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

FIGURE 2

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Steady-state hiring and vacancy-filling rates
Notes: Parameter values are based on the model calibrated as described in Section 3.

This completes the m-solution that we sought to derive. We now show how the elements of Proposition 3 can be used to characterize steady-state equilibrium.

Steady-state equilibrium. The matching structure implies that all outcomes of the model described thus far in Propositions 1, 2, and 3 depend on a single endogenous aggregate state, labor market tightness θ⁠, via the contact rates λ(θ) and χ(θ)⁠. Given this, we can characterize steady-state equilibrium in terms of two conditions reminiscent of the standard search model of Mortensen and Pissarides (1994).
Proposition 4
There exists15 a steady-state equilibrium in which unemployment U and labor market tightness θ satisfy (i) a Beveridge curve condition,
UBC(θ)=ς(θ)ς(θ)+λ(θ)L;
(28)
and (ii) a job creation condition,
UJC(θ)=L−[X/∫m11−αg(m;θ)dm],
(29)
where X≡E[(αx)1/(1−α)]⁠.
The Beveridge curve condition emerges from the law of motion for unemployment. Making explicit the dependence of the separation rate on tightness, this reads
dUdt=ς(θ)(L−U)−λ(θ)U.
(30)
In steady state, unemployment is stationary, and we obtain the Beveridge curve (28).
The job creation condition is implied by aggregation of firms’ labor demand. Aggregate employment is the mean of employment across firms, N=E[(αx/m)1/(1−α)]⁠. Observing that the latter is equal to the ratio of the mean of (αx)1/(1−α) across firms and the employment-weighted mean of m1/(1−α) gives rise to the job creation condition in (29).
This completes our constructive characterization of labor market equilibrium. Since it is founded on an m-solution for labor demand and turnover, we label it an m-equilibrium
Definition An m-equilibrium is a collection of optimal worker acceptance, and firm hiring and firing decisions {a∗;dH∗,dS∗}⁠; worker and firm (marginal) values {W,J}⁠; quit, layoff, hiring and vacancy-filling rates {δ,ς,η,q}⁠; aggregate unemployment and job-finding rate {U,λ}⁠; and offer and worker distributions {F,G}such that (i) optimal worker and firm decisions, their associated (marginal) values, and the quit rate satisfy Lemma 1 and Propositions 1 and 2; (ii) layoff, hiring and vacancy-filling rates are given by Proposition 3; (iii) aggregate unemployment and labor market tightness are given by Proposition 4; and (iv) the offer and worker distributions solve (15). Vacancies V, market tightness θ⁠, and the vacancy contact rate χcan then be inferred from the matching function and λ⁠.
Figure 3 illustrates the steady-state job creation and Beveridge curves, and depicts the upward shift of the job creation curve induced by a decline in aggregate labor productivity. Specifically, it plots the effect of modifying the production function to pxnα⁠, such that p falls by 1%. The positive slope of the job creation curve reflects the nature of search equilibrium in the model: Labor market tightness equilibrates the labor market via its effects on firms’ turnover costs, in contrast to its effects on the cost of hiring, as in many conventional search models.