This paper considers equilibrium and stability analysis using the dynamics of a cap-and-trade system between firms considering corporate values from carbon dioxide (CO₂) reductions. It turns out that the total amount of CO₂ reduction increases by increasing the initial emission allowance of each agent, and the necessary conditions for achieving carbon neutrality are given by the relation between the sum of the initial CO₂ emissions and the parameters characterizing the reduction cost and the corporate values due to CO₂ emission reduction. In the simplest case with 2 agents, the dynamics can be expressed as a piecewise nonlinear system, and the analysis is facilitated by nonlinear coordinate transformation and dimensionality reduction. Finally, we derive a stability condition for a piecewise linearized system around the origin, and provide numerical examples to validate the stability condition.