We study a singular mean-field control problem motivated by systemic risk, in which a central planner allocates losses from defaults to maximize a concave measure of the system’s terminal health. We prove that the optimal allocation follows a cutoff rule, or “taxing-the-richest” policy, by solving a time-discretized problem and passing to the continuous-time limit. We then characterize the resulting optimal flow through a novel reflected free-boundary problem and establish well-posedness in a probabilistic solution class. The main technical difficulty lies in the dependence of the optimal feedback control on a quantile-type moving boundary, which is unstable under standard convergence of probability laws; we address it through stochastic-order arguments, compactness estimates for the cutoff boundaries, and an augmented call-function formulation of the dynamics. Joint work with Qinxin Yan.
