Optimal Pricing and Carbon Emission Abatement Allocation Decisions in Supply Chains With Option Contract

With the continuous increase of carbon emissions and the deterioration of environment, governments have passed various laws and regulations to force enterprises to take carbon emission abatement measures. As one of the regulations, carbon tax plays an important and positive role in environmental protection. For enterprises, the allocation of carbon emission abatement is a common means to promote emission reduction. In this paper, under the uncertain market demand, the option contract is adopted to study the retailer’s optimal retail price and order quantity, as well as the manufacturer’s optimal ratio of total carbon emission abatement and production quantity under the carbon tax. In addition, We analyze the optimal decisions and expected profits of supply chain members with respect to the carbon tax, total carbon emission abatement and option prices by analytical and numerical study, and find that (1) when the carbon tax increases, the optimal ratio of total carbon emission abatement and the optimal retail price increase, while the optimal order and production quantity decrease, which is contrary to the situation of the total carbon emission abatement; (2) when the option prices increase, the optimal retail price increases while the ratio of total carbon emission abatement and the optimal order and production quantity decrease; (3) the expected profits of the manufacturer and retailer show the same trend with the increase of carbon tax and total carbon emission abatement, while have the opposite trend with the increase of option prices.


I. INTRODUCTION
Carbon emissions have led to climate change, with serious consequences for human beings and their environment. In response to climate change issues, in addition to government subsidy [1], many countries have passed various regulations to curb carbon emissions, such as emission taxes, cap-and-trade policies, and deposit-refund schemes [2]. Among these mechanisms, carbon taxes are a means of price adjustment to guide the behavior of enterprises [3]. In this respect, enterprises pay a fixed tax rate on each unit of their carbon emissions [4]. For example, British Columbia in Canada introduced a carbon tax in 2008. The tax is currently The associate editor coordinating the review of this manuscript and approving it for publication was Hao Luo . CA$30/t CO2 and is applicable to almost all fossil fuel combustion in the province [5]. Since 2012, Australia has imposed a carbon tax on industrial enterprises at a rate of AU$23/t CO2 [3]. Intergovernmental Panel on Climate Change (IPCC) suggested that severe climate change can be prevented by imposing a carbon tax at $80 per metric ton imposed on large carbon emitters of CO 2 [6]. Carbon tax is proven to be an effective policy tool that provides financial incentives for enterprises to take measures to increase resource utilization and reduce emissions. At the same time, it has little negative effect on economic growth, which is strongly advocated by experts and international organizations [7]. Faced with the power of carbon tax regulation, enterprisers have the motivation to control and reduce carbon emissions through incentives and innovation [8]. For example, Hewlett-Packard (HP) reported that its toxic emissions into the air dropped from 26.1 tons to 18.3 tons in 2010 [9], and after many suppliers reduced their packaging, Wal-Mart committed itself to cut 20 million metric tons of greenhouse gas emissions by 2015 [10]. In addition, many reductions in emissions are the result of joint effort by multiple parties, for example, Tesco has proposed to reduce the carbon emissions of the products they sell by 30% by 2020, jointly with their suppliers [11]. As the total carbon emission control target can reduce the overall emission reduction cost via the market mechanism [12], it is necessary for supply chain members to make joint efforts to achieve the goal of total carbon emission abatement. Therefore, fostering supply chain cooperation has become a critical issue in carbon emission abatement.
Under a cooperative framework, researchers have proposed many contracting forms, such as revenue-sharing, buyback, quantity discount, and two-part tariff contracts, to provide guidance to supply chain members in a competitive environment and to remain committed to cooperation [13]. Since market demand usually appears to be seasonal and random, in order to avoid demand uncertainty, buyers prefer to place flexible orders to meet market demand, while suppliers prefer to receive fixed orders to determine production quantities. Option contract allows the buyer to decide the order quantities according to the realized market demand, and provides the supplier with the buyer's commitment and advance payment to guarantee the order quantities, which can ease the conflict between the buyer and the supplier when the market demand is uncertain, and it has been applied in many fields, including IT, telecommunications, semiconductors, electricity, apparels industry, agriculture, etc. ( [14]- [18]). For example, many giant companies, such as IBM, Sun Microsystems Hewlett-Packard have taken a portfolio procurement strategy with options [19].
Option mechanism can alleviate the conflict between the buyer and the supplier under the uncertain demand, however, the existing research on the option mechanism usually focuses on the coordination of supply chain and the management problems of production and procurement, but does not study how supply chain members make joint efforts to achieve carbon emission reduction target when the option contract is adopted. To fill this gap, in this paper we consider a supply chain with one retailer and one manufacturer, in which the manufacturer is the leader and the so-called allocator. In the case of uncertain demand, we adopt the option contract to cooperate the supply chain, and study the allocation strategy of total carbon emission abatement between the manufacturer and the retailer under carbon tax policy. This paper tries to answer four important questions: (i) How to allocate the carbon emission reduction target among the supply chain members? (ii) How to set the retail price and order quantity to maximize the retailer's profit? (iii) How to set the optimal production quantity to maximize the manufacturer's profit? (iv) How do carbon tax, carbon emission abatement and option prices affect the optimal strategies and profits of supply chain members?
The rest of the paper is organized as follows. In Section 2, we review the related literature. In section 3, the optimal decisions of the retailer and manufacturer are studied. Section 4 examines the impacts of carbon tax, total carbon emission abatement and option prices on the optimal decisions through numerical studies, and Section 5 contains concluding remarks. We relegate relevant proofs to the Appendix.

