Firm Dynamics, On-the-Job Search, and Labor Market Fluctuations

5.2. The structure of frictions

Our baseline analyses assume frictions take the form of a per-worker hiring cost c⁠. Here we summarize a few examples of deviations from this baseline; formal details are provided in the Supplementary Appendix. Key to tractability is the availability of an m-solution. The simplest such extension is thus a state-dependent linear hiring cost, c(m)⁠; for example, a training cost whereby the cost of generating productive new hires depends on the marginal product of existing employees required to train them. All results generalize in this case.
But it is also possible to extend our results to cases in which there is convexity in the recruitment technology, which are widely implemented in models of firm dynamics. Here, an m-solution is preserved provided that the marginal hiring (vacancy) cost is a function solely of the hiring (vacancy) rate. The key insight is that the aggregation results in Proposition 3 continue to hold, in particular for the hiring rate η(m)⁠, and the vacancy rate ν(m)≡η(m)/q(m)⁠. We show in the Supplementary Appendix that this observation distils the problem into a system of differential equations—in the marginal value J(m)⁠, the quit rate δ(m)⁠, the hiring rate η(m)⁠, and the vacancy-filling rate q(m)—that is straightforward and efficient to solve numerically. The relative simplicity of this result further widens the scope of application of the current framework.
A final ingredient common to models of firm dynamics is firm entry, exit, and growth. We show how it is also possible to embed these in the model, and its various extensions considered above. Again, the key is that, upon entry, firm entrants can be summarized by a distribution of marginal products. The Supplementary Appendix shows how this, together with exogenous firm exit, can be accommodated straightforwardly into our framework.