In this paper we consider joint optimization of rate and power for communication systems that use multilayer superposition source coding with successive refinement of information. We assume a Rayleigh fading channel, where rates and power are jointly and optimally allocated between the source layers based on channel statistics information, with the objective of maximizing the expected total received rate at the end user. We show that the optimization problem possesses a strong duality, and hence we use the dual form to show that the optimal solution can be obtained using a two-dimensional bisection search for any number of layers. The outer bisection search is over the Lagrangian dual variable and the inner bisection search is over the decoding SNR threshold of the last layer. Moreover, we show that with a small number of layers, we can approach the performance upper bound, that is achieved by transmitting an infinite number of layers.