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

Proposition 2
There is a unique hiring region in which the firm’s marginal value is constant, J(m)=c⁠, and in which there is a unique solution for the quit rate given by
δ(m)=sλ+1c{(1−ω1)(m−mh)α−[(1−ω1)mhα−ω0−(r+sλ)c][(mmh)11−α−1]},
(23)
for all m∈(mh,mu)⁠, where the upper boundary solves δ(mu)=0 and is unique. Furthermore, δ(m) is strictly decreasing and concave for all m∈(mh,mu).
Proposition 2 is an important result. By establishing the equilibrium quit rate δ(m)⁠, it in turn implies a solution for the equilibrium offer distribution of marginal products, F(m) in (15). Proposition 2 thus provides a key part of the solution to the challenge of how to determine equilibrium turnover, and thereby the equilibrium distributions of marginal products. We will see that this provides a key building block to the determination of steady-state aggregate equilibrium, as well as out-of-steady-state aggregate dynamics.
Proposition 2 also has a surprising implication: Hiring firms that face a homogeneous per-worker hiring cost c nonetheless allow their marginal products to vary over an interval, giving rise to a non-degenerate distribution of worker values across hiring firms.
The intuition for why is as follows. Consider a firm at the middle boundary mh⁠. Following a positive innovation to its productivity x⁠, and thereby its marginal product m⁠, the firm faces a novel trade-off in the presence of on-the-job search.
On one hand, the firm values the net additional output generated by new hires. If this were the firm’s only consideration, it would simply hire until its marginal product returns to the middle boundary; formally, mh would become a reflecting barrier. This is the force captured in canonical models of firm dynamics (such as Bentolila and Bertola 1990). On the other hand, in the presence of on-the-job search, this is not the firm’s only consideration: it also values reductions in turnover costs afforded by the lower quit rate that accompanies a higher marginal product.
The economy resolves this trade-off in a novel way. Firms no longer hire until their marginal products are reflected back to mh⁠. Instead, they hire less aggressively, allowing their ms to diffuse across an interval (mh,mu)⁠. This policy is supported by a quit rate δ(m) in (23) that declines in m at an appropriate rate to maintain firms’ desire to hire. Intuitively, the rate at which δ(m) declines in m is shaped by the density of offers f(m)⁠. Suppose the density at some m were higher than implied by (23), any firms at that m would face excessive turnover costs. Their marginal value of labor would be below the hiring cost c⁠, they would not find it optimal to hire, and the density of offers at that m would be zero—a contradiction. Conversely, suppose the density at some m were lower than implied by (23), any firms at that m would face excessively low turnover costs, and value labor on the margin strictly in excess of the hiring cost c⁠. Such firms would seek to hire a mass of workers, inducing a mass point in the distribution of offers at that m—a contradiction. The density prescribed by Proposition 2 exactly balances these forces.
Finally, note that δ(m) is declining throughout the hiring region, and so the implied offer distribution F(m) is rising in m⁠. By Lemma 1, it follows that the quit rate in (23) is consistent with optimal worker turnover.