# Joint Pricing and Product Carbon Footprint Decisions and Coordination of Supply Chain with Cap-and-Trade Regulation

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## Abstract

**:**

## 1. Introduction

- (1)
- How do the manufacturer and the retailer make optimal product carbon footprint and pricing decisions under the carbon cap and trade regulation?
- (2)
- What effects does the cap-and-trade regulation have on the product carbon footprint, selling price, consumer demand and the profit of the supply chain system?
- (3)
- How does the manufacturer design coordinating contract (such as the wholesale price and fixed fee in the two-part tariff) to ensure the retailer to cooperate based on the change of the carbon cap and the carbon trading price?
- (4)
- For the government, what policies and measures are also implemented to encourage the manufacturer to invest in low-carbon technology under cap-and-trade regulation, and then to reduce the product carbon footprint more effectively.

## 2. Literature Review

## 3. Models and Decision Analysis

#### 3.1. Problem Statement and Assumption

- (1)
- The manufacturer produces the product with unit cost $c$ and sells it at wholesale price $w$ to the retailer, which is a decision variable of the manufacturer.
- (2)
- Under cap-and-trade regulation, the manufacturer is allocated a predetermined emission quota $N$ (hereinafter referred to as the carbon cap, which is similar to the studies of Hua et al. [7], Benjaafar et al. [8], Zhou et al. [16] and Xu et al. [21]), and the carbon trading price in an outside market is denoted by $\theta $. If the actual emissions exceed the carbon cap $N$, the manufacturer needs to buy carbon permits. Conversely, if the emissions are less than the carbon cap $N$, the manufacturer can sell the extra carbon permits. We assume that $X$ is traded quantity of carbon emissions: if $X>0$, it means that the manufacturer can sell $X$ units of carbon credit; if $X<0$, it means that the manufacturer need to buy $\left|X\right|$ units of carbon credit; and if $X=0$, it means that the manufacturer neither buys nor sells any carbon emission credits.
- (3)
- If the manufacturer carries out investment in low-carbon technology, the initial carbon footprint of the product ${f}_{0}$ can be reduced to $f(0\le f<{f}_{0})$ with an investment cost $C(f)=K{({f}_{0}-f)}^{2}$, which is similar to the study of Yalabik and Fairchild [28], where $K>0$ represents the cost factor related to low-carbon technology, $f$ is the product carbon footprint, which is decision-making variable for the manufacturer.
- (4)
- Following the study of Goyal and Netessine [29] and Xu et al. [17], we assume that consumer demand for the product is $Q=a-bp$, where $a$($a>0$) represents the potential market size, $b$ $(b>0)$ is the price sensitivity of consumer, and $p$($p>w>c+\theta {f}_{0}$) is the selling price, which is a decision-making variable for the retailer.

#### 3.2. The Centralized Decision-Making Model

**Proposition**

**1.**

**Note.**With centralized decision-making, if the carbon cap allocated by the government is relatively large ($N>{N}_{0}^{CTC}$), the manufacturer can sell $N-{N}_{0}^{CTC}$ emission permits to the carbon trading market. If the carbon cap is relatively small ($N<{N}_{0}^{CTC}$), the manufacturer needs to buy ${N}_{0}^{CTC}-N$ carbon rights to emit additional emissions from the carbon trading market. If the carbon cap satisfies $N={N}_{0}^{CTC}$, the manufacturer neither sells nor buys carbon permits.

**Corollary**

**1.**

- (a)
- If the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy any of the following two conditions, the optimal product carbon footprint will decrease with increasing carbon trading price:$$\u24600{f}_{0}{\gamma}^{CTC},0\theta \frac{2\sqrt{Kb}}{b};\u2461{\gamma}^{CTC}{f}_{0}\frac{a-bc}{b\theta},0\theta {\theta}_{1}^{CTC}.$$
- (b)
- If the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy, respectively, ${\gamma}^{CTC}<{f}_{0}<\frac{a-bc}{b\theta}$ and ${\theta}_{1}^{CTC}<\theta <\frac{2\sqrt{Kb}}{b}$, the optimal product carbon footprint will increase with increasing carbon trading price.$$\u24600{f}_{0}{\gamma}^{CTC},0\theta \frac{2\sqrt{Kb}}{b};\u2461{\gamma}^{CTC}{f}_{0}\frac{a-bc}{b\theta},0\theta {\theta}_{1}^{CTC};$$$$\mathrm{where}{\gamma}^{CTC}=\frac{(a-bc)\sqrt{Kb}}{2Kb},{\theta}_{1}^{CTC}=\frac{4Kb{f}_{0}-2\sqrt{Kb\left[4Kb{f}_{0}{}^{2}-{(a-bc)}^{2}\right]}}{(a-bc)b}.$$

