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

4. Aggregate dynamics

The analysis thus far has addressed the first analytical challenge posed by the interaction of firm dynamics with on-the-job search—namely that of inferring the steady-state offer and worker distributions, F(m) and G(m)⁠. We begin this section by showing how these results also inform the solution to the second analytical challenge—inferring the out-of-steady-state dynamics of the distributions. We then use this approach to solve for the transition path of model outcomes, including the distributions F(m) and G(m)⁠, following an MIT shock to aggregate productivity. The section concludes with a comparison of the amplitudes of labor market stocks and flows, and wages, with empirical counterparts.

4.1. Solution method

The key insight of our approach is that the form of the quit rate in Proposition 2 also will hold out of steady state, subject to the modification that the middle boundary and the job offer arrival rate will vary over time, mht and λt⁠. The intuition is simple. Out-of-steady-state dynamics give rise to additional capital gains in the firm’s marginal value relative to its steady-state form in (16),
rJt(m)=(1−ω1)m−ω0−[δt(m)−(1−α)mδ′t(m)]Jt(m)+[μ+(1−α)δt(m)]mJ′t(m)+12σ2m2J′′t(m)+∂Jt(m)∂t.
(37)
Optimality in the hiring region, however, requires that the firm’s marginal value of labor is a constant, equal to the marginal hiring cost, Jt(m)=c for all m∈(mht,mut)⁠. As before, this implies that J′t(m)=0=J′′t(m) in the hiring region. But, crucially, it also implies that any such out-of-steady-state capital gains are zero in the hiring region, ∂Jt(m)/∂t=0⁠. Thus, the quit rate shares the same functional form as in Proposition 2. This is a considerable simplification, as the solution for the dynamic path of the quit rate—or, equivalently, the offer distribution Ft(m)—is thus known up to the path of two scalars, mht and λt⁠, a much simpler prospect.
This in turn aids the solution for the time path of the worker distribution. Just as the quit rate informs the steady-state vacancy-filling rate in (27), and thereby the steady-state worker distribution, its time path induces the dynamics of Gt(m) via the out-of-steady-state Fokker–Planck (Kolmogorov Forward) Equation. Thus, the dynamic path of the worker distribution Gt(m) is known up to the path of three scalars mlt, mht, and λt⁠.
Finally, consider the natural wastage region. Here, the quit rate is maximal and equal to sλt⁠. The job offer arrival rate λt is thus the sole aggregate state in this region. Given a time path for λt⁠, the firm’s marginal value Jt(m)⁠, and the boundaries mlt and mht⁠, can then be inferred out of steady state. This implies a further simplification: the path of λt is also sufficient to determine the paths of mlt and mht⁠.
The upshot is that the dimensionality of the problem of inferring the model’s transition dynamics is greatly reduced by the analytical results developed earlier in the paper. Absent these results, solving the model out of steady state would involve forecasts of the unknown functions δt(m) and qt(m)⁠. With these results, we can distil the problem to one which requires a forecast of the dynamic path of just one scalar, λt⁠.