Joint Emission Reduction Dynamic Decisions and Coordination in a Supply Chain Considering Altruistic Behavior and Reference Low-Carbon Effect

: In the context of low-carbon economy, supply chain members’ joint emission reduction 8 issue has become a research hotspot, while there are few researches which synthetically studies the 9 effect of consumers ’ reference low-carbon effect and supply chain members’ altruistic behavior on 10 their decisions. To study the impact of supply chain members’ altruistic behavior and consumers ’ 11 reference low-carbon effect on their joint emission reduction decisions and profits, we build 12 optimization models under four decision scenarios, in which we solve the manufacturer’s and the 13 retailer’s optimal emission reduction strategies and other equilibrium solutions by differential game 14 theory. We obtain some findings. First, consumers' reference low-carbon effect will harm the profits of 15 the manufacturer and the retailer, discourage the manufacturer's enthusiasm to reduce emissions and 16 retailer's enthusiasm for low-carbon publicity. Second, the altruistic behavior of the manufacturer and 17 the retailer can not only weaken the negative impact of the reference low-carbon effect, but also 18 promote both parties to actively reduce emissions, help achieve Pareto improvement of their own 19 profits and utilities, and obtain additional social welfare. Third, the cost-sharing contract can encourage 20 the manufacturer to increase emission reduction investment without affecting the retailer’ s low-carbon 21 publicity investment, and can achieve a Pareto improvement of both parties ’ profits and utilities. In 22 addition, the cost-sharing ratio is only proportional to the marginal profits and altruistic intensity of 23 both parties, and is not affected by the reference low-carbon effect. Meanwhile, the cost-sharing ratio 24 will de crease as the manufacturer’s marginal profit and altruistic intensity increase, and will increase as 25 the retailer’s marginal profit and altruistic intensity increase. In particular, when the retailer is 26 completely altruistic, the cost-sharing contract can achieve perfect coordination of the supply chain.


