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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 66, 2018 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Songying Zhao, Yougang Sun, Ye Zhou 
Copyright © 2018, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-63-1; ISSN 2283-9216 

Carbon Reduction Technology Investment of Supply Chain 

under Quota Trading and Government Subsidy 

Qiang Hou*, Jing Sun 

School of Management, Shenyang University of Technology, Shenyang 110870, China 

syuthouqiang@126.com 

This paper aims to improve government and enterprise decision-making on emission reduction. Focusing on 

the carbon emission reduction (CER) technology investment of manufacturers and the cost sharing of 

retailers, the government policies on carbon quota, carbon trading and emission reduction subsidy were taken 

into consideration. The game theory was adopted to build an expected profit model between the retailer and 

the manufacturer, and the impacts of the government policies on four decision-making scenarios for emission 

reduction, namely, decentralized decision-making, cooperative game, centralized decision-making and social 

planning. Through modelling, comparative analysis and parametric sensitivity analysis, it is concluded that the 

CER level, the optimal order quantity and the optimal social benefit are highest under social planning, followed 

by centralized decision-making, decentralized decision-making and cooperative game. The research findings 

shed new light on energy conservation and emission reduction in two-level supply chain. 

1. Introduction 

Global warming, a common challenge to the world, concerns the survival and development of mankind. To 

solve the problem, the Kyoto Protocol was adopted in 1997 to control greenhouse gas emissions through the 

concerted efforts of countries around the world. Since then, some countries have stepped up their efforts on 

energy conservation and emission reduction, while some have laid down a series of policies to support low-

carbon development and emission reduction among enterprises. These efforts and policies have won an 

active response from the enterprises. Much research has been done on the carbon emission reduction (CER) 

behaviours of the supply chain in the low-carbon environment. For instance, Liu et al. (2012) suggested that 

enterprises on a two-level low-carbon supply chain generally invest a lot on emission reduction technologies. 

Jaber et al. (2013) investigated the cooperative emission reduction in a two-level supply chain. Luo et al. 

(2014) discussed the effect of consumer carbon footprint on optimal decision-making after modelling the 

decentralized, centralized and cooperative decision-making processes of supply chain enterprises. Yang and 

Wang (2016) constructed a Stackelberg game model to analyse the economic effect of supply chain emission 

reduction under three conditions. Based on low carbon preference of consumers, Wang and Zhao (2014) 

examined the optimal decision-making of supply chain members. Zhang et al. (2015) created a game model 

for cooperative emission reduction in supply chain and applied the model to analyse the optimal emission 

reduction under carbon tax. From the angle of carbon tax, Yang et al. (2016) probed into the optimal decision-

making of CER among supply chain enterprises. Meng (2010) explored the impact of government subsidy on 

emission reduction, considering exogenous carbon emission tax. Cao et al. (2013) verified the effectiveness of 

government subsidy and enterprise cooperation with a game model between government and enterprises. 

Gong and Zhou (2013) identified the most cost-effective decision-making behaviours of enterprises on the 

purchase of carbon quota. Yang and Yu (2016) proposed and solved a game model of two-level low-carbon 

supply chain, laying the basis for determining the optimal emission reduction rate. To improve government and 

enterprise decision-making, this paper targets the emission reduction technology investment of manufacturers 

and the cost sharing of retailers, considering the government policies on carbon quota, carbon trading and 

emission reduction subsidy.  

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1866089 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Hou Q., Sun J., 2018, Carbon reduction technology investment of supply chain under quota trading and 
government subsidy, Chemical Engineering Transactions, 66, 529-534  DOI:10.3303/CET1866089   

529



2. Problem description and hypotheses 

It is assumed that the low-carbon supply chain consists of only one manufacturer and one retailer, the carbon 

trading price is determined by exogenous variables in the carbon market, and the unit carbon emission at the 

current technology level is constant. Besides, the manufacturer produces products at the unit cost of 𝑐𝑚, while 

the retailer purchases the intermediate products from the manufacturer at the wholesale price of 𝑝𝑚 according 

to the market demand. The quantity of the purchased products is denoted as q, and the production cost of 

these products is denoted as 𝑐𝑑. There is no overstock or understock. In addition, the consumers are willing to 

