The economic dispatch problems (EDPs) in a microgrid (MG) have been extensively investigated by a variety of emerging algorithms. In this paper, we propose two newly distributed dynamic optimization algorithms to respectively study the EDPs under both cases without and with generation constraints under a directed topology network. Two novel dynamic optimization algorithms are based on the distributed incremental cost consensus (ICC), where the mismatch between total demand and power generation is considered. Our algorithms only require the weight matrix of the directed network to be row stochastic. The theoretical analysis on the convergence of the proposed algorithms is presented by using the small gain theorem. It can be found that the algorithms are convergent at the geometric rate. Meanwhile, the power output of the generators are proved to achieve the optimal solution of EDPs based on the proposed algorithms. Finally, the corresponding conditions are also derived, and simulation studies illustrate the correctness of our results.