This would certainly not appear in models built around the representative agent, because the latter, fully insured over time, would save and consume regardless of their employment status.
Responsibility for crisis resolution lies outside the domain of monetary policy and is up to supervisory and fiscal authorities. More precisely, while the SSM is to be entrusted with the analytical role performed inside the central bank, the SRM needs to take the financial and redistributive decisions that belong to the political sphere, when it comes to winding down non-viable banks.