buy low-carbon products if the retail price is 𝑝𝑑 = 𝑎 + 𝛼 − 𝑏𝑞 (𝑎>0, 𝑏>0 and 𝑎>𝑐𝑚). To support the emission 

reduction among enterprises, the government grants them a certain amount of carbon quota 𝑔𝑚, allows them 

to trade the quota at the unit price of 𝑝𝑐, and subsidizes their investment on emission reduction technologies 

at the rate oft. Considering the consumers’ preference to low-carbon products and the 𝑔𝑚 𝑝𝑐 introduction of 

carbon trading system, the manufacturer is more inclined to produce low-carbon products, increase 

investment in emission reduction technologies. The unit initial carbon emissions and the unit CER are denoted 

as 𝑒𝑚  and ∆𝑒𝑚 , respectively. Theemission reduction technology investment is a function of CER 𝐶(∆𝑒𝑚 ) =
𝛽∆𝑒𝑚

2

2
, 𝐶′(∆𝑒𝑚 ) ⩾ 0with 𝛽 (𝛽>0) is the emission reduction cost coefficient. 

3. Model analysis of emission reduction technology investment 

There are four different decision-making scenarios for CER technology investment of supply chain enterprises: 

decentralized decision-making, the cooperative game, centralized decision-making and social planning. 

3.1 Model of decentralized decision-making 

Under decentralized decision-making, the manufacturer and the retailer make their own decisions to maximize 

their own profits. The decision-making process is as follows: the government determines the subsidy rate for 

emission reduction; the manufacturer determines the wholesale price and the CER level; the retailer 

determines the order quantity. The three-stage process was discussed below by backward induction.  

In the third stage, the retailer determines the order quantity based on the market demand and the wholesale 

price 𝑝𝑚 to maximize its profit. The retailer’s strategy can be expressed as: 

𝑝𝑚 𝑚𝑎𝑥𝜋𝑑 = [𝑝𝑑 − (𝑝𝑚 + 𝑐𝑑 )]𝑞                                                                                                                          (1) 

The optimal order quantity is: 

𝑞 =
𝐴+𝑐𝑚−𝑝𝑚

2𝑏
                                                                                                                                                       (2) 

where𝐴 = 𝑎 + 𝛼 − 𝑐𝑚 − 𝑐𝑑. 

In the second stage, the manufacture makes the decision (𝑝𝑚 , 𝛥𝑒𝑚) to maximize its profit. The manufacturer’s 

strategy can be expressed as: 

𝑝𝑚 𝑚𝑎𝑥
𝑝𝑚,𝛥𝑒𝑚

𝜋𝑚 = (𝑝𝑚 − 𝑐𝑚 )𝑞 + 𝑝𝑐 [𝑔𝑚 − (𝑒𝑚 − 𝛥𝑒𝑚 )]𝑞 −
1

2
(1 − 𝑡)𝛽𝛥𝑒𝑚

2
                                                             (3) 

Substituting equation (2) into equation (3), the optimal wholesale price can be derived from the first-order 

optimal condition:𝑝𝑚 =
𝐴+2𝑐𝑚−𝑝𝑐(𝑔𝑚−𝑒𝑚+𝛥𝑒𝑚)

2
,the optimal CER level 𝛥𝑒𝑚 =

𝑝𝑐(𝐴+𝑝𝑐 𝑔𝑚−𝑝𝑐𝑒𝑚)

4𝑏𝛽(1−𝑡)−𝑝𝑐
2

  . 

In the first stage, the government determines the subsidy rate for CER investment to maximize the social 

benefit. The government’s strategy can be expressed as: 

𝑚𝑎𝑥𝐺 =
1

2
𝑏𝑞2 + 𝜋𝑚 + 𝜋𝑑 −

1

2
𝑡𝛽𝛥𝑒𝑚

2
                                                                                                                 (4) 

Substituting equation (5) into equations (1), (2), (3), (4) and (6), the optimal strategy for social benefit can be 

derived from the first-order optimal condition. Thus, it is possible to put forward Proposition 1. 

