In general, how does price relate to quality?

Good sound need only cost more for certain when you need more of it like in rare cases with very large rooms in a handful of houses.
Then there are the cases of truly large public venues that also target good sound quality. They need the most and also are most motivated to find ways to do it cost effectively. As opposed to hobbyists who get enjoyment as much out of tinkering and throwing money into the hobby in pursuit of “better sound” as much as they do merely enjoying music. It’s all good. Just not equally cost effective. Not to mention it’s hard to get people to even agree on what better sound is when they hear it. That’s why there are so many options and people do it so many different ways. Variety is the spice of life whereas the ultimate sound is a pipe dream except each may have their own conception of what the ultimate sound is for them. If they care about the best sound possible (for them) they need to. You can’t hit the target if you don’t know where it is.

@mapman " You can’t hit the target if you don’t know where it is."

Truer words were never spoken. How do you know what a stereo is capable of sounding like if you have never heard one perform at that level? Just as you have stated sound quality can be all over the map for stereos and live performances. But, in my experience when faced by a really excellent system everyone will agree that it is an excellent system.
Once you hear it you are in serious trouble. I remember vividly the first time I heard such a system. My system at the time was no slouch but it did not come close to doing what this system did and it was not at the store either, but at the apartment of a hermit high school teacher in Miami who stubbornly like driving Alphas. I thought my system was imaging at the state of the art until I heard this. It took me ten years and lord knows how much money to replicate it. It can certainly be done for about $100,000 in todays money and that includes a turntable. I suspect it can be done for considerably less. There are people who pay more for just a turntable! If there is better I have not heard it yet. I have never heard it at a show either. 

I'd say there is a loose relationship between price and build quality (e.g. quality of parts used, level of construction, tolerance...etc), but this doesn't equate to sound quality, which is most important to us.  So overall, there isn't a clear correlation between price and overall quality as judged by the end user.  Is a 10K phono stage 5x better than a 2K phono stage?  How would one even determine or quantify it is objectively better?

Think it really boils down to if it is worth the money.
IME, a much more pricey piece is usually well worth it.

Money can't buy you happiness but it sure can my a lot of misery!  I have said this so many times here on Agon.  Get to know what parts make a difference in sound.  Price has nothing to do with sound or quality unless the component is designed well.  I have a old Lafayette KT-550 tube power amp that probably sounds better than most amps built today.  No special parts but transformers are excellent.  Go Figure. 

Happy Listening.