The operation of the multiple input multiple output (MIMO) Gaussian broadcast channel (BC) with K transmit antennas and K/r users with r receive antennas each is analyzed when channel state information (CSI) is obtained at the transmitter from the i(th) user via finite-rate feedback link of capacity B-i bits. It is proved that if Bi is scaled at the rate of alpha(i)r(K - r) log(2) P, where P is the transmit power and alpha(i) is an element of [0, 1], then using dirty paper coding (DPC), the multiplexing gain of air can be achieved for user i. This result on the feedback scaling rate rests on an elegant achievability scheme, the construction of which brings to light several new insights on the manner in which DPC achieves the complete multiplexing gain in the perfect-CSIT case.