The demands of creating an immersive Virtual Reality (VR) experience often exceed the raw capabilities of graphics hardware. Perceptually-driven techniques can reduce rendering costs by directing effort away from features that do not significantly impact the overall user experience while maintaining a high level of quality where it matters most. One such approach is foveated rendering, which allows for a reduction in the quality of the image in the peripheral region of the field-of-view where lower visual acuity results in users being less able to resolve fine details. 6 Degrees of Freedom tracking allows for the exploration of VR environments through different modalities, such as user-generated head or body movements. The effect of self-induced motion on rendering optimization has generally been overlooked and is not yet well understood. To explore this, we used Variable Rate Shading (VRS) to create a foveated rendering method triggered by the translational velocity of the users and studied different levels of shading Level-of-Detail (LOD). We asked 10 participants in a within-subjects design to report whether they noticed a degradation in the rendering of a rich environment when performing active ego-movement or when being passively transported through the environment. We ran a psychophysical experiment using an accelerated stochastic approximation staircase method and modified the diameter and the LOD of the peripheral region. Our results show that self-induced walking can be used to significantly improve the savings of foveated rendering by allowing for an increased size of the low-quality area in a foveated algorithm compared to the passive condition. After fitting psychometric functions showcasing the percentage of correct responses related to different shading rates in the two types of movements, we also report the threshold severity (75%) point for when participants are able to detect such degradation. We argue such metrics can inform the future design of movement-dependent foveated techniques that could reduce computational load and increase energy savings.