Stackelberg equilibrium strategies between insurance demand and government interventions

This paper investigates Stackelberg equilibrium strategies between insurance demand and government interventions—ex ante premium subsidies and ex post disaster relief—in catastrophe risk management. We develop a continuous-time framework where policyholders' losses follow a compound Poisson process, integrating dual government mechanisms to analyze their interplay. By solving (extended) Hamilton-Jacobi-Bellman (HJB) equations, we derive equilibrium insurance strategies for policyholders under mean-variance preferences and optimize government expenditure policies. First, we demonstrate that subsidies and relief have opposing effects on risk retention: higher subsidies tend to reduce retention, whereas increased relief expectations incentivize retention due to anticipated post-disaster compensation. Specifically, within the framework of a linear relief function, we characterize the relative growth rate of the subsidy relative to the relief coefficient to ensure that the effect of the premium subsidy on the retention level either dominates or is dominated by the impact of the relief payment. Second, for risks exhibiting decreasing mean residual life (DMRL), we derive optimal subsidies in closed-form solutions under proportional or truncated relief structures, depending on disaster probability and relief trends. Third, we innovatively prove that the cost-minimizing relief function adopts a proportional form relative to pre-retention claims, aligning with empirical practices such as FEMA's aid caps. Fourth, extensions to scenarios with loss-increasing and ambiguous relief probabilities are explored within the policyholders' optimization framework, demonstrating robustness and consistency with static expected-utility results. Additionally, we analyze the government's optimal strategy when incorporating policyholders' welfare constraints and conduct a comprehensive social welfare assessment under various intervention scenarios. Our work advances policy design by quantifying the trade-offs between subsidies and relief, providing actionable insights for enhancing societal resilience. Governments can strategically balance interventions to stabilize insurance markets, mitigate fiscal exposure, and incentivize proactive risk management. This study bridges theoretical rigor with practical relevance under dynamic risk models, offering a comprehensive framework for optimizing public-private catastrophe risk-sharing mechanisms.