The have-nots will steal from the haves. Not all of them, not even most of them, but some of them will give in to temptation.
For some that is caused by envy. They have it. I want it. I'll take it.
For some of them it's their *need* to support their habit (a very powerful motivator).
For some, it isn't greed that drives them, it's necessity. If your children are starving you will do *anything* to feed them, anything at all - including selling your body or stealing from your weaker neighbours (Your essay needs to note that prostitution and theft/low level crime, go hand in hand with poverty).
Stealing from each other makes poverty worse. The poor can't afford good door locks or window locks (or insurance) but the rich can. So the thieves steal from the easy target, the poor. That makes the poor poorer and doesn't bother the rich at all.
If you haven't already, then read "Les Miserable" (the original book, not the abridged book of the play). Then read up on the Irish Potato Famine. See how the have-very-little (police, army, paid thugs) seriously oppressed the have-nots (peasants).