Turnpike Theorems and Thresholds for Carbon-Priced Modal Split on Capacity-Constrained Transport Networks

Abstract We study carbon-priced modal split on capacity-constrained transport networks with piecewise-linear tariff schedules (“tariff blocks”) and quadratic adjustment frictions. We establish turnpike theorems showing that, for any finite horizon, optimal policies spend all but $$O(1)$$ time near the minimizers of a static average-cost program that aggregates logistics costs and carbon charges; the resulting $$O(1)$$ value gap is uniform in the horizon. We then analyze comparative statics of the steady modal mix with respect to the carbon weight and other parameters. Because the steady problem is a parametric convex program with a polyhedral feasible set and piecewise-linear objective, there exists a finite set of policy thresholds at which the optimal steady mix can change discontinuously, while both the steady value and selected minimizers vary Lipschitz-continuously away from those thresholds. We further give envelope identities (e.g., the derivative of the steady value with respect to the carbon weight equals steady emissions) and show that receding-horizon (economic MPC) implementations are near-optimal with bounded, horizon-independent loss. The analysis is purely theoretical; a small synthetic network illustrates threshold locations, jump magnitudes, and boundary-layer dynamics. The results provide transparent diagnostics for pricing vs. capacity instruments and support regime-aware planning on multimodal networks.