Actually, people should starve to death if they are capable of work and simply won't. If they are incapable or are trying to find work, then we should have a minimum standard of living (but it shouldn't be cushy, not should prison be, the workhouses of Dicken's day would probably be more along the lines of my view).
As far as the gap, this is a misnomer. I am in the top 20% of household incomes off my salary alone. I qualify as rich. I make just over 100k in salary. The bottom 20% average about 24k in income (either earned or handouts) right now. So 80k difference between my W-2 earnings and theirs. The problem isn't the gap between the defined "rich" and the defined "poor" (top 20% versus bottom 20%). The problem is the demotivating aspect of handouts in the form of "TV/cable" and "private apartments" and "appetizing food" and "designer clothes". Our prisoners even live high on the hog. Turn living on welfare and prison into something truly undesirable (basic sustenance and prison style jumpsuits with no entertainment) and you will see people pull themselves out of poverty.
The only exception to my view should be the handicapped, elderly, or children.
Now, should a person be trying to get work or be working but still not quite making it, that is where subsidies should kick in to make their life a little better than the workhouses. I've always been a fan of a 5 to 1 ratio. You get X dollars per year to live on. For every 5 dollars you make, we only take away 1 from the subsidy. So if you get say 10k per year to live on (requiring you to live in the workhouse most likely) then when you are making 25k per year, we still give you 5k. if you are making 40k, you still get an extra 2k subsidy. When you hit 50k you are on your own.
Or maybe make it 4 to 1, but give you the first 10k of earning without subsidy penalty. So make 10k and get the full subsidy, make 20k and get a 7.5k subsidy, 30k and get a 5k subsidy...
Both encourage people who can to go out and work.
There would, of course, be a separate set for handicapped, elderly, and children guaranteeing them a decent (not middle class, but private housing, TV, better than jumpsuit clothes and better than gruel food) but it would be closely monitored and stringently controlled to prevent abuse.