# Financing Strategy and Carbon Emission Abatement in a Supply Chain considering Retailers’ Competition

Authors: Man Yu; Tuo Li; Zhanwen Shi
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper investigates the issues of financing channels (bank credit financing, trade credit financing, and dual-channel financing) and carbon emission abatement in a supply chain consisting of one capital-constrained manufacturer and two capital-constrained retailers. Compared with bank credit, we find that every member can make more profit under trade credit when only one financing channel is available. When both bank credit and trade credit are available, the retailers’ financing strategy highly depends on the interest rates charged by the creditors. In addition, we also examine the impact of financing channels on emission abatement. It shows that the manufacturer reduces more carbon emissions under trade credit. Interestingly, the emission abatement has nothing to do with trade credit interest rate when retailers only adopt trade credit, whereas it is closely related to trade credit interest rate under dual-channel financing.


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## Body

## 1. Introduction

Recently, global natural disasters and extreme weather occur frequently. The frequency of floods, high temperature, drought, freezing, typhoons, and other disasters has increased significantly. One of the main causes of these environmental crises is global warming, which is trigged by greenhouse emissions, such as carbon dioxide emissions. Since the Kyoto Protocol came into effect in 2005, governments that signed the agreement have taken various measures to reduce carbon emissions. However, the carbon emissions around the world have not decreased considerably [1]. To curb carbon emissions effectively, nearly 200 countries signed the Paris Agreement and committed to realize climate goals in 2030 [2]. As the main contributor of carbon emissions, firms need to make efforts to reduce carbon emissions in the manufacturing process. On the other hand, consumers pay more attention to environmental protection nowadays [3], which also pushes firms to reduce carbon emissions.Capital constraint is a very common problem faced by many firms [4]. To solve this problem, the capital-constrained enterprises may borrow from a supply chain partner by trade credit or a commercial bank by trade credit [5]. For the start-up and small-and medium-sized enterprises, it may be easier to access trade credit due to lack of credit information and limited liability structure [6, 7]. Some firms may avail a loan from a bank or from their upstream firms. Then, an interesting issue that arises from the above phenomenon is what financing method a capital-constrained firm should choose when both financing channels are available. What if adopting two financing ways? How does financing channel affect carbon emission abatement? Yet, little attention has been paid to the above issues.To fill the aforementioned gap, we establish a green supply chain consisting of one capital-constrained manufacturer and two capital-constrained retailers. The manufacturer has to invest in carbon emission reduction due to consumer’s environmental awareness and environmental policies. The manufacturer borrows from a bank when necessary. The retailers borrow either from the manufacturer by trade credit or from a bank by bank credit, or both channels at the same time. By comparing each member’s optimal decision under different financing methods, we find that the retailers place larger orders under trade credit. The reason behind this result is that the manufacturer offers a lower wholesale place in this scenario. The result also shows that the capital-constrained firms can make more profits under trade credit than that under bank credit when only one financing method is available. On the other hand, the manufacturer reduces more carbon emissions under trade credit. Interestingly, under trade credit, the retailers’ optimal order quantities are not affected by the interest rate and the emission abatement is immune to trade credit interest. We also investigate the conditions under which each financing method is adopted when both financing methods are available. The findings show that, when the trade credit interest is lower than bank interest rate, the retailers borrow more money from the manufacturer instead of crediting only from the manufacturer. Only when the trade credit interest is much lower than bank credit interest, the retailers will borrow all needed funds from the manufacturer. In addition, we further discuss the impact of dual-channel financing on carbon emission reduction and find that carbon emission reduction is closely related to trade credit interest and bank credit interest charged on the manufacturer and retailers. It shows that when trade credit interest is twice higher than the bank credit interest on the manufacturer, the carbon emission reduction level increases in bank interest rate on retailers. When the bank interest rate on the manufacturer is relatively larger, the carbon emission reduction level increases in trade credit interest.The remainder of this paper is organized as follows. Section2 reviews the relevant literature. Section 3 describes the basic modeling framework. Section 4 analyzes each member’s optimal decision in different financing cases. Section 5 does some numerical experiments to investigate how various parameters affect the participators’ optimal decisions. Section 6 presents the insights and conclusions of this study.


## 2. Literature Review

This research is relevant to carbon emission abatement. Firms abate carbon emissions due to governmental regulations in some countries and areas. Krass et al. [8] show that firms have to adopt green emission reduction technology since governments charge environmental taxes on carbon emissions. Drake et al. [9] investigate as to how emission tax influences an enterprise’s technology choice and capacity decision by constructing a two-stage model. Xu et al. [10] investigate the effectiveness of different low carbon policies on reducing carbon emissions. Lee [11] considers a supply chain consisting of two manufacturers, one of whom produces green products and another who produces nongreen products, and shows that the competition brings more profit for them. Considering the impact of GHG emissions, Marchi et al. [12] compare the order quantity and production rate decisions in different scenarios. Hong and Guo [13] study the optimal retail price decision in a supply chain in which the supplier decides the product greenness. Cohen et al. [3] investigate how demand uncertainty affects each participator when the government provides subsidy to end consumers to stimulate adopting green technology. Under carbon cap-and-trade regulation, Choi [14] establish a newsvendor model to investigate the emission abatement issue in a supply chain where the decision-maker is loss-averse.The carbon emission abatement is also driven by consumers’ eco-friendly awareness. Du et al. [15] discuss the decision of the optimal greening level of each firm when it makes decisions separately. Yang and Chen [16] study how supply chain contracts affect a manufacturer’s carbon emission reduction when considering consumers’ environmental awareness. Swami and Shah [17] examine whether a two-part tariff contract can coordinate the supply chain when considering firms investing in carbon emission reduction. Xu et al. [10] suggest that a cost-sharing contract is able to achieve supply chain coordination. Dong et al. [18] examine firms’ optimal decisions considering carbon emission abatement investment and suggest that a revenue sharing contract can coordinate a sustainable supply chain. Tong et al. [19] suggest that, when consumers pay attention to technologies of reducing carbon emissions, more firms are willing to invest in R&D.Some literature also suggests that firms may actively make a costly effort to reduce carbon emission to make more profit or to take social responsibility. Ghosh and Shah [20] analyze how channel structures impact the green levels, price, and profits of supply chain members. Shi et al. [21] examine the investment strategy in green product and find that a supplier makes more profit when both retailers invest in green technology. Zhu and He [22] discuss how green product design was affected by supply chain structures in competing supply chains. Luo et al. [23] investigate how the emission abatement affects the performance of a supply chain comprising two competitive manufacturers in competition and cooperation scenarios, respectively. Wang and Sun [24] suggest that a manufacturer is willing to invest in green products when the sales volume is high in the direct channel.Our work also considers consumers’ environmental awareness, whereas our work differs from the literature above in at least two dimensions. First, we consider a supply chain consisting of capital-constrained retailers, who may finance by bank credit or by trade credit. Under some conditions, the manufacturer may also borrow from a bank. Second, this paper investigates the impacts of different financing methods on carbon emission reduction, and finds that carbon emission reduction is immune to trade credit.This work is also closely related to enterprise financing. According to funding sources, financing can be divided into external financing, such as bank credit financing, and internal financing, such as trade credit financing.In recent years, researchers have paid considerable attention to trade credit and investigated the roles of trade credit. Tiwari et al. [25] establish an EOQ model to analyze the optimal decisions about deteriorating products. Otrodi et al. [26] discuss the optimal pricing and ordering decisions in a supply chain where the supplier offers trade credit to the retailer and the retailer provides trade credit to customers. Marchi et al. [27] study how trade credit that is offered from a buyer to a vendor affects a vendor’s production capability and the performance of a supply chain. When retailers compete in terminal market, Wu et al. [28] show that a manufacturer can use trade credit to respond to a dominant retailer’s bargaining and the weak retailer can benefit from trade credit. Li et al. [29] establish a multi-period model to analyze the relationship between a supplier’s trade credit term strategy and a retailer’s optimal order quantity decision by using game theory. Zou and Tian [30] examine the retailer’s optimal payment strategy and ordering decision when two-part trade credit is available. Chod [6] suggest that when a retailer orders multiple products from different suppliers, trade credit can mitigate the inventory distortion. Yang and Birge [31] suggest that trade credit improves supply chain performance because a creditor can share the distress risk with a borrower. Ren et al. [32] study how trade credit affects each member’s pricing decision and profit in a supply chain. Yan et al. [33] examine how each party makes a decision in a supply chain with capital-constrained retailer and suggest that a partial credit guarantee is able to achieve channel coordination considering bank credit financing.This work is also related bank to credit financing. Ries et al. [34] study how financial conditions affect a retailer’s optimal order quantity and the payment decision. Boyabatli and Toktay [35] inspect a case in which a manufacturer lacks capital and borrows from a bank aiming to maximize its profit. Yang et al. [36] show the impact of bank financing on each member’s optimal decision by establishing a supply chain comprising a supplier and two competitive retailers who borrow money from a bank. Kouvelis and Zhao [37] study the supply chain coordination considering retailer’s bankruptcy cost. Researchers also pay attention to compare trade credit with bank credit and investigate the preferences of the financing channel. For example, Bing et al. [38] inspect the financing equilibriums when a capital-constrained retailer uses bank credit financing and trade credit financing, respectively. Kouvelis and Zhao [39] also examine the impact of interest rate on firms’ financing choices. Deng et al. [40] provide the equilibriums under buyer finance and bank finance, respectively, and show that a buyer may offer suppliers finance even if its capital opportunity cost is higher than bank interest rate. Tang et al. [41] compare purchase ordering financing and bank credit financing and suggest that purchase ordering financing is efficient in lowering performance risk.This paper complements the abovementioned literature by investigating different financing channels considering consumers’ environmental awareness. Different from the above studies, first, we consider the case in which two capital-constrained retailers compete for items in the market. Second, we not only analyze the scenarios in which either trade credit or bank credit is adopted but also examine the scenario in which both trade credit and bank credit are availed under a competing situation. Third, this paper also inspects the impact of financing methods on carbon emission abatement.


