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

Acknowledgments

We are grateful to the Editor Dirk Krueger, and anonymous referees for their constructive suggestions. We also thank Alex Clymo, Karl Harmenberg, Rasmus Lentz, Guido Menzio, John Moore, and Jonathan Thomas as well as seminar participants at numerous institutions for helpful comments. All errors are our own. We gratefully acknowledge financial support from the UK Economic and Social Research Council (ESRC), Award reference ES/L009633/1.

Supplementary Data

Supplementary data are available at Review of Economic Studies online. And the replication packages are available at https://dx.doi.org/10.5281/zenodo.5221647.

Data Availability Statement

The data and code underlying this research is available on Zenodo at https://doi.org/10.5281/zenodo.5221647.

APPENDIX

A. Proofs of Lemmas and Propositions

Proof of Lemma 1
We first verify that, under an m-solution, the worker surplus in (11) is a function solely of m⁠. Denoting the firm’s hiring and separation rates by dH∗/n=η(m;dt) and dS∗/n=ς(m;dt)⁠, we can rewrite (11) as a function only of m
rW(m)dt={βm1−β(1−α)+ω0−b−λ∫˜WdΦ(˜W)+sλ∫W(m)[˜W−W(m)]dΦ(˜W)+[μ+(1−α)δ(W(m))]mW′(m)+12σ2m2W′′(m)}dt−(1−α)[η(m;dt)−ς(m;dt)]mW′(m)−ς(m;dt)W(m).
(A.1)
Likewise, one can confirm that the firm’s marginal value J in (14) is a function only of m⁠.
We now establish monotonicity of W in m⁠. First, we verify that all separations into unemployment occur at a lower reflecting boundary for m⁠. Suppose, to the contrary, that the firm implements strictly positive fires such that the marginal product m diffuses over an interval [m1,m2]⁠. By optimality, it must be that the firm’s marginal value J(m)=0 for all m∈[m1,m2]⁠, and thus that J′(m)=J′′(m)=0 for all m∈(m1,m2)⁠. Inserting the latter into (14) yields J(m)={(1−β)/[1−β(1−α)]}m−ω0=0 for all m∈(m1,m2)⁠, a contradiction. Thus, all fires occur at a lower reflecting barrier; the firm’s firing rate ς is thus weakly decreasing in m⁠.
Now consider two firms with different initial marginal products m′>m⁠. Fix, for both firms, a given sample path for changes in idiosyncratic productivity, arrivals of job offers, and layoff shocks. Furthermore, suppose that the worker employed in firm m′ implements, for all future periods, the same job acceptance policy as the optimal policy for the worker employed in firm m⁠. Denote by T the first time one of the following events occurs for the worker in firm m⁠: the worker is fired; the worker accepts a job; the marginal product equals that in firm m′⁠. Further denote by VT the continuation value thereafter. Since we have fixed the same sample paths of shocks and job acceptance strategy, T and VT are the same in firm m′⁠. Since the worker surplus is based on expectations over sample paths, and since the worker in firm m′ implements a weakly suboptimal job acceptance policy, W(m′)≥E[∫T0e−rtw(m′t)dt+e−rTVT|m0=m′]>E[∫T0e−rtw(mt)dt+e−rTVT|m0=m]=W(m)⁠, as required. □