**Corollary**

**2.**

- (a)
- If the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy any of the following two conditions, the optimal selling price will increase with increasing carbon trading price:$$\u24600{f}_{0}{\gamma}^{CTC},0\theta {\theta}_{3}^{CTC}\u2461{\gamma}^{CTC}{f}_{0}\frac{a-bc}{b\theta},0\theta \frac{2\sqrt{Kb}}{b}.$$
- (b)
- If the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy, respectively, $0<{f}_{0}<{\gamma}^{CTC}$ and ${\theta}_{3}^{CTC}<\theta <\frac{2\sqrt{Kb}}{b}$, the optimal selling price will decrease with increasing carbon trading price,$$\mathrm{where}{\theta}_{3}^{CTC}=\frac{a-bc-\sqrt{{(a-bc)}^{2}-4Kb{f}_{0}{}^{2}}}{b{f}_{0}}.$$

#### 3.3. The Decentralized Decision-Making Model

**Proposition**

**2.**

**Note.**With decentralized decision-making, if the carbon cap allocated by the government is relatively large ($N>{N}_{0}^{CTD}$), the manufacturer can sell $N-{N}_{0}^{CTD}$ emission permits to the outside market. If the carbon cap is relatively small ($N<{N}_{0}^{CTD}$), then the manufacturer needs to buy ${N}_{0}^{CTD}-N$ carbon rights to emit additional carbon emissions from a carbon trading market. If the carbon cap satisfies $N={N}_{0}^{CTD}$, then the manufacturer neither sells nor buys carbon permits.

**Corollary**

**3.**

- (a)
- If the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy any of the following two conditions, then the optimal product carbon footprint will decrease with increasing carbon trading price:$$\u24600{f}_{0}{\gamma}^{CTD},0\theta \frac{2\sqrt{Kb}}{b};\u2461{\gamma}^{CTD}{f}_{0}\frac{a-bc}{b\theta},0\theta {\theta}_{1}^{CTD}.$$
- (b)
- If the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy, respectively, ${\gamma}^{CTD}<{f}_{0}<\frac{a-bc}{b\theta}$ and ${\theta}_{1}^{CTD}<\theta <\frac{2\sqrt{Kb}}{b}$, then the optimal product carbon footprint will increase with increasing carbon trading price,$$\mathrm{where}{\gamma}^{CTD}=\frac{(a-bc)\sqrt{2Kb}}{4Kb},{\theta}_{1}^{CTD}=\frac{8Kb{f}_{0}-2\sqrt{2Kb\left[8Kb{f}_{0}{}^{2}-{(a-bc)}^{2}\right]}}{(a-bc)b}.$$

**Corollary**

**4.**

- (a)
- If the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy any of the following two conditions, then the optimal sales price will increase with increasing carbon trading price:$$\u24600{f}_{0}{\gamma}^{CTD},0\theta {\theta}_{3}^{\mathit{CTD}};\u2461{\gamma}^{CTD}{f}_{0}\frac{a-bc}{b\theta},0\theta \frac{2\sqrt{Kb}}{b}.$$
- (b)
- If the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy, respectively,$0<{f}_{0}<{\gamma}^{CTD}$ and ${\theta}_{3}^{CTD}<\theta <\frac{2\sqrt{Kb}}{b}$, then the optimal selling price will decrease with increasing carbon trading price,$$\mathrm{where}{\theta}_{3}^{CTD}=\frac{a-bc-\sqrt{{(a-bc)}^{2}-8Kb{f}_{0}{}^{2}}}{b{f}_{0}}.$$

#### 3.4. The Comparative Analysis of Models

**Proposition**

**3.**

- (a)
- When the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy any of the following three conditions, regardless of the manufacturer and retailer taking centralized or decentralized decision-making, the optimal product carbon footprint will decrease with increasing carbon trading price:$$\u24600{f}_{0}{\gamma}^{CTD},0\theta \frac{2\sqrt{Kb}}{b};\u2461{\gamma}^{CTD}{f}_{0}{\gamma}^{CTC},0\theta {\theta}_{1}^{CTD};\phantom{\rule{0ex}{0ex}}\u2462{\gamma}^{CTC}{f}_{0}\frac{a-bc}{b\theta},0\theta {\theta}_{1}^{CTC}.$$
- (b)
- When the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy, respectively, ${\gamma}^{CTD}<{f}_{0}<{\gamma}^{CTC}$ and ${\theta}_{1}^{CTD}<\theta <\frac{2\sqrt{Kb}}{b}$, if the manufacturer and retailer take centralized decision-making, the optimal product carbon footprint will decrease with increasing carbon trading price; and, if the manufacturer and the retailer take decentralized decision-making, the optimal product carbon footprint will increase with increasing carbon trading price.
- (c)
- When the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy, respectively, ${\gamma}^{CTC}<{f}_{0}<\frac{a-bc}{b\theta}$ and ${\theta}_{1}^{CTC}<\theta <{\theta}_{1}^{CTD}$, if the manufacturer and the retailer take centralized decision-making, the optimal product carbon footprint will increase with increasing carbon trading price; and, if the manufacturer and retailer take decentralized decision-making, the optimal product carbon footprint will decrease with increasing carbon trading price.
- (d)
- When the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy, respectively, ${\gamma}^{CTC}<{f}_{0}<\frac{a-bc}{b\theta}$ and ${\theta}_{1}^{CTD}<\theta <\frac{2\sqrt{Kb}}{b}$, regardless of the manufacturer and retailer taking centralized or decentralized decision-making, the optimal product carbon footprint will increase with increasing carbon trading price.