Proposition 1: Under decentralized decision-making, the maximum social benefit is the goal of setting the 

subsidy rate for CER investment. If0 < 7𝑝𝑐
2 < 16𝑏𝛽, the optimal values of relevant parameters are as follows: 

The optimal subsidy rate for CER investment 𝑡1
∗ =

3

7
 ,the optimal CER level 𝛥𝑒𝑚1

∗
=

7𝑝𝑐(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐 𝑒𝑚)

16𝑏𝛽−7𝑝𝑐
2

   ,the 

optimal wholesale price: 𝑝𝑚1
∗ =

8𝑏𝛽[𝐴+2𝑐𝑚−𝑝𝑐(𝑔𝑚−𝑒𝑚)]−7𝑝𝑐
2(𝐴+𝑐𝑚)

16𝑏𝛽−7𝑝𝑐
2

 , and the optimal order quantity 𝑞1
∗ =

4𝛽(𝐴+𝑝𝑐 𝑔𝑚−𝑝𝑐𝑒𝑚)

16𝑏𝛽−7𝑝𝑐
2

 . 

530



The manufacturer’s profit 𝜋𝑚1
∗ =

2𝛽(𝐴+𝑝𝑐 𝑔𝑚−𝑝𝑐𝑒𝑚)
2

16𝑏𝛽−7𝑝𝑐
2  ,The retailer’s profit 𝜋𝑑1

∗ =
16𝑏𝛽2(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

2

(16𝑏𝛽−7𝑝𝑐
2)

2 ,the supply 

chain profit 𝜋𝑇1
∗ =

(48𝑏𝛽2−14𝛽𝑝𝑐
2)(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚 )

2

(16𝑏𝛽−7𝑝𝑐
2)

2 ,and the social benefit 𝐺1
∗

=
7𝛽(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

2

2(16𝑏𝛽−7𝑝𝑐
2)

. 

The following conclusion can be drawn from Proposition 1: 

Conclusion 1: When CER cost coefficient and carbon trading price remain constant, both the CER level and 

order quantity increase with carbon quota, while the wholesale price decreases. When the carbon quota and 

carbon trading price remain constant, the CER level is negatively correlated with the CER cost coefficient. 

3.2 Model of cooperative game 

Under the cooperative game, the manufacturer and the retailer look for cooperation in that the CER 

investment cost is partially allocated to the retailer at the ratio of k. The decision-making process is as follows: 

the government determines the subsidy rate for emission reduction investment; the retailer determines the 

cost-sharing ratio; the manufacturer determines the wholesale price and the CER level; the retailer determines 

the order quantity. The four-stage process was discussed below by backward induction. 

In the fourth stage, the retailer determines the order quantity based on the market demand and the wholesale 

price𝑝𝑚to maximize its profit. The retailer’s strategy can be expressed as: 

𝑝𝑚 𝑚𝑎𝑥𝜋𝑑 = [𝑝𝑑 − (𝑝𝑚 + 𝑐𝑑 )]𝑞 −
1

2
𝑘(1 − 𝑡)𝛽𝛥𝑒𝑚

2
                                                                                             (5) 

The optimal order quantity is 𝑞 =
𝐴+𝑐𝑚−𝑝𝑚

2𝑏
   

In the third stage, the manufacture makes the decision (𝑝𝑚 , 𝛥𝑒𝑚) to maximize its profit. The manufacturer’s 

strategy can be expressed as: 

𝑚𝑎𝑥
𝑝𝑚,𝛥𝑒𝑚

𝜋𝑚 = (𝑝𝑚 − 𝑐𝑚 )𝑞 + 𝑝𝑐 [𝑔𝑚 − (𝑒𝑚 − 𝛥𝑒𝑚 )]𝑞 −
1

2
(1 − 𝑘)(1 − 𝑡)𝛽𝛥𝑒𝑚

2
                                                              (6) 

The optimal wholesale price can be derived from the first-order optimal condition:𝑝𝑚 =
𝐴+2𝑐𝑚−𝑝𝑐(𝑔𝑚−𝑒𝑚+𝛥𝑒𝑚)

2
, 

the optimal CER level 𝛥𝑒𝑚 =
𝑝𝑐(𝐴+𝑝𝑐 𝑔𝑚−𝑝𝑐𝑒𝑚)

4𝑏𝛽(1−𝑘)(1−𝑡)−𝑝𝑐
2
 . 

In the second stage, the retailer determines the cost-sharing ratio of the CER investment to maximize its own 

profit.  

