Abstract: The emission of greenhouse gas is increasingly harmful to the environment. How to develop the economy while controlling greenhouse gas emissions has become an important issue in countries all over the world. Under immense pressure from climate change, governments and environmental agencies have successively issued a series of laws and regulations. In order to meet the requirements of government policies and regulations, manufacturing enterprises have begun to implement low-carbon management strategies and improve emission reduction technologies. At the same time, many retailers adopt advertising strategies to enhance consumers' low-carbon awareness and encourage their purchase of low-carbon products. Further, the extensive application of internet technology has greatly changed traditional consumption patterns. Many manufacturers have begun selling products directly to consumers through online channels. In this way, the online channel can help the manufacturer to enlarge market needs. However, for the retailer, the manufacturer is not only the supplier but also the competitor, thus, the channel conflict remains. This means studying supply chain members' optimal strategies is much more complicated under the dual-channel structure. In addition, the operational production of manufacturers is usually multi-period rather than single period, thus the long-term decisions of supply chain members should be closely examined. On account of the above issues, by putting the characteristics of low-carbon products into the dual-channel supply chain, and using the knowledge in the fields of optimal control theory, the pricing and joint emission reduction strategies are explored in a dual-channel supply chain from the long-term and dynamic perspectives, and the results can provide a theoretical basis for supply chain members to make optimal decisions and a policy basis for the government to formulate regulations. In this paper, a differential game is present that involves one manufacturer and one retailer in a dual-channel supply chain, where emission reduction and goodwill are state variables. Supply chain members' optimal strategies are discussed under different channel structures and analyzes how the cooperative advertising and emission reduction cost sharing contract affect supply chain's coordination is analyzed. The results suggest that: (1) The emission reduction trajectory is monotonic while the goodwill trajectory may change direction. The degree of customer loyalty to the retail channel and the manufacturers' competition intensity exert a significant influence on optimal strategies, specifically, when the degree of customer loyalty to the retail channel is lower and the manufacturers' competition intensity is higher than a threshold, the retail price in the centralized supply chain will be higher than the decentralized supply chain, which is totally different from the result of "double marginalization" in a single channel. (2) The retailer's advertising effort is always lower in a dual-channel supply chain than in a single channel supply chain, therefore, the manufacturer can share a part of the advertising cost with the retailer to stimulate the retailer's advertising investment and eliminate channel conflicts. In addition, when the degree of customer loyalty to the retail channel is lower or the manufacturers' competition intensity is higher than a threshold, the manufacturer's emission reduction effort in a dual-channel supply chain will be higher than in a single channel supply chain, so the government should provide manufacturers with emission reduction subsidies to encourage them to adopt a dual-channel structure to promote sustainable production. (3) Compared with single channel, retailer's profit is lower, while manufacturer's profit is higher in a dual-channel structure until the degree of customer loyalty to the retail channel is higher or the manufacturers' competition intensity is lower than a threshold, so the manufacturer will not open online channel. Since the dual-channel structure always hurts retailer's benefit, so when the degree of customer loyalty to the retail channel and the manufacturers' competition intensity is high, the retailer can rebate the manufacturer, which enables the manufacturer to make large profits in the retail channel, and would not open an online channel. (4) The cooperative advertising and emission reduction cost sharing contract can achieve Pareto improvement for the manufacturer, the retailer and the whole supply chain system, and the higher consumer's low-carbon and brand preference, the more likely manufacturer and retailer would cooperate with each other.

Key words: dual-channel supply chain, goodwill, dynamic pricing, joint emission reduction, coordination contract, differential game