Bit rate and power are two commonly used optimality criteria for MIMO transceiver design. In the literature, bit rate maximization and power minimization problems are viewed as different problems and solved independently. In this paper, we derive the duality between these two problems for both the cases with and without integer constraint on bit allocation. We will show that if a transceiver is optimal for the power-minimizing problem, it is also optimal for the rate maximizing problem, and the converse is true. Such a duality has not been stated and proved in the literature to the best of our knowledge. The derivation does not involve any existing optimal solution and we can establish duality result even for the rate maximization problem with integer bit constraint, which the optimal solution is not known. The duality also allows us to develop an algorithm for finding the rate-maximizing transceiver with integer bit allocation using the solution of power-minimizing system. We will also consider some possible generalizations of the problem, for example, when there is a constraint on the maximal constellation size and when the subchannel bit error rates (BERs) are constrained. For each of these cases, we will see that the duality between the two problems continued to hold. In the simulations, we will compute the optimal solutions for these two problems and demonstrate the duality between these two.