To maintain a competitive advantage, manufacturers of household appliances should promote the product’s energy efficiency, considering the impact on customer purchasing behavior. Since the product’s energy efficiency and pricing policies influence customers’ purchasing decisions, manufacturers confront significant challenges in balancing costs and demand since they must consider their profit-maximizing objective and government regulations. The Stackelberg game framework represents the interactions between the government, the leader, and a manufacturer, the follower, incorporating the government’s involvement in environmentally dependent social welfare under a tax structure. This paper proposes closed-form equilibrium using a game theory approach and geometric programming (GP) to solve the government’s and manufacturers’ non-linear decision models. The analytical results offer insight into the management’s approach to the product’s energy efficiency. The findings demonstrate that when clients’ concerns about energy-saving grow, the net payoff to the total manufacturer revenue ratio continuously decreases. The outcomes imply that the manufacturer must allocate significant revenue to tax expenditures in markets with more price-sensitive clients. As a motivation for research, this paper explores the application of the proposed model by examining a numerical example of a real-world refrigerator manufacturer case to obtain further managerial insight.