Enhancing financial viability and social welfare in public transportation: A study of subsidy schemes for urban rail transit systems

Subsidization is vital for supporting the full-life-cycle operation and sustainable development of urban rail transit (URT) systems. It is crucial for ensuring the financial viability of transit agencies, boosting ridership, and facilitating sustainable access to public transportation for urban populations. This paper analyzes three different subsidy schemes (namely, fixed subsidy, fare-based subsidy, and distance-based subsidy) for a URT system to maximize social welfare. Firstly, a benchmark model is formulated with a fixed subsidy scheme and the optimal conditions for fares, operating frequency, and subsidy are derived and determined by the Lagrange multipliers method. Two additional subsidy models are then developed based on fare and trip distance, respectively. Compared with the fixed subsidy scheme, the results show that (i) the effects and system performance of the fixed subsidy scheme are worse than for the other two subsidy schemes; (ii) the fare-based subsidy scheme has the highest social welfare and passenger utility among the three subsidy schemes; and (iii) the distance-based subsidy scheme is the most profitable and requires the least subsidy. In this paper, we explicitly derive the Karush-Kuhn-Tucker (KKT) Conditions for the three different subsidy models, providing valuable insights into the precise subsidization of URT systems. Additionally, computational experiments demonstrate the significance of adopting an appropriate subsidy scheme to maximize social welfare in public transportation.