In this paper, we investigate the problem of user rate balancing for the downlink transmission of multiuser multicell Multiple-Input-Multiple-Output (MIMO) systems with per cell power constraints. Due to the multiple streams per user, user rate balancing involves both aspects of balancing and sum rate optimization. We exploit the rate Mean Squared Error (MSE) relation, formulate the balancing operation as constraints leading to Lagrangians in optimization duality, allowing to transform rate balancing into weighted MSE minimization with Perron Frobenius theory. The Lagrange multipliers for the multiple power constraints can be formulated as a single weighted power constraint in which the weighting can be optimized to lead to the satisfaction with equality of all power constraints. Actually, various problem formulations are possible, including single cell full power transmission leading to a dual norm optimization problem, and per cell rate balancing which breaks the balancing constraint between cells. Simulation results are provided to validate the proposed algorithms and demonstrate their performance improvement over e.g. unweighted MSE balancing.