"So the point, as I get it, is that zero-lower-bound models work very well in practice, but have a problem in theory: it’s not at all clear how to reconcile them with Diamond-Mortenson-Pissarides-type models of the labor market."
I think one of the solutions is to recognize that in a lot of ways a labor contract is more like an investment than a consumer good. Labor isn't bought on a spot market for the most part - it is rented for an indeterminate period of time (as long as both the renter and the rentee find it to their advantage). The marginal product of labor, in that sense, is going to be compared to interest rates in the same way that the marginal efficiency of capital is, and we would expect to see high marginal products of labor with low hiring for the same reason that only high MEC investments are being made right now.
I am not well placed - right now - for integrating this into a search and matching model. But this is precisely the sort of thing that I want to do doctoral work on. I have two main project ideas I've been noodling over:
(1.) Taking Bob Shimer's model that combines wage rigidity with search and matching to describe fluctuations and letting firms react to wage rigidity by increasing turnover (wages for existing labor contracts are rigid, but new contracts aren't - job turnover is a wage adjustment strategy), and
(2.) Precisely the point I made above: the implications of labor as an investment for macroeconomic performance at the zero lower bound.
I'm hoping to do something like "Three Essays on the Macroeconomics of Labor Market Dynamics" so I can treat somewhat disparate topics and get a few papers out of it.