II. LITERATURE
In recent years, increasing attention has been paid to the research on carbon emissions. Since this paper is mainly devoted to study the optimal decisions of supply chain members under carbon tax and option contract, only the most relevant papers are reviewed. The related literature can be grouped into two categories: (i) carbon emission abatement; (ii) the decisions of supply chain members with option contract.

A. CARBON EMISSION ABATEMENT
To date, many scholars have recognized the importance of carbon emission abatement in operational management, and begun to study the effects of carbon emission policies. Carbon tax, and allocation of carbon emissions and carbon trading policy are the practiced carbon regulatory methods world widely. As one of the important policies proposed by the government, carbon tax's impact has attracted more and more attention. Andrew mentioned that carbon emission tax is considered to be more transparent and visible, and thus harder to evade or avoid compared to the emission trading scheme [20]. Oreskes noted that the carbon tax levy has been considered as one of the most common market-based approaches from the aspect of economic incentives in carbon emission regulation [21]. Therefore, based on the effectiveness and importance of carbon tax, Fahimnia et al. studied the potential impacts a carbon tax policy scheme on the financial and emission reduction performance of supply chains at the tactical planning level [22]. Yang et al. investigated the role of revenue sharing and first-mover advantage on manufacturer's carbon emission abatement effort and studied firms' profitability when the government imposes some carbon taxes on the manufacturer [23], and Yang and Chen expanded this research and analyzed the impact of revenue-sharing and cost-sharing contract on a manufacturer's carbon emission abatement efforts [10]. In addition to analyzing the impact of carbon tax on the optimal decisions of supply chain members, many scholars have recognized the effectiveness of cooperation among enterprises in reducing emissions and improving profits under the carbon emission regulations, e.g., Zu et al. focused on the difference influence of the emission control strategy and the emission reduction strategy on the supply chain performance, and found strong incentives from the manufacturer to the supplier on both emission control and emission reduction can promote two partner firms reach maximum profit and lowest emission [24]. Therefore, a large number of studies have proposed coordination contracts to effectively allocate the cost of carbon tax among members to promote the coordination of the supply chain. Xiao et al. considered a two-echelon supply chain with the government levied carbon tax on the supplier [25]. They analyzed the optimal decisions of the supplier and retailer under centralized and decentralized conditions and proposed a tax-sharing contract to coordinate the supply chain. Taking into account manufacturer's and retailer's fairness concerns, Liu et al. studied the impacts of members' fairness concerns on the abatement emission efforts and pricing decisions, and explored the coordination problem under carbon tax regulation [3]. Yi and Li studied the cooperation of a two-echelon supply chain with the cost-sharing contract on the energy saving subsidies and carbon tax emission reduction [26], and He et al. investigated a service supply chain consisting of a service provider and a service integrator to explore the optimal decisions under three types of cost sharing contracts to realize carbon emission reduction [27]. Furthermore, some scholars make decisions on carbon tax from the perspective of the government. Yenipazarli explored the influences of the carbon tax on the production and pricing decisions of the manufacturer who could remanufacture its own product, and investigated how a government should formulate its carbon tax, so as to realize an all-win outcome for the economy, the environment, and society as a whole [2]. Wang et al. also treated the carbon emission tax as an endogenous variable and examined its impact on the optimal decisions of supply chain enterprises and government by developing a three-stage Stackelberg game model with a decentralized supply chain and a two-stage Stackelberg game model with a centralized supply chain [28]. Aside from research on single supply chains, Choi assumed the existence of a domestic manufacturer and an overseas manufacturer, and investigated and compared the impacts brought by the carbon tax on two-tiered fashion supply chain systems under the wholesale pricing contract and the markdown money contract [29]. Similar to the study by Yang et al. [23], this paper aims to analyze the carbon emission abatement effort under option contract when the government imposes some carbon taxes on both the manufacturer and the retailer.
Another stream of carbon emission abatement is the allocation of carbon emissions and carbon trading policy. From a macro perspective, Chang et al. estimated the carbon emissions in the Bohai Rim Economic Circle from 2005 to 2017, and established a two-stage allocation model to allocate the carbon emission quotas in 2030 [30]. Based on the analysis of the seven carbon emission trading pilots implemented in China since 2013, Zhang et al. evaluated the impact of emission trading system on the carbon emission reduction and economic growth, and the operating efficiency of the carbon emission trading markets. They found that the implementation of the carbon trading policy can increase economic dividends and reduce emissions [31]. From the perspective of operation management, Wang et al. studied the effect of capand-trade policy on the vehicle optimization problem, and analyzed the impacts of over-emission intensity, carbon emission quota, emission price on the operational performance of the company [32]. Zhang et al. discussed the impact of two kinds of trading policies, i.e., the carbon price floor and the revenue floor on carbon emission reduction investment in coal-fired power plants [33], and in a joint effort to reduce carbon emissions, Gan et al. proposed a revenue sharing contract to find the optimal pricing strategies for both manufacturer and retailer who engaged in remanufacturing under the cap and trade policy [34]. Our paper is mainly about the allocation of carbon emission abatement target. Bagchi et al. considered the issue of emission abatement allocation among the partners to achieve the required abatement target over the supply chain [35]. Feng et al. provided improved DEA-based centralized allocation models under the assumptions of constant returnsto-scale and variable returns-to-scale, respectively, and two compensation schemes for centralized allocation plans were developed [36]. These studies are mainly done in centralized supply chains. Close to our work, Caro et al. analyzed the allocation of carbon footprint and carbon emission reduction target in a decentralized supply chain. It is assumed that there is an allocator in the supply chain who has the power to determine the carbon emission abatement target for each firm [11]. Ren et al. expanded this research and focusing on a product level rather than on the firm level to study the allocation of carbon footprint and carbon emission abatement target [37]. This paper attempts to study the optimal allocation strategy of total carbon emission abatement target among supply chain members with option contract.

