With the further worsening of global warming,and decreasing of some exhausting resources,reducing carbon emission and promoting resources recycling have become one of the hot issues from the world-wide scope.The majority of the countries have taken several restricted measures on the greenhouse gases,in which carbon tax is the common policies used by different government.Considering that the manufacturing and remanufacturing of production enterprises is the main source of carbon emission,the manufacturers can improve the degree of greenness for the products via investing on the low-carbon technology.In general,the higher the degree of greenness for products,the lower carbon emissions of one product.Therefore,how to balance the increase of investment and the reduction of carbon emissions,and then arrive at the optimal production,and thus to obtain the optimal strategy for the manufacturing,remanufacturing and pricing becomes the significant decision problem for the enterprises.Based on variational inequality(VI),complement theory and Lagrangian dual theory,and considering the competition between same type of the players on the supply chain network and the manufacturersdecision for the degree of greenness during the product design,we established the closed-loop supply chain network equilibrium with consumerslow-carbon preference under the carbon tax and subsidy policies of government.Furthermore,we designed the solution algorithm based on the modified projection and contraction algorithm of VI.We analyzed the impact of carbon tax and subsidies of government,recycling ratio and consumer preference for low-carbon products on the decision for the degree of product greenness,decision for production capacity and pricing,and the profit of enterprises,and further summarized the corresponding insights of the management.Several results go as follows.Firstly,on the one hand,government should set the reasonable subsidy extent because the lower and higher subsidy can both prevent the effect of the policy.On the other hand,government should provide some guidance(e.g.,combining the level of cost reduction during remanufacturing and product green degree)for the enterprises when they set the recycling ratio so that the ratio can be within the reasonable interval.Secondly,the carbon tax is a good mechanism for the carbon reduction deserved to be spread,which can render the enterprises to choose the production technologies with low-carbon emission and can improve their profits slightly.Lastly,although the profit of the enterprises can be improved with the increasing of consumers low-carbon consciousness,this consciousness should be promoted via the external power,e.g.,the government.Results in this paper can provide some reference for the research of low-carbon supply chain and the design of government policy.