Extensive Labor Margin and Full Insurance
I'm not very well versed in macro-labor, but I'm aware of a few papers that might be close to what you mean. One could make unemployment uninsurable, and instead let agents partially self-insure by saving capital, like in Chang &amp; Kim (2007) AER paper. Also Krusell &amp; coauthors have some papers (like 2010 ReStud) where they combine uninsurable unemployment with search/matching framework.

The obvious problem is that dealing with heterogeneity is painful, because one must keep track of distribution of variables across agents (e.g. employment status and savings). If you're interested in computation and what can go wrong with it, check out comment on Chang &amp; Kim paper by Takahashi (as well as reply by original authors in the same issue).