In the first stage, the government determines the subsidy rate for CER investment to maximize the social 

benefit. The government’s strategy can be expressed as: 

𝑚𝑎𝑥𝐺 =
1

2
𝑏𝑞2 + 𝜋𝑚 + 𝜋𝑑 −

1

2
𝑡𝛽𝛥𝑒𝑚

2
                                                                                                                 (7) 

The optimal strategy for social benefit can be derived from the first-order optimal condition. Thus, it is possible 

to put forward Proposition 2. 

Proposition 2: Under cooperative game, the maximum social benefit is the goal of setting the subsidy rate for 

CER investment. If 0 < 7𝑝𝑐
2 < 16𝑏𝛽, the optimal values of relevant parameters are as follows: 

The optimal subsidy rate for CER investment 𝑡2
∗ =

24𝑏𝛽−7𝑝𝑐
2

56𝑏𝛽
  ,the optimal cost-sharing ratio 𝑘2

∗
=

7𝑝𝑐
2

32𝑏𝛽+7𝑝𝑐
2
 , 

the optimal CER level 𝛥𝑒𝑚2
∗

=
7𝑝𝑐(𝐴+𝑝𝑐 𝑔𝑚−𝑝𝑐𝑒𝑚)

16𝑏𝛽−7𝑝𝑐
2

    ,the optimal wholesale price 𝑝𝑚2
∗ =

8𝑏𝛽[𝐴+2𝑐𝑚−𝑝𝑐(𝑔𝑚−𝑒𝑚)]−7𝑝𝑐
2(𝐴+𝑐𝑚)

16𝑏𝛽−7𝑝𝑐
2

,and the optimal order quantity 𝑞2
∗ =

4𝛽(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

16𝑏𝛽−7𝑝𝑐
2

  . 

The manufacturer’s profit 𝜋𝑚2
∗ =

2𝛽(𝐴+𝑝𝑐 𝑔𝑚−𝑝𝑐𝑒𝑚)
2

16𝑏𝛽−7𝑝𝑐
2

 ,the retailer’s profit 𝜋𝑑2
∗ =

(16𝑏𝛽+7𝑝𝑐
2)(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

2

16𝑏(16𝑏𝛽−7𝑝𝑐
2)

 ,the 

supply chain profit 𝜋𝑇2
∗ =

(48𝑏𝛽+7𝑝𝑐
2)𝐵2

16𝑏(16𝑏𝛽−7𝑝𝑐
2)

   , and the social benefit 𝐺2
∗

=
7𝛽(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐 𝑒𝑚)

2

2(16𝑏𝛽−7𝑝𝑐
2)

   

The following conclusion can be drawn from Proposition 2: 

Conclusion 2: When CER cost coefficient and carbon trading price remain constant, both the CER level and 

order quantity increase with carbon quota, while the wholesale price decreases. When the carbon quota and 

carbon trading price remain constant, the cost-sharing ratio and CER level are negatively correlated with the 

CER cost coefficient. 

3.3 Model of centralized decision-making 

Under centralized decision-making, the manufacturer and the retailer aim to maximize the overall profit of the 

supply chain. The decision-making process is as follows: the government determines the subsidy rate for 

emission reduction; the supply chain enterprises determine the CER level and the order quantity. The two-

531



stage process was discussed below by backward induction. 

In the second stage, the order quantity 𝑞 and the CER level 𝛥𝑒𝑚  are the bases for decision-making. The 

strategy of the two enterprises can be expressed as: 

𝑚𝑎𝑥𝜋𝑇 = 𝜋𝑚 + 𝜋𝑑 = [𝑝𝑑 − (𝑐𝑚 + 𝑐𝑑 )]𝑞+𝑝𝑐 [𝑔𝑚 − (𝑒𝑚 − 𝛥𝑒𝑚 )]𝑞 −
1

2
(1 − 𝑡)𝛽𝛥𝑒𝑚

2
                                             (8) 

Find the partial derivatives of𝑞and𝛥𝑒𝑚according to equation (9). Then, the following equations can be derived 

from the first-order optimal condition: 

𝜕𝜋𝑇

𝜕𝑞
= 𝐴 − 2𝑏𝑞 + 𝑝𝑐 (𝑔𝑚 − 𝑒𝑚 ) + 𝑝𝑐 𝛥𝑒𝑚 = 0                                                                                                        (9) 

𝜕𝜋𝑇

𝜕𝛥𝑒𝑚
= 𝑝𝑐 𝑞 − 𝛽(1 − 𝑡)𝛥𝑒𝑚 = 0                                                                                                                         (10) 

Then, the optimal CER level 𝛥𝑒𝑚 =
𝑝𝑐(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

2𝑏𝛽(1−t)−𝑝𝑐
2

 ,the optimal order quantity 𝑞 =
𝛽(1−𝑡)(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

2𝑏𝛽(1−t)−𝑝𝑐
2

. 

