Joint Economic–Environmental Benefit Optimization by Carbon-Abatement Cost Sharing in a Capital-Constrained Green Supply Chain

Abstract

This paper studies the potential of carbon-abatement cost-sharing contracts in optimizing the joint economic–environmental benefit of a green supply chain. One-way and two-way cost-sharing contracts were investigated, respectively, in scenarios in which a capital-constrained manufacturer has a dominant downstream retailer or a dominant upstream supplier. The manufacturer obtains financing from a competitively priced bank to fulfill its production, carbon-abatement investment, and even insufficient emission permit purchase given the fact that the cap-and-trade regulation exists. Results show that in both one-way and two-way cost-sharing cases, cost sharing of carbon abatement has no effect on the manufacturer’s output or its counterparty’s wholesale price decisions; however, it improves the carbon abatement level of the supply chain. As a result, such cost-sharing of carbon abatement is proven to hamper the profit of the overall supply chain, but it improves the joint “economic-environmental” benefit of the supply chain if the cost-sharing coefficient is properly chosen. Furthermore, this problem is studied in the case of consumers’ green preferences, and carbon-abatement cost sharing is also verified to have the potential to optimize joint economic–environmental benefits.

1. Introduction

With the rapid development of the manufacturing industry, the carbon dioxide emissions problem becomes a worldwide issue that has brought a variety of climate anomalies and disasters. Since the agreement on reducing greenhouse gas emissions has already been reached in the late 20th century by the Kyoto Protocol, many countries have been sparing no efforts in lowering carbon emissions [1,2,3,4]. When it comes to China, at the 75th United Nations General Assembly held in September 2020, President Xi Jinping solemnly promised the world that China would strive to peak carbon dioxide emissions by 2030 and achieve carbon neutrality by 2060 (accessed on 22 September 2021, http://www.qstheory.cn/yaowen/2020-09/22/c_1126523612.htm).

In order to effectively implement the task of energy saving and emission reduction, carbon cap, cap and trade, carbon tax, and other policies have been widely used, among which, the cap-and-trade regulation has been proven to be efficient [5,6,7,8]. Apart from government policies, consumers also play a role in stimulating enterprises to make carbon reduction efforts due to their green preferences [9,10]. Therefore, the task of building green supply chains is imminent. As an innovative measure to build a green supply chain, collaborative emission abatement in supply chains is becoming a global consensus and widely practiced, e.g., as a retail giant, Walmart plans to make a joint carbon abatement with suppliers for 1 billion tons by 2030 through “Project Gigaton” (accessed on 6 April 2022, https://corporate.walmart.com/esgreport/esg-issues/climate-change), and BMW announced to cooperate with suppliers such as CATL and Shougang Group to reduce emissions by 20% in 2030 (accessed on 27 September, 2021, https://www.shougang.com.cn/sgweb/html/sgyw/20190927/3856.html).

The above shows that joint emission abatement has become an essential action for more and more supply chains while it may bring huge financial pressure for the small- and medium-sized enterprises (SMEs) among these chains [11,12]. This paper aims to study how cost sharing of carbon abatement affects the operational decisions as well as the performances of supply chains with a capital-constrained manufacturer. One-way and two-way cost-sharing contracts were investigated in scenarios in which the manufacturer has a dominant downstream retailer and upstream supplier, respectively. The following questions were addressed: (1) What are the equilibriums of supply chain financing systems consisting of supply members and a bank in the cases of one-way and two-way cost sharing? (2) How does cost sharing of carbon abatement affect the operational decisions as well as the benefit of the supply chain in each case? (3) Can and in what circumstance can cost-sharing mechanisms bring satisfactory joint economic–environmental benefits to capital-constrained supply chains? (4) When considering consumers’ green preferences, how are the above problems influenced? Our main results show that in each of the one-way and two-way cost-sharing cases, cost-sharing of carbon abatement will hamper the profit of the overall supply chain but enhance the joint “economic-environmental” benefit of the supply chain if the parameter of the contract is properly chosen. Consumers’ green preferences along with cost-sharing will further enhance the carbon abatement of the supply chain, but a similar role of cost-sharing on the joint “economic-environmental” benefit of the supply chain is concluded.

The rest of this article is organized as follows: Section 2 reviews the literature to clarify research gaps. Section 3 and Section 4 present the one-way and two-way cost-sharing cases, respectively. In Section 5, the problem is extended to consider consumers’ green preferences. Section 6 concludes the paper and provides discussions.

2. Literature Status

This paper examines the role of carbon-abatement cost-sharing contracts in low-carbon supply chains with a capital-constrained manufacturer. Two streams of research, i.e., “operations and financing of the low-carbon supply chain” and “cost-sharing in supply chain operations”, are most related to the current paper. Below, we give a summary of the literature and highlight our innovations.

2.1. Operations and Financing of Low-Carbon Supply Chain

Aside from various research on traditional supply chain operations, i.e., ordering, pricing, production, etc., more and more scholars have been paying attention to the operations of low-carbon supply chains where the carbon abatement decision of the chain members is further emphasized [13,14,15]. Impacts of different low-carbon policies, e.g., carbon trading [8,16,17,18], carbon tax [19,20,21], etc., on the emission reduction efficiency of low-carbon supply chains are widely studied. Xu et al. [22] examined the impact of carbon trading price on manufacturer carbon abatement policies in a make-to-order (MTO) system while Bai et al. [23] studied manufacturers’ integrated decisions on selling prices and carbon reduction level in the context of considering two kinds of products. Bian and Guo [20] investigated the roles of different environmental policies on emissions reduction in manufacturing and indicated that compared to a carbon tax, a carbon abatement subsidy is more effective in abating carbon emissions, but it may lead to lower social welfare.

