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In this paper, the rate region of the two user MIMO broadcast channel (BC) under linear filtering at high signal-to-noise ratio (SNR) is investigated when time sharing is not available and the transmitter has more antennas than the sum of the receiving antennas. To reach the rate region's boundary, the sum rate is maximized subject to a given ratio between the users rates. The sum rate is first considered asymptotically when the SNR tends to infinity and taken as an affine function of the logarithm of the SNR, with the multiplicative and the additive parameters called the multiplexing gain (MG) and the rate offset (RO), respectively. The maximal MG and the maximal RO are obtained for every rate ratio constraint. Additionally, the asymptotic optimal stream allocations that achieve those values are also derived. Analytical inner and outer bounds, which offer a rough approximation of the boundary but are extremely easy to evaluate even in fading channels, are then developed. The maximization of the rate subject to a rate ratio constraint is then studied at finite SNR. Algorithmic inner and outer bounds for the rate region boundary are derived and shown to be very close to each other and accurate even at intermediate SNR.