This paper addresses the problem of distributed rate allocation for a class of multicast networks employing linear network coding. The goal is to minimize the cost (for example, the sum rates allocated to each link in the network) while satisfying a multicast rate requirement for each destination in the network. In essence, this paper aims to achieve network capacity while ensuring that the cost of operation (equivalently, the rate allocated per link in the network) is minimal. This paper uses a belief propagation framework to obtain a distributed algorithm for the rate allocation problem. Simulation results are presented to demonstrate the convergence of this algorithm to the optimal rate allocation solution.