Specifically, I am looking for the standard neoclassical growth model, but the labour decision is based on either search or matching theory instead of a Walrasian labour mechanism.
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$\begingroup$ I am currently looking for a similar model. Did you ever find some references? The closest I could come to is A three state model of worker flows in general equilibrium (Krusell, Mukoyama, Rogerson, Sahin), but they fail to include a matching function. $\endgroup$– Alex RuizCommented Nov 11, 2021 at 23:31
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Search and matching models are fairly standard in macroeconomics. I guess this is fair to say since Peter Diamond, Dale Mortensen, and Christopher Pissarides won the Nobel prize in 2010. These are the names behind the canonical DMP model that most graduate students have to study in their first year. Although I am not a macroeconomist, I am confident to say that search models recognized as a key ingredient to explain unemployment in equilibrium. I suggest to watch the interview with Per Krusell on the Nobel prize homepage and read the scientific background.