\right)\\ & +s\left(e_0-e_1-e_2-k\right){\int }_m^{w_b}G\left(t\right)dt]\end{array}$$

(10)

$${\pi }_2^{IV}\left(e_2\right)=P_2-C_2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\lambda \left[\phi s\left(e_1+e_2\right)-w_0{q}_0-w_{b}s\left(e_0-e_1-e_2-k\right)+s\left(e_0-e_1-e_{2}k\right){\int }_m^{w_b}G\left(t\right)dt\right]$$

(11)

$${\pi }^{IV}=P_1-C_1-C_2-{\displaystyle \frac{1}{2}}{\epsilon }_1{e}_1^2-{\displaystyle \frac{1}{2}}{\epsilon }_2{e}_2^2+\phi s\left(e_1+e_2\right)-w_0{q}_0-w_{b}s\left(e_0-e_1-e_2-k\right)+s\left(e_0-e_1-e_2-k\right){\int }_m^{w_b}G\left(t\right)\mathrm{d}t$$

(12)

4.2. The Optimal Decisions

This paper constructs a construction supply chain game model consisting of a general contractor and subcontractors. The game sequence is as follows: the general contractor first decides its own carbon emission reduction, carbon option ordering, and profit distribution ratio. Subcontractors, as followers, decide their own carbon emission reduction after the general contractor decides. The fixed profit of the general contractor stems from its dominant position in the Stackelberg game, where strategic risk transfer ($\mathsf{\lambda }=1$) is an option to avoid uncertainties in the carbon market. Within the scope of normal contract agreements, general contractors usually use fixed-price contracts to separate core profits from cost fluctuations. This mechanism does not deny the role of general contractors in risk management. The introduction of two-way options is a management measure adopted by general contractors to enable risk-bearing subcontractors to act boldly and make stable decisions. Options serve the overall risk coordination of the supply chain rather than directly altering the fixed income of the general contractor. This extreme outcome highlights the boundary of the model rather than real practice. Therefore, the result that λ converges to 1 in all four cases should be interpreted as a boundary benchmark under complete risk transfer, not as a universal rule. In practice, construction supply chains often operate under long-term contracts, stepwise incentives, or partial cost-sharing mechanisms, which can naturally lead to equilibria where (0 < $\mathsf{\lambda }$ < 1). In real projects, contractors may still undertake emission reductions due to reputational concerns, strategic positioning, or the prospect of long-term cooperation. Such drivers are beyond the scope of our current short-term, decentralized setting but deserve further study.

(1)

Case 1: $\mathbf{0}\le \mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)\le {\mathit{q}}_{\mathbf{0}}$

Proposition 1.

Under the cap-and-trade policy, the optimal carbon emission reduction in subcontractors in the construction supply chain is $e_2^{\ast }={\displaystyle \frac{\lambda sA}{{\epsilon }_2}}$, and the optimal carbon emission reduction, carbon option ordering quantity, and profit distribution ratio of the general contractor are $e_1^{\ast }=0$, $q_0^{\ast }={\displaystyle \frac{s}{w_0}}\left(k-e_0\right)\left(\mu +{\int }_m^{w_b}G\left(t\right)dt\right)+{\displaystyle \frac{s^2{A}^2}{{\epsilon }_2{w}_0}}, {\lambda }^{\ast }=1$, where $A=\phi +\mu +{\int }_m^{w_b}G\left(t\right)dt$.

Proposition 1 implies that if the general contractor and subcontractor achieve a surplus of carbon quotas through carbon emission reduction, and the amount does not exceed the carbon emission rights option purchase quantity of the two-way option, the general contractor will compare the exercise price of the two-way option with the immediate purchase price and choose whether to execute the two-way option.

The proof is provided in Appendix A.

(2)

Case 2: ${\mathit{q}}_{\mathbf{0}}\le \mathit{s}\left({\mathit{e}}_{\mathbf{1}}+{\mathit{e}}_{\mathbf{2}}+\mathit{k}-{\mathit{e}}_{\mathbf{0}}\right)$

Proposition 2.