In the first stage, the government determines the subsidy rate for CER investment to maximize the social 

benefit. The government’s strategy can be expressed as: 

𝑚𝑎𝑥𝐺 =
1

2
𝑏𝑞2 + 𝜋𝑇 −

1

2
𝑡𝛽𝛥𝑒𝑚

2
                                                                                                                        (11) 

We can obtain the optimal strategy for social benefit according to the first-order optimal condition. Thus, it is 

possible to put forward Proposition 3. 

Proposition 3: Under centralized decision-making, the maximum social benefit is the goal of setting the 

subsidy rate for CER investment. If 0 < 3𝑝𝑐
2 < 4𝑏𝛽, the optimal values of relevant parameters are as follows: 

The optimal subsidy rate for CER investment 𝑡3
∗ =

1

3
 ,the optimal CER level 𝛥𝑒𝑚3

∗
=

3𝑝𝑐(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚 )

4𝑏𝛽−3𝑝𝑐
2

 ,the 

optimal order quantity 𝑞3
∗ =

2𝛽(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

4𝑏𝛽−3𝑝𝑐
2

 .The supply chain profit 𝜋𝑇3
∗ =

𝛽(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐 𝑒𝑚)
2

4𝑏𝛽−3𝑝𝑐
2

, the social benefit 

𝐺3
∗

=
3𝛽(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

2

2(4𝑏𝛽−3𝑝𝑐
2)

 . 

The following conclusion can be drawn from Proposition 3: 

Conclusion 3: When CER cost coefficient and carbon trading price remain constant, both the CER level and 

order quantity increase with carbon quota, while the wholesale price decreases. When the carbon quota and 

carbon trading price remain constant, the CER level is negatively correlated with the CER cost coefficient. 

3.4 Model of social planning 

Under social planning, the manufacturer and the retailer aim to achieve the best social benefit through 

implementing CER activities. In other words, the two enterprises pursue the optimal social benefit in their 

decisions on order quantity 𝑞 and the CER level 𝛥𝑒𝑚.The strategy of the two enterprises can be expressed as: 

𝑒𝑚 𝑚𝑎𝑥𝜋 = [𝑝𝑑 − (𝑐𝑚 + 𝑐𝑑 )]𝑞+𝑝𝑐 [𝑔𝑚 − (𝑒𝑚 − 𝛥𝑒𝑚 )]𝑞 −
1

2
(1 − 𝑡)𝛽𝛥𝑒𝑚

2
                                                          (12) 

𝑚𝑎𝑥𝐺 =
1

2
𝑏𝑞2 + 𝜋 −

1

2
𝑡𝛽𝛥𝑒𝑚

2
                                                                                                                          (13) 

Find the partial derivatives of𝑞 and 𝛥𝑒𝑚 according to equation (13). Then, the following equations can be 

derived from the first-order optimal condition: 

𝜕𝐺

𝜕𝑞
= 𝐴 − 𝑏𝑞 + 𝑝𝑐 (𝑔𝑚 − 𝑒𝑚 ) + 𝑝𝑐 𝛥𝑒𝑚 = 0                                                                                                         (14) 

𝜕𝐺

𝜕𝛥𝑒𝑚
= 𝑝𝑐 𝑞 − 𝛽𝛥𝑒𝑚 = 0                                                                                                                                    (15) 

The optimal solution can be determined by solving two equations. The solution is summed up as Proposition 4. 

Proposition 4: 𝑝𝑐
2Under social planning, the maximum social benefit is the goal of both the manufacture and 

the retailer. If0 < 𝑝𝑐
2 < 𝑏𝛽, the optimal values of relevant parameters are as follows: 

The optimal CER level ∆𝑒𝑚4
∗

=
𝑝𝑐(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

𝑏𝛽−𝑝𝑐
2

  ,the optimal order quantity 𝑞4
∗ =

𝛽(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

𝑏𝛽−𝑝𝑐
2

,the supply 

chain profit 𝜋𝑇4
∗ = −

𝛽𝑝𝑐
2(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐 𝑒𝑚)

2

2(𝑏𝛽−𝑝𝑐
2)2

 , and the social benefit 𝐺4
∗

=
𝛽(𝐴+𝑝𝑐𝑔𝑚−𝑝𝑐𝑒𝑚)

2

2(𝑏𝛽−𝑝𝑐
2)

. 