Introduction 30
At present, facing the severe global climate change issue, reducing greenhouse gas emissions to 31 reduce air pollution and greenhouse effect has become a global consensus, and a low-carbon economic 32 development mode characterized by low pollution, low energy consumption and sustainable 33 development has become the focus of attention of all countries. Subsequently, the European Union 34 pledged in 2014 to reduce carbon emissions by 40% from 1990 levels by 2030. China promised to reduce 35 carbon emissions per unit of GDP by 40% to 50% by 2020 (compared to 2005). Meanwhile, affected by 36 the deterioration of the ecological environment, the public's awareness of environmental protection has 37 been further improved, more and more consumers are willing to buy low-carbon products (Zhang et 38 al.2015). According to a consumer survey, 80% of American consumers would like to buy low-carbon 39 products even though they have to pay an additional fee (Zhang et al.2020). Similarly, more and more 40 Chinese consumers are willing to purchase low-carbon products on the Ali platform, and the compound 41 growth rate of annual sales of low-carbon products was greater than 80% between 2011 and 2015 (Xia et 42 al.2020). In the context of low-carbon economy, to obtain sustainable competitive advantages, actively 43 carry out emission reduction activities will be an important way for companies around the world. 44 Facing the pressure of emission reduction and market competition, more and more manufactures 45 in the supply chain have also started to cooperate with retailers to carry out emission reduction 46 activities. On the one hand, manufacturers produce low-carbon products with higher low-carbon level 47 by increasing emission reduction investment and strengthening the low-carbon technology research and 48 development. For instance, in order to reduce carbon emissions, Haier, a home appliance manufacturer 49 that has been committed to practicing the low-carbon concept. Since the launch of the photovoltaic 50 power generation project, it has successively launched energy-saving and low-carbon central air 51 conditioners with a haze removal rate of 99.8% and above. On the other hand, retailers implement 52 low-carbon promotion to draw consumers in the market to buy low-carbon products. For instance, JD 53 and Taobao in China promote low-carbon products by using poster display and merchandise exhibition. 54 Wal-Mart requires its suppliers to increase their emissions reduction investment and join the carbon 55 emission disclosure program as required. At the same time, it will also use its strong network to 56 provide special product promotion for the suppliers in the plan (Zhou et al.2016). Many examples in 57 reality and a large number of previous studies have shown that manufacturers and retailers conducting 58 joint emission reduction activities has been an effective way when one party faces greater pressure to 59 reduce emissions. Thus, discussing how supply chain members can effectively carry out joint emission 60 reductions will be of great significance. 61 In the process of practicing low-carbon economy, consumers' low-carbon consumption awareness 62 is a factor that cannot be ignored for companies. Companies must fully study consumers' behavior 63 characteristics and carefully consider them when making operational decisions to obtain stronger market 64 competitiveness and greater market share. After a lot of researches, scholars find that when consumers 65 are going to buy products, both the current attributes and the expected attributes formed by their previous 66 purchase experiences will affect their purchasing behavior. This kind of purchasing psychology is 67 defined as reference effect by scholars (Dye et al.2018; Liu et al.2016). Accordingly, when consumers 68 choose to buy low-carbon products, they will not only be affected by the present low-carbon level of the 69 products, but also be affected by the reference point of low-carbon level formed from their past purchase 70 experience. Based on the definition of the reference effect, we define it as reference low-carbon effect. 71 Due to the significant impact of the reference low-carbon effect on consumers' low-carbon buying 72 behavior, it is very necessary to introduce it into the issue of joint emission reduction and study its impact 73 on supply chain members' emission reduction decisions. 74 In addition, many behavioral economics experiments and studies have shown that people not only 75 pay attention to their own interests, but also the interests of partners in the transaction process, which is 76 manifested as altruistic attributes (Loch and Wu.2008). Ignoring the altruistic behavior of individuals, 77 the research results often deviate from reality when researches are based on individual rationality 78 (Bendoly et al.2006). At present, with the continuous development of economic globalization, altruistic 79 behavior among supply chain members is becoming more and more obvious. For example, IBM 80 establishes the "IBM China Channel University" to share the latest technology and management 81 experience with channel partners; General Motors in the United States contributes funds to help its 82 suppliers achieve technological upgrades, and Toyota provides a large amount of technical and 83 management support for its parts suppliers to help them improve product productivity; Lenovo helps its 84 channel provider Teng Chuang Technology to raise funds to achieve a win-win situation. There is no 85 doubt that with the development of globalization, supply chain members will pay more and more 86 attention to mutual benefit with their partners, and their altruistic preferences will become more and more 87 obvious, which will have a significant impact on their decision-making. Therefore, in the context of a 88 low-carbon economy, it is of great practical significance to explore the impact of supply chain members' 89 altruistic behavior on their cooperative emission reduction decisions. 90 Under the above background and based on Stackelberg differential game method, this paper 91 explores the issue of joint emission reduction of supply chain with the consideration of supply chain 92 members' altruistic behavior and consumers' reference low-carbon effect from a long-term perspective. 93 The following problems will be solved in this paper. 94 Under the impact of supply chain members' altruistic behavior and consumers' reference 95 low-carbon effect and different scenarios, what are the supply chain members' equilibrium strategies. 96 What are the optimal evolution paths for product low-carbon level, reference low-carbon level and  97  low-carbon goodwill over time under different scenarios?  98   what impact will the cost-sharing contract have on the supply chain members' equilibrium  99  strategies?  100  What conditions should be met to achieve supply chain coordination with or without cost-sharing  101  contract if supply chain members have altruistic behavior.  102  By answering the above questions, this paper mainly has the following contributions.  103 First, this paper simultaneously integrates altruistic behavior and consumers' reference low-carbon 104 effect these two factors to study supply chain members' joint emission reduction decisions and supply 105 chain coordination. Second, this paper establishes supply chain emission reduction dynamic optimization 106 models under four different decision scenarios, and obtains the joint emission reduction strategies of 107 supply chain members from a long-term perspective. In short, through mathematical derivation and 108 numerical analysis, this paper has deeply discussed the impact of these two factors on the supply chain 109 members' joint emission reduction decisions, and can provide theoretical support for supply chain 110 members' long-term cooperative emission reduction under different decision scenarios. 111 This paper is organized as follows. Section 2 presents the literature review. Section 3 provides 112 problem description. The equilibrium solutions under four decision scenarios are provided in Section 4. 113 Section 5 is the analysis and comparison section. Section 6 gives some numerical examples to verify the 114 findings. Section 7 summarizes conclusions and limitations of this paper. 115