In recent years, the problem of financing is becoming a new issue for sustainable operations of supply chains since low-carbon activities, such as carbon abatement investment, may increase the financing requirements of supply chain members. Then, many studies began to explore issues such as integrated operations (e.g., ordering, pricing, etc.), financing, and carbon abatement decisions in supply chains [12,24,25]. At the same time, various motivational factors for low-carbon investment, e.g., consumers’ green preferences [26,27], etc., have also been widely characterized. Representatively, Wu et al. [26] studied downstream retailers’ ordering and financing decisions as well as their suppliers’ pricing and carbon abatement decisions while Xu and Fang [28] and Zou et al. [12] examined manufacturers’ integrated ordering, carbon abatement, and financing decisions. For the scenario of upstream financing, Qin et al. [24] researched green suppliers’ choices between bank financing and advance payment while Huang et al. [29] explored how government subsidy strategies affect low-carbon supply chain operations and financing.

Different from the above studies, this paper mainly focuses on examining how cost sharing of carbon abatement affects the operations as well as the benefit of low-carbon supply chains given the manufacturer in the chain faces capital constraints.

2.2. Cost-Sharing in Supply Chain Operations

Cost-sharing is a common contract or mechanism in supply chain management. When it comes to low-carbon supply chains, Ghosh and Shah [30] studied the coordination effect of a cost-sharing contract for a supply chain consisting of a manufacturer and a retailer. Wang et al. [9] combined “cost-sharing” and “altruistic preference” to improve the profit of a retailer-led, low-carbon supply chain. Under the cap-and-trade regulation, Wang et al. [31] implemented a two-way cost-sharing contract to enhance the carbon abatement level as well as the profit of a supply chain. In the context of capital constraint, Wu et al. [26] proposed a contract portfolio of revenue sharing, cost sharing, and transfer payment to coordinate a low-carbon supply chain in which a capital-constrained retailer may apply bank credit financing or trade credit financing. Lai et al. [32] explored the production, financing, and green investment decisions of a manufacturer who lacks money and can obtain financing from a bank or its supplier’s investment. They found that a portfolio of cost-sharing and quantity discount contracts can achieve supply chain coordination.

Different from the above studies, this paper focuses on exploring how cost sharing of carbon abatement affects the operational decisions of a supply chain where the capital-constrained manufacturer has a dominant downstream retailer or upstream supplier. One-way and two-way cost-sharing contracts were investigated in these two supply chain settings, respectively, and the joint “economic-environmental” benefit of the contracts is confirmed in this research.

3. The One-Way Cost-Sharing Case

3.1. Problem Statement

In this case, we consider the supply chain to consist of a capital-constrained manufacturer and a creditworthy downstream retailer. The retailer is a retail giant who acts as the leader in the chain, and the manufacturer who encounters capital constraint acts as the follower. In response to the retailer’s request, the manufacturer develops a single kind of green product and sells them to the end market through the retailer’s supermarkets. This follows the common practice in retailer-led, low-carbon supply chains, e.g., Walmart has developed Project Gigaton to engage its suppliers to reduce carbon emissions by one billion metric tons by 2030. Due to its capital constraint, the manufacturer is assumed to obtain funds from a competitively priced bank to fulfill its needs of production and a low-carbon investment. The carbon abatement occurs as a cost for the manufacturer, which follows a quadratic form to the abatement level $e$, i.e., $\frac{1}{2}k{(e)}^2$, where $k$ is a cost coefficient [12,26]. There also exists a carbon trading market wherein the manufacturer can obtain a free permit quota from the government and buy (sell) insufficient (excess) carbon emission permits at a certain price [2,3,12,28]. Assuming $G$ is the free quota, $e_0$ is the initial carbon emissions for unit production, and $p_e$ is the carbon trading price, we have $(e_{0}q-e-G)p_e$ denoting the carbon trading cost or revenue of the manufacturer in the cases of over emitting or under emitting, where $q$ is the output of the manufacturer which is a decision variable.

Given the manufacturer jointly deciding its output $q$ and carbon abatement level $e$, the retailer decides the wholesale price of the product that it would like to purchase, i.e., $w$. After that, the bank in the competitive banking market sets a suitable interest rate $r$ to obtain an expected zero profit after examining the manufacturer’s repayment abilities in the random market, where $X$ is the random demand whose PDF, CDF, complementary CDF, and failure rate are $f\left(x\right)$, $F\left(x\right)$, $\overline{F}\left(x\right)$ and $h\left(x\right)=f\left(x\right)/\overline{F}\left(x\right)$, respectively. In addition, without loss of generality, the manufacturer’s self-capital is normalized to zero [28,33]. The time value of funds and the salvage value of leftovers are not considered [34,35].

To incentivize the manufacturer to invest in carbon emission reduction, the retailer may promise to share a proportion of the carbon-abatement cost for the manufacturer, and our intention is to investigate whether such a cost-sharing mechanism can benefit the whole supply chain. We summarize the main notations in Notations section.

