Job search is not macro..it's micro
The treatment in the book we are discussing is decidedly macro, and looks generally similar to the treatment in a previous Sargent book titled "Recursive Macroeconomic Theory."
Disregarding the fact that job search is a subtopic of unemployment, one of the main concepts in macro, you'll notice their model parameters are human capital, investment, and wage. The only quasi micro flavor of that model is search effort as these variables will almost always be in equilibrium based on some kind of an optimal stopping rule. Whether that rule has to do with finding a new job or discovering information about, say, the prices in the market dictates whether it would be a micro or macro use case. Again, if you actually looked at the model, their optimal stopping rule is when the person finds another job. And once again, the idea of jobs are the main ingredient in the concept of unemployment.... a macro topic. Not to mention, whether or not I'm searching for a job dictates whether I'm factored into the unemployment rate or not.
So..... yeah, job search is definitely not micro.
In microeconomics, search theory studies buyers or sellers who cannot instantly find a trading partner, and must therefore search for a partner prior to transacting.
Search theory has been influential in many areas of economics. It has been applied in labor economics to analyze frictional unemployment resulting from job hunting by workers.
I definitely studied these types of models in macro classes from Pissarides, one of the pioneers in the job search field. I think he'd be amused to learn that he had become a microeconomist.
Also, if you look at the example, it is taken directly out of Sargent's textbook: Recursive Macroeconomic Theory.