Literature Review 116
Based on the content of our research, there are two main streams of literature related to the issue 117 of supply chain operation that we will need to review. One is the issue of joint emission reduction in the 118 supply chain considering altruistic behavior or not, another one is the issue of operational decision in 119 the supply chain with consideration of consumers' reference effect. 120 Currently, joint emission reduction issue in the supply chain has been deeply studied in the 121 context of low-carbon economy (Benjaafar et al.2013 introduced reciprocal preferences and consumers' low-carbon awareness into the dyadic supply chain, 157 and investigated how reciprocity and CLA affect the emission reduction decisions and performances of 158 the supply chain members. 159 Although the above studies discuss the impact of altruistic behavior of supply chain members on 160 their emission reduction decisions, the emission reduction optimization models they establish are all 161 static optimization models. The strategies got by the static decision-making models can just report the 162 supply chain members' short-term optimal decisions, but the dynamic decision-making models can 163 better report the long-term effects of supply chain members' emission reductions. At present, there are 164 few literatures that consider the altruistic behavior of supply chain members and use dynamic 165 optimization models to discuss the issue of joint emission reduction in the supply chain. As far as we 166 know, only Xia et al. (2020) studied joint emission reduction issues by using differential game theory 167 considering social preferences of supply chain members. However, Xia et al. (2020) only considered 168 supply chain members' social preferences in the emission reduction model, they do not consider 169 consumers' reference low-carbon effect and product low-carbon goodwill, simultaneously. Ignoring 170 consumers' reference low-carbon effect and product low-carbon goodwill will inevitably cause models 171 and research conclusions to deviate from reality, so the research content needs to be supplemented. 172 Consumer behavior studies have shown that the difference between the actual product attribute 173 level experienced by consumers after shopping and the product reference attribute level expected by 174 consumers before shopping will have an important impact on product goodwill and market demand, that 175 is, consumers' reference effect. Recently, there have been some studies that introduce consumers' considering reference quality effect. 187 The above literature shows that reference price effect and reference quality effect have been 188 extensively and in-depth studied, but there are still few studies on consumers' reference low-carbon 189 effect. With the growing prosperity of the low-carbon economy, the awareness of low-carbon 190 consumption has become increasingly popular, there is no doubt that consumers' reference low-carbon 191 effect will have a remarkable influence on their low-carbon consumption behavior. At the same time, 192 as supply chain members carrying out joint emission reduction activities is becoming increasingly 193 common, studying the joint emission reduction issue considering their altruistic behavior will be very 194 necessary. In Table 1, we have summarized some literatures that have strong relevance to this paper to 195 further reveal the innovation and contribution of this paper. 196 Table 1 The main contributions of this paper impact of supply chain members' altruistic behavior and consumers' reference low-carbon effect, this 199 paper will study how these two factors impact the supply chain members' optimal emission reduction 200 strategies, profits and utilities under different decision scenarios through the differential game theory. 201 Furthermore, we will also explore the conditions for the cost-sharing contract to achieve supply chain 202 coordination when supply chain members have altruistic behavior. 203

Problem Description 204
To simplify the model and highlight the research focus, we consider a supply chain system 205 including a manufacturer and a retailer. The manufacturer produces a sort of low-carbon product by 206 emission reduction investment, and the low-carbon level of the product will be higher when the 207 manufacturer's emission reduction level is higher. Meanwhile, the retailer will also implement 208 low-carbon promotion will also be implemented by the retailer to draw consumers to purchase this sort of 209 low-carbon product. In addition, since manufacturers often need to pay greater emission reduction costs 210 to reduce emissions, retailers often share their emission reduction costs to encourage manufacturers to 211 reduce emissions. Some related parameters and decision variables involved in this paper are explained 212 in Table 2. 213 To better serve follow-up research, we propose some assumptions as follows: 214 Assumption 1 It is assumed that the manufacturer is the leader and the retailer is the follower in the 215 supply chain system. Both manufacturers and retailers grasp symmetrical and clear market information. 216 When they do not have altruistic behavior, they play Stackelberg game to maximize their own profits. 217 But when they have altruistic behavior, both of them will play Stackelberg game to maximize their own 218 altruistic utilities. 219 Assumption 2 Manufacturers can increase the low-carbon level of their products by investing in 220 emission reductions, and due to various factors such as equipment aging and backward technology, the 221 low-carbon level of the product has a natural decline over time. Learn from previous 222 literature (Bertinelli et al.2014), we use the differential equation shown in equation (1) to describe the 223 dynamic change process of the product's low carbon level over time. 224 Assumption 3 According to Hellofs et al.(1999), before purchasing low-carbon products, 226 consumers will make a basic judgment on the low-carbon level of the product based on the product's 227 low-carbon goodwill, and believe that the product's reference low-carbon level is positively correlated 228 with its low-carbon goodwill. Therefore, we follow Hellofs's point of view and assume that the product 229 reference low-carbon level and low-carbon goodwill have the following relationship: 230 Assumption 4 Refer to He et al.(2017), assuming that the product's low-carbon goodwill is 232 affected by both the consumer's reference low-carbon effect and the retailer's low-carbon publicity 233 investment. When the actual low-carbon level of the product purchased by the consumer is higher than 234 the reference low-carbon level, the low-carbon goodwill will be improved. When the actual low-carbon 235 level of the product purchased by consumers is lower than expected, it will have a negative impact on the 236 low-carbon goodwill of the product.
Substituting equation (2) Assumption 5 According to the literature (Kopalle and Winer 1996), we assume that the 244 low-carbon product's demand function is a deterministic and linear function. Products with higher 245 low-carbon level, low-carbon promotion level and low-carbon goodwill level are most popular among 246 consumers. At the same time, consumers will choose to purchase such product when its reference 247 low-carbon level is lower than the actual low-carbon level at present, otherwise, they will be reluctant to 248 buy the product because of the impact of reference low-carbon effect. What needs special explanation is 249 that with the current product prices becoming more and more transparent, consumers are no longer as 250 sensitive to product prices as they used to be. Instead, they pay more attention to non-price factors. 251 Therefore, this paper does not consider the impact of product prices. The low-carbon product's demand 252 function is shown as equation (5). 253 Assumption 6 Refer to the literature (Ghosh and Shah 2015), we assume that the emission 255 reduction cost of the manufacturer is , the low-carbon promotion cost of the retailer is 256 Assumption 7 Without considering the impact of price, the manufacturer's marginal revenue is 258 assumed to be 1  , the retailer's marginal revenue is assumed to be 2  . 1  and 2  are both assumed to be 259 constants. It is also assumed that both parties have the equal discount rate  ( 0   )at any time during 260 a business scope of unlimited time. Besides, the transportation and inventory costs of the product will 261 not be considered. 262 Table 2. Related parameters and decision variables involved in this paper Emission reduction cost coefficient and low-carbon promotion cost coefficient.

