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Governments and policymakers around the world put their best efforts to control the pollutions and climate change. Thus, they set various regulations to reduce greenhouse gases and carbon footprints. It is expected that firms should follow these regulations while maintaining their profitability. In this regard, firms can manage their carbon emissions across their supply chain (SC) by inventory management, since operational adjustments may affect the amount of carbon emissions, such as changing the production quantity and the frequency of transportations. This study applies a Stackelberg game between the government and a multi-stage green supply chain (GSC), in which the government's goal is to maximize social welfare and that of the GSC is to minimize its cost. First, we formulate the inventory cost and carbon emission of a multi-stage GSC under two decision-making structures: non-coordinated and coordinated GSCs. Second, we develop eight bi-level mathematical programming models considering the structure of the GSC and the regulations of the government including carbon cap, carbon tax, carbon trade, and carbon offset. Then two solution approaches will present for solving problems based on the type of the GSC structure. This study examines the effect of coordination and carbon regulations on inventory cost, carbon emission and the objective function of the government.