In this paper, we investigate the problems of sum power minimization and sum rate maximization for multi-cell networks with non-orthogonal multiple access. Considering the sum power minimization, we obtain closed-form solutions to the optimal power allocation strategy and then successfully transform the original problem to a linear one with a much smaller size, which can be optimally solved by using the standard interference function. To solve the nonconvex sum rate maximization problem, we first prove that the power allocation problem for a single cell is a convex problem. By analyzing the Karush-Kuhn-Tucker conditions, the optimal power allocation for users in a single cell is derived in closed form. Based on the optimal solution in each cell, a distributed algorithm is accordingly proposed to acquire efficient solutions. Numerical results verify our theoretical findings showing the superiority of our solutions compared to the orthogonal frequency division multiple access and broadcast channel.