Impulse Control and Multi-Dimensional Differential Games: Optimal Timing of Recycling and Substitution under Strategic Interaction

This paper develops a dynamic game framework for environments in which recycling
and substitution technologies emerge endogenously. We formulate the interaction as
a Markovian subgame-perfect equilibrium with impulse controls, derive the associated
Hamilton–Jacobi–Bellman systems, and establish smooth-pasting conditions governing
regime transitions. Departing from classical exhaustible-resource models, our setting introduces
recycling as an additional state variable and allows virgin resource prices to
depend jointly on substitution and recycling. This structure generates a two-dimensional
state space with interconnected regimes, leading to a switching fixed-curve rather than
a single threshold and creating new challenges for theoretical characterization. Under
broad convex cost functions and CES-type preferences, we characterize the resulting equilibrium and the geometry of the switching regions, thereby providing general insights into multi-dimensional impulse-control problems in dynamic games.