Search and matching model using Dynare

It’s actually more complicated. There are conditions under which you can handle a unit root with drift, despite not existing. See

Having a unit root itself without drift is usually not a problem. In this case, there are infinitely many steady states. The issue is there is no unique steady state so that you cannot use numerical techniques to search for it. Therefore, you have to provide the exact values of the one steady state you want to use to Dynare.