3.2. The Case without Cost-Sharing

In the Stackelberg game between the manufacturer and the retailer, by backward induction, we start with solving the problem of the manufacturer to identify its response function and then derive the optimal strategy of the retailer to find the equilibrium of the game. Given the loan interest rate announced by the bank and the wholesale price quoted by the retailer, the supplier decides on its production quantity and carbon abatement level to maximize its expected profit. The total amount of capital that the manufacturer needs to borrow from the bank for production, carbon abatement investment, and carbon trading activities would be $cq+\frac{1}{2}k{(e)}^2+(e_{0}q-e-G)p_e$. In a competitive banking market, by considering the bank’s competitively-priced equation as a constraint, the manufacturer’s decision-making problem can be formulated as follows:

$$\begin{array}{l}Max{\pi }_m\left(e,q_{;}w,r\right)=E{\left\{w\min \left[X,q\right]-\left[cq+\frac{1}{2}k{e}^2+(e_{0}q-e-G)p_e\right](1+r)\right\}}^{+}\\ s.t.E\min \left\{w\min \left[X,q\right],\left[cq+\frac{1}{2}k{e}^2+(e_{0}q-e-G)p_e\right](1+r)\right\}=cq+\frac{1}{2}k{e}^2+(e_{0}q-e-G)p_e\end{array}$$

(1)

Lemma 1. 

In the case without cost-sharing of carbon abatement, the manufacturer’s optimal carbon abatement and production decisions are

$$\{\begin{array}{l}{e}^{*}=\frac{p_e}{k}\\ q^{*}={\overline{F}}^{-1}\left(\frac{c+e_0{p}_e}{w}\right)\end{array}$$

(2)

Proof. 

See Appendix A. □

Next, we discuss the retailer’s problem. By manipulating the manufacturer’s best response, the retailer determines the optimal wholesale price to maximize its profit:

$$Max{\pi }_r(w;q^{*})=(p-w)\min \left[X,q^{*}\right]$$

(3)

Lemma 2. 

In the case without cost-sharing of carbon abatement, the retailer’s optimal wholesale price quotation is

$$w^{*}=\frac{p\overline{F}(q^{*})-c-e_0{p}_e}{h(q^{*})\left[q^{*}-{\displaystyle {\int }_0^{q^{*}}F(x)dx}\right]}$$

(4)

Proof. 

See Appendix A. □

As for the bank, if $q^{*}$, $e^{*}$, and $w^{*}$ are obtained, the competitively priced bank’s interest rate can be determined with the constraint condition in Equation (1). Then, we can obtain the equilibrium of the supply chain financing game.

Proposition 1. 

In the case without cost-sharing of carbon abatement, the equilibrium of the supply chain financing system, i.e., $\left(e^{*},q^{*},w^{*},r^{*}\right)$ , solves the following:

$$\{\begin{array}{l}{e}^{*}=\frac{p_e}{k}\\ q^{*}={\overline{F}}^{-1}(\frac{c+e_0{p}_e}{w^{*}})\\ w^{*}h(q^{*})\left[q^{*}-{\displaystyle {\int }_0^{q^{*}}F(x)dx}\right]=p\overline{F}(q^{*})-c-e_0{p}_e\\ E\min \left\{w^{*}\min \left[X,q^{*}\right],\left[c{q}^{*}+\frac{1}{2}k{(e^{*})}^2+(e_0{q}^{*}-e^{*}-G)p_e\right](1+r^{*})\right\}=c{q}^{*}+\frac{1}{2}k{(e^{*})}^2+(e_0{q}^{*}-e^{*}-G)p_e\end{array}$$

(5)

Proof. 

See Appendix A. □

From Equation (5), it is obvious that compared to the centralized supply chain where $p\overline{F}(q^{*})-c-e_0{p}_e=0$ holds, $w^{*}h(q^{*})\left[q^{*}-{\displaystyle {\int }_0^{q^{*}}F(x)dx}\right]$ is the efficiency loss of the supply chain attributed to the double marginalization.

3.3. The Case with Cost-Sharing

In this part, we consider the case where the retailer shares a proportion of carbon abatement investment, i.e., $\beta$, for the manufacturer. The manufacturer’s decision-making problem is thus transformed into:

$$\begin{array}{l}Max{\pi }_m(e,q;w,r)=E{\left\{w\min \left[X,q\right]-\left[cq+\frac{1}{2}k(1-\beta )e^2+(e_{0}q-e-G)p_e\right]\left(1+r\right)\right\}}^{+}\\ s.t.E\min \left\{w\min \left[X,q\right],\left[cq+\frac{1}{2}k(1-\beta )e^2+(e_{0}q-e-G)p_e\right](1+r)\right\}=cq+\frac{1}{2}k(1-\beta )e^2+(e_{0}q-e-G)p_e\end{array}$$

(6)

Lemma 3. 

In the case of cost-sharing of carbon abatement, the manufacturer’s optimal carbon abatement and production decisions are

$$\{\begin{array}{l}{e}^{**}=\frac{p_e}{k\left(1-\beta \right)}\\ q^{**}={\overline{F}}^{-1}\left(\frac{c+e_0{p}_e}{w}\right)\end{array}$$

(7)

Proof. 

See Appendix A. □

Next, we discuss the retailer’s problem. By manipulating the manufacturer’s best response, the retailer determines the optimal wholesale price to maximize its profit:

$$Max{\pi }_r(w;e^{**},q^{**})=(p-w)\min \left[X,q^{**}\right]-\frac{1}{2}\beta k{(e^{**})}^2$$

(8)

Lemma 4. 

