When introducing wage stickiness in labor search framework, costs of posting a vacancy must no longer be linear but convex, for the following reason. Thomas (2008, p. 940) says:
In the case of staggered wage bargaining, the resulting wage dispersion creates dispersion in the marginal benefit of posting vacancies. If vacancy-posting costs were linear, then the marginal cost of posting vacancies would be the same for all firms and only one firm (the one with the lowest wage) would post vacancies. In order to have an equilibrium where all firms post vacancies, I assume convexity in vacancy-posting costs.
In the same matter Gertler, Sala and Trigari (2008, p. 1720) say:
Because the contract structure leads to temporary wage dispersion and because (to simplify the bargaining problem) we have constant returns at the firm level, quadratic costs are required to keep capital and labor from shifting in mass to the low-wage firms.
How can I see it mathematically? I ask since I don’t fully comprehend how you realize of that, what would happen if I formulate a search labor model with staggered wage bargaining (using Nash solution) and assume that costs of posting a vacancy are linear as usual? Would not there be existence of equilibrium?
Also Galí (2010, p.20) (although not a fully search labor model, but an arguably very similar one, since he uses the same reason as the previous authors for argumenting the changes) introduce instead of those convex cost in vacancy posting, diminishing returns to scale to labor production:
The assumption of a decreasing returns technology is required in order for wage differentials across firm to be consistent with equilibrium, given the assumption of price taking
behavior (otherwise only the firm with the lowest wage would not be priced out of the
I find Galí’s way more appealing, but I don’t recall where I think having read that introducing diminishing or increasing returns to scale to the production function where the labor that is involved in search frictions enters, causes flaws in Nash bargaining solution since produces different importance of the marginal worker that enters the firm. I mean can Galí’s way be also used in a full labor search model?
If you may help me in understanding this better I’d be very grateful!