Model Solution 264
According to the research assumptions in the former section, in order to conduct a comparative 265 analysis of altruistic behavior, this section will first analyze the strategies and performances of the 266 members under centralized scenario and decentralized scenario without altruistic behavior. Then this 267 section will analyze decentralized scenario with altruistic behavior and decentralized scenario with 268 altruistic behavior and a cost-sharing contract these two decision scenarios. In order to distinguish among 269 different scenarios, superscripts c , d , a , y respectively represent centralized decision scenario, 270 decentralized decision scenario without altruistic behavior, decentralized decision scenario with 271 altruistic behavior, decentralized decision scenario with altruistic behavior and a cost-sharing contract 272 these four decision scenarios. In addition, subscripts m , r , sc respectively represent the manufacturer,

273
retailer and the supply chain system these three decision makers. For the convenience of writing, the time 274 will not be listed below. Here, the Hamilton-Jacobi-Bellman equation which also expressed as the HJB 275 equation is used to solve the equilibrium solutions of the different models. 276

4.1.Centralized Scenario 277
To achieve the optimal profit of the supply chain system, both the manufacturer and the retailer will 278 jointly determine their emission reduction investment decisions. Although the centralized decision 279 scenario is the most ideal supply chain decision scenario, it is difficult to achieve in actual enterprise 280 operations. Here we use it as the upper limit of reference for analyzing altruistic behavior. At this time, 281 the supply chain system's objective function is as follows: 282 Theorem 1 (1) The manufacturer's and retailer's optimal emission reduction decisions are as 284 follows： 285 (2) The optimal evolution path of low-carbon level is as follows： 287

289
(3) The optimal evolution paths of low-carbon goodwill and reference low carbon level are as 290 follows：: 291

295
(4) The optimal total profit of the supply chain system is as follows: 296

300
The proof of Theorem 1 is similar to the proof of Theorem 3 and is omitted. 301 Theorem 1 shows that the manufacturer's equilibrium emission reduction investment, the retailer's 302 equilibrium low-carbon publicity investment, the steady-state low-carbon level, the steady-state 303 low-carbon goodwill and the overall profit of the supply chain are all positively correlated with the 304 marginal profits of the manufacturer and the retailer. This means that in order to further highlight the 305 advantages of centralized decision-making and improve the overall profit level of the supply chain, 306 manufacturers and retailers should take a variety of measures to increase their own marginal profits. For 307 example, manufacturers can improve production efficiency, reduce production costs and transportation 308 costs through technological upgrades, and retailers can improve marketing efficiency through accurate 309 advertising. In addition, both parties should strengthen communication and cooperation, improve the 310 overall operating efficiency of the supply chain and reduce the overall operating costs of the supply 311 chain, thereby further improving the overall performance of the supply chain. 312

Decentralized Scenario 313
Under decentralized decision scenario, both the manufacturer and the retailer aim at maximizing 314 their own profits to make decisions. This decision-making model is often used in the actual operation of 315 enterprises, but the inherent double marginal effect of the decentralized decision-making model often 316 leads to the loss of the supply chain system's profit. Therefore, here, we will use it as the lower limit of 317 the reference for analyzing altruistic behavior. 318 At this time, the manufacturer's objective function is: 319 The retailer's objective function is: 321 Theorem 2 (1) The manufacturer's and retailer's optimal emission reduction decisions are as 323 follows： 324 (2) The optimal evolution path of low-carbon level is as follows： 326

328
(3) The optimal evolution paths of low-carbon goodwill and reference low carbon level are as 329 follows：: 330

334
(4) The optimal profits of the manufacturer and retailer are as follows: 335 The proof of Theorem 2 is similar to the proof of Theorem 3 and is omitted. 342 Theorem 2 shows that the manufacturer's emission reduction investment has a positive correlation 343 with its marginal revenue and a negative relationship with its emission reduction cost coefficient. The 344 retailer's low-carbon publicity investment also has a positive correlation with its marginal revenue and a 345 negative correlation with its low-carbon publicity cost coefficient. This shows that under decentralized 346 decision scenario, manufacturers and retailers often make decisions based on their own interests, while 347 ignoring the overall benefits of the supply chain, which will result in a double marginal effect and 348 damage the overall profit of the supply chain. In the face of this situation, first of all, both parties should 349 abandon individualism and think more from the perspective of the overall supply chain. On the other 350 hand, they should actively improve their own marginal revenue through various measures such as 351 technological upgrades and accurate advertising.
At this time, the manufacturer's objective function is: 362 .
The retailer's objective function is: 364 .
Theorem 3 (1) The manufacturer's and retailer's optimal emission reduction decisions are as 366 follows： 367 (2) The optimal evolution path of low-carbon level is as follows： 369

