Dual reference effect and dynamic control of quality improvement and low-carbon effort under nonlinear demand

Businesses often engage in quality improvement efforts to enhance their competitiveness in the market and increase their product sales. However, in a low-carbon economy, it is crucial for enterprises to incorporate green development into their business strategies. This paper uses optimal control theory to study the dynamic control problem of enterprise quality improvement and low carbon input under consumer reference quality and reference low carbon, analyzes and compares the input decisions under two conditions of profit maximization by the enterprise and social welfare maximization by the government, explores the system steady-state equilibrium and performs simulation analysis. It is shown that (i) there is a unique saddle-point equilibrium of the system in both cases and it depends on the discount rate and memory parameters (ii) the higher the consumer loyalty to product quality or low-carbon attributes, the corresponding instantaneous investment rate increases (iii) there is a cross-effect of reference quality and reference low-carbon on the firm's decision, and the impact of product quality on the firm's profit is higher than that of low-carbon attributes (iv) the social welfare maximization conditions are all higher than the investment in the firm's profit maximization conditions. Based on this, enterprises can make investment decisions more rationally, while also helping the government conduct macroeconomic regulation and control.