In the case of cost-sharing of carbon abatement, the retailer’s optimal wholesale price quotation is

$$w^{**}=\frac{p\overline{F}(q^{**})-c-e_0{p}_e}{h(q^{**})\left[q^{**}-{\displaystyle {\int }_0^{q^{**}}F(x)dx}\right]}$$

(9)

Proof. 

See Appendix A. □

As for the bank, if $q^{**}$, $e^{**}$, and $w^{**}$ are obtained, the competitively priced bank’s interest rate can be determined by the constraint condition in Equation (6). Then, we can obtain the equilibrium of the supply chain financing game.

Proposition 2. 

In the case of cost-sharing of carbon abatement, the equilibrium of the supply chain financing system, i.e., $\left(e^{**},q^{**},w^{**},r^{**}\right)$, solves the following:

$$\{\begin{array}{l}{e}^{**}=\frac{p_e}{k\left(1-\beta \right)}\\ q^{**}={\overline{F}}^{-1}(\frac{c+e_0{p}_e}{w^{**}})\\ w^{**}h(q^{**})\left[q^{**}-{\displaystyle {\int }_0^{q^{**}}F(x)dx}\right]=p\overline{F}(q^{**})-c-e_0{p}_e\\ E\min \left\{w^{**}\min \left[X,q^{**}\right],\left[c{q}^{**}+\frac{1}{2}k(1-\beta ){(e^{**})}^2+(e_0{q}^{**}-e^{**}-G)p_e\right](1+r^{**})\right\}=c{q}^{**}+\frac{1}{2}k(1-\beta ){(e^{**})}^2+(e_0{q}^{**}-e^{**}-G)p_e\end{array}$$

(10)

Proof. 

See Appendix A. □

In the case of cost-sharing of carbon abatement, the efficiency loss attributed to the double marginalization still exists and remains the same as that in the case without the cost-sharing contract.

3.4. The Value of Cost-Sharing

By comparing the equilibriums in Propositions 1 and 2, we obtain the following corollary:

Corollary 1. 

The one-way cost-sharing contract on carbon abatement provided by the retailer has no influence on the manufacturer’s output or the retailer’s wholesale price quotation decisions; however, it will undoubtedly improve the carbon abatement level of the manufacturer, i.e., $q^{**}=q^{*}$, $w^{**}=w^{*}$, $e^{**}>e^{*}$

Proof. 

See Appendix A. □

We are more interested in how the cost-sharing contract of the retailer affects the profits of the chain members. As Corollary 1 shows, the contract has positive environmental externalities. Then, we further examine the joint “economic-environmental” benefit of the contract. The following Proposition 3 summarizes all these results.

Proposition 3. 

The one-way cost-sharing contract for carbon abatement provided by the retailer can:

(a)

Increase the profit of the manufacturer while reducing the profits of the retailer as well as the overall supply chain;

(b)

Enhance the joint “economic-environmental” benefit only when the proportion of cost-sharing is smaller than a threshold, i.e., $0<\beta <\frac{2}{3}$.

Proof. 

See Appendix A. □

From Proposition 3, the one-way cost-sharing contract for carbon abatement cannot benefit all members in the supply chain, but it undoubtedly reduces the carbon emissions of the manufacturer, which in turn may bring positive environmental externalities and enhance the joint “economic-environmental” benefit, showing the potential value of the contract.

4. The Two-Way Cost-Sharing Case

4.1. Problem Statement

Different from that in Section 3, in this section, we consider a supply chain consisting of a capital-constrained manufacturer and a creditworthy upstream supplier. The supplier (acting as the leader) and the manufacturer (acting as the follower) jointly complete the production process of green products. Both parties are assumed to put efforts into carbon abatement, and their respective levels of carbon abatement are $e_s$ and $e_m$. Following the similar quadratic form as in Section 3, the corresponding carbon-abatement costs for both parties are $\frac{1}{2}{k}_s{(e_s)}^2$ and $\frac{1}{2}{k}_m{(e_m)}^2$, where $k_s$ and $k_m$ are cost coefficients. In order to incentivize the joint emissions reduction among the supply chain members, we assume the supplier and manufacturer promise to share a proportion of the carbon-abatement costs with each other, where $\beta$ ($\alpha$) is the proportion of the cost that the supplier (manufacturer) shares with the manufacturer (supplier). What we are interested in is whether the two-way cost-sharing mechanism could benefit the supply chain.

4.2. The Case without Cost-Sharing

In this case, the supplier and the manufacturer invest in their carbon abatement, respectively. We first solve the problem of the follower, i.e., the manufacturer, who makes joint decisions on an output $q$ (production quantity) and carbon abatement level $e_m$. Given that it is capital-constrained, the manufacturer should raise money from the bank to cover its cost of production, carbon abatement, and even insufficient carbon emission permit purchase. Its total financing amount can be formulated as $(c_m+w)q+\frac{1}{2}{k}_m{(e_m)}^2+(e_0^{m}q-e_m-G_m)p_e$. Considering the bank’s competitively priced equation as a constraint, the manufacturer’s decision-making problem is as follows:

$$\begin{array}{l}Max{\pi }_m\left(q,e_m;w,r\right)=E{\left\{p\min \left[q,X\right]-\left[(c_m+w)q+\frac{1}{2}{k}_m{(e_m)}^2+(e_0^{m}q-e_m-G_m)p_e\right](1+r)\right\}}^{+}\\ s.t.E\min \left\{p\min \left[q,X\right],\left[(c_m+w)q+\frac{1}{2}{k}_m{(e_m)}^2+(e_0^{m}q-e_m-G_m)p_e\right](1+r)\right\}\\ =(c_m+w)q+\frac{1}{2}{k}_m{(e_m)}^2+(e_0^{m}q-e_m-G_m)p_e\end{array}$$

(11)

Lemma 5. 