## 3. Model Description

We consider a supply chain consisting of one manufacturer (called “she”) and two capital-constrained retailers (called “he”). The manufacturer and the retailers are engaged in a Stackelberg game in which the manufacturer is the leader. The retailers compete for sales quantities in the market. The capital-constrained retailers do not have enough initial working capital to purchase items from the manufacturer, and have to borrow money from an external bank or from the upstream manufacturer. For simplicity, we assume that the retailers’ initial working capital is zero [40]. In this work, we consider three different financing methods: bank credit financing, trade credit financing, and dual-channel financing. Dual-channel financing refers to the case in which the retailers use both bank credit and trade credit. Driven by consumers’ environmental awareness, the manufacturer has incentives to reduce emissions. Thus, the manufacturer’s decisions are the wholesale price w and emission abatement level of products e. Define the cost of emission abatement he2 ( [10, 13]), where h stands for the cost coefficient of emission abatement and a small h implies that the manufacturer is efficient in abating emissions. In line with An et al. [4], we assume that the manufacturer has enough capital to invest in green improvement of products but has less money in production. This is because a firm needs to meet the carbon emission standards first before its normal operation. For example, the Ministry of Environmental Protection of China set a limit that enterprises that cannot reach emission reduction standard have to close in a regulated time period (http://china.cnr.cn/gdgg/20170714/t20170714_523850712.shtml). Then, when trade credit is adopted, the manufacturer has to borrow money from the bank to cover the production cost. In this work, we assume that the manufacturer’s production capital is zero to focus on the financing issue ( [31, 33]).The sequence of events is as follows. First, the manufacturer provides the wholesale price and decides the emission abatement level. Second, the retailers borrow money from a bank, or a manufacturer, or from both and order products from the manufacturer. Third, the manufacturer borrows money from a bank if necessary and produces products by investing in emission abatement at the same time. After the demand is realized, the firms repay their debts. The relevant notation is expressed in Table1.Table 1
Notation.
wWholesale price charged by the manufacturerhCost coefficient of carbon emission abatementeCarbon emission abatement levelpiRetail price charged by retaileriqiOrder quantity of retaileriaMarket sizevRetailer 1’s underlying market shareϕSubstitutability of productsλConsumers’ price sensitivity to the emission abatementr0Basic bank interest rateηRImpact factor of bank interest rate related to retailersrRiThe interest rate charged by the bank on retailerirTThe interest rate charged by the manufacturer under trade creditrMThe interest rate charged by the bank on the manufacturerTiRetaileri’s loan size borrowed from the manufacturerConsidering consumers’ eco-friendly awareness, they are willing to pay higher price for green products. Then, when the retailers decide selling quantity, in line with Zhu and He [22], the inverse demand functions are written as(1)p1=va−q1−ϕq2+λe,p2=1−va−q2−ϕq1+λe,v stands for retailer 1’s underlying market share, a is the potential demand of the market. ϕ represents the substitutability of products selling by different retailers, and a larger ϕ implies fierce competition in the market. λ is consumers’ price sensitivity to the emission abatement level.


## 4. Model Analysis

In this section, we analyze three financing models (bank credit financing, trade credit financing, and dual channel financing) in detail. It is assumed that all the members are risk-neutral and seek maximized profit. Since the demand is certain and depends on the retail prices, the retailers always repay the entire loan. Otherwise, they will order nothing. Therefore, we assume that all the firms do not go bankrupt and the bank does not face firms’ default risk.


### 4.1. Bank Credit Financing

Assume that the bank credit cost is convex in the borrowers’ book leverage. The rational reasons include managerial risk aversion [42], adverse selection [43], and microeconomic foundations [44]. For simplicity, in line with Chod et al. [45], we adopt the bank credit cost by assuming that the bank interest rate increases linearly in the borrower’s book leverage. Thus, the interest rate charged by the bank on the retailer over loan size y is given by(2)rRi=r0ηRywqi,r0 is the basic bank interest rate, ηR is the impact factor of bank interest rate related to the retailers. When only bank financing is adopted by retailers, the bank interest rate equals(3)rR1=rR2=r0ηR.Given the wholesale price, the retailers’ expected profit are written as(4)πR1=va−q1−ϕq2q1−wq11+r0ηR,πR2=1−va−q2−ϕq1q2−wq21+r0ηR.The next proposition fully characterizes the retailers’ optimal ordering strategy when adopting bank credit financing.Proposition 1.
Under bank credit financing, the retailers’ optimal order quantities areq1B∗=av2+ϕ−ϕ−2−ϕw2+r0ηR−λe/4−ϕ2, q2B∗=a2−v2+ϕ−2−vw2+r0ηR−λe/4−ϕ2.Proof.
Taking the first derivative ofπR1 on q1, we get(5)∂πR1∂q1=−2q1+λe+av−w−ϕq2−wr0ηR.
Taking the second derivative ofπR1 on q1, we get(6)∂2πR1∂q12=−2<0.
Thus,πR1 is concave on q1. Similarly,(7)∂πR2∂q2=−2q2+λe+a1−v−w−ϕq1−wr0ηR,∂2πR2∂q22=−2<0,
Thus,πR2 is concave on q2.
By solving equations∂πR1/∂q1=0 and ∂πR2/∂q2=0, denote q1B∗ and q2B∗ the optimal order quantities under bank credit, we can get(8)q1B∗=av2+ϕ−ϕ−2−ϕw2+r0ηR−λe4−ϕ2,q2B∗=a2−v2+ϕ−2−vw2+r0ηR−λe4−ϕ2.
Then, Proposition1 is proved.
Proposition1 shows that the retailers’ optimal order quantities increase in an eco-friendly level. The intuition behind this result is that consumers’ environmental awareness promotes them to buy green products. To meet consumers’ demand, the retailers enhance their order quantity. It also shows that the optimal order quantities decrease in the wholesale price and the bank interest rate. This is because a higher wholesale price increases the retailers’ ordering cost and a higher interest rate of bank credit increases the retailers’ financing costs. From Proposition 1, we know that retailers can benefit from consumers’ eco-friendly awareness. As a result, in practice, retailing firms can offer different kinds of items, such as low-carbon item and ordinary item, to satisfy different kinds of consumers, which may bring more profits for them.
The manufacturer’s revenuewq1+q2 is enough to cover the production cost cq1+q2, so she does not to need borrow money from the bank. In this case, the manufacturer’s expected profit is(9)πM=w−cq1+q2−12he2.
Then, we can get the following proposition.Proposition 2.
Under bank financing, the optimal wholesale price decided by the manufacturer iswB∗=ah2+ϕ+2ch2+ϕ2+r0ηR−4cλ2/4h2+ϕ2+r0ηR−4λ2, and the optimal emission abatement level is eB∗=aλ−2cλ2+r0ηR/2h2+ϕ2+r0ηR−2λ2.Proof.
Proof. Substitutingq1 and q2 into equation (9), then taking the first derivatives, we get(10)∂πM∂w=a+4c+2λe−8w+2r0ηRc−2w2+ϕ,∂πM∂e=−eh+4λw−c2+ϕ.
By taking the second derivatives, we get(11)∂2πM∂w2=−8−4r0ηR2+ϕ,∂2πM∂e2=−h,∂2πM∂e∂w=4λ2+ϕ,−8−4r0ηR2+ϕ4λ2+ϕ4λ2+ϕ−h=h8+4r0ηR−16λ22+ϕ2>0.
Thus,πMis concave on w and e. Let ∂πM/∂w=0 and ∂πM/∂e=0, denote wB∗ the optimal wholesale price and eB∗ the optimal carbon emission abatement level under bank financing, we get(12)wB∗=ah2+ϕ+2ch2+ϕ2+r0ηR−4cλ24h2+ϕ2+r0ηR−4λ2,eB∗=aλ−2cλ2+r0ηR2h2+ϕ2+r0ηR−2λ2.
Proposition2 shows that the wholesale price the trade credit increases in r0ηR, λ and c, decreases in ϕ and h. When the bank interest rate is high, the retailers tend to reduce the order quantity to save cost. In this case, the manufacturer charges a higher wholesale price to offset the revenue loss caused by reduced sales volume. λ represents consumers’ sensitivity of eco-friendly products, and a higher λ means consumers are willing to pay higher price for eco-friendly products. Then, charging a higher wholesale price does not reduce the retailers’ order quantity. Thus, w increases in λ. Contrary to our intuition, w decreases in h. h implies the manufacturer capability in reducing emission, and a higher h followed by a lower emission abatement level, which lowers the retailers’ order quantities. To induce the retailers to order more products, the manufacturer has to provide a lower wholesale price. In conclusion, when deciding wholesale price, the manufacturer should comprehensively consider bank interest rate, market competition, and its efficiency in abating carbon emissions.
From Proposition2, we can also find that the optimal emission abatement level increases in λ but decreases in r0ηR, ϕ, c, and h. With λ increasing, consumers are willing to pay higher price for eco-friendly products, which induce the manufacturer to invest in a higher emission abatement level. When r0ηR is high, the retailers have to pay more extra interest, which lowers their purchasing ability. As a result, the manufacturer has no incentive to invest in emission reduction. Thus, eB∗ decreases in r0ηR. With ϕ increasing, the competition between retailers is intensified, which enhances the total order quantities. In this scenario, the manufacturer has no incentive to invest in a high emission abatement to increase sales volume. Therefore, eB∗ decreases in ϕ. It implies that market competition hurts carbon emission reduction. To reduce more carbon emissions to meet government regulatory requirements, the manufacturer should try its best to weak competition. Obviously, either c or h increases the manufacturer’s operating cost, so eB∗decreases in c and h. From the analysis, it shows that both wholesale price and emission abatement affected by interest rate were charged on the retailers. Hence, to improve carbon emission abatement level and induce retailers to order more products, it is better for the upstream-firms to cooperate with the retailers to strive for lower interest rate from banks.
SubstitutingwB∗, eB∗, q1B∗, and q2B∗ into πR1, πR2, and πM, we can get every member’s maximized profit.(13)πR1B∗=2aλ21−2v+ah8v+4vϕ−3ϕ−22+r0ηR−2ch2−ϕ1+r0ηR2162−ϕ2h2+ϕ2+r0ηR−λ22,πR2B∗=2aλ21−2v+ah8v+4vϕ−6−ϕ2+r0ηR−2ch2−ϕ1+r0ηR2162−ϕ2h2+ϕ2+r0ηR−λ22,πMB∗=ha−4c−2cr0ηR28h2+ϕ2+r0ηR−8λ2.


