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This paper studies spatiotemporal pricing and fleet management for autonomous mobility-on-demand (AMoD) systems while taking elastic demand into account. We consider a platform that offers ride-hailing services using a fleet of autonomous vehicles and makes pricing, rebalancing, and fleet sizing decisions in response to demand fluctuations. A network flow model is developed to characterize the evolution of system states over space and time, which captures the vehicle-passenger matching process and demand elasticity with respect to price and waiting time. The platform's objective of maximizing profit is formulated as a constrained optimal control problem, which is highly nonconvex due to the nonlinear demand model and complex supply-demand interdependence. To address this challenge, an integrated decomposition and dynamic programming approach is proposed, where we first relax the problem through a change of variable, then separate the relaxed problem into a few small-scale subproblems via dual decomposition, and finally solve each subproblem using dynamic programming. Despite the nonconvexity, our approach establishes a theoretical upper bound to evaluate the solution optimality. The proposed model and methodology are validated in numerical studies for Manhattan. We find that compared to the benchmark case, the proposed upper bound is significantly tighter. We also find that compared to pricing alone, joint pricing and fleet rebalancing can only offer a minor profit improvement when demand can be accurately predicted. However, during unanticipated demand surges, joint pricing and rebalancing can lead to substantially improved profits, and the impacts of demand shocks, despite being more widespread, can dissipate faster.