371
(3) The optimal evolution paths of low-carbon goodwill and reference low carbon level are as 372 follows：: 373

377
(4) The optimal profits of the manufacturer and retailer are as follows: 378

Decentralized Scenario with Altruistic Behavior and a Cost-Sharing Contract 391
At this time, the retailer is willing to share a certain percentage of emission reduction cost  for the 392 manufacturer. Similarly, the manufacturer's objective function is: 393 The retailer's objective function is: 395 Theorem 4 (1) The manufacturer's and retailer's optimal emission reduction decisions are as 397 follows： 398 (2) The optimal evolution path of low-carbon level is as follows： 400

402
(3) The optimal evolution paths of low-carbon goodwill and reference low carbon level are as

408
(4) The optimal profits of the manufacturer and retailer are as follows: 409 The proof of Theorem 4 is similar to the proof of Theorem 3 and is omitted. 419 Theorem 5 (1) The optimal emission reduction cost sharing ratio provided by the retailer is as 420 follows: 421 (2) When the retailer is totally altruistic ( ,this cost-sharing contract can achieve perfect coordination of the supply chain.

425
Theorem 5 is easy to proof and is omitted.

Corollary 1 The relationship between related parameters under decentralized decision scenario 428
with altruistic behavior is shown in the Table 3. 429 : "+" denotes positive correlation; "-" denotes negative correlation; "×" denotes irrelevant; "±" 431 denotes positive or negative correlation depends on the situation. 432 Corollary 1 shows:(1) As the marginal revenue of manufacturers and retailers continues to 433 increase, manufacturers' investment in emission reduction and retailers' investment in low-carbon 434 publicity will also continue to increase, which will positively promote the low-carbon goodwill, further 435 stimulate consumer demand, increase product market demand and finally bring good benefits to the 436 supply chain. Therefore, when the supply chain members' altruism intensity is certain, in order to 437 further highlight the positive effects of altruistic behavior, the supply chain members should start from 438 their own point of view and increase their own marginal profits through a variety of ways, thereby 439 enhancing the positive impact of supply chain members' altruistic behavior on the entire supply chain.

440
(2) As both parties' altruism intensity continues to increase, manufacturers' investment in emission 441 reduction and retailers' investment in low-carbon publicity will continue to increase, which will 442 promote the increase in low-carbon level, low-carbon goodwill and market demand. This means that, 443 within the reasonable scope of their own altruism intensity, supply chain members showing a higher 444 altruistic intensity to the other party can effectively promote the cooperation between the two parties, 445 which is not only conducive to improving their own benefits, but also bring products with higher 446 low-carbon level and low-carbon goodwill to consumers, thereby further enhance the market 447 competitive advantage of the overall supply chain.

448
(3) With the increasing impact of the reference low-carbon effect on product demand, retailers' 449 low-carbon publicity investment will continue to decrease, while manufacturers' investment in 450 emission reduction and product low-carbon level will continue to increase, low-carbon goodwill and 451 market demand will not change significantly. This shows that when the reference low-carbon effect has 452 a significant impact on the market demand, supply chain members will adjust their measures 453 accordingly. The specific manifestation is that they will pay more attention to the low-carbon level of 454 products, so manufacturers will increase investment in emission reduction to improve the low-carbon 455 level of products. At the same time, since product low-carbon goodwill and market demand are jointly 456 affected by product low-carbon level and retailers' low-carbon publicity investment, there is no 457 significant change between product low-carbon goodwill and market demand at this time.

458
(4) As the relevance degree between the reference low-carbon level and product low-carbon 459 goodwill continues to increase, manufacturers' investment in emission reduction and retailers' 460 investment in low-carbon publicity will continue to decline, and the low-carbon goodwill will also 461 continue to decline. This means that consumers ignorantly judging the low-carbon level of products 462 based on the low-carbon goodwill level will not only bring losses to the overall profit of the supply 463 chain, but also adversely affect the low-carbon level of products they obtain.