**Proposition**

**4.**

- (a)
- When the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy any of the following two conditions, regardless of the manufacturer and retailer taking centralized or decentralized decision-making, the optimal selling price of the product will increase with increasing carbon trading price:$$\u24600{f}_{0}{\gamma}^{CTC},0\theta {\theta}_{3}^{CTC};\u2461{\gamma}^{CTC}{f}_{0}\frac{a-bc}{b\theta},0\theta \frac{2\sqrt{Kb}}{b}.$$
- (b)
- When the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) satisfy any of the following two conditions, if the manufacturer and retailer take centralized decision-making, the optimal selling price of the product will decrease with increasing carbon trading price; and if the manufacturer and retailer take decentralized decision-making, then the optimal selling price of the product will increase with increasing carbon trading price.$$\u24600{f}_{0}{\gamma}^{CTD},{\theta}_{3}^{CTC}\theta {\theta}_{3}^{CTD};\u2461{\gamma}^{CTD}{f}_{0}{\gamma}^{CTC},{\theta}_{3}^{CTC}\theta \frac{2\sqrt{Kb}}{b}.$$
- (c)
- When the initial carbon footprint (${f}_{0}$) and carbon trading price ($\theta $) are satisfied, respectively, $0<{f}_{0}<{\gamma}^{CTD}$ and ${\theta}_{3}^{CTD}<\theta <\frac{2\sqrt{Kb}}{b}$, regardless of the manufacturer and retailer taking centralized or decentralized decision-making, the optimal selling price of the product will decrease with increasing carbon trading price.

**Proposition**

**5.**

**Corollary**

**5.**

**Note.**The conclusions of Corollary 5 are similar to the research of Hua et al. [7], Benjaafar et al. [8] and Chen et al. [9]: given a fixed carbon trading price, the carbon emissions quota allocated by the government does not affect the retailer’s order quantities and carbon footprints. In fact, in Corollary 5, if the government reduces the carbon cap for the manufacturer, then the manufacturer has less carbon credit to sell, but to buy additional carbon credit through outside market, so the profits for the supply chain will be reduced.

#### 3.5. The Design of Coordinating Mechanism

**Proposition**

**6.**

## 4. Numerical Examples

#### 4.1. The Effects of Carbon Trading Price on Product Carbon Footprint

#### 4.2. The Effects of Carbon Trading Price on Selling Price

#### 4.3. The Effects of Carbon Trading Price on Consumer Demand

#### 4.4. The Effects of Carbon Trading Price on Supply Chain Profits

#### 4.5. The Effects of Carbon Trading Price on the Coordination Mechanism

#### 4.6. The Effects of Carbon Cap on Product Carbon Footprint

- (1)
- For the product with small initial carbon footprint (${f}_{0}=6$), whether it is centralized or decentralized, with the increasing of carbon trading price, the government should cut down the carbon cap to impel the manufacturer to reduce the product carbon footprint.
- (2)
- For the product with large initial carbon footprint (${f}_{0}=9$), with the decreasing of the carbon trading price, if the manufacturers and retailer take the centralized decision-making, the government should cut the carbon cap to impel the manufacturer to reduce the product carbon footprint. However, if the manufacturer and retailer take the decentralized decision-making, the government should first cut and then increase the carbon cap to motivate the manufacturer to reduce the product carbon footprint.
- (3)
- For the product with great initial carbon footprint (${f}_{0}=11$), whether it is centralized or decentralized, with the increasing of the carbon trading price, the government should first cut and then increase the carbon cap to motivate the manufacturer to reduce the product carbon footprint.

