And of course, that's natural; people DO value their own well-being and that of their families and friends (and especially children) more than that of strangers. It's a very rare person who's willing to completely sacrifice their own standard of living in the name of global utilitarianism. (Mother Teresa comes to mind, but that's about it...) But it's also pretty rare for a person not to feel any selfless compassion for others either; it's just harder when one feels one's own needs aren't being met.
So maybe that's the real solution to your problem: find a way to address people's needs without resorting to a zero-sum mentality. When people feel they're getting by alright, they will naturally be less begrudging of others. Easier said than done of course, but not impossible. Much more difficult when the focus is on this zero-sum game instead though.
The crux of the issue. I appreciate you taking the time, and employing the insight, to identify it. In my experience of such discussions, few do.