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Regarding carbon emission as a principal cause of global warming, governments begin to take note of the urgency of emission reduction and propose feasible carbon regulations. Consumers also become more caring about the carbon emission factor when purchasing product. In response, the firms have exerted considerable efforts to reduce their emissions. In this paper, we investigate a supply chain facing carbon-sensitive demand market. Two major carbon regulations, absolute-cap (AC) and intensity-cap (IC), are considered. We explore the optimization problems for the centralized and decentralized models, and derive the optimal solutions for the production quantity as well as the emission reduction effort. We find that under regulation IC, a higher emission cap will lower the carbon emission of unit product but raise the total production quantity, while the emission cap is irrelevant to these operational decisions under regulation AC. In addition, the production quantity and the emission reduction effort under regulation AC are lower than those under regulation IC. We further show double marginalization within this supply chain manifests as not only a lower production quantity but also a higher unit carbon emission. Finally, numerical experiments are conducted to obtain more insights.