### 4.2. Trade Credit

We now tend to see how the manufacturer and retailers make decisions when retailers adopt trade credit financing. As the retailers have more opportunities to reach terminal consumers, they can obtain more information about the market demand. Therefore, the manufacturer does not worry about the retailers’ bankruptcy risk and is willing to provide the loans that the retailers need. Since the retailers’ initial working capital is zero [40], the manufacturer permits retailers to pay for items at the end of the sales season, and charges retailers the extra interest rate rT. According to the assumption, the manufacturer does not have capital to invest in production and borrows from a bank at an interest rate rM. Similar to the retailers, the manufacturer confronts an interest rate that increases linearly in her total loan size yM,(14)rM=r0ηMyMcq1+q2.After the manufacturer announces the wholesale price and the interest rate of trade credit, the retailers order products from the manufacturer. Then, the retailers’ problems are to maximize their expected profits by determiningq1 and q2, which are described as(15)πR1=a−q1−ϕq2q1−wq11+rT,πR2=a−q2−ϕq1q2−wq21+rT.Then, we can get the retailers’ optimal order quantitiesq1T∗=av2+ϕ−ϕ−2−ϕw2+rT−λe/4−ϕ2 and q2T∗=a2−v2+ϕ−2−ϕw2+rT−λe/4−ϕ2.As the manufacturer’s loan size isyM=cq1+q2 in this scenario, the interest rate of bank financing on the manufacturer is rM=r0ηM. For the manufacturer, her profit consists of four parts: the sales profit that equals sales revenue minus production cost; the interest earned from trade credit; the emission abatement cost; and the bank financing cost. Then, the manufacturer’s expected profit is given by(16)πM=w−cq1+q2+rTq1+q2−12he2−cq1+q2r0ηM.Then, we can get the next proposition that describes the manufacturer’s optimal decision under trade credit financing.Proposition 3.
When the retailers use trade credit financing, the optimal wholesale price iswT∗=ah2+ϕ+2c1+r0ηMh2+ϕ−2λ2/41+rTh2+ϕ−λ2, and the optimal emission abatement level is eT∗=aλ−2cλ1+r0ηM/22h+hϕ−λ2.Proof.
The proof is similar to Proposition2.
Proposition3 shows that the optimal wholesale price increases in rM but decreases in rT. As bank interest rate increases the manufacturer’s cost, the manufacturer tends to charge retailers higher wholesale price to cover the cost. On the other hand, the trade credit offered to retailers brings extra financing profit to the manufacturer. With rT increasing, the retailers’ financing cost is enhanced. Thus, the manufacturer lowers the wholesale price to enlarge the retailers’ demand for products. Hence, the manufacturer needs to balance trade credit interest rate charged on retailers and the wholesale price to make more profit. From Proposition 3, we can also see that the emission reduction level decreases in the coefficient of emission reduction h. To enhance emission abatement level, the manufacturer should find an effective way to lower coefficient h. The findings also show that the emission abatement level eT∗ decreases in rM due to the increased extra financing cost for the manufacturer. Interestingly, we find that eT∗ has nothing to do with rT, which implies that the emission abatement level is not affected by the retailers’ financial status.
By substitutingwT∗ and eT∗ into q1T∗ and q2T∗, we can get(17)q1T∗=2aλ21−2v+ah8v+4vϕ−3φ−2−2ch2−ϕ1+r0yM42−ϕ2h+hϕ−λ2,q2T∗=2aλ21−2v+ah8v+4vϕ−ϕ−6−2ch2−ϕ1+r0yM42−ϕ2h+hϕ−λ2.
The forms of order quantities show that the retailers’ final optimal order quantity is immune to the trade credit interest raterT. This is because the manufacturer has to lower the wholesale price when charging a higher interest rate rT. The side negative effect of charging a higher rT is offset by the positive effect of a lower wholesale price. As a result, the optimal order quantities are not affected by the interest rate rT. It also shows that the optimal order quantities decrease in the bank interest rM due to the increased wholesale price caused by the manufacturer’s bank financing.
By substitutingwT∗, eT∗, q1T∗, and q2T∗ into πR1, πR2, and πM, we can get each participator’s optimal expected profit under trade credit:(18)πR1T∗=2aλ21−2v+ah8v+4vϕ−3ϕ−2−2ch2−ϕ1+r0yM2162−ϕ22h+hϕ−λ22,(19)πR2T∗=2aλ21−2v+ah8v+4vϕ−ϕ−6−2ch2−ϕ1+r0yM2162−ϕ22h+hϕ−λ22,(20)πMT∗=ha−2c−2cr0ηM28h2+ϕ−8λ2.
From equations (18)–(20), we find that the trade credit interest rate does not affect all the members’ profit when the retailers use trade credit financing.
By comparing the equilibriums under bank credit and trade credit, the following proposition holds.Proposition 4.
Trade credit can bring in more profits for all the supply chain members and contributes more to environmental protection, i.e.,qiT∗>qiB∗, eiT∗>eiB∗, wiT∗<wiB∗, πRiT∗>πRiB∗, and πMT∗>πMB∗.Proof.
(a)
By solvingqiT∗−qiB∗=0, we get ηM1=1+r0ηRaλ2+2ch2+ϕ2+r0ηR−3cλ21+r0ηR/2cr0h2+ϕ2+r0ηR−λ2. Then, rM1=1+r0ηRaλ2+2ch2+ϕ2+r0ηR−3cλ21+r0ηR/2ch2+ϕ2+r0ηR−λ2>1+r0ηRWhenrM=rM1, q1B∗=q1T∗. As q1T∗ decreases in rM, so q1T∗>q1B∗ when rM<1.In a similar way, we can proveeiT∗>eiB∗, wiT∗<wiB∗, πRiT∗>πRiB∗, and πMT∗>πMB∗.(b)
Comparing the numerators ofeiT∗ and eiB∗, we find that the numerator of eiT∗ is larger than that of eiB∗. Comparing the denominators of the two expressions, we find denominator of eiT∗ is smaller than that of eiB∗. Thus, eiT∗>eiB∗.(c)
wT∗=ah2+ϕ+2c1+r0ηMh2+ϕ−2λ2/41+rTh2+ϕ−λ2<ah2+ϕ+2c1+r0ηRh2+ϕ−2λ2/4h2+ϕ−λ2, wiB∗−ah2+ϕ+2c1+r0ηRh2+ϕ−2λ2/4h2+ϕ−λ2>0.SowiT∗<wiB∗(d)
Taking the first derivative ofπR1T∗ on rM, we get(21)∂πR1T∗∂rM=chr02h2+ϕ−λ2>0.SoπR1T∗ increases in rM.By solvingπR1T∗−πR1B∗=0, we get ηM2=1+r0ηRaλ2+2ch2+ϕ2+r0ηR−2cλ23+r0ηR/2cr0h2+ϕ2+r0ηR−λ2.Then,rM2=1+r0ηRaλ2+2ch2+ϕ2+r0ηR−2cλ23+r0ηR/2ch2+ϕ2+r0ηR−λ2>1+r0ηR.soπR1T∗>πR1B∗ when rM<1.In a similar way, we can proveπR2T∗>πR2B∗, and πMT∗>πMB∗.
Proposition4 shows that each member of the supply chain can make more profit under trade credit. The managerial insight we can get from this proposition is that offering financial assistance is an effective way for firms to earn more profit. As a result, when the retailers have no access for borrowing money from external financing, it is better for the manufacturer to offer trade credit to the downstream partners. On the other hand, Proposition 4 also shows that the emission abatement level under trade credit is higher than that under bank credit, which implies that trade credit contributes more in curbing carbon emissions. In addition, both retailers’ order quantities under trade credit are greater than those under bank credit. It implies that trade credit brings more items for consumers and enlarges consumption. This finding shows that it is a better choice for all the supply chain members to use trade credit. Therefore, the upstream firms should try their best to offer trade credit to downstream firms to earn more profits. To promote the development of enterprises and reduce carbon emissions more effectively, governments can promulgate policy to encourage firms to use trade credit.