In the case without cost-sharing of carbon abatement, the manufacturer’s optimal carbon abatement and production decisions are

$$\{\begin{array}{l}{e}_m^{*}=\frac{p_e}{k_m}\\ q^{*}={\overline{F}}^{-1}\left(\frac{c_m+w+e_0^m{p}_e}{p}\right)\end{array}$$

(12)

Proof. 

See Appendix A. □

By manipulating the manufacturer’s best response in Equation (12), as the leader, the supplier decides its optimal wholesale price and carbon abatement level for producing the semifinished products to maximize its profit:

$$Max{\pi }_s(w,e_s;q^{*},e_m^{*})=(w-c_s)q^{*}-\frac{1}{2}{k}_s{(e_s)}^2-(e_0^s{q}^{*}-e_s-G_s)p_e$$

(13)

Lemma 6. 

In the case without cost-sharing of carbon abatement, the supplier’s optimal wholesale price quotation and carbon abatement level are

$$\{\begin{array}{l}{w}^{*}=p{q}^{*}f(q^{*})+c_s+e_0^s{p}_e\\ e_s^{*}=\frac{p_e}{k_s}\end{array}$$

(14)

Proof. 

See Appendix A. □

If $q^{*}$, $e_m^{*}$, $w^{*}$, and $e_s^{*}$ are obtained, the competitively priced bank interest rate can be determined by the constraint condition in Equation (11). Then, we obtain the equilibrium of the supply chain financing game.

Proposition 4. 

In the case without cost-sharing of carbon abatement, the equilibrium of the supply chain financing system, i.e., $\left(q^{*},e_m^{*},w^{*},e_s^{*},r^{*}\right)$ , solves the following.

$$\{\begin{array}{l}{q}^{*}={\overline{F}}^{-1}(\frac{c_m+w^{*}+e_0^m{p}_e}{p})\\ e_m^{*}=\frac{p_e}{k_m}\\ w^{*}=p{q}^{*}f(q^{*})+c_s+e_0^s{p}_e\\ e_s^{*}=\frac{p_e}{k_s}\\ E\min \left\{p\min \left[q^{*},X\right],\left[(c_m+w^{*})q^{*}+\frac{1}{2}{k}_m{(e_m^{*})}^2+(e_0^m{q}^{*}-e_m^{*}-G_m)p_e\right](1+r^{*})\right\}\\ =(c_m+w^{*})q^{*}+\frac{1}{2}{k}_m{(e_m^{*})}^2+(e_0^m{q}^{*}-e_m^{*}-G_m)p_e\end{array}$$

(15)

Proof. 

See Appendix A. □

From Equation (15), it is obvious that both the carbon abatement decisions of the supplier and manufacturer are decoupled from their prospective operational decisions (output and pricing). By some transformations, we have $p\overline{F}\left(q^{*}\right)=c_m+c_s+e_0^s{p}_e+e_0^m{p}_e+p{q}^{*}f(q^{*})$, where the term $p{q}^{*}f(q^{*})$ corresponds to the efficiency loss of the supply chain attributing to the double marginalization.

4.3. The Case with Cost-Sharing

In this part, we consider the case where the supplier (manufacturer) shares a proportion of carbon abatement investment, i.e., $\beta$ ($\alpha$), for the manufacturer (supplier). Under such a two-way cost-sharing contract, the manufacturer’s decision-making problem becomes

$$\begin{array}{l}Max{\pi }_m\left(q,e_m;w,e_s,r\right)=E{\left\{p\min \left[q,X\right]-\left[(c_m+w)q+\frac{1}{2}{k}_m(1-\beta ){(e_m)}^2+\frac{1}{2}{k}_s\alpha {(e_s)}^2+(e_0^{m}q-e_m-G_m)p_e\right](1+r)\right\}}^{+}\\ s.t.E\min \left\{p\min \left[q,X\right],\left[(c_m+w)q+\frac{1}{2}{k}_m(1-\beta ){(e_m)}^2+\frac{1}{2}{k}_s\alpha {(e_s)}^2+(e_0^{m}q-e_m-G_m)p_e\right](1+r)\right\}\\ =(c_m+w)q+\frac{1}{2}{k}_m(1-\beta ){(e_m)}^2+\frac{1}{2}{k}_s\alpha {(e_s)}^2+(e_0^{m}q-e_m-G_m)p_e\end{array}$$

(16)

Lemma 7. 

In the case of two-way cost-sharing of carbon abatement, the manufacturer’s optimal carbon abatement and production decisions are

$$\{\begin{array}{l}{e}_m^{**}=\frac{p_e}{\left(1-\beta \right)k_m}\\ q^{**}={\overline{F}}^{-1}\left(\frac{c_m+w+e_0^m{p}_e}{p}\right)\end{array}$$

(17)

Proof. 

See Appendix A. □

By manipulating the manufacturer’s best response, as the leader, the supplier decides its optimal wholesale price and carbon abatement level for producing the semifinished products to maximize its expected profit:

$$Max{\pi }_s(w,e_s;q^{*},e_m^{*})=(w-c_s)q^{*}-\frac{1}{2}(1-\alpha )k_s{(e_s)}^2-\frac{1}{2}\beta k_m{(e_m^{*})}^2-(e_0^s{q}^{*}-e_s-G_s)p_e$$

(18)

Lemma 8. 

