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It is a pretty safe bet (but not always) when you have buildings with lower AMI levels such as 30%, 50%, and even 70% or 80% there is going to be a some amount of homeless, voucher holders etc... Since city treats lottery apartment scheme as a resource it can tap to find homeless permanent housing it stands to reason that is where they will go.
Households usually end up homeless because income just doesn't enable them to afford an apartment in NYC. A single parent household working part or even full time may not be able to swing anything but lowest rent. Ditto for those living on small pensions, veterans benefits, low monthly SS checks and so on.
Supportive housing comes into play for those who have "issues" such as mental illness, recovering from drugs and so on. For those with income enough to pay some sort of rent any sort of housing will do. Back in day SROs and boarding houses provided cheap rooms, but they're all largely gone.
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