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

2.5. Aggregation and steady-state equilibrium

We now infer the implications of the preceding microeconomic structure for equilibrium labor market dynamics. An important first step toward this end is to aggregate individual firm and worker behavior for a given aggregate state. Given a solution to this aggregation problem, we then characterize conditions for steady-state labor market equilibrium.
To aid these steps, we restrict the drift of the stochastic process for idiosyncratic shocks in (1) to ensure that aggregate labor demand is stationary. This obtains when frictionless employment, which is proportional to x1/(1−α)⁠, has no drift. Applying Ito’s lemma, this requires that
μ+12α1−ασ2=0.
(24)
This assumption is made purely to simplify the analysis by abstracting from growth.14
Aggregation. Steady-state aggregate labor market stocks and flows in the model are summarized by solutions for the separation rate into unemployment (denoted ς⁠), the hiring rate (denoted η⁠), and the density of employees g⁠, at each marginal product m⁠.
Proposition 3
In steady state, (i) the separation rate into unemployment is given by
ς=σ2/21−αmlg(ml).
(25)
(ii) The hiring rate is given by
η(m)=−σ2/21−αmδ′(m)δ(m).
(26)
(iii) The vacancy-filling rate is given by
q(m)=χexp[−1−ασ2/2∫mumδ(˜m)˜md˜m],
(27)
with ψ=exp[1−ασ2/2∫mumlδ(˜m)˜md˜m]⁠. Using (15), this yields the worker distribution G(m)⁠.
The most standard element of Proposition 3 is the solution for the separation rate into unemployment ς⁠. All such separations arise at the lower boundary ml⁠. There, a density of g(ml) employees receives shocks to their log marginal product of variance σ2⁠. Following negative shocks, employees are shed into unemployment until the marginal product is replenished, at a rate proportional to the elasticity of labor demand, 1/(1−α)⁠.
The remaining results in Proposition 3 are novel features of this environment, however, and are illustrated in Figure 2. The economic content of these will be explained in more detail in the next section. For now, we simply note that the hiring rate η(m)⁠, the worker distribution G(m) and thereby the vacancy-filling rate q(m) are inferred from the Fokker–Planck (Kolmogorov Forward) equation that describes the flow of workers across marginal products. Intuitively, δ(m) determines the flow of hires at each m (via F(m))⁠, and the outflow of employees from each m⁠. In this way it shapes the hiring rate η(m)⁠, and the worker distribution G(m)⁠. An important implication is that the solution for the equilibrium quit rate δ(m) in Proposition 2 is sufficient to solve for all of these aggregate outcomes, which in turn are functions solely of the marginal product m⁠.