464
(5) As the discount rate continues to increase, manufacturers' investment in emission reduction 465 and retailers' investment in low-carbon publicity will continue to decline, resulting in a decline in 466 product low-carbon level, low-carbon goodwill and ultimately a reduction in product market demand. 467 The higher the discount rate, the more the company cares about short-term profits. At this time, the 468 manufacturer's emission reduction investment and the retailer's low-carbon publicity investment are 469 difficult to be perceived by the market in a short time, so the higher discount at this time will inhibit the 470 enthusiasm of manufacturers and retailers to invest in emissions reduction. 471 Corollary 2 By comparing manufacturers' emission reduction investment, retailers' low-carbon 472 publicity investment, steady-state low-carbon level, low-carbon goodwill and product demand under 473 four decision scenarios, we can obtain: c It can be seen from Corollary 2 that under centralized decision scenario, the manufacturer's 476 emission reduction investment and the retailer's low-carbon publicity investment are both the largest. 477 The altruistic behavior of the manufacturer and the retailer can effectively promote both parties to 478 increase emission reduction investment, make the emission reduction investment of both parties no less 479 than that when they are fully self-interested, but not higher than the emission reduction investment of Corollary 3 Comparing the evolution path of product low-carbon level under four decision 489 scenarios, we can obtain: 490 It can be seen from Corollary 3 that the low-carbon level in centralized decision scenario is the 496 highest, which is consistent with the conclusion that the manufacturer's investment in emission 497 reduction is the largest under centralized decision scenario in Corollary 2. The altruistic behavior of the 498 manufacturer can effectively improve the low-carbon level, and the emission reduction cost-sharing 499 contract provided by the retailer can further enhance the low carbon level. In addition, as the intensity 500 of the altruistic behavior of manufacturers continues to increase, the difference between the low-carbon 501 level under centralized decision scenario and the low-carbon level under two altruistic decentralized 502 decision scenarios will continue to shrink, which again verifies that the altruistic behavior of the 503 manufacturer can effectively promote its increase in emissions reduction investment. If the initial 504 low-carbon level of the product is less than the steady-state low-carbon level under each decentralized 505 decision scenario, the low-carbon level of the product will continue to rise over time, otherwise, the 506 low-carbon level of the product will continue to decline over time. 507 Corollary 4 Comparing the evolution trajectory of product low-carbon goodwill under four 508 decision scenarios, we can obtain: 509 It can be seen from Corollary 4 that similar to low-carbon level, the low-carbon goodwill in 516 centralized decision scenario is the highest, which is consistent with the conclusion that the 517 manufacturer's investment in emission reduction and the retailer's investment in low-carbon publicity 518 are both the largest under centralized decision scenario in Corollary 2. Both parties' altruistic behavior 519 can effectively improve the low-carbon goodwill, and the emission reduction cost-sharing contract 520 provided by the retailer can further enhance the low carbon goodwill. In addition, as the intensity of the 521 altruistic behavior of both parties continues to increase, the difference between the low-carbon 522 goodwill under centralized decision scenario and the low-carbon goodwill under two altruistic decentralized decision scenarios will continue to shrink, which again verifies that the altruistic behavior 524 of the manufacturer can effectively promote its increase in emissions reduction investment and the 525 altruistic behavior of the retailer can effectively promote its increase in low-carbon publicity 526 investment. If the initial low-carbon goodwill of the product is less than the steady-state low-carbon 527 goodwill under each decentralized decision scenario, the low-carbon goodwill of the product will 528 continue to rise over time, otherwise, the low-carbon goodwill of the product will continue to decline 529 over time. 530 Corollary 5 Under decentralized decision scenario with altruistic behavior and a cost-sharing 531 contract, we can obtain: It can be seen from Corollary 5 that the greater the sharing ratio of emission reduction costs 534 provided by the retailer to the manufacturer, the greater the incentive for the manufacturer to invest in 535 emission reductions. The sharing ratio of emission reduction costs provided by the retailer to the 536 manufacturer will decrease with the increase in the intensity of the manufacturer's altruistic behavior 537 and the manufacturer's marginal revenue, and increase with the increase in the intensity of the retailer's 538 altruistic behavior and the retailer's marginal revenue. This is consistent with the reality. When retailers 539 pay more attention to manufacturers' profits and their marginal revenue increases, they will naturally 540 strengthen their incentives for manufacturers to invest in emission reductions. On the contrary, when 541 manufacturers pay more attention to retailers' profits and their marginal revenue increases, it is not 542 difficult to understand that at this time manufacturers have a strong economic power to invest in 543 emission reductions, so the retailer will reduce its economic support for the manufacturer's emission 544 reduction investment. In addition, the manufacturer's profit will decrease as its intensity of the 545 altruistic behavior increases, and the retailer's profit will increase as the manufacturer's intensity of the 546 altruistic behavior increases. Furthermore, as the intensity of the altruistic behavior of both parties 547 continues to increase, both parties' utilities will continue to increase. 548 Corollary 6 Under decentralized decision scenario with altruistic behavior, we can obtain: 549 It can be seen from Corollary 6 that the profits of the manufacturer and the retailer will increase 552 with the increase in the other parties' altruistic intensity. Regardless of the retailer's altruistic intensity, 553 the retailer's profit will decrease with the increase in the altruistic intensity, while the manufacturer's 554 profit may increase or decrease with the increase in its altruistic intensity, depending on its altruistic 555 intensity. In particular, when its altruistic intensity is lower than a certain value, the manufacturer's 556 profit will increase as its altruistic intensity increases, otherwise it will decline. In addition, the utilities 557 of both the manufacturer and the retailer will increase as the altruistic intensity of its own and the other 558 party increases. 559 It can be seen from Corollary 7 that compared with when the two parties do not have altruistic behavior, the altruistic behavior of the manufacturer and the retailer can increase the profits of both 562 parties. When the retailer chooses to share the emission reduction costs for the manufacturer, it can 563 further improve the profit level of both parties. In addition, the emission reduction cost sharing contract 564 provided by the retailer to the manufacturer can effectively improve the utilities of both parties, so the 565 emission reduction cost-sharing contract can promote the coordination of the supply chain. 566

