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The rate loss of a multiresolution source code (MRSC) describes the difference between the rate needed to achieve distortion Di in resolution i and the rate-distortion function R(Di). This paper generalizes the rate loss definition to multiple description source codes (MDSCs) and bounds the MDSC rate loss for arbitrary memoryless sources. For a two-description MDSC (2DSC), the rate loss of description i with distortion Di is defined as Li=Ri-R(Di), i=1,2, where Ri is the rate of the ith description; the joint rate loss associated with decoding the two descriptions together to achieve central distortion D0 is measured either as L0=R1+R2-R(D0) or as L12=L1+L2. We show that for any memoryless source with variance σ2, there exists a 2DSC for that source with L1≤1/2 or L2≤1/2 and a) L0≤1 if D0≤D1+D2-σ2, b) L12≤1 if 1/D0≤1/D1+1/D2-1/σ2, c) L0≤LG0+1.5 and L12≤LG12+1 otherwise, where LG0 and LG12 are the joint rate losses of a Gaussian source with variance σ2.