McCall Job Search

Here, we model a person looking for a job. This person gets random job offers which they can choose to accept or reject.

¶ Details of the Model

In this model, a person is either employed or unemployed.

• If they are unemployed, then:
  • They receive some unemployment benefit $b$,
  • and they search for a job:
    • There is some probability $p$ that they get a job offer.
    • If they do get a job offer, then this offer has a randomly-drawn wage $w$
    • They can choose to accept the job, or reject the job (which means they will still be unemployed tomorrow).
• If they are employed in a job with wage $w$, then:
  • They earn that wage $w$,
  • and then there is a random chance $s$ that they lose their job.

¶ The Reservation Wage

When should the person accept an offer with wage $w$? Only when the offered wage is high enough to be ‘worth it’.

The Reservation Wage, labelled $w^\ast$, is the lowest wage which the person is willing to accept.

Label the value¹ of being unemployed as $V_u$, and the value of being employed with wage $w$ as $V_e(w)$. Then:

The person will accept a job offer only if $w \geq w^\ast$, and the probability that they accept a random offer $w$ is therefore $\Pr(w \geq w^\ast)$

¶ Steady-state Unemployment Rate

Let’s simplify the above picture by noting the transition probabilities:

• There is a probability $s$ that an employed person becomes unemployed.
• There is a probability $p \cdot \Pr(w \geq w^\ast)$ that an unemployed person becomes employed.

Let $U$ be the percentage of people who are unemployed, and $1-U$ the percentage employed. Then for the flows between these two categories to be equal, it must be that:

The amount of employed people who lose their job equals the amount of unemployed people who start a new job.

Solving for $U$, we find the steady-state unemployment rate is

¶ Other Search Models

Many other variants of search and matching models exist.

For example, one can build a Two-sided model of search, where unemployed workers are looking for jobs, and where firms are looking for workers to fill vacant positions.

These kinds of two-sided search models are concerned not just with the unemployment rate, but also the vacancy rate, which is the portion of job positions which currently lack a worker filling them. (The BLS also uses the term “job openings rate”. This is synonymous with “vacancy rate”.)

¶ Links

• Some Graphs from the BLS’ Job Openings and Labor Turnover Survey:
  • Hires and Seperations rates
  • A curve plotting Unemployment Rates vs Vacancy Rates (aka “The Beveridge Curve”)