With the aid of differential game, the government subsidies strategies and the long-term dynamic problem about reducing emissions in cooperation in a supply chain which consists of a single supplier and manufacturer is studied under the background of low-carbon, in which the carbon emission reduction of products is used as a state variable. This paper establishes centralized and decentralized without cost-sharing contract differential game models, also the cost-sharing contract is introduced to achieve Pareto improvement for the profits of the supply chain. As for the three decision-making situations, the supplier's and manufacturer's optimal feedback equilibrium strategies and their own optimal trajectory of the carbon emission reduction of products over time can be obtained and compared respectively. Then, the interaction game between government subsidies behaviors and companies' choice of reducing emissions in cooperation is discussed. It is found that the introduction of government subsidies for the carbon emission reduction changes the optimal emission reduction efforts and the profits of supply chain members under the three decision-making situations. Finally, the sensitivity analysis of the relevant parameters is conducted by using the proposed method to verify the validity of conclusions, which provides scientific basis for long-term reducing emissions in cooperation and subsidies made by government in a supply chain.