In the case of two-way cost-sharing of carbon abatement, the supplier’s optimal wholesale price quotation and carbon abatement level are

$$\{\begin{array}{l}{w}^{**}=p{q}^{**}f\left(q^{**}\right)+c_s+e_0^s{p}_e\\ e_s^{**}=\frac{p_e}{\left(1-\alpha \right)k_s}\end{array}$$

(19)

Proof. 

See Appendix A. □

Then, the competitively priced bank chooses an interest rate based on the constraint condition in Equation (16). The equilibrium of the supply chain financing game is yielded as follows:

Proposition 5. 

In the case of two-way cost-sharing of carbon abatement, the equilibrium of the supply chain financing system, i.e., $\left(q^{**},e_m^{**},w^{**},e_s^{**},r^{**}\right)$ solves the following:

$$\{\begin{array}{l}{q}^{**}={\overline{F}}^{-1}(\frac{c_m+w^{**}+e_0^m{p}_e}{p})\\ e_m^{**}=\frac{p_e}{\left(1-\beta \right)k_m}\\ w^{**}=p{q}^{**}f(q^{**})+c_s+e_0^s{p}_e\\ e_s^{**}=\frac{p_e}{(1-\alpha )k_s}\\ E\min \left[p\min \left[q^{**},X\right],\left[(c_m+w^{**})q^{**}+\frac{1}{2}{k}_m(1-\beta ){(e_m^{**})}^2+\frac{1}{2}{k}_s\alpha {(e_s^{**})}^2+(e_0^m{q}^{**}-e_m^{**}-G_m)p_e\right](1+r^{**})\right]\\ =(c_m+w^{**})q^{**}+\frac{1}{2}{k}_m(1-\beta ){(e_m^{**})}^2+\frac{1}{2}{k}_s\alpha {(e_s^{**})}^2+(e_0^m{q}^{**}-e_m^{**}-G_m)p_e\end{array}$$

(20)

Proof. 

See Appendix A. □

4.4. The Value of Cost-Sharing

Comparing the two equilibriums in Equations (15) and (20), it can be easily found that the two-way cost-sharing contract does not affect the quantity or pricing decisions of the manufacturer or the supplier but undoubtedly increases the carbon abatement levels of both parties, as summarized in Corollary 2.

Corollary 2. 

The two-way cost-sharing contract for carbon abatement has no influence on the manufacturer’s output or the supplier’s wholesale price decisions; however, it will undoubtedly improve the carbon abatement levels of both parties, i.e., $q^{**}=q^{*}$, $w^{**}=w^{*}$, $e_m^{**}>e_m^{*}$, $e_s^{**}>e_s^{*}$.

Proof. 

See Appendix A. □

We then examine the joint “economic-environmental” benefit of the contract. The following Proposition 6 summarizes the results:

Proposition 6. 

The two-way cost-sharing contract for carbon abatement can

(a)

Reduce the profits of the overall supply chain;

(b)

Enhance the joint “economic-environmental” benefit only when the proportions of the cost-sharing meet ${(1-\alpha )}^2(3\beta -2{\beta }^2)k_s<{(1-\beta )}^2(2\alpha -3{\alpha }^2)k_m$.

Proof. 

See Appendix A. □

Based on Proposition 6, the two-way cost-sharing contract for carbon abatement will hamper the profit of the overall supply chain, but it can bring positive environmental externalities, and the joint “economic-environmental” benefit will be enhanced if the parameters of the contract are properly chosen.

5. Extension for the Case with Green Consumers

5.1. Game Equilibrium

Since consumers have low-carbon preferences in certain markets, in this part, we extend the problem to meet an emission-dependent demand scenario. Taking the more complicated two-way cost-sharing case in Section 4 as an example, according to the conventions in previous studies [26,28,33], the emission-dependent demand can be described as $\widehat{X}=X+a_s{e}_s+a_m{e}_m$, where $X$ is the initial demand, $e_s$ and $e_m$ are the carbon abatement levels of the supplier and the capital-constrained manufacturer, and $a_s$ and $a_m$ are two coefficients measuring the elasticity of demand for carbon abatement. In this case, the decision-making problems of the manufacturer and supplier without the two-way cost-sharing contract in Equations (11) and (13) can be rewritten as Equations (21) and (22). To distinguish the optimal results, we added a “-” to the top of the new solutions.

$$\begin{array}{l}Max{\pi }_m\left(q,e_m;w,e_s,r\right)=E{\left\{p\min \left[q,\widehat{X}\right]-\left[(c_m+w)q+\frac{1}{2}{k}_m{(e_m)}^2+(e_0^{m}q-e_m-G_m)p_e\right](1+r)\right\}}^{+}\\ s.t.E\min \left\{p\min \left[q,\widehat{X}\right],\left[(c_m+w)q+\frac{1}{2}{k}_m{(e_m)}^2+(e_0^{m}q-e_m-G_m)p_e\right](1+r)\right\}\\ =(c_m+w)q+\frac{1}{2}{k}_m{(e_m)}^2+(e_0^{m}q-e_m-G_m)p_e\end{array}$$

(21)

$$Max{\pi }_s(w,e_s;{\overline{q}}^{*})=(w-c_s){\overline{q}}^{*}-\frac{1}{2}{k}_s{(e_s)}^2-(e_0^s{\overline{q}}^{*}-e_s-G_s)p_e$$

(22)

Proposition 7. 