### 4.3. Dual Financing Channel

We now turn to the scenario in which the retailers adopt both trade credit financing and bank credit financing at the same time. Because the cost of the first dollar of bank financing is zero, the retailers use bank financing first. With the increasing marginal cost of bank credit, the retailers tend to use trade credit once the marginal cost of bank financing equals the trade credit interest rate charged by the manufacturer. In this scenario, the retailers decide the order quantity and the loan size of each financing channel. Assuming retaileri’s loan size borrowed from the manufacturer is Ti, then, the loan size of bank credit is wqi−Ti. The interest rate of bank credit is(22)rRi=r0ηRwqi−Tiwqi.Given the wholesale price and the interest rates, the retailers’ profits consist of two parts: the operating profit and the financing cost, which are expressed as follows:(23)πR1=a−q1−ϕq2−wq1−wq1−T12r0ηRwq1−T1rT,πR2=a−q2−ϕq1−wq2−wq2−T22r0ηRwq2−T2rT.By solving the retailers’ maximizing problem, we can get the following proposition.Proposition 5.
When both trade credit and bank credit are used by the retailers, the optimal order quantities areq1TB∗=rT2w2−ϕ+4r0ηR2av+λe2−ϕ−w1+rT2−ϕ−aϕ1−v/4r0ηR4−ϕ2 and q2TB∗=rT2w2−ϕ+4r0ηRλe2−ϕ−w1+rT2−ϕ+a2−2v−vϕ/4r0ηR4−ϕ2. The optimal loan sizes of trade credit are T1∗=wq1TB∗2r0ηR−rT/2r0ηR and T2∗=wq2TB∗2r0ηR−rT/2r0ηR.Proof.
By taking the first derivatives, we can get(24)∂πR1∂q1=−2q1+λe+av−w−ϕq2−wr0ηR+r0ηRT12wq12,∂πR1∂T1=−rT+2r0ηRwq1−T1wq1,∂πR2∂q2=−2q2+λe+a1−v−w−ϕq1−wr0ηR+r0ηRT22wq22,∂πR2∂T2=−rT+2r0ηRwq2−T2wq2.
By taking the second derivatives, we can get(25)∂2πR1∂q12=−2−2r0ηRT12wq13,∂2πR1∂T12=−2r0ηRwq1.
Thus,πR1 is concave on q1 and T1.(26)∂2πR2∂q22=−2−2r0ηRT22wq23,∂2πR2∂T22=−2r0ηRwq2.
Thus,πR2 is concave on q2 and T2.
By solving the equations∂πR1/∂q1=0, ∂πR1/∂T1=0, ∂πR2/∂q2=0, and ∂πR2/∂T2=0, we can get q1TB∗=rT2w2−ϕ+4r0ηR2av+λe2−ϕ−w1+rT2−ϕ−aϕ1−v/4r0ηR4−ϕ2, q2TB∗=rT2w2−ϕ+4r0ηRλe2−ϕ−w1+rT2−ϕ+a2−2v−vϕ/4r0ηR4−ϕ2, T1∗=wq1TB∗2r0ηR−rT/2r0ηR, T2∗=wq2TB∗2r0ηR−rT/2r0ηR.
Proposition5 demonstrates the retailers’ optimal decisions under dual-channel financing. It shows that the order quantities decrease in interest rate and the bank credit interest rate. This is because financing cost directly increases the retailers’ operating cost. As for the loan size of trade credit, it highly depends on the interest rate charged by the manufacturer and that charged by the bank. The form of Ti shows that when rT≥2r0ηR, the retailers will not borrow from the manufacturer. When rT<2r0ηR, the retailers borrow from the manufacturer and the bank. Specifically, if rT=r0ηR, the retailers borrow the same loan size under the two financing channels; if rT<r0ηR, the retailers borrow more money from the manufacturer. The result shows that even if the trade credit interest rate is smaller than the bank interest rate, the retailers still borrow from the bank. From Proposition 4, we know that trade credit can bring more profit to a manufacturer. Thus, retailers can use trade credit to maintain a long-term and stable partnership with the manufacturer. These findings imply that when both financing channels are available, firms should properly allocate the loan amount upon trade credit and bank credit, so as to minimize the loan cost and maximize the profit.
Anticipating the retailers’ optimal order quantitiesq1TB∗, q2TB∗ and the loan size of trade credit T1∗+T2∗, the manufacturer chooses the optimal wholesale price and emission abatement level to maximize her expected profit. In this scenario, the loan size borrowed from the bank by the manufacturer is cq1+q2−wq1+q2+T1+T2. Thus, the manufacturer’s expected profit can be written as(27)πM=w−cq1+q2+rTT1+T2−12he2−cq1+q2−wq1+q2+T1+T2r0ηM.
Then, we can get the following conclusion that describes the manufacturer’s optimal decision under dual-channel financing.Proposition 6.
Under dual-channel financing, the optimal wholesale price offered by the manufacturer iswTB∗=−r0ηRah2+ϕ+2cλ21+r0ηM/D−c1+r0ηM/A, and increases in rM but decreases in rT and rR, where rR=r0ηR, rM=r0ηM, A=rTrT−r0ηM−2rR1+rT, and D=h2+ϕrT2−4rR1+rT+2λ2rTr0ηM−rT+2r0ηR1+rT. The optimal emission abatement level is eTB∗=λrTar0ηM−arT+crT+crTr0ηM+2r0ηRλ1+rTa−2c−2cr0ηM/−D, and it increases in rM but decreases in ϕ. If rT>2rM, eTB∗ increases in rR; if rT<2rM, eTB∗ decreases in rR. If rM>2rTrR2+rT/rT2+4rR, eTB∗ increases in rT; if rM<2rTrR2+rT/rT2+4rR, eTB∗ decreases in rT.Proof.
Substitutingq1TB∗ and q2TB∗ into equation (27), then taking the first derivative of on w and e, we can get(28)∂πM∂w=rT3wr0ηM−rT−r0ηRrTrTa+c+2λe−61+rTw2r02ηR22+ϕ+r0ηR−a+crT−2λe+41+rTwηM1+rTa+2c+λe−21+rTw+cr0ηM2r02ηR22+ϕ,∂πM∂e=rTλwr0ηM−rT−r0ηReh2+ϕ+2λc−w1+rT+cr0ηM2+ϕr0ηR.
By taking the second derivatives, we can get(29)∂2πM∂w2=rT3wr0ηM−rT−r0ηRrT4r0ηM−6rT1+rT−81+rTr02ηR22r02ηR22+ϕ,∂2πM∂e2=−h,∂2πM∂w∂e=λ−rT2+2r0ηR+r0rTηM+2ηRr0ηR2+ϕ,∂2πM∂w2∂2πM∂w∂e∂2πM∂w∂e∂2πM∂e2>0.
Thus,πM is concave on w and e. Let ∂πM/∂w=0 and ∂πM/∂e=0 denote wTB∗ the optimal wholesale price and eTB∗ the optimal carbon emission abatement level under bank financing, we get(30)wTB∗=−r0ηRah2+ϕ+2cλ21+r0ηMD−c1+r0ηMA,eTB∗=λrTar0ηM−arT+crT+crTr0ηM+2r0ηRλ1+rTa−2c−2cr0ηM−D,∂e∂rT=λr0ah2+ϕ−2cλ21+r0ηMrT2ηM−2rT2+rTηR+4r0ηMηRh2+ϕrT2−4r0ηR1+rT+2λ2r0ηM−rTrT+2r0ηR1+rT2.
By solvingrT2ηM−2rT2+rTηR+4r0ηMηR=0, we get rM=2rTrR2+rT/rT2+4rR. When rM>2rTrR2+rT/rT2+4rR, ∂e/∂rT>0; when rM<2rTrR2+rT/rT2+4rR, ∂e/∂rT<0.
Then, Proposition6 is proved.
When retailers adopt both bank financing and trade credit, the wholesale price and the emission abatement level decided by the manufacturer highly depend on the bank credit interest rate charged on retailers and the trade credit interest rate. It shows that the wholesale price increases inrM but decreases in rT and rR. As the manufacturer does not have enough capital to invest in producing, she borrows from the bank. With rM increasing, the financing cost of the manufacturer increases. To offset the financing cost, the manufacturer tends to charge a higher wholesale price to increase the sales revenue. Under dual-channel financing, the manufacturer should also pay attention to the bank interest rate charged on retailers. When bank charges a higher interest rate on retailers, the manufacturer should lower the wholesale price to promote retailers to order more products.
Interestingly, we find that the optimal emission abatement level increases inrM. As we know, a larger rM increases the manufacturer’s financial cost. To cover the increased cost, the manufacturer either sets a higher wholesale price or promotes the sales volume. Given that the consumers are eco-friendly aware, the manufacturer chooses to invest in a higher emission abatement level to increase the market demand, whereas when ϕ is large, it implies that the competition between retailers is fierce, which lowers the retail prices. In this case, the marginal profit of the manufacturer is reduced. To save the operating cost, the manufacturer sets a lower emission abatement level. The relationship between eTB∗ and rT and rR is not conclusive. When rT is relatively large, i.e., rT>2rM, the financial profit earned from the retailers is larger than the financial cost paid to the bank. With a larger bank interest rate rR, the retailers’ profits decrease. To maintain the retailers’ normal operation, the manufacturer has incentive to invest in a higher emission abatement level to increase the retail prices. When rM>2rTrR2+rT/rT2+4rR, the financial cost exceeds the financing profit for the manufacturer. In this scenario, the manufacturer is better off by charging a higher trade credit interest rate. However, the retailers are worse off due to the increased financial cost. To drive retailers to order more products, the manufacturer is willing to invest in a higher emission abatement level. Thus, there exits eTB∗ increases in rT when rM>2rTrR2+rT/rT2+4rR. Under carbon emission regulation, if the bank interest rate is relatively large, firms either charge a higher interest rate on the borrowers to guarantee carbon emissions meeting the requirements or they can charge a higher wholesale price to increase revenue and use it to buy carbon emission permits in the carbon trading market.Proposition 7.
IfrT<2rM, there exits eTB∗>eT∗; if rT>2rM, there exits eTB∗<eT∗.Proposition7 demonstrates the relationship of the optimal emission abatement levels under different financial channels. It shows that when rT is relatively small, i.e., rT<2rM, the emission abatement level under dual-channel financing is larger. In this scenario, bank credit is not only a possible way to finance capital-constrained firms but also helps to control the emission abatement level and brings more environmental benefit. When rT is relatively large, trade credit contributes more in abating carbon emissions than dual-channel financing. The managerial insight behind this result is that choosing an appropriate financing way can not only solve the financing difficulties of enterprises but also help to control carbon emissions.As the forms of the wholesale price and each member’s profit under dual-channel financing are too complicated, we compare them with single-channel financing in the numerical experiment section.Proof.
eTB∗−eT∗=λrTrT−2r0ηMah2+ϕ−2cλ21+r0ηM/2h2+ϕ−λ2h2+ϕrT2−4r0ηR1+rT+2λ2r0ηM−rTrT+2r0ηR1+rT. Thus, when rT<2rM, eTB∗>eT∗; when rT>2rM, eTB∗<eT∗.


