In other words, if there were no way to reduce inequality without reversing the gains that everybody made, then reducing inequality in that situation would be harmful.
At the low end, at some point you can afford to have a roof over your head, and an extra $20 might mean that you're eating much better for a whole week, or you can buy new shoes, or whatever. But if you're homeless, $20 won't feed you for very long or get you out of the rain for very long.
At the mid range, a salary bump from $30k to $50k might make a world of difference if you're trying to raise a family. It might mean that you're not worrying about how you're going to pay for groceries, or it might mean that your kids are always wearing reasonable, well-fitting clothing.
A few years ago, I met a beggar at a bus stop, who said that the store across the street had a sleeping bag for 50% off until the end of the week, and he was hoping that he would be able to buy it for the winter. I didn't have much money on me myself (basically just the bus fare), but I gave him what I had and walked home instead. To him, those five-ish euros meant not freezing in his sleep, while to me, they only meant a bit of unplanned exercise.
If you have nothing at all, every little bit of money gives you access to some tangible improvement of your situation. But if you have much less than nothing, and every cent of additional income just serves to lower a gigantic debt, there isn't really much incentive to earn more when it makes no significant difference in the end. This is probably one of the reasons bankruptcy law allows discharging unpayable debts, because otherwise debtors might not be motivated to do something about the debts they could repay.
I believe that utility is logarithmic with rate of expenditure. Hence doubling your spending rate gets you a constant increment in utility. This is why we measure pay rises in percent rather than absolute value. Another $100/week would make a huge difference to someone on the breadline, but is lost in the noise for a millionaire.
This means that money transferred from a rich person to a poor person will increase utility, because the decrement in the rich person's utility is more than outweighed by the incremenent in the poor person's utility. This difference can also accomodate the transaction costs (i.e. running the IRS) and the "deadweight cost" of decreasing the rich person's incentive to earn more.
This would seem to suggest that money should be redistributed so that everyone has the same amount, as that would maximise utility if the total amount of money is constant. But of course that condition isn't true: in a world with zero economic incentives there would be no economic activity. In reality of course such a world would go horribly wrong in any number of ways.
So there would seem to be an optimum amount of inequality in a society. Too little, and you destroy incentives to create new stuff, thereby making everyone poorer than they would have been. Too much and you are wasting valuable resources on gold plated baths that would be better spent giving poor people a tolerable standard of living, education, health care etc.