B. THE DECISIONS OF SUPPLY CHAIN MEMBERS WITH OPTION CONTRACT
Our study is also related to strategic decisions with an option contract under uncertainties. Since option contracts are an efficient instrument to help hedge against the uncertainties and to reduce double marginalization in supply chains, they have attracted substantial attention in the area of supply chain management. With respect to the usage of option contract in the context of supply chain coordination, Zhao et al. emphasized the impact of option contract on supply chain coordination based on the wholesale pricing mechanism. They illustrated that option contract could coordinate the supply chain and reach Pareto Optimality. At the end of the research, according to different risk appetite and bargaining power of the members in the supply chain, the scenario of choosing option contract also been discussed [38]. Liang et al. considered a two-echelon supply chain with one supplier and one buyer, they established an option pricing model in order to improve the efficiency and performance management of the relief supply chain, and evaluated the value of supply chain members [39]. Zhao et al. developed a bidirectional option contract to coordinate a two-echelon supply chain and the optimal ordering decision for the retailer has been derived [15]. Arani et al. introduced a novel mixed revenuesharing option contract to coordinate a retailer-manufacturer supply chain [40]. With respect to the impact of option contract on the context of optimal strategic decisions, Burnetas and Ritchken analyzed the impact of option contract when the demand curve is downward sloping, and showed that when considering the option contract, the wholesale price would increase, while the retail price would decrease [41]. Hu et al. considered a decentralized supply chain where both the production and demand are stochastic [42]. They studied the optimal ordering policy for the retailer and the corresponding production decision for the manufacturer. Wang and Chen researched the impact of option contract on a Newsvendor problem of jointly replenishment and pricing [43]. Zhao et al. investigated the option contract in a twoechelon supply chain in the presence of both stochastic spot market and demand information updating. A new concept expected unit opportunity saving (EUOS) was developed and used to obtain analytical results to characterize the optimal strategies for portfolio procurement via the option contract and spot markets [18]. Biswas and Avittathur developed an option contract model and demonstrated that pure-strategy Nash equilibrium exists in a single supplier-multiple buyer supply chain network [44]. Current research on option contract primarily studies its impact on the pricing, production, procurement, and inventory decisions, therefore, this paper tries to study its impact on the allocation of carbon emission abatement among supply chain members.
Note from the above reviewed literature, carbon emission abatement and option contract have been widely studied. However, under the uncertainty of market demand and carbon tax policy, the allocation of carbon emission abatement target among supply chain members with option contract has not yet been studied. To fill the gap, this paper considers a twoechelon supply chain consisting of a manufacturer and a retailer, in which the manufacturer is the leader and allocator. The optimal production, procurement and carbon emission abatement allocation strategy of supply chain members are analyzed, and some important results are obtained through numerical analysis, which can provide some suggestions for enterprise management.

