How much poverty are you willing to create to reduce inequality? That's the fundamental question. For me the answer is zero.
I also don't assign much value to proportionate representation. I think I'd prefer the rich to have more political power, they seem to be the least insane. Unfortunately, in our system we do have proportionate representation. The only way a rich person can influence electoral outcomes is by persuading poor and middle class people to vote for his candidate. Sadly, as Trump demonstrates, this can be done just as easily with nonsensical emotional appeals as with intellectual ones.
Unfortunately, inequality is, empirically, one of the strongest sources of disutility. So, as much as you might think you can "make everyone better off, but increase inequality in the process" when you use poor proxies for utility, you probably can't.
> I also don't assign much value to proportionate representation.
Comparative studies (at least among nominal democracies), show that proportionality is fairly strongly correlated with experienced utility of government. So, this may be incompatible with your claimed preference for utility. (see, e.g., Lijphart's Patterns of Democracy.)
> I think I'd prefer the rich to have more political power, they seem to be the least insane.
Maybe they just have more power to shape perceptions through media, image management, etc.
Note that if you pull out one of the ever popular left wing scatterplots putting cross-country inequality on the x-axis, I'm going to pull out one with a higher r^2 putting something politically incorrect like single mothers or black people on the x-axis.
As for proportional repersentation, if it indeed is instrumentally useful in creating utility that's fine. But I only care about it insofar as it actually gets us other good outcomes - it's not something I intrinsically value.
Without proportionate representation, it's unlikely that you can achieve "everyone better off", because your measure will be biased - such a system practically ensures that the rich/powerful are better off at the expense of the poor, because their desires are weighted more heavily!
I'm asking about values, not specific world conditions. Sometimes the world lets us get everything we want - e.g., globalization and capitalism so far have reduced poverty and inequality together.
But sometimes the world doesn't give us the easy case. That's when we actually need to understand our values, and answer questions like "how many people will I push into poverty to reduce GINI by 1%?" If your answer is zero, you agree with me completely and don't care about inequality. Are you agreeing with me that inequality is irrelevant now?
Of course, it would be even better if I/we could simply reduce poverty and improves lives directly (without the detour of fighting inequality), but currently I don't know how (and neither does anyone else).
One of the best, if unrealistic, solutions I see are robots; but between now and when we invent really useful robots, I think it's important to strive to prevent outcomes as displayed in the movie Elysium (or the alternative future in the story Manna).