Numerical Analysis 567
To more intuitively reveal the impact of key parameters on the equilibrium solutions under different 568 scenarios and obtain some valuable management enlightenments, we will conduct numerical analysis to 569 further test the previous conclusions in this section. Based on the related researches and the specific 570 research hypothesis of this paper, the parameters involved in this paper are set as follows: 1 6

580
It can be seen from Figure 1 that under the four decision scenarios, product low-carbon level, 581 low-carbon goodwill, reference low-carbon level and product market demand all have a time-stabilizing 582 trend, showing that they tend to a certain stable value when time tends to infinity. In addition, it can be 583 seen that since the initial product low-carbon level and low-carbon goodwill are both assumed to be 0 at 584 this time, which is less than the stable value, the product low-carbon level and low-carbon goodwill both 585 show a monotonous increasing trend over time. At the same moment, each state quantity under 586 centralized decision scenario is the highest, and it is much larger than that under each decentralized 587 decision scenario. Each state quantity under decentralized decision scenario with altruistic behavior is 588 higher than that under decentralized decision scenario without altruistic behavior. The above 589 phenomenon shows that the altruistic behavior of manufacturers and retailers can help improve product 590 low-carbon level and low-carbon goodwill, and when retailers share the cost of reducing emissions for 591 manufacturers, it will help further improve product low-carbon level and low-carbon goodwill, and 592 ultimately increase the market demand of low-carbon products. 593 (2) The optimal evolution paths of the manufacturer's profit, retailer's profit, overall supply chain 594 profit, manufacturer's utility, retailer's utility and utility difference over time under different decision 595 scenarios. 596  This shows that from a  611   long-term perspective, if both the manufacturer and the retailer hold an altruistic attitude towards each  612 other, it will not only contribute to the sustained and stable growth of the both parties' profits, but also 613 produce lasting and stable social welfare. From Figure 2(f), it can be seen that the difference between the 614 utilities of the manufacturer and the retailer under the two decision scenarios will increase over time, and 615 eventually tends to a certain stable value. This means that when the manufacturer and the retailer both 616 have altruistic behavior, if the retailer chooses to share the emission reduction costs for the manufacturer, 617 the utilities of both parties will not only be further improved, but from a long-term perspective, both 618 parties can obtain a sustained and stable utility increment, bringing lasting and stable social welfare. 619 6.2 Impact of Altruistic Behavior 620 (1) The impact of manufacturer's altruistic behavior on the manufacturer's profit, retailer's profit, 621 manufacturer's utility and retailer's utility. 622 manufacturer's altruistic intensity, the utilities of the manufacturer and the retailer under the two 631 decision scenarios will both continue to increase, and when the retailer chooses to share the emission 632 reduction cost for the manufacturer, that is, when the manufacturer's altruistic intensity is less than a 633 certain value, the utility level of both parties has been further improved compared with when the 634 emission reduction cost is not shared. In addition, it can be clearly seen that as the manufacturer's 635 altruistic intensity continues to increase, the differences between the manufacturer's profit, the retailer's 636 profit, the manufacturer's utility and the retailer's utility under the two decision scenarios will constantly 637 decrease. The above phenomenon shows that if the manufacturer's altruistic intensity is high enough, 638 which means that the manufacturer is very concerned about the retailer's profit, then compared with the 639 retailer sharing the manufacturer's emission reduction cost, the manufacturer can gain more profits, 640 greater utility, and the retailer can also obtain greater utility when the retailer does not share the 641 manufacturer's emission reduction cost. 642 (2) The impact of retailer's altruistic behavior on the manufacturer's profit, retailer's profit, 643 manufacturer's utility and retailer's utility. 644 retailer's altruism intensity, both parties' utilities under the two decision scenarios will continue to 652 increase, and when the retailer chooses to share emission reduction cost for the manufacturer, the utility 653 level of both parties has been further improved. In addition, it can be clearly seen that with the continuous 654 increase of the retailer's altruistic intensity, the differences between the manufacturer's profit, the 655 retailer's profit, the manufacturer's utility and the retailer's utility under the two decision scenarios will 656 constantly increase. The above phenomenon shows that if the retailer's altruistic intensity is high 657 enough, which means that the retailer pays much attention to the manufacturer's profit, then compared 658 with the retailer not sharing the manufacturer's emission reduction cost, the manufacturer can gain more 659 profits, greater utility, and retailers can also obtain greater profits and utility when the retailer shares the 660 manufacturer's emission reduction cost. 661 According toTheorem 5 (2), the cost-sharing contract can achieve a win-win situation for supply 662 chain members, because compared with no cost-sharing contract, both the manufacturer and the retailer 663 can obtain a certain utility increment under the cost-sharing contract. Particularly, when the retailer is 664 totally altruistic ( 1 r   ), both the manufacturer's emission reduction investment and the retailer's 665 low-carbon publicity investment can reach the investment levels under centralized decision scenario, and the supply chain can be fully coordinated. Combining Figure 3 and Figure 4, it can be seen that 667 when the retailer is totally altruistic ( 1 r   ),the less the altruistic intensity of the manufacturer, the 668 more the profit and utility increment the manufacturer and retailer can obtain. 669 6.3 Impact of Reference Low-Carbon Effect 670 (1) The impact of reference low-carbon effect on the manufacturer's emission reduction 671 investment ,the retailer's low-carbon publicity investment, steady-state low-carbon level and 672 low-carbon goodwill. 673 increase of the reference low-carbon effect, the manufacturer's emission reduction investment will 678 continue to decrease, but the reduction magnitude shows a marginal diminishing law. In addition, it can 679 be clearly seen that compared with when both parties do not have altruistic behavior, the final stable 680 value of the manufacturer's emission reduction investment is higher when both parties have altruistic 681 behavior, and when the retailer shares the emission reduction costs for the manufacturer, the final stable 682 value will be further improved. From Figure 5(b), it can be seen that under different decision scenarios, 683 the retailer's low-carbon publicity investment also decreases with the continuous increase of the 684 reference low-carbon effect, and the reduction magnitude also shows a marginal diminishing law. When 685 both parties have altruistic behavior, the stable value of the retailer's low-carbon publicity investment is 686 much higher. Directly affected by the manufacturer's investment in emission reduction and the retailer's 687 low-carbon publicity investment, the steady-state low-carbon level and low-carbon goodwill also 688 decrease as the reference low-carbon effect continues to increase, and the reduction magnitude also 689 shows a marginal diminishing law. In particular, when consumers rely solely on low-carbon goodwill to 690 determine the low-carbon level of the product ( 1   ), the low-carbon goodwill of the product will be 691 reduced to a minimum level, but it will still be higher than the initial low-carbon goodwill level. 692 (2) The impact of reference low-carbon effect on the manufacturer's profit and the retailer's profit. continuous increase of the reference low-carbon effect, both parties' profits will continue to decrease, but 697 the reduction magnitude shows a marginal diminishing law. When the reference low-carbon effect 698 reaches its maximum ( 1   ), both parties' profits will stabilize at a minimum level. In addition, it can 699 also be clearly seen that compared with when both parties do not have altruistic behavior, the final stable 700 values of both parties' profits are higher when both parties have altruistic behavior, and when the retailer 701 shares the emission reduction costs for the manufacturer, the final stable values will be further improved. 702 Combining Figures 5 and 6, it can be found that when consumers' low-carbon consumption 703 behavior is affected by the reference low-carbon effect, it will damage the profits of both manufacturers 704 and retailers, forcing manufacturers to reduce emissions reduction investment and retailers to reduce 705 low-carbon publicity investment, which will directly do harm to low-carbon level and low-carbon 706 goodwill and eventually adversely affect the low-carbon product market. However, when consumers' 707 low-carbon consumption behavior is affected by the reference low-carbon effect, the altruistic behavior 708 of both parties can weaken the adverse effect of the reference low-carbon effect to a certain extent, and 709 the weakening ability is proportional to the altruistic intensity of both parties. In addition, if the retailer 710 shares the emission reduction costs for the manufacturer on this basis, this weakening effect can be 711 further strengthened, and the damaging effect on the profits of both parties due to consumers' reference 712 low-carbon effect can be minimized. 713