III. MODEL DESCRIPTION
We consider a two-echelon supply chain with one manufacturer and one retailer. The manufacturer distributes its product via the retailer to the end market, and the market demand is uncertainty. The option mechanism is employed to facilitate the production and procurement in the supply chain, which is characterized by the reservation price and exercise price. The reservation price is paid by the retailer to the manufacturer for reserving the production at the beginning of the production season, while the execution price is paid by the retailer for product purchased after the demand is realized. Considering the current industry environment in which many manufacturers are more powerful than the retailers, we assume the manufacturer is the leader and allocator in our study.
The event sequence of our model can be described as follows: At the beginning of the production season, the manufacturer offers the retailer an option contract, and decides its ratio of total carbon emission abatement and production quantity. Then the retailer reserves an order quantity of the product at the reservation price and decides its retail price.
During the selling season, depending on the realized market demand for the product, the retailer purchases a quantity of the product up to the order quantity Q at the execution price to satisfy the demand, and any unsatisfied demand is lost with no penalty cost. Fig.1 illustrates the sequence of the game.

A. NOTATIONS AND ASSUMPTIONS
We present the notations in Table 1.
The assumptions made in the paper are as follows: First, it is assumed that the allocator has the priority to allocate the ratio θ to itself, and the rest ratio of carbon emission abatement must be done by its upstream and downstream companies in the supply chain.
Second, following the literatures ( [28], [36]), we define the carbon emission abatement cost function as TC i (a i ) = k i a 2 i /2, the quadratic form captures the diminishing returns of a carbon emission abatement expenditures Third, assuming that during production, the retailer orders only under the option contract.
Forth, the manufacturer is assumed to be the leader and allocator in the supply chain, and the government charges the companies who emit the carbon dioxide directly. We focus on the reasonable and non-trivial case where v < c < c o + c e < p. In order to make the option contract reasonable, we set v < c e , and 0 < c o < c − v. In fact, assuming ν < c e avoids the trivial case where the retailer always exercises all the options purchased by it while assuming 0 < c o < c − v avoids the unreasonable case where the manufacturer is risk-free for its production. In this paper, we normalize the shortage cost of the retailer and the manufacturer to zero.
Fifth, assuming that the market demand information is symmetric between the manufacturer and the retailer. The market demand is uncertainty and can be affected by the retail price, i.e., D = d − bp + x, in which d is the potential scale of the market, b (0 < b < 1) represents the elasticity of price, x is a random variable characterized by probability density function f (·), cumulative distribution function F(·), and has a support of [A, B] where B > A> 0. Furthermore, PDF f (·) and CDF F(·) satisfy the following conditions: (1) In the scope of [A, B], F (x) is continuous and the second-order derivable; ( is derivative and f (·) > 0.

B. THE OPTIMAL STRATEGIES OF RETAILER
The retailer's expected profit function under the option contract is given by where p is the retail price and Q is the order quantity. The first term of the function is the retailer's sales revenue, the second term is the allowance payout for the reserved capacity, the third term is the procurement cost, the fourth term is the payment for carbon tax, and the last term is the carbon emission abatement investment cost. Constraints ensure that the retailer is willing to participate in the game with the manufacturer. To maximize the retailer's expected profit function with respect to p and Q, we can get the following theorem. Theorem 1: For the given ratio of carbon emission reduction θ , the optimal order quantity of option Q * and the optimal retail price p * satisfy the following equations: Theorem 1 implies that the optimal order quantity and the optimal retail price can be derived from the above equation. Based on the optimal decisions, we can get the retailer's optimal expected profit. Then the impact of carbon tax, total carbon emission abatement and option prices on the order quantity, retail price and expected profit are analyzed, see the following Corollaries for details.
Corollary 1: For the given ratio of carbon emission abatement θ , (1) When the carbon tax s increases, the optimal ordering quantity of option Q * decreases; (2) When the total carbon emission abatement per unit product a increases, the optimal ordering of option Q * increases; (3) When the unit reservation price c o increases, the optimal ordering quantity of option Q * decreases; (4) When the unit execution price c e increases, the optimal ordering quantity of option Q * decreases.
Corollary 1(1) implies that, the increase of carbon tax leads to higher tax payment. Therefore, in order to reduce the total expenditure, the retailer tends to cut the procurement cost and thus the order quantity.
Corollary 1 (2) states that, the increase of total carbon emission abatement leads to the increase of investment cost and reduction of carbon tax expenditure. Thus, in order to offset the loss of investment cost, the company will order more quantity in advance to ensure the market demand for more profit.
According to Corollary 1(3), the optimal order quantity decreases with the increase of the reservation price c o . As the higher the unit reservation price, the more the retailer will pay to reserve the product. In order to reduce the reservation cost, the order quantity will be reduced.
Similar to Corollary 1(4), the increase of the execution price c e causes the manufacturer to charge the retailer more purchase fees. Therefore, retailer tends to reduce the order quantity to reduce procurement cost.
Corollary 2: For the given ratio of carbon emission reduction θ , (1) The expected profit of retailer E(π r ) decreases in carbon tax s; (2) When the total carbon emission abatement per unit product locates in the interval a ∈ 0, s Q * − , the expected profit of retailer E(π r ) increases with a; otherwise, E(π r ) decreases with a; (3) When the unit reservation price c o increases, the expected profit of retailer E(π r ) decreases; (4) When the unit execution price c e increases, the expected profit of retailer E(π r ) decreases.
Corollary 2(1) states that the retailer's expected profit decreases with the carbon tax s. As the retailer tries to reduce the expenditure by cutting the procurement cost, this will lead to the decrease of order quantity and a loss of revenue, thus the retailer's expected profit decreases.
Based on Corollary 1(2), the order quantity increases with the total carbon emission abatement a. Therefore, the increase of a brings the increase of sales revenue and reduction of carbon tax payment, but also leads to the increase of carbon emission reduction investment and procurement cost. Corollary 2(2) implies that when the total carbon emission abatement a is kept low, the retailer's carbon tax savings and increased sales revenue are greater than the investment and procurement costs, and the retailer prefers to reduce carbon emissions to obtain more profit, thus the retailer's expected profit increases with the total carbon emission abatement. However, when the total carbon emission abatement a is large enough, the savings in carbon tax and the increased sales revenue cannot offset the losses caused by investment and procurement, and the retailer is not inclined to reduce carbon emissions, thus the retailer's expected profit drops when the total carbon emission abatement goes higher.
According to Corollary 1(3), the increase of reservation price will lead to the decrease of order quantity, which means that the retailer tries to reduce the procurement cost to keep the profit, and this will result in a loss of revenue. The savings part cannot mitigate the lost revenue, so the retailer's profit will decline. Corollary 2(4) has a similar explanation.

C. THE OPTIMAL STRATEGIES OF MANUFACTURER
After forecasting the reactions of the retailer, the manufacturer makes decisions of the ratio of carbon emission abatement θ and the quantity of its production quantity Q m .
The manufacturer's expected profit function under the option contract is given by: where, the first term of the function is the manufacturer's revenue, including the reservation part and execution part, the second term is the production cost, the third term is the carbon tax cost, the fourth term is the salvage value, and the last term is the investment cost of carbon emission abatement. Constraints ensure that the output is greater than the option order quantity and the manufacturer is willing to participate in the game with the retailer. To maximize the manufacturer's expected profit, we can get the following theorem. Theorem 2: The optimal quantity of production Q * m equals to Q * and the optimal ratio of carbon emission abatement θ * of the manufacturer satisfy the following equations: Theorem 2 implies that the manufacturer should produce according to the order quantity. Since the salvage value of the product is less than the production cost, i.e., v < c, and the reservation price is less than the difference between the production cost and the salvage value, i.e., 0 < c o < c − v, the manufacturer cannot profit from the excessive production and should produce according to the order quantity. In addition, the optimal ratio of carbon emission abatement can be derived from the equation, and based on the optimal decisions, we can get the manufacturer's optimal expected profit. We will analyze the impact of carbon tax, total carbon emission abatement and option prices on the optimal ratio θ * and manufacturer's expected profit by numerical analysis in the next part.

IV. NUMERICAL ANALYSIS
In this section, we analyze the influences of carbon tax, total carbon emission abatement and option prices on the manufacturer's and retailer's optimal decisions and expected profits. The demand is D = d −bp+x. Let d = 880, b = 0.3. We assume that x follows uniform distribution in [-50,50].

A. THE IMPACT OF CARBON TAX
In this subsection, we mainly analyze the optimal decisions and expected profits of supply chain members with respect to the carbon tax s.

1) THE OPTIMAL DECISIONS OF SUPPLY CHAIN MEMBERS WITH RESPECT TO CARBON TAX s
We have Figure 2 to illustrate the optimal retail price p * , the optimal order quantity Q * , the optimal ratio of carbon emission abatement θ * and the optimal production quantity Q * m under different carbon tax s.
(1) When s subjects to [0, 0.08], we have Figure 2(a) to illustrate the optimal ratio of carbon emission abatement θ * under carbon tax s; (2) When s subjects to [0, 0.08], we have Figure 2(b) to illustrate the optimal order quantity of option Q * under carbon tax s; (3) When s subjects to [0, 0.08], we have Figure 2(c) to illustrate the optimal retail price p * under carbon tax s; (4) When s subjects to [0, 0.08], we have Figure 2(d) to illustrate the optimal production quantity Q * m under carbon tax s. Observation 1: When the carbon tax s increases, both the optimal ratio of carbon emission abatement θ * and the optimal retail price p * increase.
When the carbon tax remains a low level, the manufacturer is more sensitive to the investment cost of carbon emission reduction. Thus, as the allocator, the manufacturer will allocate a lower ratio of carbon emission abatement to itself to gain more profit. With the increase of carbon tax, the manufacturer is willing to increase the ratio of carbon emission abatement to meet emission abatement target and reduce the carbon tax cost. When the carbon tax is higher than a threshold, the manufacturer will allocate all carbon emission abatement to itself, i.e., θ * = 1. Furthermore, when the carbon tax increases, the retailer faces higher carbon tax cost and prefers to increase the retail price to reduce profit losses. However, the increase in price will lead to a decline in demand for products, so the number of orders will follow, which also matches with Corollary 1(1). For the manufacturer, it sets the production quantity according to the retailer's order quantity.

2) THE EXPECTED PROFITS OF SUPPLY CHAIN MEMBERS WITH RESPECT TO CARBON TAX s
We have Figure 3 to illustrate the optimal expected profits of supply chain members under different carbon tax s.
(1) When s subjects to [0, 0.08], we have Figure 3(a) to illustrate the optimal expected profit of the manufacturer E(π m ) under carbon tax s; (2) When s subjects to [0, 0.08], we have Figure 3(b) to illustrate the optimal expected profit of the retailer E(π r ) under carbon tax s.  It can be seen from Figure 3(a) that the optimal expected profit of manufacturer E(π m ) decreases with carbon tax s. As the order quantity decreases with the carbon tax, the higher the carbon tax imposed by the government, the lower the sales revenue will get. As a result, the manufacturer's expected profit will fall. For the retailer, as the manufacturer's ratio of total carbon emission abatement increases with the carbon tax, the retailer faces a higher tax payment, which cannot be offset by the reduction in investment cost. Hence, the expected profit of the retailer will decrease, which is consistent with Corollary 2(1).

B. THE IMPACT OF TOTAL CARBON EMISSION ABATEMENT
In this subsection, we mainly analyze the optimal decisions and expected profits of supply chain members with respect to the total carbon emission abatement a.

1) THE OPTIMAL DECISIONS OF SUPPLY CHAIN MEMBERS WITH RESPECT TO TOTAL CARBON EMISSION ABATEMENT a
We have Figure 4 to illustrate the optimal retailing price p * , the optimal order quantity Q * , the optimal ratio of carbon emission abatement θ * and the optimal production quantity Q * m under different total carbon emission abatement a.
(1) When a subjects to [170,240], we have Figure 4(a) to illustrate the optimal ratio of carbon emission abatement θ * under total carbon emission abatement a; (2) When a subject to [170,240], we have Figure 4(b) to illustrate the optimal order quantity of option Q * under total carbon emission abatement a; (3) When a subjects to [170,240], we have Figure 4(c) to illustrate the optimal retail price p * under total carbon emission abatement a; (4) When a subjects to [170,240], we have Figure 4(d) to illustrate the optimal production quantity Q * m under total carbon emissions abatement a.
Observation 2: When the total carbon emission abatement a increases, both the optimal ratio of carbon emission abatement θ * and the optimal retail price p * decrease.
When the total carbon emission abatement remains a low level, the manufacturer is more sensitive to the cost of carbon tax. As the allocator, the manufacturer will allocate a higher ratio of carbon emission to itself to save the cost of carbon tax and gain more profit. With the increase of total carbon emission abatement, the investment cost increases gradually, the manufacture becomes more sensitive to the investment cost and thus has to decrease the ratio in order to mitigate the increasing cost, that is to say, the retailer faces a higher investment cost. Therefore, the retailer tends to decrease the retail price to attract more customers and to increase the order quantity to meet market demand, which is match with Corollary 1 (2). For the manufacturer, it sets the production quantity based on the retailer's order strategy, which matches with Figure 4(d).

2) THE EXPECTED PROFITS OF SUPPLY CHAIN MEMBERS WITH RESPECT TO TOTAL CARBON EMISSION ABATEMENT a
We have Figure 5 to illustrate the optimal expected profits of supply chain members under different total carbon emission abatement a.
(1) When a subjects to [170,240], we have Figure 5(a) to illustrate the optimal expected profit of the manufacturer E(π m ) under total carbon emission abatement a; (2) When a subjects to [170,240], we have Figure 5(b) to illustrate the optimal expected profit of the retailer E(π r ) under total carbon emission abatement a. Figure 5(a) implies that the manufacturer's optimal expected profit E(π m ) is concave with total carbon emission abatement a. Since the manufacturer's ratio of total carbon emission abatement decreases with a, when a is less than a certain threshold, the manufacturer bears a higher ratio of carbon emission reduction, so does a lower carbon tax payment. Therefore, as a increases, the manufacturer can benefit from the lower carbon tax payment and increased sales revenue. However, when the total carbon emission abatement a is large enough and the ratio is sufficiently low, the manufacturer faces a higher carbon tax cost, and this loss cannot be mitigated from the increased sales revenue, thus the manufacturer's profit decreases. Figure 5(b) states that the optimal expected profit of retailer E(π r ) increases with total carbon emission abatement a. By numerical calculation we have a ∈ 0, s Q * − (1 − θ ) , thus the result is consistent with Corollary 2(2).