The following conclusion can be drawn from Proposition 4: 

Conclusion 4: When CER cost coefficient and carbon trading price remain constant, both the CER level and 

order quantity increase with carbon quota, while the wholesale price decreases. When the carbon quota and 

carbon trading price remain constant, the CER level is negatively correlated with the CER cost coefficient. 

532



4. Comparative Analysis 

To disclose the utility differences among the above four different decision-making scenarios, the optimal 

solutions of these scenarios were subjected to comparative analysis. Under 0 < 𝑝𝑐
2 < 𝑏𝛽 and 𝐵 = 𝐴 + 𝑝𝑐 𝑔𝑚 −

𝑝𝑐 𝑒𝑚, we have the following proposition: 

Proposition 5: When 16𝑏𝛽 < 21𝑝𝑐
2,  the four scenarios can be ranked in descending order below by the 

government subsidy: decentralized decision-making, centralized decision-making, social planning and 

cooperative game; when 16𝑏𝛽 > 21𝑝𝑐
2 , the ranking can be expressed as: decentralized decision-making, 

cooperative game, social planning and centralized decision-making. In other words, if 16𝑏𝛽 < 21𝑝𝑐
2,then 𝑡1

∗ >

𝑡3
∗ > 𝑡2

∗; if 16𝑏𝛽 > 21𝑝𝑐
2, then 𝑡1

∗ > 𝑡2
∗ > 𝑡3

∗. 

Proof: 𝑡1
∗ =

3

7
=

24𝑏𝛽

56𝑏𝛽
> 𝑡2

∗ =
24𝑏𝛽−7𝑝𝑐

2

56𝑏𝛽
, and 𝑡1

∗ =
3

7
> 𝑡3

∗ =
1

3
,
𝑡2

∗

𝑡3
∗ =

72𝑏𝛽−21𝑝𝑐
2

56𝑏𝛽
= 1 +

16𝑏𝛽−21𝑝𝑐
2

56𝑏𝛽
 

If 16𝑏𝛽 < 21𝑝𝑐
2,

𝑡2
∗

𝑡3
∗ < 1, then𝑡2

∗ < 𝑡3
∗; If 16𝑏𝛽 > 21𝑝𝑐

2,
𝑡2

∗

𝑡3
∗ > 1, then𝑡2

∗ > 𝑡3
∗. 

Hence, if 16𝑏𝛽 < 21𝑝𝑐
2, then 𝑡1

∗ > 𝑡3
∗ > 𝑡2

∗; if 16𝑏𝛽 > 21𝑝𝑐
2, then 𝑡1

∗ > 𝑡2
∗ > 𝑡3

∗.Q.E.D. 

Proposition 6: The CER level is the highest under social planning, followed by that under centralized decision-

making, decentralized decision-making and cooperative game. In other words, ∆𝑒𝑚4
∗

> ∆𝑒𝑚3
∗

> ∆𝑒𝑚1
∗

=

∆𝑒𝑚2
∗
. 

Proof: Since ∆𝑒𝑚4
∗

− ∆𝑒𝑚3
∗

=
𝑝𝑐𝐵

𝑏𝛽−𝑝𝑐
2

−
3𝑝𝑐 𝐵

4𝑏𝛽−3𝑝𝑐
2

=
𝑏𝛽𝑝𝑐𝐵

(𝑏𝛽−𝑝𝑐
2)(4𝑏𝛽−3𝑝𝑐

2)
> 0, we have ∆𝑒𝑚4

∗
> ∆𝑒𝑚3

∗
 

Whereas ∆𝑒𝑚3
∗
−∆𝑒𝑚1,2

∗
=

3𝑝𝑐𝐵

4𝑏𝛽−3𝑝𝑐
2

−
7𝑝𝑐 𝐵

16𝑏𝛽−7𝑝𝑐
2

=
20𝑏𝛽𝑝𝑐𝐵

(4𝑏𝛽−3𝑝𝑐
2 )(16𝑏𝛽−7𝑝𝑐

2)
> 0, 

Thus, ∆𝑒𝑚3
∗

> ∆𝑒𝑚1,2
∗
, Therefore, ∆𝑒𝑚4

∗
> ∆𝑒𝑚3

∗
> ∆𝑒𝑚1

∗
= ∆𝑒𝑚2

∗
. Q.E.D. 

