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Though federated learning (FL) well preserves clients' data privacy, many clients are still reluctant to join FL given the communication cost and energy consumption in their mobile devices. It is important to design pricing compensations to motivate enough clients to join FL and distributively train the global model. Prior pricing mechanisms for FL are static and cannot adapt to clients' random arrival pattern over time. We propose a new dynamic pricing solution in closed-form by constructing the Hamiltonian function to optimally balance the client recruitment time and the model training time, without knowing clients' actual arrivals or training costs. During the client recruitment phase, we offer time-dependent monetary rewards per client arrival to trade-off between the total payment and the FL model's accuracy loss. Such reward gradually increases when we approach to the recruitment deadline or have greater data aging, and we also extend the deadline if the clients' training time per iteration becomes shorter. Further, we extend to consider heterogeneous client types in training data size and training time per iteration. We successfully extend our dynamic pricing solution and develop an optimal algorithm of linear complexity to monotonically select client types for FL. Finally, we also show the robustness of our solution against estimation error of clients' data sizes and run numerical experiments to validate our conclusion.