Social Planner’s Carbon Tax Problem Revisited: A New Mathematical Approach

The carbon tax, a mechanism that penalises carbon dioxide-emitting energy sources, has regained policy relevance amid the global debate over the Carbon Border Adjustment Mechanism (CBAM). This study aims to develop and analyse a novel dynamic model to explicitly assess the impact of carbon taxes on firm emissions and the speed of transition to renewable energy sources. The model is based on a social planner maximising lifetime utility, incorporating short-run profit maximisation by firms regarding energy demand. A CES-Cobb-Douglas mixed production function is utilised to capture the substitutability and complementarity of labour, capital, and renewable and non-renewable energy sources. This function's application is a key contribution, enabling the dynamic characterisation of interactions between energy types and production factors through specific parameterisation. The model follows a conceptual system-dynamics approach developed for explanatory purposes, without empirical data, and emphasises the structural feedback mechanisms that drive the dynamics of the energy transition. Simulations of the model indicate that, following the imposition of a carbon tax, non-renewable energy demand decreases over time, accompanied by a corresponding increase in renewable energy demand. The model further demonstrates a reduction in cumulative emissions with increased tax levels. Additionally, the time-dependent elasticity of output with respect to renewable energy input shows a notable increase, indicating the growing competitiveness of renewable sources. These findings suggest that strategically designed carbon taxes within the proposed dynamic framework can effectively accelerate the shift to renewable energy and reduce long-term carbon emissions. This model provides policymakers with a valuable tool for assessing the temporal impacts of carbon tax policies on energy transitions.