C. THE IMPACT OF OPTION PRICES
In this subsection, we mainly analyze the optimal decisions and expected profits of supply chain members with respect to the reservation price c o and the execution price c e .

1) THE OPTIMAL DECISIONS OF SUPPLY CHAIN MEMBERS WITH RESPECT TO OPTION PRICES
We have Figure 6 and Figure 7 to illustrate the optimal retail price p * , the optimal order quantity Q * , the optimal ratio of carbon emission abatement θ * and the optimal production quantity Q * m under different reservation price c o and execution price c e , respectively.
(1) When c o subjects to [15,35], we have Figure 6(a) to illustrate the optimal ratio of carbon emission abatement θ * under reservation price c o ; 2) When c o subjects to [15,35], we have Figure 6(b) to illustrate the optimal order quantity of option Q * under reservation price c o ; (3) When c o subjects to [15,35], we have Figure 6(c) to illustrate the optimal retail price p * under reservation price c o ; (4) When c o subjects to [15,35], we have Figure 6(d) to illustrate the optimal production quantity Q * m under reservation price c o .
(5) When c e subjects to [150, 220], we have Figure 7(a) to illustrate the optimal ratio of carbon emission abatement θ * under execution price c e ;    Observation 3: When the reservation price c o and execution price c e increase, the optimal ratio of carbon emission abatement θ * decreases while the optimal retail price p * increases.
Consistent with Corollary 1(3) and 1(4), Figure 6(b) and 7(b) illustrate that the order quantity decreases with the reservation price and execution price, which may result in the loss of manufacturer's sales revenue. Therefore, as the allocator, manufacturer tends to reduce its ratio of carbon emission abatement to cut its investment cost. Furthermore, manufacturer's lower ratio of carbon emission abatement means a higher ratio for the retailer, thus the retailer is inclined to raise its retail price to alleviate the loss caused by reduced order quantity and increased investment cost.

2) THE EXPECTED PROFITS OF SUPPLY CHAIN MEMBERS WITH RESPECT TO OPTION PRICES
We have Figure 8 and Figure 9 to illustrate the optimal expected profits of supply chain members under different reservation price c o and execution price c e , respectively. (1) When c o subjects to [15,35], we have Figure 8(a) to illustrate the optimal expected profit of the manufacturer E(π m ) under reservation price c o ; (2) When c o subjects to [15,35], we have Figure 8(b) to illustrate the optimal expected profit of the retailer E(π r ) under reservation price c o .
(3) When c e subjects to [150, 220], we have Figure 9(a) to illustrate the optimal expected profit of the manufacturer E(π m ) under execution price c e ; (4) When c e subjects to [150, 220], we have Figure 9(b) to illustrate the optimal expected profit of the retailer E(π m ) under execution price c e .
It can be seen from Figure 8(a) and 9(a) that the optimal expected profit of manufacturer E(π m ) increases with reservation price c o and execution price c e . As the ratio of total carbon emission abatement decreases can lead to the reduction of the manufacturer's investment cost, the investment savings and increased option prices can mitigate the loss caused by the reduction of order quantity, so that the manufacturer's expected profit increases with option prices. In addition, the reduction in retailer's order quantity and increase in investment cost can result in a decline in profit, which matches with Corollary 2(3) and 2(4).

V. CONCLUSION
In this paper, we consider a two-echelon supply chain consisting of one manufacturer and one retailer, in which the manufacturer is the leader and allocator. Based on carbon tax and option contract, we first analyze the retailer's optimal retail price and order quantity when the ratio of carbon emission abatement is given, and then derive the manufacturer's optimal ratio of total carbon emission abatement and production quantity. By analytical and numerical analysis, we illustrate the optimal decisions and expected profits of supply chain members with respect to carbon tax, total carbon emission abatement and option prices, and find that the retailer's optimal order quantity and retail price have the opposite trend with the change of these parameters. In addition, since retailer tends to mitigate the increased expenditure, its optimal order quantity decreases with the increase of carbon tax and option prices, while increases with total carbon emission abatement as the retailer tends to get more profit from sales revenue and tax savings. For the manufacturer, the optimal ratio of carbon emission abatement has an opposite trend to the order quantity under the changes of carbon tax and total carbon emission abatement as the manufacturer tends to reduce its VOLUME 8, 2020 tax payments or investment cost, but has the same trend with order quantity under the changes of option prices because the manufacturer prefers to cut its investment cost. Besides, the manufacturer's and retailer's expected profits show the same trend with the increase of carbon tax and total carbon emission abatement, while have the opposite trend with the increase of option prices, and these findings can provide some management insights for the enterprises.
We assume the manufacturer is the leader and allocator in the two-echelon supply chain in this work. As a natural extension of our work, future research could be done by considering coordination issues in a manufacturer-retailer supply chain with option contract. The carbon emission abatement allocation should be considered in the model and using the wholesale price mechanism as a benchmark, to study if the option contract could coordinate the supply chain and achieve Pareto-improvement, or how the option contract should be set to attain supply chain coordination. Furthermore, making decisions on option execution price, reservation price, and the ratio of carbon emission abatement, or considering supply chain with multiple manufacturers and multiple retailers is also a research direction. Besides, the paper only analyzes the optimal decisions with option contract. As an extension, the spot market could be taken into consideration in the future study.