Conclusions and Discussion 714
In the context of the low-carbon economy, when consumers buy low-carbon products, they are 715 always affected by the reference low-carbon effect. At the same time, supply chain members are not 716 completely rational when making emission reduction decisions, and they are always affected by altruistic 717 behavior. Based on the above background, this paper introduces consumers' reference low-carbon effect 718 and the supply chain members' altruistic behavior into the joint emission reduction issue of a supply 719 chain, and respectively constructs decision-making models under four decision scenarios： centralized 720 scenario, decentralized scenario, decentralized scenario with altruistic behavior and decentralized 721 scenario with altruistic behavior and a cost-sharing contract. Then this paper uses differential game 722 theory and Bellman continuous dynamic programming theory to solve the equilibrium solutions under 723 the four decision scenarios in turn, and discusses the impact of the supply chain members' altruistic 724 behavior, consumers' reference low-carbon effect, cost-sharing contract and related key parameters on 725 supply chain members' equilibrium strategies and performance from a long-term perspective. After 726 mathematical derivation and numerical analysis, the following conclusions can be obtained: 727 Compared with both parties not having altruistic behavior, manufacturers' emission reduction investment, retailers' low-carbon publicity investment, steady-state low-carbon level, steady-state 729 low-carbon goodwill and both parties' profits will increase when both parties have altruistic behavior. Figure 1 Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption