I am a current graduate student at a research-oriented university. Nonetheless, for me, it is a big mystery why and how the professors are incentivised to recruit and train the graduate students.
In an experiment-oriented research group, graduate students are the work-force of the principal investigator, so the professors train the graduate students to conduct experiments and research so that they contribute to the research group. However, in theory-oriented research areas such as pure mathematics, theoretical computer science, and theoretical physics, in many situations, graduate students do not contribute to the professor's research. They receive training and guidance for their research career unilaterally.
It should depend on the system for each nation, but what are the designed incentives for the professors in such areas to hire and train graduate students other than volunteer service spirit? In addition, I have heard that there exists an incentive system to push the professor to let their graduate students graduate within a reasonable time. What is such a system?