APPENDIX
In this section, we provide the detailed proofs of the main results.
Proof of Theorem 1: We use Kuhn-Tucker method to solve this problem. Since Lagrange multiplier µ * 1 , µ * 2 is adopted to transfer a problem with constraint conditions into another one without constrain condition. We can get For any given θ , solving ∂L/∂p = 0 and ∂L/∂Q = 0, Then, And, Substituting the retailing price p and the ordering of option Q in E(π r ), we can get E(π r ) = −k r (1 − θ ) 2 a 2 /2 < 0. If the profit of the retailer is negative, the retailer will not produce and operate, thus the solution is not the K − T point.
thus the profit of the retailer E(π r ) = −k r (1 − θ ) 2 a 2 /2 < 0 is negative. Similar to (2), the retailer will not produce and operate, and the solution is not the K − T point.
(4) When µ * 1 = 0 and µ * 2 = 0, the ordering quantity of option and the retailing price of retailer (p, Q) satisfy: We can see p − c e − s(e r − (1 − θ )a) > 0 and Q > 0. So, the ordering quantity of option and the retailing price of retailer (p, Q) is the K − T point.
To proof the ordering quantity and the retail price (p, Q) which satisfy equation (A.5) is the optimal decisions of the retailer, we first calculate the Hessian matrix The second-order derivation of E(π r ) with respect to p and Q can be derived. Then, When H 2 > 0, the Hessian matrix H is negative definite, and E(π r ) is concave in (p, Q). Then the optimal retail price and ordering quantity (p * , Q * ) can be derived by the following equation:

Proof of Corollary 1:
(1) Taking the first-order derivative of equation (A.6) and (A.7) with respect to the carbon tax s, then getting the solution from the simultaneous equations, we can get is obtained by the equation as shown at the bottom of this page.
(2) Taking the first-order derivative of equation (A.6) and (A.7) with respect to the total carbon emission abatement per unit a, getting the solution from the simultaneous equations, we can obtain is obtained by the equation as shown at the bottom of next page. Proof of Corollary 2: So, when a ∈ 0, s Q * − , ∂E(π r )/∂a < 0, the expected profit of retailer E(π r ) is decreasing in a.
Proof of Theorem 2: It can be easily verified that ∂E(π m ) ∂Q m = v − c − s(e m − θ a) < 0, thus E(π m ) decreases in Q m . From the constraints we know that Q m ≥ Q * , so the optimal output is equal to Q * , i.e., Q * m = Q * . Placing Q * m by Q * in the manufacturer's objective function, we get Similar to the proof of Theorem 1, we also use Kuhn-Tucker method to solve this problem.
Lagrange multiplier µ * 3 is adopted to transfer a problem with constraint conditions into another one without constrain condition. We can obtain θ a)). Solving ∂L/∂θ = 0, then, The expected profit of the manufacturer is negative, and the industry will not produce and operate, so this point is not K − T point.
(2) When µ * 3 = 0, Solving equation (A.10), we can see c o + c e − c − s(e m − θ a) > 0, Substituting it into equation (A.9), we obtain the ratio of carbon emission abatement θ satisfies the following equations: Based on Theorem 1, taking the first-order derivation of Q * and p * with respect to the ratio of carbon emission abatement θ . After simplification, we can get and Thus, we can derive that if c o +c e −c−s(e m −θ a) > 0 and the ratio of carbon emission abatement θ is the K − T point.
To proof θ satisfying (A.11) and (A.14) is the optimal solution, we taking the second-order derivative of E (π m ) with respect to the ratio of carbon emission abatement θ and derive that Then, taking the second-order derivation of p * and Q * with respect to the ratio of carbon emission abatement θ respectively, we can get in which, g = ∂Q * ∂θ + b ∂p * ∂θ , m =  From 2016 to 2018, she worked at the Gansu Branch, China Construction Bank. Her research interests include supply chain coordination, green supply chain, inventory management, and machine scheduling. VOLUME 8, 2020