With the growing severity of global environmental issues, the international community has reached a consensus on the importance of reducing and controlling carbon emissions. As a result, an increasing number of consumers are opting to purchase green products in order to reduce the emission of greenhouse gases. This trend has prompted supply-chain enterprises to invest in green innovation. Simultaneously, carbon tax policies have gained significant attention from governments worldwide due to their dual role as environmental laws and fiscal-policy tools. Considering consumers’ preference for green products and the risk of R&D failure associated with them, this study focuses on the effects on emissions reductions and profits associated with different carbon tax policies for a green supply chain consisting of a manufacturer and a retailer. The results reveal that (1) increases in the carbon tax per unit of product motivate the manufacturer to increase R&D efforts; (2) wholesale and retail prices follow a pattern of initial increase and subsequent decrease as the carbon tax per unit of product rises; (3) higher carbon taxes per unit of product generally lead to lower manufacturer profits, while both carbon emissions and retailer profits can increase with a per-unit carbon tax under certain circumstances; and (4) the increase in the proportion of the population with green preferences can yield long-term benefits for both the retailer and the manufacturer, yielding an inverted U-shaped relationship with carbon emissions.