## 4.1. Bank Credit Financing

Assume that the bank credit cost is convex in the borrowers’ book leverage. The rational reasons include managerial risk aversion [42], adverse selection [43], and microeconomic foundations [44]. For simplicity, in line with Chod et al. [45], we adopt the bank credit cost by assuming that the bank interest rate increases linearly in the borrower’s book leverage. Thus, the interest rate charged by the bank on the retailer over loan size y is given by(2)rRi=r0ηRywqi,r0 is the basic bank interest rate, ηR is the impact factor of bank interest rate related to the retailers. When only bank financing is adopted by retailers, the bank interest rate equals(3)rR1=rR2=r0ηR.Given the wholesale price, the retailers’ expected profit are written as(4)πR1=va−q1−ϕq2q1−wq11+r0ηR,πR2=1−va−q2−ϕq1q2−wq21+r0ηR.The next proposition fully characterizes the retailers’ optimal ordering strategy when adopting bank credit financing.Proposition 1.
Under bank credit financing, the retailers’ optimal order quantities areq1B∗=av2+ϕ−ϕ−2−ϕw2+r0ηR−λe/4−ϕ2, q2B∗=a2−v2+ϕ−2−vw2+r0ηR−λe/4−ϕ2.Proof.
Taking the first derivative ofπR1 on q1, we get(5)∂πR1∂q1=−2q1+λe+av−w−ϕq2−wr0ηR.
Taking the second derivative ofπR1 on q1, we get(6)∂2πR1∂q12=−2<0.
Thus,πR1 is concave on q1. Similarly,(7)∂πR2∂q2=−2q2+λe+a1−v−w−ϕq1−wr0ηR,∂2πR2∂q22=−2<0,
Thus,πR2 is concave on q2.
By solving equations∂πR1/∂q1=0 and ∂πR2/∂q2=0, denote q1B∗ and q2B∗ the optimal order quantities under bank credit, we can get(8)q1B∗=av2+ϕ−ϕ−2−ϕw2+r0ηR−λe4−ϕ2,q2B∗=a2−v2+ϕ−2−vw2+r0ηR−λe4−ϕ2.
Then, Proposition1 is proved.
Proposition1 shows that the retailers’ optimal order quantities increase in an eco-friendly level. The intuition behind this result is that consumers’ environmental awareness promotes them to buy green products. To meet consumers’ demand, the retailers enhance their order quantity. It also shows that the optimal order quantities decrease in the wholesale price and the bank interest rate. This is because a higher wholesale price increases the retailers’ ordering cost and a higher interest rate of bank credit increases the retailers’ financing costs. From Proposition 1, we know that retailers can benefit from consumers’ eco-friendly awareness. As a result, in practice, retailing firms can offer different kinds of items, such as low-carbon item and ordinary item, to satisfy different kinds of consumers, which may bring more profits for them.
The manufacturer’s revenuewq1+q2 is enough to cover the production cost cq1+q2, so she does not to need borrow money from the bank. In this case, the manufacturer’s expected profit is(9)πM=w−cq1+q2−12he2.
Then, we can get the following proposition.Proposition 2.
Under bank financing, the optimal wholesale price decided by the manufacturer iswB∗=ah2+ϕ+2ch2+ϕ2+r0ηR−4cλ2/4h2+ϕ2+r0ηR−4λ2, and the optimal emission abatement level is eB∗=aλ−2cλ2+r0ηR/2h2+ϕ2+r0ηR−2λ2.Proof.
Proof. Substitutingq1 and q2 into equation (9), then taking the first derivatives, we get(10)∂πM∂w=a+4c+2λe−8w+2r0ηRc−2w2+ϕ,∂πM∂e=−eh+4λw−c2+ϕ.
By taking the second derivatives, we get(11)∂2πM∂w2=−8−4r0ηR2+ϕ,∂2πM∂e2=−h,∂2πM∂e∂w=4λ2+ϕ,−8−4r0ηR2+ϕ4λ2+ϕ4λ2+ϕ−h=h8+4r0ηR−16λ22+ϕ2>0.
Thus,πMis concave on w and e. Let ∂πM/∂w=0 and ∂πM/∂e=0, denote wB∗ the optimal wholesale price and eB∗ the optimal carbon emission abatement level under bank financing, we get(12)wB∗=ah2+ϕ+2ch2+ϕ2+r0ηR−4cλ24h2+ϕ2+r0ηR−4λ2,eB∗=aλ−2cλ2+r0ηR2h2+ϕ2+r0ηR−2λ2.
Proposition2 shows that the wholesale price the trade credit increases in r0ηR, λ and c, decreases in ϕ and h. When the bank interest rate is high, the retailers tend to reduce the order quantity to save cost. In this case, the manufacturer charges a higher wholesale price to offset the revenue loss caused by reduced sales volume. λ represents consumers’ sensitivity of eco-friendly products, and a higher λ means consumers are willing to pay higher price for eco-friendly products. Then, charging a higher wholesale price does not reduce the retailers’ order quantity. Thus, w increases in λ. Contrary to our intuition, w decreases in h. h implies the manufacturer capability in reducing emission, and a higher h followed by a lower emission abatement level, which lowers the retailers’ order quantities. To induce the retailers to order more products, the manufacturer has to provide a lower wholesale price. In conclusion, when deciding wholesale price, the manufacturer should comprehensively consider bank interest rate, market competition, and its efficiency in abating carbon emissions.
From Proposition2, we can also find that the optimal emission abatement level increases in λ but decreases in r0ηR, ϕ, c, and h. With λ increasing, consumers are willing to pay higher price for eco-friendly products, which induce the manufacturer to invest in a higher emission abatement level. When r0ηR is high, the retailers have to pay more extra interest, which lowers their purchasing ability. As a result, the manufacturer has no incentive to invest in emission reduction. Thus, eB∗ decreases in r0ηR. With ϕ increasing, the competition between retailers is intensified, which enhances the total order quantities. In this scenario, the manufacturer has no incentive to invest in a high emission abatement to increase sales volume. Therefore, eB∗ decreases in ϕ. It implies that market competition hurts carbon emission reduction. To reduce more carbon emissions to meet government regulatory requirements, the manufacturer should try its best to weak competition. Obviously, either c or h increases the manufacturer’s operating cost, so eB∗decreases in c and h. From the analysis, it shows that both wholesale price and emission abatement affected by interest rate were charged on the retailers. Hence, to improve carbon emission abatement level and induce retailers to order more products, it is better for the upstream-firms to cooperate with the retailers to strive for lower interest rate from banks.
SubstitutingwB∗, eB∗, q1B∗, and q2B∗ into πR1, πR2, and πM, we can get every member’s maximized profit.(13)πR1B∗=2aλ21−2v+ah8v+4vϕ−3ϕ−22+r0ηR−2ch2−ϕ1+r0ηR2162−ϕ2h2+ϕ2+r0ηR−λ22,πR2B∗=2aλ21−2v+ah8v+4vϕ−6−ϕ2+r0ηR−2ch2−ϕ1+r0ηR2162−ϕ2h2+ϕ2+r0ηR−λ22,πMB∗=ha−4c−2cr0ηR28h2+ϕ2+r0ηR−8λ2.


