why do expensive amplifiers produce a better soundstage

"What's coming out of the speakers is the power supply being modulated by the source (music)"

on my variables importance pareto chart, speakers+room-acoustics are higher than amp. PA system, including amp+speakers+room, has to be characterized combined, where time and frequency domain audio transfer function is defined between amp input and sound at listener ears points. 

The recording, components, speakers, speaker placement, and all the variables associated with the room dimensions, construction materials, and acoustic treatment all affect a listener’s perception of soundstage. However, the OP specifically asked,

“why do expensive amplifiers produce a better soundstage”?

I interpreted the question as being specific to the effect of the amplifier on soundstage, all else being equal. In the case of expensive amplifiers (whatever the threshold for “expensive”), the power delivery is usually (but not always) less affected by speaker impedance variations with frequency and therefore the sound heard by the listener is usually more consistent with the information on the recording than with less expensive amplifiers, that presumably are less capable of delivering consistent power across all frequencies, and particularly with harder to drive speakers that present more severe impedance fluctuations, due to cost compromises in the amplifier’s power supply and other design compromises. However, I am sure there are exceptions to the rule.

“expensive” amp term has no technical merit, thus irrelevant to SQ! 

"In the case of expensive amplifiers (whatever the threshold for “expensive”), the power delivery is usually (but not always) less affected by speaker impedance variations with frequency and therefore the sound heard by the listener is usually more consistent with the information on the recording than with less expensive amplifiers, that presumably are less capable of delivering consistent power across all frequencies, and particularly with harder to drive speakers that present more severe impedance fluctuations, due to cost compromises in the amplifier’s power supply and other design compromises." 

Whew! One of the longest sentences I have ever read ; )

@tcutter +1 yep.. “expensive” things money could go to marketing - sales team instead of improving engineering and production line!