Depending on workers’ application decisions, there may be more competition for some jobs than others. To capture this, we let qj be the ratio of workers who apply for jobs at firms offering wage Wj to the number firms posting that wage. We refer to this as the job’s expected queue length, an endogenous measure of the extent of competition for jobs offering Wj. We assume that a worker applying to wage wj is hired with probability //(fy), where ^ : E+ Uoo —> [0,1] is decreasing and continuously differentiable; if many workers apply for one type of job, each has a low employment probability website.
Symmetrically, the probability that firm j hires a worker is r](qj), where 77 : 1R+ U 00 —» [0,1] is increasing and continuously differentiable.2 This implies that holding constant the number of workers applying for a given wage, if more firms post that wage, each has a lower hiring probability. We impose the boundary conditions т/(0) = д(оо) == 0 and 77(00) = /x(0) = 1.
This formulation of the matching technology encompasses many reasonable possibilities. One can think of firms opening jobs in different geographic regions or industries, and workers directing their search towards one of these labor markets (see Acemoglu, 1997). In labor market j, all firms offer a common wage wj, and the ratio of workers to firms, qj, determines the matching probabilities.
Standard matching frictions ensure that within an individual labor market, unemployment and vacancies coexist. Moen (1997) offers a model where labor markets with different wages are created by a competitive “market making” sector, Montgomery (1991), Peters (1991) and Burdett, Shi, and Wright (1997) offer another story. Workers use identical mixed strategies in making their applications. A firm may receive multiple applications, in which case it hires one applicant and the others remain unemployed; or it may receive no applications, in which case its capital remains idle. One can prove that if in expectation q workers apply to each firm posting a wage of w, then rj(q) = 1 — exp(—q) and fi(q) = r}(q)/q. In this case, search frictions are due to a lack of coordination among workers and firms.
Another possibility is the “frictionless” matching process, where the shorter side of the market is fully employed, T}F(q) = min(l, q) and fiF(q) = min(l/^, 1). This matching process does not satisfy the differentiability assumption at q = 1, but serves as a useful limiting case. Although our results obtain under any smooth approximation to this matching process, they do not hold in this frictionless limit, demonstrating that search frictions are crucial to our analysis.