Differential game model of joint emission reduction strategies and contract design in a dual-channel supply chain

Joint emission reduction is widely used in low-carbon supply chain practices. It consists of the manufacturer's emission reduction and the retailer's advertising campaign. In this paper, we present a differential game that involves one manufacturer and one retailer in a dual-channel supply chain under low-carbon environment. We analyze the optimal equilibrium strategies in centralized and decentralized dual-channel supply chains, and compare the results of the single channel and dual-channel supply chains. We also discuss how the cooperative advertising contract and the cooperative advertising and emission reduction cost sharing contract affect the optimal strategies and coordination of dual-channel supply chains. Our results suggest that the emission reduction level trajectory is monotonic, whereas the goodwill trajectory changes direction once at most. Wholesale price and the degree of customer loyalty to the retail channel exert a significant influence on optimal strategies. Manufacturer's profit and emission reduction effort are higher and retailer's profit and advertising effort are lower in a dual-channel supply chain than in a single channel supply chain. The cooperative advertising and emission reduction cost sharing contract is more efficient than the cooperative advertising contract under certain conditions, and a high low-carbon and brand preference of consumers indicates a greater likelihood that the supply chain members will adopt the contract. The results can provide a theoretical basis for supply chain members to make optimal decisions and choose the appropriate contract in a dual-channel supply chain under low-carbon environment.