Under the cap-and-trade policy, the optimal carbon emission reduction in subcontractors in the construction supply chain is $e_2^{\ast }={\displaystyle \frac{\lambda s(\phi +\mu )}{{\epsilon }_2}}$, and the optimal carbon emission reduction, carbon option ordering quantity, and profit distribution ratio of the general contractor are $e_1^{\ast }=0$, $q_0^{\ast }={\displaystyle \frac{s^2{(\phi +\mu )}^2}{{\epsilon }_2(w_0-{\int }_m^{w_b}G\left(t\right)dt)}}+{\displaystyle \frac{\mu s(k-e_0)}{w_0-{\int }_m^{w_b}G\left(t\right)dt}} \mathrm{and} {\lambda }^{\ast }=1$ respectively.

Proposition 2 demonstrates that if the general contractor and subcontractor achieve a carbon quota surplus through carbon emission reduction, it is not less than the carbon emission rights option purchase quantity of the two-way option. At this stage, the general contractor has no incentive to undertake any emission reduction efforts. That is, the optimal emission reduction level for the general contractor is zero, and the entirety of the emission reduction rewards is allocated to the subcontractors. In other words, the profit-sharing ratio given by the general contractor to the subcontractor reaches 1.

The proof is provided in Appendix A.

(3)

Case 3: ${\mathit{q}}_{\mathbf{0}}\le \mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)$

Proposition 3.

Let $B={\int }_m^{w_b}G\left(t\right)dt+\mu -w_0-w_b$, under the cap-and-trade policy, the optimal carbon emission reduction in subcontractors in the construction supply chain is $e_2^{\ast }={\displaystyle \frac{\lambda s(\phi +\mu )}{{\epsilon }_2}}$, and the optimal carbon emission reduction, carbon option ordering quantity and profit distribution ratio of the general contractor are $e_1^{\ast }=0$, $q_0^{\ast }={\displaystyle \frac{\mu s(e_0-k)}{B}}-{\displaystyle \frac{s^2{(\phi +\mu )}^2}{{\epsilon }_{2}B}} and {\lambda }^{\ast }=1$.

Proposition 3 implies that if the general contractor and subcontractor still exceed emissions after carbon emission reduction, and the missing quota is not less than the number of carbon emission rights options purchased under the two-way option. At this stage, the general contractor has no incentive to undertake any emission reduction efforts. That is, the optimal emission reduction level for the general contractor is zero, and the entirety of the emission reduction rewards is allocated to the subcontractors. In other words, the profit-sharing ratio given by the general contractor to the subcontractor reaches 1.

The proof is provided in Appendix A.

(4)

Case 4: $\mathbf{0}\le \mathit{s}\left({\mathit{e}}_{\mathbf{0}}-{\mathit{e}}_{\mathbf{1}}-{\mathit{e}}_{\mathbf{2}}-\mathit{k}\right)\le {\mathit{q}}_{\mathbf{0}}$

Proposition 4.

Let $\mathrm{D}=\mathsf{\phi }+{\mathrm{w}}_{\mathrm{b}}-{\int }_{\mathrm{m}}^{{\mathrm{w}}_{\mathrm{b}}}\mathrm{G}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}$, under the cap-and-trade policy, the optimal carbon emission reduction in subcontractors in the construction supply chain is ${\mathrm{e}}_2^{\ast }={\displaystyle \frac{\mathsf{\lambda }\mathrm{s}\mathrm{D}}{{\mathsf{\epsilon }}_2}}$, and the optimal carbon emission reduction, carbon option ordering quantity and profit distribution ratio of the general contractor are ${\mathrm{e}}_1^{\ast }=0$, ${\mathrm{q}}_0^{\ast }={\displaystyle \frac{\mathrm{s}}{{\mathrm{w}}_0}}\left({\mathrm{e}}_0-\mathrm{k}\right)\left({\int }_{\mathrm{m}}^{{\mathrm{w}}_{\mathrm{b}}}\mathrm{G}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}-{\mathrm{w}}_{\mathrm{b}}\right)+{\displaystyle \frac{{\mathrm{s}}^2{\mathrm{D}}^2}{{\mathsf{\epsilon }}_2{\mathrm{w}}_0}} and {\mathsf{\lambda }}^{\ast }=1$.

Proposition 4 indicates that if the general contractor and subcontractor still exceed emissions after carbon emission reduction, the missing quota does not exceed the number of carbon emission rights options purchased under the two-way option. At this stage, the general contractor has no incentive to undertake any emission reduction efforts. That is, the optimal emission reduction level for the general contractor is zero, and the entirety of the emission reduction rewards is allocated to the subcontractors. In other words, the profit-sharing ratio given by the general contractor to the subcontractor reaches 1.

The proof is provided in Appendix A.

Under the incentive mechanism of government carbon emission quotas and owner carbon emission rewards, general contractors and subcontractors in the construction supply chain deal with carbon trading risks through carbon options. In the case of decentralized decision-making, since the general contractor is in a dominant position, to maximize its own interests, the general contractor will refuse to make emission reduction efforts and allocate all carbon emission reduction rewards to subcontractors, that is, the reward ratio allocated to subcontractors is 1. In this case, the emission reduction investment is entirely borne by the subcontractor, who must bear all the costs and carbon trading risks of achieving the emission reduction target alone. This risk-only allocation rarely achieves emission reduction targets in practice: subcontractors lack motivation when bearing full risks, leading to behaviors that deviate from the project’s overall goals. In addition, this decision-making method undermines the cooperative relationship between the general contractor and the subcontractor and is not conducive to the formation of long-term and stable synergy between the two parties. From the perspective of corporate management, under the carbon quota policy and option trading mechanism, enterprises should make full use of the advantages of supply chain partners based on dynamic profit distribution and responsibility-sharing mechanisms to achieve efficient and sustainable carbon emission targets, thereby enhancing the competitiveness of enterprises in the low-carbon economy. Based on this, it is recommended that the general contractor balance the interests of both parties in the distribution of carbon emission reduction rewards and enhance the enthusiasm of subcontractors through a reasonable incentive mechanism. Establish a common responsibility and interest-sharing mechanism to avoid the incentive failure caused by unilateral decision-making and promote collaborative emission reduction within the supply chain. By encouraging subcontractors to undertake emission reduction tasks, the general contractor can optimize the overall carbon trading revenue and improve the overall carbon emission reduction efficiency of the project. However, if this strategy is only adopted for a single cooperation, it may cause subcontractors to have negative expectations for long-term cooperation. At the same time, considering both short-term profits and long-term synergy benefits in the distribution mechanism, and adopting a reasonable distribution ratio, performance rewards or step-by-step incentive mechanism can ensure the achievement of carbon emission reduction goals and achieve a sustainable win-win situation in multiple cooperation.

Proposition 5.

Substituting the optimal decision of the general contractor and subcontractor in the above four cases into the corresponding expected profit model, we can obtain that the maximum profit of the general contractor is the same in the four cases, that is, ${\mathsf{\pi }}_1^{\ast }={\mathrm{P}}_1-{\mathrm{C}}_1-{\mathrm{P}}_2$. The maximum profit of the subcontractor in case 1 is ${\mathsf{\pi }}_2^{\mathrm{I}}={\mathrm{P}}_2-{\mathrm{C}}_2-{\displaystyle \frac{{\mathrm{s}}^2{(\mathsf{\phi }+\mathsf{\mu }+{\int }_{\mathrm{m}}^{{\mathrm{w}}_{\mathrm{b}}}\mathrm{G}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t})}^2}{{2\mathsf{\epsilon }}_2}}$, the maximum profit of the subcontractor in cases 2 and 3 is ${\mathsf{\pi }}_2^{\mathrm{I}\mathrm{I}}={\mathsf{\pi }}_2^{\mathrm{I}\mathrm{I}\mathrm{I}}={\mathrm{P}}_2-{\mathrm{C}}_2-{\displaystyle \frac{{\mathrm{s}}^2{(\mathsf{\phi }+\mathsf{\mu })}^2}{{2\mathsf{\epsilon }}_2}}$, and the maximum profit of the subcontractor in case 4 is ${\mathsf{\pi }}_2^{\mathrm{I}\mathrm{V}}={\mathrm{P}}_2-{\mathrm{C}}_2-{\displaystyle \frac{{\mathrm{s}}^2{(\mathsf{\phi }+{\mathrm{w}}_{\mathrm{b}}-{\int }_{\mathrm{m}}^{{\mathrm{w}}_{\mathrm{b}}}\mathrm{G}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t})}^2}{{2\mathsf{\epsilon }}_2}}$. If the maximum profit of the subcontractor in the four cases is ${\mathsf{\pi }}_2^{\ast }$, then the profit of building a supply chain is ${\mathsf{\pi }}^{\ast }={\mathsf{\pi }}_1^{\ast }+{\mathsf{\pi }}_2^{\ast }$.

5. Sensitivity Analysis

In this section, a sensitivity analysis is carried out regarding the relationship between the emission reduction cost coefficient and carbon quotas, along with the carbon emission reduction per unit of building area. This analysis is conducted under both decentralized decision-making scenarios.

Proposition 6.

(1) case 1: ${\displaystyle \frac{\mathbf{\partial }{\mathit{q}}_{\mathbf{0}}^{\mathbf{\ast }}}{\mathbf{\partial }{\mathit{\epsilon }}_{\mathbf{2}}}}$ < 0, ${\displaystyle \frac{\mathbf{\partial }{\mathit{e}}_{\mathbf{2}}^{\mathbf{\ast }}}{\mathbf{\partial }{\mathit{\epsilon }}_{\mathbf{2}}}}$ < 0. (2) case 2: ${\displaystyle \frac{\mathbf{\partial }{\mathit{q}}_{\mathbf{0}}^{\mathbf{\ast }}}{\mathbf{\partial }{\mathit{\epsilon }}_{\mathbf{2}}}}$ < 0, ${\displaystyle \frac{\mathbf{\partial }{\mathit{e}}_{\mathbf{2}}^{\mathbf{\ast }}}{\mathbf{\partial }{\mathit{\epsilon }}_{\mathbf{2}}}}$ < 0. (3) case 3: ${\displaystyle \frac{\mathbf{\partial }{\mathit{q}}_{\mathbf{0}}^{\mathbf{\ast }}}{\mathbf{\partial }{\mathit{\epsilon }}_{\mathbf{2}}}}$ > 0, ${\displaystyle \frac{\mathbf{\partial }{\mathit{e}}_{\mathbf{2}}^{\mathbf{\ast }}}{\mathbf{\partial }{\mathit{\epsilon }}_{\mathbf{2}}}}$ < 0. (4) case 4: ${\displaystyle \frac{\mathbf{\partial }{\mathit{q}}_{\mathbf{0}}^{\mathbf{\ast }}}{\mathbf{\partial }{\mathit{\epsilon }}_{\mathbf{2}}}}$ < 0, ${\displaystyle \frac{\mathbf{\partial }{\mathit{e}}_{\mathbf{2}}^{\mathbf{\ast }}}{\mathbf{\partial }{\mathit{\epsilon }}_{\mathbf{2}}}}$ < 0.

The results can be easily obtained, so the proof is omitted.

In the decentralized decision-making cases 1, 2 and 4 the subcontractor’s emission reduction cost coefficient is inversely proportional to the number of carbon quotas purchased by the carbon option contract and the subcontractor’s carbon emission reduction per unit building area. However, in case 3 of decentralized decision-making, the subcontractor’s emission reduction cost coefficient is inversely proportional to the carbon emission reduction per unit building area and is directly proportional to the number of carbon quotas purchased by the subcontractor’s carbon option contract.

6. Numerical Analysis

In this section, we perform some numerical analysis to determine the impact of different parameter changes on supply chain decisions and profits under four scenarios. The parameters in this example are set as following: ${\mathrm{P}}_1=\mathrm{3,200,000}$, ${\mathrm{P}}_2=\mathrm{1,200,000},\mathrm{s}=2700 {\mathrm{m}}^2$, ${\mathrm{C}}_1=\mathrm{1,820,000}$, ${\mathrm{C}}_2$ = 1,080,000, The total construction area and contract prices are calibrated to the average scale of mid-sized commercial building projects in Sichuan Province, China. For fixed costs, labor, materials and management costs account for 50–60% of the total contract value. For the area, we referred to the scale of common small and medium-sized residential or public buildings. ${\mathsf{\epsilon }}_1$ = 90,000, ${\mathsf{\epsilon }}_2$ = 95,000, based on the relationship between the investment cost and the reduction volume of building emission reduction technologies, we make assumptions about ${\mathsf{\epsilon }}_1$ and ${\mathsf{\epsilon }}_2$, where ${\mathsf{\epsilon }}_2$ > ${\mathsf{\epsilon }}_1$ reflects the higher costs that subcontractors may face in specific emission reduction technologies. $\mathrm{k}=0.78 \mathrm{t}\mathrm{C}{\mathrm{O}}_2/{\mathrm{m}}^2$, which is the value obtained by weighted averaging the values of residential buildings and public buildings. e₀ = 1.5 $\mathrm{t}\mathrm{C}{\mathrm{O}}_2/{\mathrm{m}}^2$, referring to the carbon emission level per unit area of general building construction. ${\mathrm{w}}_{\mathrm{b}}=60$, ${\mathrm{w}}_0=10 \mathrm{y}\mathrm{u}\mathrm{a}\mathrm{n}/\mathrm{t}\mathrm{C}{\mathrm{O}}_2$, $\mathsf{\phi }=14 \mathrm{y}\mathrm{u}\mathrm{a}\mathrm{n}/\mathrm{t}\mathrm{C}{\mathrm{O}}_2$. ${\mathrm{w}}_{\mathrm{b}}$ refers to the fluctuation range of the implementation price of the national carbon emission quota in 2025. The incentive factor φ is determined based on the green building carbon reduction subsidy policy of Chengdu City, Sichuan Province, China. In addition, the instant purchase price of carbon emission rights follows a uniform distribution on $\mathrm{t}$ (20, 80), then ${\mathsf{\mu }}_{\mathrm{t}}=50 \mathrm{y}\mathrm{u}\mathrm{a}\mathrm{n}/\mathrm{t}\mathrm{C}{\mathrm{O}}_2$.

6.1. The Impact of the Introduction of Two-Way Options on the Decision-Making and Profits of Supply Chain Enterprises

To verify the theoretical findings and illustrate the impact of key parameters on supply chain decisions and profits, we conducted numerical simulations using a set of representative parameters based on the models established in Section 3 and Section 4. These parameters have been calibrated to reflect the average characteristics of medium-sized commercial building projects in Sichuan Province, China, and combined with industry benchmarks that take into account cost, emission factors, and policy factors such as carbon prices and subsidies. The specific parameter values used in this analysis are detailed in

Table 3. Optimal carbon option order quantity and expected profit under four scenarios.

Variable	Case 1	Case 2	Case 3	Case 4
$e_1$	0	0	0	0
$q_0$	2900	2500	1300	1800
$\lambda$	1	1	1	1
$e_2$	0.282	0.182	0.182	0.138
${\pi }_1$	300,000	300,000	300,000	300,000
${\pi }_2$	82,166.857	104,284.295	104,284.295	110,912.651
$\pi$	382,166.857	404,284.295	404,284.295	410,912.651

below.

6.1.1. The Impact of Initial Carbon Emissions per Unit Building Area of the Project on Supply Chain Decisions and Profits

As shown in Figure 2, after introducing carbon options, the construction supply chain’s option order volume first increases and then decreases with rising initial carbon emissions per unit area. At low initial emission levels, subcontractors can control abatement costs through their own efforts, reducing the demand for carbon options. In contrast, the purchase of carbon options at this stage requires additional payment, so the subcontractor’s demand for carbon options is low, which in turn affects the general contractor’s decision on the overall carbon option ordering volume and reduces it. As the initial carbon emissions continue to increase, the difficulty and cost of the subcontractor’s own emission reduction will rise sharply.

At this time, purchasing carbon options can reduce the high-cost risk faced by excessive carbon emissions to a certain extent. Therefore, the subcontractor will feedback to the general contractor on the demand for increasing the carbon option ordering volume, prompting the general contractor to adjust its strategy. At the same time, since the emission reduction investment is entirely borne by the subcontractor, the general contractor does not make emission reduction efforts and will not receive emission reduction rewards and emission reduction benefits or bear emission reduction costs, so it will not affect its expected profit. It can be seen from this that after the introduction of carbon options, the expected profit of the enterprise can be unaffected by emission reduction technology. As initial carbon emissions continue to rise, the challenges and costs associated with subcontractors’ own emission reduction efforts are expected to increase significantly. In this context, purchasing carbon options can mitigate the financial risks linked to excessive carbon emissions to some extent. On the one hand, subcontractors will communicate their demand for an increased number of carbon option orders back to the general contractor, prompting a strategic adjustment on the part of the general contractor. On the other hand, under a decentralized decision-making model, the cooperative relationship between the general contractor and subcontractor may be adversely affected, leading to a decline in subcontractors’ motivation for emission reduction. As initial carbon emissions escalate, it becomes imperative for the general contractor to increase their order volume of carbon options to incentivize subcontractors towards greater emission reduction efforts. However, since all investment in emission reductions is solely borne by subcontractors—while general contractors do not engage in any emission reduction initiative, they will neither receive rewards nor benefits from such reductions nor incur related costs. Consequently, this lack of involvement does not impact on their anticipated profits. This analysis indicates that following the introduction of carbon options, an enterprise’s expected profit remains unaffected by advancements in emission reduction technology.

6.1.2. The Impact of Carbon Emission Benchmark per Unit Building Area of Engineering Projects (Advanced Carbon Emission Level) on Supply Chain Decisions and Profits

As illustrated in Figure 3, following the introduction of carbon options by construction supply chain enterprises, an increase in Carbon emission benchmark per unit building area of engineering projects (advanced carbon emission level) initially leads to a rise in the orders for carbon options, which is subsequently followed by a decline. When the carbon emission baseline is low, subcontractors can fulfill carbon emission requirements through their own reduction measures and are therefore less inclined to actively seek an increase in carbon option orders. However, as the carbon emission baseline rises, it becomes increasingly challenging for subcontractors to ensure adequate carbon emission quotas solely through their own efforts. Consequently, they may seek to augment their orders for carbon options as a means of compensating for potential quota shortages. When the carbon emission baseline escalates to an exceedingly high level, market fluctuations in carbon prices can result in significant increases in the costs associated with purchasing carbon options or lead to supply shortages. This scenario renders reliance on these instruments imprudent due to elevated purchase costs that may exceed demand capabilities. At this juncture, it would be unwise for subcontractors to depend excessively on carbon options as a strategy for managing price volatility risks, such dependence could ultimately diminish demand for these options and consequently reduce overall order volumes.

At the same time, since only subcontractors are engaged in emission reduction efforts, the general contractor occupies a dominant position within this model and opts not to pursue any emission reduction initiatives. The decision-making of the general contractor remains unaffected by variations in the Carbon emission benchmark per unit building area of engineering projects (advanced carbon emission level). Consequently, it receives neither rewards nor benefits associated with emission reductions nor incur costs related to these efforts. Fluctuations in Carbon emission benchmark per unit building area of engineering projects (advanced carbon emission level) do not influence the expected profit of the general contractor. It is evident that following the introduction of carbon options, the anticipated profits for enterprises remain uninfluenced by the carbon emission limits established by government authorities.

6.1.3. The Impact of the Option Price per Unit of Carbon Quota on a Carbon Option Contract on Supply Chain Decisions and Profits

As illustrated in Figure 4, the relationship between option price and carbon option ordering volume exhibits a distinct pattern: initially, as the option price increases, the ordering volume decreases. However, it begins to rise beyond a certain threshold. When the option price is low, an increase in this price results in a reduction in order volume. This phenomenon occurs because companies tend to decrease their purchases to mitigate costs when faced with higher option expenses. Conversely, as the option price continues to escalate and reaches a specific level, market participants may anticipate that the market value of carbon emission rights will also rise. In such scenarios, purchasing options becomes a strategic approach for hedging against potential future increases in carbon prices. Consequently, ordering volume starts to increase.

Furthermore, factors such as the general contractor’s decisions regarding carbon emission reductions, the distribution ratio of profits from carbon trading, subcontractors’ decisions on emissions reductions, and expected profits for subcontractors remain unaffected by these dynamics. This observation underscores that reasonable pricing of carbon options plays a crucial role in influencing carbon option ordering volumes.

7. Conclusions and Future Research

7.1. Conclusions

The cap-and-trade policy has commoditized carbon quotas, spurring the development of carbon financial instruments. Carbon options have become a key tool for hedging carbon price uncertainty due to their flexibility. Based on the Stackelberg game, this paper constructs a supply chain decision model consisting of general contractors and subcontractors. Among them, the general contractor first signs a general contract with the owner and then subcontracts the part that it cannot complete to a professional subcontractor and signs a subcontract contract with it. The general contractor and the subcontractor obtain the trading rights of carbon quotas by signing a carbon option contract. This paper first establishes a decentralized decision-making model, analyzes the optimal decisions of both parties in the supply chain, and then analyzes the role of carbon option contracts in supply chain coordination. Based on the theoretical derivation, sensitivity analysis, and numerical simulations in Section 4, Section 5 and Section 6, the main conclusions are as follows:

(1)

Under decentralized decision-making, enterprises face complex market environments and decision-making choices. Carbon options can give enterprises flexibility in the face of price fluctuations in carbon emission rights. Enterprises can choose favorable times to trade carbon emission rights. Through analysis of the model, it is found that the profits of enterprises under the execution of carbon options are higher than those under the non-execution of carbon options. As shown in Table 2 and Figure 2, Figure 3 and Figure 4, the expected profits with carbon options are consistently higher than without them. The option mechanism enables enterprises to better hedge carbon price risks and choose optimal trading timing.

Therefore, it is recommended that policymakers promote the popularization and application of carbon option tools in the construction industry, and provide fiscal and tax incentives for enterprises that adopt carbon options for emission reduction risk management, such as reducing or exempting option transaction fees, incorporating option costs into the scope of emission reduction subsidy accounting, etc., to enhance enterprises’ enthusiasm for participation.

(2)

Due to the complexity and uncertainty of the market, subcontractors cannot determine the optimal price and exercise price of carbon options at the same time. However, when either price is determined, subcontractors can maximize their expected profits by deciding on the other price. Therefore, whether it is the exercise price of carbon options or the option price, subcontractors can adjust another corresponding uncertain pricing strategy based on their circumstances and market forecasts to adapt to the market and achieve profit optimization. Subcontractors can achieve a balance between risk and return through option transactions. They can use the profit share allocated by the general contractor to participate in option investments, prioritizing the selection of option products that align with their own emission reduction capabilities, and sharing both emission reduction costs and option benefits with the general contractor. For example, when subcontractors’ actual emission reductions exceed the agreed targets, they can share the profits obtained by the general contractor through option transactions in proportion, which enhances the willingness for long-term cooperation.

(3)

Under certain conditions, carbon option contracts can achieve supply chain coordination, which can prompt members of the construction supply chain to form synergy in carbon emission management and jointly deal with carbon emission constraints and market risks. However, when the supply chain is coordinated, general contractors pay more attention to coping with carbon emission challenges through collaboration with subcontractors rather than relying on the external carbon option market. Therefore, the decision-making of general contractors shows a specific pattern: general contractors often choose not to exercise carbon options.

(4)

The increase in the carbon emission cap and the initial carbon emissions per unit building area are key factors affecting the construction supply chain. First, the increase in the carbon emission cap means that companies have more carbon emission quotas, and companies can maintain normal production operations without increasing too much emission reduction costs, thereby reducing the cost pressure caused by emission reduction investment and increasing the expected profit of the supply chain. Secondly, the reduction in the initial carbon emissions per unit building area means that companies gradually have higher carbon emission levels, indicating that improvements in production or construction technology by companies will help reduce the total amount of carbon emissions and reduce emission reduction costs, thereby increasing the expected profit level of the supply chain. Therefore, considering the impact of carbon emission benchmarks on decision-making in the model, it is suggested that the government refine the carbon emission benchmark values for different building types, such as residential buildings and public buildings. This will provide an accurate basis for setting the strike price of carbon options and accounting for quotas and avoid option trading disputes caused by ambiguous benchmarks.

(5)

Under the carbon option contract, the general contractor tends to purchase carbon emission rights in the form of options because it provides a risk-controlled purchase method for the general contractor, which can lock in the transaction price of carbon emission rights in the future. In addition, the general contractor will no longer make up for the insufficient carbon emission rights by immediate purchase. The missing carbon quota will be achieved through its own emission reduction, which reflects the general contractor’s emphasis on long-term development. By improving its own emission reduction technology and carbon emission reduction efficiency, it can reduce dependence on external carbon emission rights, enhance market competitiveness and help promote emission reduction actions throughout the supply chain, thereby achieving low-carbon goals. Given the hierarchical characteristics of the construction supply chain, it is proposed to design a “stepwise incentive policy”. To address insufficient emission reduction motivation among small and medium-sized subcontractors, a stepwise incentive policy tailored to the construction supply chain’s hierarchical structure is proposed, with core mechanisms and implementation considerations as follows: Initially, general contractors adopting two-way carbon option cooperation with subcontractors could qualify for a 10–15% increase in initial carbon quota allocation. In the medium term, subcontractors’ emission reduction performance and option execution records should be linked to low-carbon credits, which can be converted into tax preferences or government project priorities. In the long run, mature option-based models can be promoted as industry guidelines via construction associations. We can adopt blockchain technology to record option trading and emission data, alleviate information asymmetry, and establish a joint working group involving the environmental, construction and financial sectors to align carbon quota rules with project approval procedures. At the same time, we can provide government guarantees for the option premiums of subcontractors to reduce the financial burden in the early stage.

These results offer theoretical insights and practical guidance for carbon emission management in the construction industry. For general contractors, while there is a lack of financial incentive for autonomous emissions reduction under short-term collaborative decentralized decision-making, long-term collaboration can enhance emissions reduction capabilities and competitiveness through investment in green technologies and materials. For subcontractors, establishing fair R&D cost-sharing and revenue allocation mechanisms is key to fostering long-term cooperation. For policymakers, enriching carbon financial instruments and refining carbon trading regulations can improve market liquidity and reduce implementation barriers.

7.2. Limitations and Future Work

This study has several limitations that should be acknowledged. First, the assumptions of perfect information and a single-shot interaction simplify the analysis but also amplify opportunistic behavior by the general contractor. Supply chains often face information asymmetry and engage in long-term repeated collaborations. Relaxing these assumptions—by introducing asymmetric information, repeated games, or reputation effects—would likely produce more balanced outcomes and stronger incentives for collaboration. Second, while the profit-sharing ratio ($\lambda$) is modeled as a decision variable, in our baseline setting it consistently converges to one. This extreme result is a consequence of the complete risk-transfer assumption and should be interpreted as a boundary benchmark rather than a universal prediction. In extended scenarios involving partial cost-sharing, stepwise incentive contracts, or long-term partnerships, equilibria with 0 < $\lambda$ < 1 are expected to emerge, offering a more realistic representation of contracting practices. Third, the parameter values employed in the numerical analysis are illustrative in nature. They are designed to demonstrate the internal mechanism of two-way carbon options under varying conditions rather than to provide precise quantitative guidance. Finally, while we recommend the use of stepwise incentive policies to promote cooperative emission reduction, we also acknowledge the challenges of implementation in practice. Administrative complexity, enforcement costs, and the need for transparent performance measurement may hinder their adoption.

Further research could consider the carbon trading and option selection game of the construction supply chain under the stochastic variation in the engineering construction area. In addition, the participation of project owners can be incorporated to expand the existing model, since owner incentives and performance-based rewards may significantly influence contractors’ behavior. This article only considers the construction supply chain game under carbon option contracts; future work could also investigate the case of immediate procurement and conduct a comparative analysis to highlight the relative advantages and limitations of different trading mechanisms. Moreover, future work can relax the assumption of complete risk transfer ($\lambda$ = 1) and explore scenarios where general contractors partially share emission reduction costs or bear residual risks, thereby capturing how their profits respond to fluctuations in carbon parameters in more complex real-world environments. Finally, future extensions should introduce more realistic constraints by combining project-specific data. Through parameter calibration and case-based validation, the practical adaptability of the model can be gradually improved, thereby bridging the gap between theoretical modeling and actual construction practice.