What are the economic incentives for professors to hire and train graduate students? 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?
Since you mention economic incentives, one obvious answer might be for teaching, at least this was the way it was in Germany. Even the most theoretical professor is supposed to teach a certain number of classes each semester. Generally the work involved in this is too much for one person alone, that is why the professor is supposed to do the lectures and the PhD students handle exercise classes and organisation. So its either supervise a bit on a research level or spend twice the time dealing with undergrad students having missed the deadline for their homework again because their grandfather died for the fifth time in this semester. Personally I'd know what I'd choose...
To this end each professor has a number of positions available, which can only be used to hire PhD students or postdocs. In particular it is not possible to hire someone to just help with the teaching without supervising them scientifically (Not that many qualified people would be willing anyway), or use the money for something else. Also like with any budget item, not using positions for too long is a great way to have them reallocated to someone else.
Regarding incentives to have them graduate on time, that one is also simple, if a bit unrefined: There is a maximum amount of years one can be employed in such positions. So in order to supervise the next student, the old one should be finished.
But I have to say that homo economicus is a spectacularly bad model for professors in theoretical subjects. Most of the ones I know are primarily motivated by science and would gladly work with a gifted student even without receiving any compensation.