When should homeless families get subsidized apartments? A theoretical inquiry

How and when should operators of homeless shelters place families from these shelters into subsidized housing? I apply the tools of contract theory to this problem, especially some approaches that have been taken to optimal unemployment insurance. The problem combines moral hazard and adverse selection. When all families are drawn from the same distribution, placement should occur immediately. When families are heterogeneous, the optimal strategy creates a separating equilibrium. Good searchers self-select into a contract with high probability of immediate placement and no probability of later placement; poorer searchers self-select into a contract with a lower probability of immediate placement but a positive probability of later placement.