## 4.2. Trade Credit

We now tend to see how the manufacturer and retailers make decisions when retailers adopt trade credit financing. As the retailers have more opportunities to reach terminal consumers, they can obtain more information about the market demand. Therefore, the manufacturer does not worry about the retailers’ bankruptcy risk and is willing to provide the loans that the retailers need. Since the retailers’ initial working capital is zero [40], the manufacturer permits retailers to pay for items at the end of the sales season, and charges retailers the extra interest rate rT. According to the assumption, the manufacturer does not have capital to invest in production and borrows from a bank at an interest rate rM. Similar to the retailers, the manufacturer confronts an interest rate that increases linearly in her total loan size yM,(14)rM=r0ηMyMcq1+q2.After the manufacturer announces the wholesale price and the interest rate of trade credit, the retailers order products from the manufacturer. Then, the retailers’ problems are to maximize their expected profits by determiningq1 and q2, which are described as(15)πR1=a−q1−ϕq2q1−wq11+rT,πR2=a−q2−ϕq1q2−wq21+rT.Then, we can get the retailers’ optimal order quantitiesq1T∗=av2+ϕ−ϕ−2−ϕw2+rT−λe/4−ϕ2 and q2T∗=a2−v2+ϕ−2−ϕw2+rT−λe/4−ϕ2.As the manufacturer’s loan size isyM=cq1+q2 in this scenario, the interest rate of bank financing on the manufacturer is rM=r0ηM. For the manufacturer, her profit consists of four parts: the sales profit that equals sales revenue minus production cost; the interest earned from trade credit; the emission abatement cost; and the bank financing cost. Then, the manufacturer’s expected profit is given by(16)πM=w−cq1+q2+rTq1+q2−12he2−cq1+q2r0ηM.Then, we can get the next proposition that describes the manufacturer’s optimal decision under trade credit financing.Proposition 3.
When the retailers use trade credit financing, the optimal wholesale price iswT∗=ah2+ϕ+2c1+r0ηMh2+ϕ−2λ2/41+rTh2+ϕ−λ2, and the optimal emission abatement level is eT∗=aλ−2cλ1+r0ηM/22h+hϕ−λ2.Proof.
The proof is similar to Proposition2.
Proposition3 shows that the optimal wholesale price increases in rM but decreases in rT. As bank interest rate increases the manufacturer’s cost, the manufacturer tends to charge retailers higher wholesale price to cover the cost. On the other hand, the trade credit offered to retailers brings extra financing profit to the manufacturer. With rT increasing, the retailers’ financing cost is enhanced. Thus, the manufacturer lowers the wholesale price to enlarge the retailers’ demand for products. Hence, the manufacturer needs to balance trade credit interest rate charged on retailers and the wholesale price to make more profit. From Proposition 3, we can also see that the emission reduction level decreases in the coefficient of emission reduction h. To enhance emission abatement level, the manufacturer should find an effective way to lower coefficient h. The findings also show that the emission abatement level eT∗ decreases in rM due to the increased extra financing cost for the manufacturer. Interestingly, we find that eT∗ has nothing to do with rT, which implies that the emission abatement level is not affected by the retailers’ financial status.
By substitutingwT∗ and eT∗ into q1T∗ and q2T∗, we can get(17)q1T∗=2aλ21−2v+ah8v+4vϕ−3φ−2−2ch2−ϕ1+r0yM42−ϕ2h+hϕ−λ2,q2T∗=2aλ21−2v+ah8v+4vϕ−ϕ−6−2ch2−ϕ1+r0yM42−ϕ2h+hϕ−λ2.
The forms of order quantities show that the retailers’ final optimal order quantity is immune to the trade credit interest raterT. This is because the manufacturer has to lower the wholesale price when charging a higher interest rate rT. The side negative effect of charging a higher rT is offset by the positive effect of a lower wholesale price. As a result, the optimal order quantities are not affected by the interest rate rT. It also shows that the optimal order quantities decrease in the bank interest rM due to the increased wholesale price caused by the manufacturer’s bank financing.
By substitutingwT∗, eT∗, q1T∗, and q2T∗ into πR1, πR2, and πM, we can get each participator’s optimal expected profit under trade credit:(18)πR1T∗=2aλ21−2v+ah8v+4vϕ−3ϕ−2−2ch2−ϕ1+r0yM2162−ϕ22h+hϕ−λ22,(19)πR2T∗=2aλ21−2v+ah8v+4vϕ−ϕ−6−2ch2−ϕ1+r0yM2162−ϕ22h+hϕ−λ22,(20)πMT∗=ha−2c−2cr0ηM28h2+ϕ−8λ2.
From equations (18)–(20), we find that the trade credit interest rate does not affect all the members’ profit when the retailers use trade credit financing.
By comparing the equilibriums under bank credit and trade credit, the following proposition holds.Proposition 4.
Trade credit can bring in more profits for all the supply chain members and contributes more to environmental protection, i.e.,qiT∗>qiB∗, eiT∗>eiB∗, wiT∗<wiB∗, πRiT∗>πRiB∗, and πMT∗>πMB∗.Proof.
(a)
By solvingqiT∗−qiB∗=0, we get ηM1=1+r0ηRaλ2+2ch2+ϕ2+r0ηR−3cλ21+r0ηR/2cr0h2+ϕ2+r0ηR−λ2. Then, rM1=1+r0ηRaλ2+2ch2+ϕ2+r0ηR−3cλ21+r0ηR/2ch2+ϕ2+r0ηR−λ2>1+r0ηRWhenrM=rM1, q1B∗=q1T∗. As q1T∗ decreases in rM, so q1T∗>q1B∗ when rM<1.In a similar way, we can proveeiT∗>eiB∗, wiT∗<wiB∗, πRiT∗>πRiB∗, and πMT∗>πMB∗.(b)
Comparing the numerators ofeiT∗ and eiB∗, we find that the numerator of eiT∗ is larger than that of eiB∗. Comparing the denominators of the two expressions, we find denominator of eiT∗ is smaller than that of eiB∗. Thus, eiT∗>eiB∗.(c)
wT∗=ah2+ϕ+2c1+r0ηMh2+ϕ−2λ2/41+rTh2+ϕ−λ2<ah2+ϕ+2c1+r0ηRh2+ϕ−2λ2/4h2+ϕ−λ2, wiB∗−ah2+ϕ+2c1+r0ηRh2+ϕ−2λ2/4h2+ϕ−λ2>0.SowiT∗<wiB∗(d)
Taking the first derivative ofπR1T∗ on rM, we get(21)∂πR1T∗∂rM=chr02h2+ϕ−λ2>0.SoπR1T∗ increases in rM.By solvingπR1T∗−πR1B∗=0, we get ηM2=1+r0ηRaλ2+2ch2+ϕ2+r0ηR−2cλ23+r0ηR/2cr0h2+ϕ2+r0ηR−λ2.Then,rM2=1+r0ηRaλ2+2ch2+ϕ2+r0ηR−2cλ23+r0ηR/2ch2+ϕ2+r0ηR−λ2>1+r0ηR.soπR1T∗>πR1B∗ when rM<1.In a similar way, we can proveπR2T∗>πR2B∗, and πMT∗>πMB∗.
Proposition4 shows that each member of the supply chain can make more profit under trade credit. The managerial insight we can get from this proposition is that offering financial assistance is an effective way for firms to earn more profit. As a result, when the retailers have no access for borrowing money from external financing, it is better for the manufacturer to offer trade credit to the downstream partners. On the other hand, Proposition 4 also shows that the emission abatement level under trade credit is higher than that under bank credit, which implies that trade credit contributes more in curbing carbon emissions. In addition, both retailers’ order quantities under trade credit are greater than those under bank credit. It implies that trade credit brings more items for consumers and enlarges consumption. This finding shows that it is a better choice for all the supply chain members to use trade credit. Therefore, the upstream firms should try their best to offer trade credit to downstream firms to earn more profits. To promote the development of enterprises and reduce carbon emissions more effectively, governments can promulgate policy to encourage firms to use trade credit.


## 4.3. Dual Financing Channel

We now turn to the scenario in which the retailers adopt both trade credit financing and bank credit financing at the same time. Because the cost of the first dollar of bank financing is zero, the retailers use bank financing first. With the increasing marginal cost of bank credit, the retailers tend to use trade credit once the marginal cost of bank financing equals the trade credit interest rate charged by the manufacturer. In this scenario, the retailers decide the order quantity and the loan size of each financing channel. Assuming retaileri’s loan size borrowed from the manufacturer is Ti, then, the loan size of bank credit is wqi−Ti. The interest rate of bank credit is(22)rRi=r0ηRwqi−Tiwqi.Given the wholesale price and the interest rates, the retailers’ profits consist of two parts: the operating profit and the financing cost, which are expressed as follows:(23)πR1=a−q1−ϕq2−wq1−wq1−T12r0ηRwq1−T1rT,πR2=a−q2−ϕq1−wq2−wq2−T22r0ηRwq2−T2rT.By solving the retailers’ maximizing problem, we can get the following proposition.Proposition 5.
When both trade credit and bank credit are used by the retailers, the optimal order quantities areq1TB∗=rT2w2−ϕ+4r0ηR2av+λe2−ϕ−w1+rT2−ϕ−aϕ1−v/4r0ηR4−ϕ2 and q2TB∗=rT2w2−ϕ+4r0ηRλe2−ϕ−w1+rT2−ϕ+a2−2v−vϕ/4r0ηR4−ϕ2. The optimal loan sizes of trade credit are T1∗=wq1TB∗2r0ηR−rT/2r0ηR and T2∗=wq2TB∗2r0ηR−rT/2r0ηR.Proof.
By taking the first derivatives, we can get(24)∂πR1∂q1=−2q1+λe+av−w−ϕq2−wr0ηR+r0ηRT12wq12,∂πR1∂T1=−rT+2r0ηRwq1−T1wq1,∂πR2∂q2=−2q2+λe+a1−v−w−ϕq1−wr0ηR+r0ηRT22wq22,∂πR2∂T2=−rT+2r0ηRwq2−T2wq2.
By taking the second derivatives, we can get(25)∂2πR1∂q12=−2−2r0ηRT12wq13,∂2πR1∂T12=−2r0ηRwq1.
Thus,πR1 is concave on q1 and T1.(26)∂2πR2∂q22=−2−2r0ηRT22wq23,∂2πR2∂T22=−2r0ηRwq2.
Thus,πR2 is concave on q2 and T2.
By solving the equations∂πR1/∂q1=0, ∂πR1/∂T1=0, ∂πR2/∂q2=0, and ∂πR2/∂T2=0, we can get q1TB∗=rT2w2−ϕ+4r0ηR2av+λe2−ϕ−w1+rT2−ϕ−aϕ1−v/4r0ηR4−ϕ2, q2TB∗=rT2w2−ϕ+4r0ηRλe2−ϕ−w1+rT2−ϕ+a2−2v−vϕ/4r0ηR4−ϕ2, T1∗=wq1TB∗2r0ηR−rT/2r0ηR, T2∗=wq2TB∗2r0ηR−rT/2r0ηR.
Proposition5 demonstrates the retailers’ optimal decisions under dual-channel financing. It shows that the order quantities decrease in interest rate and the bank credit interest rate. This is because financing cost directly increases the retailers’ operating cost. As for the loan size of trade credit, it highly depends on the interest rate charged by the manufacturer and that charged by the bank. The form of Ti shows that when rT≥2r0ηR, the retailers will not borrow from the manufacturer. When rT<2r0ηR, the retailers borrow from the manufacturer and the bank. Specifically, if rT=r0ηR, the retailers borrow the same loan size under the two financing channels; if rT<r0ηR, the retailers borrow more money from the manufacturer. The result shows that even if the trade credit interest rate is smaller than the bank interest rate, the retailers still borrow from the bank. From Proposition 4, we know that trade credit can bring more profit to a manufacturer. Thus, retailers can use trade credit to maintain a long-term and stable partnership with the manufacturer. These findings imply that when both financing channels are available, firms should properly allocate the loan amount upon trade credit and bank credit, so as to minimize the loan cost and maximize the profit.
Anticipating the retailers’ optimal order quantitiesq1TB∗, q2TB∗ and the loan size of trade credit T1∗+T2∗, the manufacturer chooses the optimal wholesale price and emission abatement level to maximize her expected profit. In this scenario, the loan size borrowed from the bank by the manufacturer is cq1+q2−wq1+q2+T1+T2. Thus, the manufacturer’s expected profit can be written as(27)πM=w−cq1+q2+rTT1+T2−12he2−cq1+q2−wq1+q2+T1+T2r0ηM.
Then, we can get the following conclusion that describes the manufacturer’s optimal decision under dual-channel financing.Proposition 6.
Under dual-channel financing, the optimal wholesale price offered by the manufacturer iswTB∗=−r0ηRah2+ϕ+2cλ21+r0ηM/D−c1+r0ηM/A, and increases in rM but decreases in rT and rR, where rR=r0ηR, rM=r0ηM, A=rTrT−r0ηM−2rR1+rT, and D=h2+ϕrT2−4rR1+rT+2λ2rTr0ηM−rT+2r0ηR1+rT. The optimal emission abatement level is eTB∗=λrTar0ηM−arT+crT+crTr0ηM+2r0ηRλ1+rTa−2c−2cr0ηM/−D, and it increases in rM but decreases in ϕ. If rT>2rM, eTB∗ increases in rR; if rT<2rM, eTB∗ decreases in rR. If rM>2rTrR2+rT/rT2+4rR, eTB∗ increases in rT; if rM<2rTrR2+rT/rT2+4rR, eTB∗ decreases in rT.Proof.
Substitutingq1TB∗ and q2TB∗ into equation (27), then taking the first derivative of on w and e, we can get(28)∂πM∂w=rT3wr0ηM−rT−r0ηRrTrTa+c+2λe−61+rTw2r02ηR22+ϕ+r0ηR−a+crT−2λe+41+rTwηM1+rTa+2c+λe−21+rTw+cr0ηM2r02ηR22+ϕ,∂πM∂e=rTλwr0ηM−rT−r0ηReh2+ϕ+2λc−w1+rT+cr0ηM2+ϕr0ηR.
By taking the second derivatives, we can get(29)∂2πM∂w2=rT3wr0ηM−rT−r0ηRrT4r0ηM−6rT1+rT−81+rTr02ηR22r02ηR22+ϕ,∂2πM∂e2=−h,∂2πM∂w∂e=λ−rT2+2r0ηR+r0rTηM+2ηRr0ηR2+ϕ,∂2πM∂w2∂2πM∂w∂e∂2πM∂w∂e∂2πM∂e2>0.
Thus,πM is concave on w and e. Let ∂πM/∂w=0 and ∂πM/∂e=0 denote wTB∗ the optimal wholesale price and eTB∗ the optimal carbon emission abatement level under bank financing, we get(30)wTB∗=−r0ηRah2+ϕ+2cλ21+r0ηMD−c1+r0ηMA,eTB∗=λrTar0ηM−arT+crT+crTr0ηM+2r0ηRλ1+rTa−2c−2cr0ηM−D,∂e∂rT=λr0ah2+ϕ−2cλ21+r0ηMrT2ηM−2rT2+rTηR+4r0ηMηRh2+ϕrT2−4r0ηR1+rT+2λ2r0ηM−rTrT+2r0ηR1+rT2.
By solvingrT2ηM−2rT2+rTηR+4r0ηMηR=0, we get rM=2rTrR2+rT/rT2+4rR. When rM>2rTrR2+rT/rT2+4rR, ∂e/∂rT>0; when rM<2rTrR2+rT/rT2+4rR, ∂e/∂rT<0.
Then, Proposition6 is proved.
When retailers adopt both bank financing and trade credit, the wholesale price and the emission abatement level decided by the manufacturer highly depend on the bank credit interest rate charged on retailers and the trade credit interest rate. It shows that the wholesale price increases inrM but decreases in rT and rR. As the manufacturer does not have enough capital to invest in producing, she borrows from the bank. With rM increasing, the financing cost of the manufacturer increases. To offset the financing cost, the manufacturer tends to charge a higher wholesale price to increase the sales revenue. Under dual-channel financing, the manufacturer should also pay attention to the bank interest rate charged on retailers. When bank charges a higher interest rate on retailers, the manufacturer should lower the wholesale price to promote retailers to order more products.
Interestingly, we find that the optimal emission abatement level increases inrM. As we know, a larger rM increases the manufacturer’s financial cost. To cover the increased cost, the manufacturer either sets a higher wholesale price or promotes the sales volume. Given that the consumers are eco-friendly aware, the manufacturer chooses to invest in a higher emission abatement level to increase the market demand, whereas when ϕ is large, it implies that the competition between retailers is fierce, which lowers the retail prices. In this case, the marginal profit of the manufacturer is reduced. To save the operating cost, the manufacturer sets a lower emission abatement level. The relationship between eTB∗ and rT and rR is not conclusive. When rT is relatively large, i.e., rT>2rM, the financial profit earned from the retailers is larger than the financial cost paid to the bank. With a larger bank interest rate rR, the retailers’ profits decrease. To maintain the retailers’ normal operation, the manufacturer has incentive to invest in a higher emission abatement level to increase the retail prices. When rM>2rTrR2+rT/rT2+4rR, the financial cost exceeds the financing profit for the manufacturer. In this scenario, the manufacturer is better off by charging a higher trade credit interest rate. However, the retailers are worse off due to the increased financial cost. To drive retailers to order more products, the manufacturer is willing to invest in a higher emission abatement level. Thus, there exits eTB∗ increases in rT when rM>2rTrR2+rT/rT2+4rR. Under carbon emission regulation, if the bank interest rate is relatively large, firms either charge a higher interest rate on the borrowers to guarantee carbon emissions meeting the requirements or they can charge a higher wholesale price to increase revenue and use it to buy carbon emission permits in the carbon trading market.Proposition 7.
IfrT<2rM, there exits eTB∗>eT∗; if rT>2rM, there exits eTB∗<eT∗.Proposition7 demonstrates the relationship of the optimal emission abatement levels under different financial channels. It shows that when rT is relatively small, i.e., rT<2rM, the emission abatement level under dual-channel financing is larger. In this scenario, bank credit is not only a possible way to finance capital-constrained firms but also helps to control the emission abatement level and brings more environmental benefit. When rT is relatively large, trade credit contributes more in abating carbon emissions than dual-channel financing. The managerial insight behind this result is that choosing an appropriate financing way can not only solve the financing difficulties of enterprises but also help to control carbon emissions.As the forms of the wholesale price and each member’s profit under dual-channel financing are too complicated, we compare them with single-channel financing in the numerical experiment section.Proof.
eTB∗−eT∗=λrTrT−2r0ηMah2+ϕ−2cλ21+r0ηM/2h2+ϕ−λ2h2+ϕrT2−4r0ηR1+rT+2λ2r0ηM−rTrT+2r0ηR1+rT. Thus, when rT<2rM, eTB∗>eT∗; when rT>2rM, eTB∗<eT∗.


## 5. Numerical Experiment

In this section, we tend to perform several sets of numerical experiments to illustrate the managerial insight that resulted from this work. We seta=300, v=0.6, r0=0.05, c=10, h=2000, ηR=1, ηM=1.5 (ηM is variable when we investigate the impact of bank interest on the manufacturer), ϕ=0.3 (ϕ is variable when examining the impact of competition), λ=5 (λ is variable when examining the impact of consumers’ sensitivity on eco-friendly products), rT=0.05 (rT is variable when discussing the allocation of loan size of different kinds of financing).The first set of numerical examples demonstrates the impacts of consumers’ sensitivity on eco-friendly productλ. From Figure 1 we can see that the manufacturer can charge a high wholesale price when consumers are very sensitive to low-carbon products. This is because consumers are willing to pay higher price for the low-carbon items. Hence, providing a high wholesale price does not affect the retailers’ order willingness. On the other hand, if consumers pay much more attention to eco-friendly awareness, the manufacturer also has incentive to invest in a high emission abatement level to promote sales volume. Thus, the carbon emission abatement level e increases in λ, which is shown in Figure 2. It means that consumers’ environmental awareness plays a significant role in curbing carbon emissions in the world.Figure 1
Impact ofλ on w.Figure 2
Impact ofλ on e.The impacts ofλ on the retailers’ order quantity are shown in Figures 3 and 4 . As can be seen, both retailers’ order quantities increase in λ regardless of using any kind of financing methods. It implies that consumers’ eco-friendly awareness enhances the demand for low-carbon products. From Figures 3 to 4, we can also see that q1TB>q1T>q1B and q2TB>q2T>q2B. It means that the retailers order the most quantities when using dual-financing channels and they order the least order quantities when using bank credit financing.Figure 3
Impact ofλ on q1.Figure 4
Impact ofλ on q2.Figures5–7 depict the impact of λ on each supply chain member’s profit. As we can see, each member’s profit increases in λ. It shows that abating carbon emission is not only a way to protect the environment but also an effective way to increase firms’ revenue. Therefore, under the background of low-carbon economy, abating carbon emissions is a good way for firms to maintain sustainable development. On the other hand, Figures 5–7 also show that each member makes the most profit under dual-financing channel and makes the least profit under bank credit financing. It proves that trade credit is better than bank credit for all the supply chain members. If both trade credit and bank credit are available to capital-constrained firms, using dual-financing methods is the best choice.Figure 5
Impact ofλ on πR1.Figure 6
Impact ofλ on πR2.Figure 7
Impact ofλ on πM.This set of numerical experiments illustrates how competition affects each member’s decision and profit. The wholesale price under bank credit financing is relatively small, and it is difficult to show the impact ofϕ on w when drawing wT, wB, and wTB in one picture. Thus, we ignore wB in this case. As shown in Figure 8, intensified competition lowers the wholesale price. We know that the prices of products reduce due to competition. Then, the marginal profit of retailers is lowered. In this case, charging a higher wholesale price hurts retailers’ order incentive. As a result, w decreases in ϕ. In this scenario, the manufacturer does not have incentive to invest in high emission abatement level. Thus, e decreases in ϕ, which is shown in Figure 9. It implies that competition is bad for carbon emission abatement.Figure 8
Impact ofϕ on w.Figure 9
Impact ofϕ on e.In Figure10, it is interesting to note that retailer 1’s order quantity decreases in ϕ first but increases in ϕ when the order quantity reaches the lowest point. With the competition intensity increases, the price of products sold by retailer 1 decreases. In this case, retailer 1 tends to reduce the order quantity to avoid revenue loss. Whereas when ϕ is much higher, retailer 1 can make more profit by selling more products. For retailer 2, the market share is very small and he does not have competitive advantage. Thus, retailer 2’s order quantity decreases in ϕ all the time, which is shown in Figure 11.Figure 10
Impact ofϕ on q1.Figure 11
Impact ofϕ on q2.Based on the impacts ofϕ on the retailers’ order quantity and wholesale price, we can easily get to know the the relationship between retailer 1’s profit πR1 and ϕ is an inverse-U shape, as shown in Figure 12. The profits of both retailer 2 and the manufacturer decrease in ϕ, which are shown in Figures 13 and 14. From this set of numerical experiments, we find that market competition hurts the manufacturer and the weak retailer, and only benefits the strong retailer under certain conditions.Figure 12
Impact ofϕ on πR1.Figure 13
Impact ofϕ on πR2.Figure 14
Impact ofϕ on πM.This set of numerical examples examines how the impact factor of bank interest rate related to manufacturerηM affects each member’s decision and profit. As all decisions are immune to ηM when the retailers adopt bank credit financing, we ignore the impacts of ηM in bank financing scenario here. From Figure 15, we find that the emission abatement e decreases in ηM under trade credit financing. In this scenario, the manufacturer’s all productive fund is borrowed from the bank, which increases the manufacturer’s financing cost. With ηM increasing, the manufacturer prefers to lower the emission abatement level to save operational cost. However, when dual-channel financing is adopted, the bank financing cost is undertaken by the manufacturer and retailers. Then, the manufacturer can use the saved financing cost to invest in abating carbon emissions. Thus, e increases in ηM under dual-channel financing. For the wholesale price, the manufacturer tends to charge a higher price when financing cost increases, as shown in Figure 16.Figure 15
Impact ofηM on e.Figure 16
Impact ofηM on w.We know that the manufacturer transfers some operating cost to the downstream retailers by charging a larger wholesale price. The retailers will cut the order quantities due to the inverse relationship between the wholesale price and order quantity. Thus, the retailers’ order quantities decrease inηM regardless of using trade credit financing or dual-channel financing, which are shown in Figures 17 and 18. The two figures also show that each retailer’s order quantity under dual-channel financing is larger than that under trade credit financing, which once again proves that dual-channel financing is better than trade credit financing.Figure 17
Impact ofηM on q1.Figure 18
Impact ofηM on q2.WithηM increasing, the increased wholesale price enhances operating cost of retailer 1. As trade credit is better than bank credit, retailer 1 chooses to increase the loan size of trade credit instead of bank credit. Thus, T1T and T1TB increase in ηM, which are shown in Figure 19. Contrary to retailer 1, retailer 2’s trade credit loan size decreases in ηM. According to the parameter setting, retailer 2 has a small market share. When facing a large wholesale price, retailer 2 cuts the order quantity dramatically, which is followed by a small amount of operating cost. In this scenario, there is no need for retailer 2 to borrow more money. Therefore, the trade credit loan size of retailer 2 decreases in ηM, which is shown in Figure 20.Figure 19
Impact ofηM on T1.Figure 20
Impact ofηM on T2.On the one hand, the wholesale price increases inηM; on the other hand, the order quantities decrease in ηM. As a result, as shown in Figure 21, the retailers’ profits decrease in ηM. For the manufacturer, we find that the impacts of ηM on her profits are different under different financing channels. When using trade credit financing, the manufacturer’s profit decreases in ηM. This is because all the bank financing costs are borne by the manufacturer, whereas the manufacturer’s profit increases in ηM when using dual-channel financing. The logic behind this result is that the retailers will undertake some bank financing cost when they borrow money from the bank, which helps the manufacturer to save the operating cost. Thus, we can get the relationship between ηM and πM in Figure 22.Figure 21
Impact ofηM on πR.Figure 22
Impact ofηM on πM.The last set of numerical examples illustrates the retailers’ loan size allocation under dual-channel financing. Figures23 and 24 show that the loan size of trade credit decreases in trade credit interest, but the loan size of bank credit increases in rT. It also shows that the retailers’ trade credit loan size is equal to their bank credit loan size when rT=rT1. The retailers’ trade credit loan size is larger than their bank credit loan size when rT>rT1. When rT>rT1, the bank financing cost is less than the cost of trade credit. As rational individuals, the retailers tend to use more bank credit funds.Figure 23
Loan size allocation of retailer 1.Figure 24
Loan size allocation of retailer 2.


## 6. Conclusions

This paper analyzes a supply chain comprising one manufacturer and two retailers considering consumers’ eco-friendly awareness. Different from previous work, we examine each member’s optimal decision and financing strategy when different financing methods are available. Some interesting conclusion and insights are obtained in this work. The result shows that the manufacturer charges a lower wholesale price when the emission abatement cost is high. It also shows that the emission abatement level decreases the bank interest under bank credit financing, whereas it is immune to trade credit interest when only trade credit financing is adopted. When only one financing way is available, trade credit financing is better for retailers. Besides, the manufacturer also benefits from trade credit and invests in a high emission abatement level. When both financing methods are adopted, the retailers’ financing strategy highly depends on the interest rates charged by the creditors. When the trade credit interest rate is twice larger than the bank interest rate, the retailers will only use bank credit. Otherwise, they borrow both from the bank and the manufacturer. Under dual-channel financing, the emission abatement is affected by the trade credit interest, and it is larger than that in the trade credit financing scenario if the bank interest rate charged on the manufacturer is relatively large.This paper also has some limitations. For example, the demand information is common knowledge to all the members, which could be interesting to discuss the information asymmetry case in future. Besides, the retailers adopt the same financing way in this paper. More interesting results could be obtained when considering that the retailers adopt different financing channels. In addition, one supply one chain is considered in this paper; more interesting conclusions would be obtained when considering two supply chains. Another possible research direction is to investigate the case in which the decision-makers are loss-averse.



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