The asset limit for most public assistance is $2000-2500, not $5,000. Maybe if people were allowed a $5k emergency fund more poor people on assistance would have them (and admittedly a lot wouldn’t, but some would, which is arguably an improvement).
Back when I was recently destitute, a couple years ago, some friends pitched in an bought us a new air conditioner to replace our dead one. When Public Aid got wind of it we were immediately audited and subjected to questioning about how much more we were hiding, were we employed by these people, how much a month were they giving us, demands for signed statements from every contributor this was a one-time charitable donation and not a regular payment to us…
It was horrible, really - some friends doing a nice thing by getting us an air conditioner worth about $300 tops not only almost got us throw off food stamps and our medical coverage but subjected us to accusations of fraud and threats of jail time. Fortunately, we got it straightened out in about a week but it was a week of stress we all could have done without.
It’s not that all poor people are clueless about emergency funds, it’s that the system does not allow it. Should you achieve $3k in funds you are cut off all aid until you “spend down” again, and getting back to $3k, much less $5k, is nearly impossible. It is, in fact, far easier to as a person on aid to finance $10k over 10 years than to save $5k as an emergency fund. Debt payments are something the system “understands”, but being responsible and saving - or accepting a gift - can get you booted out entirely and accused of wrong-doing.
But let’s say you DO get a $5k windfall - assuming you report it promptly and you’re believed (meaning you can prove you won the lottery, a relative died and left it to you, etc.) you now go from have some assurance and security - that food stamp account being filled every month, for example - to living on dwindling resources. If you can use the windfall to jump-start to the next level that’s great but really a couple thousand isn’t likely to do that… so it trickles out the door, nothing really changes, and at the end you have to get back on aid, re-navigating the bureaucracy, perhaps spending time on waiting lists to get back to where you were before the windfall arrived.
This creates a perverse incentive. Unless you can leap over the finance zone where you’re too "wealthy’ to qualify for aid but too poor to ever get ahead there is no incentive, and perhaps disincentive, to improve. Funny, when wealthy people reject a deal that is bad for them we call them wise. When poor people do the same thing - reject a deal that, long term, is not in their interests - we call them unpleasant names.