The Matching Function

A key assumption of this class of models is that the number of jobs created at each moment is a positive function of the number of firms looking for employees and the number of unemployed job seekers.

The number of jobs created at each moment in time is given by

where L is the size of the labor force, u is the unemployment rate (the unemployed as a proportion of the labor force), v is the vacancy rate (vacancies as a proportion of the labor force), and m is the job creation rate (new jobs as a share of the labor force).

The matching function is assumed to be increasing in every one of its arguments, concave, and linearly homogeneous. Thus, it is characterized by constant returns to scale. These are the same assumptions as those made for the neoclassical production function. The higher the number of unemployed job seekers and the higher the number of vacancies, the higher will be the number of jobs that are created. If the number of the unemployed and vacancies double, then the number of jobs created will also double.

18.1.1 The Probability of Filling a Vacancy and Labor Market Tightness

Assuming that all vacancies have the same probability of being filled, and that all the unemployed have the same probability of being employed, the probability of filling a vacancy will be equal to the ratio of the number of new jobs created over all existing vacancies. If we define this probability as q, then the probability of filling a vacancy is defined by

This equation assumes that every unemployed job seeker has the same probability of finding a job and that every vacancy has the same probability of being filled.

From (18.2), the probability of filling a vacancy is a function only of the ratio of the number of unemployed to job vacancies, because of the assumption that the matching function is linearly homogeneous. The greater the number of unemployed per vacancy is, the greater will be the likelihood of filling any particular vacancy.

We define θ as the ratio of vacancies to the unemployed:

so θ measures the degree of tightness in the labor market. The higher the number of vacancies is relative to the unemployed, the greater the tightness of the labor market will be. We can thus express (18.2) as a function of labor market tightness:

From the properties of the matching function, it follows that

where η(θ) is the elasticity of q with respect to θ. Higher tightness in the labor market implies a lower probability of filling a vacancy.

Because of the assumption of a Poisson density function, note that the average expected duration of a vacancy is equal to the inverse of q, and is thus a positive function of θ.

18.1.2 The Probability of the Unemployed Finding a Job

The probability of an unemployed job seeker finding a job is correspondingly equal to

From (18.5), the elasticity of the probability of finding a job with respect to θ is given by

Therefore, the greater the tightness in the labor market (a higher θ) is, the higher will be the probability of an unemployed job seeker finding a job.

The average expected duration of unemployment is given by 1/θq(θ)) and is a negative function of θ. In models of this type, note that the price mechanism cannot cause the probability of filling a vacancy or the probability of finding a job to be unity, because the labor market does not function only via the price mechanism, but also via the degree of tightness of the labor market. This tightness determines the probabilities of firms to fill their vacancies or of the unemployed to find jobs in any particular instance.

The dependence of the probability of filling a vacancy and the probability of finding a job on the relative number of vacancies to the unemployed (tightness) is an example of a trading externality. These search externalities are important determinants of the properties of equilibrium unemployment in this model.

18.2