Proposition 7: The optimal order quantity is the highest under social planning, followed by that under 

centralized decision-making, decentralized decision-making and cooperative game. In other words, 𝑞4
∗ >

𝑞3
∗ > 𝑞1

∗ = 𝑞2
∗. 

Proof: Since 𝑞4
∗ − 𝑞3

∗ =
𝛽𝐵

𝑏𝛽−𝑝𝑐
2

−
2𝛽𝐵

4𝑏𝛽−3𝑝𝑐
2

=
𝛽(2𝑏𝛽−𝑝𝑐

2)𝐵

(𝑏𝛽−𝑝𝑐
2)(4𝑏𝛽−3𝑝𝑐

2)
> 0, we have 𝑞4

∗ > 𝑞3
∗ 

Since 𝑞3
∗−𝑞1,2

∗ =
2𝛽𝐵

4𝑏𝛽−3𝑝𝑐
2

−
4𝛽𝐵

16𝑏𝛽−7𝑝𝑐
2

=
2𝛽(8𝑏𝛽−𝑝𝑐

2)𝐵

(4𝑏𝛽−3𝑝𝑐
2)(16𝑏𝛽−7𝑝𝑐

2)
> 0, we have 𝑞3

∗ > 𝑞1,2
∗. 

Therefore, 𝑞4
∗ > 𝑞3

∗ > 𝑞1
∗ = 𝑞2

∗.Q.E.D. 

Proposition 9: The supply chain profit is the highest under centralized decision-making, followed by that under 

decentralized decision-making, cooperative game and social planning. In other words, 𝜋𝑇3
∗ > 𝜋𝑇1

∗ > 𝜋𝑇2
∗ >

𝜋𝑇4
∗. 

Proof: Since 
𝜋𝑇1

∗

𝜋𝑇3
∗

=

(48𝑏𝛽2−14𝛽𝑝𝑐
2)𝐵2

(16𝑏𝛽−7𝑝𝑐
2)

2

𝛽𝐵2

4𝑏𝛽−3𝑝𝑐
2

=
192𝑏2𝛽2−200𝑏𝛽𝑝𝑐

2+42𝑝𝑐
4

(16𝑏𝛽−7𝑝𝑐
2)

2  and 𝑏, 𝛽 > 0, we have 

192𝑏2𝛽2−200𝑏𝛽𝑝𝑐
2+42𝑝𝑐

4

(16𝑏𝛽−7𝑝𝑐
2)

2 <
192𝑏2𝛽2−180𝑏𝛽𝑝𝑐

2+42𝑝𝑐
4

(16𝑏𝛽−7𝑝𝑐
2)

2 =
(16𝑏𝛽−7𝑝𝑐

2)(12𝑏𝛽−6𝑝𝑐
2)

(16𝑏𝛽−7𝑝𝑐
2)

2 =
12𝑏𝛽−6𝑝𝑐

2

16𝑏𝛽−7𝑝𝑐
2. 

Since (12𝑏𝛽 − 6𝑝𝑐
2

) − (16𝑏𝛽 − 7𝑝𝑐
2) = 𝑝𝑐

2 − 4𝑏𝛽 < 0, we have 
12𝑏𝛽−6𝑝𝑐

2

16𝑏𝛽−7𝑝𝑐
2 < 1. Thus,

192𝑏2𝛽2−200𝑏𝛽𝑝𝑐
2+42𝑝𝑐

4

(16𝑏𝛽−7𝑝𝑐
2)

2 < 1, 

and 𝜋𝑇3
∗ > 𝜋𝑇1

∗. 