In the case of green consumers, the equilibrium of the supply chain financing system, i.e., $\left({\overline{q}}^{*},{\overline{e}}_m^{*},{\overline{w}}^{*},{\overline{e}}_s^{*},{\overline{r}}^{*}\right)$ solves the following:

$$\{\begin{array}{l}{\overline{q}}^{*}={\overline{F}}^{-1}(\frac{c_m+{\overline{w}}^{*}+e_0^m{p}_e}{p})+a_m{\overline{e}}_m^{*}+a_s{\overline{e}}_s^{*}\\ {\overline{e}}_m^{*}=\frac{a_m(p-c_m-{\overline{w}}^{*}-e_0^m{p}_e)+p_e}{k_m}\\ {\overline{w}}^{*}=\frac{p{\overline{q}}^{*}{k}_{m}f({\overline{q}}^{*}-a_m{\overline{e}}_m^{*}-a_s{\overline{e}}_s^{*})}{a_m^{2}pf({\overline{q}}^{*}-a_m{\overline{e}}_m^{*}-a_s{\overline{e}}_s^{*})+k_m}+c_s+e_0^s{p}_e\\ {\overline{e}}_s^{*}=\frac{a_s({\overline{w}}^{*}-c_s-e_0^s{p}_e)+p_e}{k_s}\\ E\min \left[p\min \left[{\overline{q}}^{*},X+a_m{\overline{e}}_m^{*}+a_s{\overline{e}}_s^{*}\right],\left[(c_m+{\overline{w}}^{*}){\overline{q}}^{*}+\frac{1}{2}{k}_m{({\overline{e}}_m^{*})}^2+(e_0^m{\overline{q}}^{*}-{\overline{e}}_m^{*}-G_m)p_e\right](1+{\overline{r}}^{*})\right]\\ =(c_m+{\overline{w}}^{*}){\overline{q}}^{*}+\frac{1}{2}{k}_m{({\overline{e}}_m^{*})}^2+(e_0^m{\overline{q}}^{*}-{\overline{e}}_m^{*}-G_m)p_e\end{array}$$

(23)

Proof. 

See Appendix A. □

Comparing the results in Equations (23) and (15), it can be found that ${\overline{e}}_m^{*}>e_m^{*}$ and ${\overline{e}}_s^{*}>e_s^{*}$. This means that the green consumers in the market can undoubtedly stimulate the carbon abatement investments of both the manufacturer and supplier. When further considering two-way cost-sharing of carbon abatement, the decision-making problems of both parties are changed into:

$$\begin{array}{l}Max{\pi }_m\left(q,e_m;w,e_s,r\right)=E{\left\{p\min \left[q,\widehat{X}\right]-\left[(c_m+w)q+\frac{1}{2}{k}_m(1-\beta ){(e_m)}^2+\frac{1}{2}{k}_s\alpha {(e_s)}^2+(e_0^{m}q-e_m-G_m)p_e\right](1+r)\right\}}^{+}\\ s.t.E\min \left\{p\min \left[q,\widehat{X}\right],\left[(c_m+w)q+\frac{1}{2}{k}_m(1-\beta ){(e_m)}^2+\frac{1}{2}{k}_s\alpha {(e_s)}^2+(e_0^{m}q-e_m-G_m)p_e\right](1+r)\right\}\\ =(c_m+w)q+\frac{1}{2}{k}_m(1-\beta ){(e_m)}^2+\frac{1}{2}{k}_s\alpha {(e_s)}^2+(e_0^{m}q-e_m-G_m)p_e\end{array}$$

(24)

$$Max{\pi }_s(w,e_s;{\overline{q}}^{**},{\overline{e}}_m^{**})=(w-c_s){\overline{q}}^{**}-\frac{1}{2}(1-\alpha )k_s{(e_s)}^2-\frac{1}{2}\beta k_m{({\overline{e}}_m^{**})}^2-(e_0^s{\overline{q}}^{**}-e_s-G_s)p_e$$

(25)

Proposition 8. 

In the case of green consumers and two-way cost-sharing, the equilibrium of the supply chain financing system, i.e., $\left({\overline{q}}^{**},{\overline{e}}_m^{**},{\overline{w}}^{**},{\overline{e}}_s^{**},{\overline{r}}^{**}\right)$ solves the following:

$$\{\begin{array}{l}{\overline{q}}^{**}={\overline{F}}^{-1}(\frac{c_m+{\overline{w}}^{**}+e_0^m{p}_e}{p})+a_m{\overline{e}}_m^{**}+a_s{\overline{e}}_s^{**}\\ {\overline{e}}_m^{**}=\frac{a_m(p-c_m-{\overline{w}}^{**}-e_0^m{p}_e)+p_e}{k_m(1-\beta )}\\ {\overline{w}}^{**}=\frac{p{k}_m[{\overline{q}}^{**}(1-\beta )+a_m\beta {\overline{e}}_m^{**}]f({\overline{q}}^{**}-a_m{\overline{e}}_m^{**}-a_s{\overline{e}}_s^{**})}{k_m(1-\beta )+a_m^{2}pf({\overline{q}}^{**}-a_m{\overline{e}}_m^{**}-a_s{\overline{e}}_s^{**})}+c_s+e_0^s{p}_e\\ {\overline{e}}_s^{**}=\frac{a_s({\overline{w}}^{**}-c_s-e_0^s{p}_e)+p_e}{k_s(1-\alpha )}\\ E\min \left[p\min \left[{\overline{q}}^{**},X+a_m{\overline{e}}_m^{**}+a_s{\overline{e}}_s^{**}\right],\left[(c_m+{\overline{w}}^{**}){\overline{q}}^{**}+\frac{1}{2}{k}_m(1-\beta ){({\overline{e}}_m^{**})}^2+\frac{1}{2}{k}_s\alpha {({\overline{e}}_s^{**})}^2+(e_0^m{\overline{q}}^{**}-{\overline{e}}_m^{**}-G_m)p_e\right](1+{\overline{r}}^{**})\right]\\ =(c_m+{\overline{w}}^{**}){\overline{q}}^{**}+\frac{1}{2}{k}_m(1-\beta ){({\overline{e}}_m^{**})}^2+\frac{1}{2}{k}_s\alpha {({\overline{e}}_s^{**})}^2+(e_0^m{\overline{q}}^{**}-{\overline{e}}_m^{**}-G_m)p_e\end{array}$$

(26)

Proof. 

See Appendix A. □

Comparing Equations (26) and (23), there is no doubt that when considering consumers’ green preferences, introducing the two-way cost-sharing contract can also enhance the carbon abatement levels of both the manufacturer and the supplier.

5.2. Numerical Analysis

In this part, we use the method of numerical analysis to intuitively show the properties of the equilibrium obtained in Proposition 8 as well as to explore how the two-way cost-sharing contract affects the benefit of the whole supply chain. Without loss of generality, we assume $p=1.0$, $c_m=c_s=e_0^m=e_0^s=0.1$, $k_m=k_s=a_s=a_m=1.0$, $p_e=0.5$, $G_s=G_m=5.0$, and the initial market demand follows $X~\exp (0.01)$ [34,35,36]. When two-way cost-sharing of carbon abatement does not exist, the initial equilibrium of the game is ${\overline{q}}^{*}=51.13$, ${\overline{e}}_m^{*}=0.89$, ${\overline{w}}^{*}=0.46$, ${\overline{e}}_s^{*}=0.81$. When cost-sharing exists, how the equilibrium changes according to the two sharing coefficients, i.e., $\alpha$ and $\beta$, is shown in Figure 1. It can be observed that as the supplier increases the proportion of cost-sharing for the manufacturer, i.e., $\beta$ rises, the manufacturer would like to abate more emissions, i.e., ${\overline{e}}_m^{*}$ increases. The influence of $\alpha$ on the supplier’s carbon abatement level has the same rule. However, offering cost-sharing for the manufacturer will increase the cost of the supplier, which leads to a rise in the supplier’s wholesale price ${\overline{w}}^{*}$. The change in the wholesale price finally decreases the output of the manufacturer ${\overline{q}}^{*}$.

We further explored the effects of two-way cost-sharing on carbon abatement on the profits of the supplier and the manufacturer, as shown in Figure 2. It can be observed that as the two sharing coefficients, i.e., $\alpha$ and $\beta$, rise, the profits of both parties will decline. Specifically, the decline of one party’s profit is more sensitive to the cost-sharing proportion that it offers to the other party. Overall, the introduction of the two-way cost-sharing contract on carbon abatement will hamper the overall profit of the supply chain, as shown in Figure 3a. However, it undoubtedly promotes the carbon abatement of both parties, and when further considering the value of positive environmental externalities, the contract can enhance the joint “economic-environmental” benefit of the supply chain as long as the two sharing coefficients fall within some favorable intervals, i.e., $\alpha \in (0,0.46)$ and $\beta \in (0,0.07)$ in the example, as shown in Figure 3b.

6. Conclusions

This paper studies the role of carbon-abatement cost-sharing contracts in supply chains with a capital-constrained manufacturer. The manufacturer obtains financing from a competitively priced bank to fulfill its production, carbon-abatement investment, and even insufficient emission permits purchase given that the cap-and-trade regulation exists. Firstly, a one-way cost-sharing contract was investigated in the scenario where the manufacturer has a dominant downstream retailer, and the retailer shares a proportion of the carbon-abatement cost for the manufacturer. The results show that the one-way cost-sharing contract has no effect on the manufacturer’s output or the retailer’s wholesale price quotation decisions; however, it improves the carbon abatement level of the manufacturer. As a result, the contract hampers the profit of the retailer as well as the overall supply chain but enhances the joint “economic-environmental” benefit of the supply chain in some circumstances. Then, a two-way cost-sharing contract was investigated in the scenario where the manufacturer has a dominant supplier, and they share a proportion of the carbon-abatement cost with each other. The results also show the potential “economic-environmental” benefit of the two-way cost-sharing contract if the parameter of the contract is properly chosen. Finally, the problem was studied in the case of consumers’ green preferences, and numerical analysis also indicates similar roles as described above.

The managerial insights of this research mainly lie in how upstream and downstream larger firms should share carbon abatement costs with capital-constrained manufacturers since, from the perspective of low-carbon supply chain management, such approaches help reduce emissions of the supply chain and achieve satisfactory joint “economic-environmental” benefits. This is also a manifestation of the social responsibility of large enterprises in the supply chain. However, such cost-sharing measures may be disadvantageous to upstream and downstream enterprises in terms of purely economic benefits, which requires further discussions of multiple contracts within the supply chain and external government incentives in future research.