This paper explains how the aggregate matching function — a tool economists use to describe how applicants and job vacancies find each other — can be derived from a network of individual connections. The authors use ideas from network theory to show how the small, granular links between applicants and vacancies shape the overall matching outcome. They show that many commonly used mathematical forms of the matching function appear as special cases of their network model, including the constant elasticity of substitution (CES) form, a widely used specification in macroeconomics.

To do this, the authors build a simple model in which applicants and vacancies form random links. Each potential link is made with some probability pij. A link means an applicant applies to a vacancy. Links are independent and are modeled as Bernoulli random variables (they either occur or do not). Jobs are assumed to be given to the most qualified applicant, with qualifications drawn from a common distribution. From this setup the paper derives a closed-form expression for the job-finding probability (or, from the vacancy side, the vacancy-filling probability).

One concrete result is a testable condition under which matching in any network from the class they study behaves “as if” it came from a CES matching function — but only up to a first-order approximation. The CES (constant elasticity of substitution) form is a flexible mathematical shape that many models use. The paper points out that this CES-like behavior can arise for many different underlying network structures. Because the network of applications is observable in application data, the authors say the condition they derive can be checked empirically.

The paper also develops a theory of match efficacy, which measures how well the matching process works beyond simple market tightness. A key finding is that inequality in search intensities matters. Search intensity is the mean number of links an agent has to the other side (how many vacancies a worker applies to, or how many applicants a vacancy attracts). The analysis shows that greater dispersion — more inequality in search intensity — lowers matching efficacy. This holds whether heterogeneity is on the applicant side, the vacancy side, or when vacancies are grouped into “locations.” The authors further show that raising the market’s average search intensity can sometimes reduce efficacy when it comes together with a higher Gini coefficient (a common measure of inequality). In some cases the job-finding probability follows an inverted-U shape with respect to mean search intensity.