A green supply chain consisting of a leading manufacturer and a following retailer is considered, whose product sales are affected by the price and the level of energy saving. Stackelberg game is applied to build a game model without premium and penalty mechanism and with three premium and penalty mechanisms. The equilibrium solutions of the four cases of retail price, wholesale price, product demand, level of energy saving, the level of government energy saving in premium and penalty and manufacturer and retailer profits are obtained. Through comparison and analysis, it is found that the government's energy saving premium and penalty mechanism can not only improve the level of product energy saving, but also improve the level of social welfare; the premium and penalty mechanism that the government takes into consideration the profit of the enterprise, the welfare of the consumer and the benefit of the environmental protection is the best; the level of premium and penalty is positively related to the product market scale, the environmental benefit coefficient and the sensitivity coefficient of energy saving, while negatively related to the cost input coefficient of the manufacturer's energy saving , the cost input coefficient of the government premium and penalty mechanism, unit production cost of the manufacturer and unit sales cost of the retailer. Finally, a numerical example is given to illustrate the effectiveness of the models.