#### 4.7. The Effects of Carbon Cap on Selling Price

#### 4.8. The Effects of Carbon Cap on Consumer Demand

#### 4.9. The Effects of Carbon Cap on the Profits of the Supply Chain

#### 4.10. The Effects of Carbon Cap Price on the Coordination Mechanism

## 5. Conclusions and Future Research

- (1)
- Under cap-and-trade regulation, when the initial carbon footprint of the product is relatively small, regardless of the manufacturer and retailer taking the centralized or decentralized decision-making, the product carbon footprint decreases with the rising of carbon trading price, and also decreases with the decreasing of the carbon cap. The selling price first increases and then decreases with the increasing of carbon trading price, and also first increases and then decreases with the decreasing of the carbon cap.
- (2)
- Under cap-and-trade regulation, when the initial carbon footprint of the product is relatively large, if the manufacturer and retailer take centralized decision-making, then the product carbon footprint decreases with the increasing of carbon trading price, and also decreases with the decreasing of the carbon cap, the selling price first increases and then decreases with the increasing of carbon trading price, and also first increases and then decreases with the decreasing of the carbon cap; and if the manufacturer and retailer take the centralized decision-making, then the product carbon footprint first decreases and then increases with the increasing of carbon trading price, and also first decreases and then increases with the decreasing of the carbon cap, the selling price increases with the increasing of carbon trading price, and also first increases and then decreases with the decreasing of the carbon cap.
- (3)
- Under cap-and-trade regulation, when the initial carbon footprint of the product is large, regardless of the manufacturer and retailer taking the centralized or decentralized decision-making, the product carbon footprint first decreases and then increases with the increasing of carbon trading price, and also first decreases and then increases with the decreasing of the carbon cap; and the selling price increases with the increasing of carbon trading price, and also increases with the decreasing of the carbon cap.
- (4)
- Under cap-and-trade regulation, when the initial carbon footprint of the product is relatively small, the manufacturer’s wholesale price first increases and then decreases with the increasing of carbon trading price, and also decreases with the decrease of the carbon cap, and the fixed fee first decreases and then increases with the increasing of carbon trading price, and also first decreases and then increases with the decreasing of the carbon cap; and when the initial carbon footprint of the product is relatively large, the manufacturer’s wholesale price increases with the increasing of carbon trading price, and also first decreases and then increases with the decreasing of the carbon cap, and the fixed fee decreases with the increasing of carbon trading price, and also decreases with the decreasing of the carbon cap, then the contract can solve the “double marginalization” problem and coordinate the supply chain system.

- (1)
- For the product with small initial carbon footprint, the increasing of carbon trading price may inspire the manufacturer to reduce the product carbon footprint, but, if the carbon trading price is low, then the government can encourage the manufacturer to reduce their carbon footprint by reducing the carbon cap.
- (2)
- For the product with large initial carbon footprint, in the centralized decision-making, the increasing of carbon trading price may inspire the manufacturer to reduce the product carbon footprint. Contrary, if carbon trading price is low, then the government can motivate the manufacturer to reduce their carbon footprint by reducing the carbon cap. In the decentralized decision-making, the increasing of carbon trading price cannot inspire the manufacturer to reduce the product carbon footprint, and then the government can first decrease and then increase the carbon cap to inspire the manufacturer to reduce the product carbon footprint.
- (3)
- For the product with great initial carbon footprint, such as some high energy consumption, high-emission product, the increaseof carbon trading price cannot promote the manufacturer to reduce product carbon footprint. Therefore, if the carbon trading price increases, the government should first decrease and then increase the carbon cap to motivate the manufacturer to reduce the product carbon footprint.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof of Corollary**

**1.**

**Proof of Corollary**

**2.**

**Proof of Corollary**

**3.**

**Proof of Corollary**

**4.**

**Proof of Proposition**

**3.**

**Proof of Proposition**

**4.**

**Proof of Proposition**

**5.**

**Proof of Corollary**

**5.**

**Proof of Proposition**

**6.**

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## Share and Cite

**MDPI and ACS Style**

Cheng, Y.; Xiong, Z.; Luo, Q.
Joint Pricing and Product Carbon Footprint Decisions and Coordination of Supply Chain with Cap-and-Trade Regulation. *Sustainability* **2018**, *10*, 481.
https://doi.org/10.3390/su10020481

**AMA Style**

Cheng Y, Xiong Z, Luo Q.
Joint Pricing and Product Carbon Footprint Decisions and Coordination of Supply Chain with Cap-and-Trade Regulation. *Sustainability*. 2018; 10(2):481.
https://doi.org/10.3390/su10020481

**Chicago/Turabian Style**

Cheng, Yonghong, Zhongkai Xiong, and Qinglin Luo.
2018. "Joint Pricing and Product Carbon Footprint Decisions and Coordination of Supply Chain with Cap-and-Trade Regulation" *Sustainability* 10, no. 2: 481.
https://doi.org/10.3390/su10020481