Hank model with searching and match

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Hi Robert. Can we solve hank model with searching and match using VFI toolkit? For example: Morten O Ravn, Vincent Sterk, Macroeconomic Fluctuations with HANK & SAM: an Analytical Approach, Journal of the European Economic Association , Volume 19, Issue 2, April 2021, Pages 1162–1202, https://doi.org/10.1093/jeea/jvaa028

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Short answer: No, unfortunately not.

Longer answer: VFI Toolkit does not currently do aggregate shocks, nor does it do search-and-matching.

Aggregate shocks in principle could be done. You can solve a transition path (a certainty-equivalent impulse response function) in a HANK model, and then BKM2018 tells us that simulating the model is just a weighted sum of this IRF (weighted by sequence of past aggregate shocks). All of this is only valid under the assumption that the model is linear in the aggregates. VFI Toolkit can solve the transition path, and you would then just be able to code all the rest by hand as a simple matrix multiplication. I hope to add some functionality around solving models with aggregate shocks next year, but currently the above is the only option.

Search-and-matching involves quite different general eqm conditions than other models. I don’t think VFI Toolkit can solve these. That said I am not that familiar with the technical details of search-and-matching (beyond the kind of basic DMP model you might teach in a first class on search-and-matching) so I don’t know precisely how you set up the general eqm conditions in such a model. I have no present plans to add search-and-matching features (if anyone out there knows search-and-matching and might be interested in developing it for the toolkit please do let me know :slight_smile: )

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I think there is a class of search models in general equilibrium that the toolkit is able to solve (maybe after a small modification), e.g.

From a purely mathematical point of view, this well-known paper of Mukoyama, Krusell and Sahin is an infinite horizon model with a semi-exogenous shock in general equilibrium.

The endgoenous state is asset holdings. There are two exogenous state variables: (1) labor productivity shock, (2) employment status, either employed or unemployed. In the simpler version of the model, without search effort, the employment status is purely random. With search effort, it becomes semi-exogenous.

I think the toolkit might not have semi-exogenous shocks for infinite horizon, but other than that, it’s ready to go! :blush:

EDIT
Actually the paper linked above does not have any search effort: the only decision is whether to work or not (0 vs 1). I think the toolkit is able to solve this model as it is

The reason why this model is simple is because the search and matching frictions are modelled in a very reduced form way: there is no matching function. Please refer to the paper for more details.

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Yeah, I think of it as a distinction between

  • search models (which use a semi-exogenous shock) which the toolkit can do.
  • search-and-matching models which involve more complicated market clearance (general equilibium) conditions. I don’t think the toolkit can solve these, but I have never really looked closely at what exactly the general eqm in search-and-matching models involves so I don’t actually know.
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