||Variable rate is a requirement for flexible and adaptable image and video compression. However, deep image compression methods (DIC) are optimized for a single fixed rate-distortion (R-D) tradeoff. While this can be addressed by training multiple models for different tradeoffs, the memory requirements increase proportionally to the number of models. Scaling the bottleneck representation of a shared autoencoder can provide variable rate compression with a single shared autoencoder. However, the R-D performance using this simple mechanism degrades in low bitrates, and also shrinks the effective range of bitrates. To address these limitations, we formulate the problem of variable R-D optimization for DIC, and propose modulated autoencoders (MAEs), where the representations of a shared autoencoder are adapted to the specific R-D tradeoff via a modulation network. Jointly training this modulated autoencoder and the modulation network provides an effective way to navigate the R-D operational curve. Our experiments show that the proposed method can achieve almost the same R-D performance of independent models with significantly fewer parameters.