I would like to know if you know any model with Search and matching framework (Mortenses-Pisarides style) with 2 types of workers (like Ricardian and rule of thumb consumers for exemple) ? I would like to see such a model featuring different wages, hours worked and unemployment rates between the 2 consumers. In the end I would like to implement such a framework to a DSGE model with Ricardian and rule of thumb consumers. Thank you.
I am not sure if this is exactly what you are looking for, but this paper has model that sounds like what you want:
Have a look at this one too: