My impression has been that people tend to dramatically underestimate the amount of power needed to accurately produce dynamic peaks. Quoted RMS power recommendations and nominal speaker impedance values don't help here as much as one might wish, and may actually mislead regarding what's sufficient. To start with, look at not only the minimum impedance, but the phase angles. One quick link that discusses the relationship and impact on required amplifier power:
For some more direct examples from an AC member with great measurement gear and the skills to use it, consider this post:
And the link he mentions in point (2) that documents some measurements obtained while using is very efficient speakers:
(I recall once that someone proposed a single variable incorporating the impact of both impedance and phase angle, but I can't find it or recall specifically the details of the math. It would certainly be a useful simplification for those seeking to match amps and speakers if were even remotely accurate.)
To address the specific question made by the OP... having more power on tap won't in and of itself alter the frequency response, including the bass, of the connected speakers. The speaker will draw what it needs up to the limit of what the amp can provide. Of course, one may receive the impression of altered bass response between amps depending on the specifics of how the amps differ from each other and how they respectively mate with the speaker. (Different damping factors/voltage slew rates, etc.) But this isn't specific to the power rating of the amp.