Since 𝜋𝑇1
∗ − 𝜋𝑇2

∗ =
(48𝑏𝛽2−14𝛽𝑝𝑐

2)𝐵2

(16𝑏𝛽−7𝑝𝑐
2)

2 −
(48𝑏𝛽+7𝑝𝑐

2)𝐵2

16𝑏(16𝑏𝛽−7𝑝𝑐
2)

=
(27𝑏𝛽+49𝑝𝑐

2)𝑝𝑐
2𝐵2

16𝑏(16𝑏𝛽−7𝑝𝑐
2)

2 > 0 , we have 𝜋𝑇1
∗ > 𝜋𝑇2

∗ and 𝜋𝑇4
∗ =

−
𝛽𝑝𝑐

2𝐵2

2(𝑏𝛽−𝑝𝑐
2)2

< 0. Therefore, 𝜋𝑇3
∗ > 𝜋𝑇1

∗ > 𝜋𝑇2
∗ > 𝜋𝑇4

∗. Q.E.D. 

Proposition 9: The social benefit is the highest under social planning, followed by centralized decision-making, 

decentralized decision-making and cooperative game. In other words, 𝐺4
∗

> 𝐺3
∗

> 𝐺1
∗

= 𝐺2
∗
. 

Proof: Since
𝐺4

∗

𝐺3
∗ =

𝛽𝐵2

2(𝑏𝛽−𝑝𝑐
2)

3𝛽𝐵2

2(4𝑏𝛽−3𝑝𝑐
2)

=
4𝑏𝛽−3𝑝𝑐

2

3𝑏𝛽−3𝑝𝑐
2

> 1 , we have 𝐺4
∗

> 𝐺3
∗

.Since
𝐺3

∗

𝐺1,2
∗ =

3𝛽𝐵2

2(4𝑏𝛽−3𝑝𝑐
2)

7𝛽𝐵2

2(16𝑏𝛽−7𝑝𝑐
2)

=
48𝑏𝛽−21𝑝𝑐

2

28𝑏𝛽−21𝑝𝑐
2

> 1 , we 

have 𝐺3
∗

> 𝐺1,2
∗
.Therefore, 𝐺4

∗
> 𝐺3

∗
> 𝐺1

∗
= 𝐺2

∗
.Q.E.D. 

5. Parametric Sensitivity Analysis 

The decision-making of supply chain enterprises hinges on carbon quota, carbon trading price and CER cost 

coefficient. Based on the previous studies, the values of the relevant parameters are configured as follows: 

𝑎 = 100, 𝛼 = 3, 𝑏 = 5, 𝑐𝑚 = 4.5, 𝑐𝑑 = 3 and 𝑒𝑚 = 2.3. With these parameters, a parametric sensitivity analysis 

was performed, and the results are recorded in Figure 1. 

As shown in Figure 1, in all four decision-making scenarios, the optimal emission reduction level ∆𝑒𝑚
∗
 

decreases with the increase in CER cost coefficient𝛽 when ∆𝑒𝑚4
∗

> ∆𝑒𝑚3
∗

> ∆𝑒𝑚1
∗

= ∆𝑒𝑚2
∗
,and increases 

with∆𝑒𝑚
∗
carbon trading price𝑃𝑐 and carbon quota𝑔𝑚when ∆𝑒𝑚4

∗
> ∆𝑒𝑚3

∗
> ∆𝑒𝑚1

∗
= ∆𝑒𝑚2

∗
. 

533



 

Figure 1: Parametric sensitivity analysis 

6. Conclusions 

Based on carbon quota, carbon trading and subsidy for emission reduction, this paper constructs an expected 

profit model between the retailer and the manufacturer under four decision-making scenarios. The modelling 

results show that the CER level, the optimal order quantity and the optimal social benefit are highest under 

social planning, followed by centralized decision-making, decentralized decision-making and cooperative 

game. When the unit carbon quota is below a certain threshold, the optimal supply chain profit is the highest 

under decentralized decision-making, followed by cooperative game, centralized decision-making and social 

planning. Under social planning, the optimal supply chain profit is always negative. When the unit carbon 

quota is above a certain threshold, the optimal supply chain profit is the highest under centralized decision-

making, followed by decentralized decision-making, cooperative game and social planning. Under social 

planning, the optimal supply chain profit is always negative. In addition, the optimal CER level is negatively 

correlated with the CER cost coefficient and positively with the unit carbon quota under all four decision-

making scenarios. 

Acknowledgments 

This research was supported by Liaoning Social Planning item (L15BJY035), Shenyang Municipal Science 

and Technology Bureau item (F16-233-5-08), Liaoning provincial financial research fund item (16C003). 

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