What are the economic incentives for professors to hire and train graduate students?
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.
A few reasons:
- It might be required by the funding agency.
- Graduate students let the professor expand his/her research program. If one is interested in the results, but don't have the time to do it yourself, then graduate students are great.
- Successful graduate students produce papers which have their advisor's name on them, and if they go on to become successful, the advisor benefits professionally (since it indirectly shows they are a great mentor).
It's a requirement of their job.
(Longer version: in the UK at least, universities seem to be increasingly setting explicit requirements of academic positions, and at least some of those lists of requirements set out minimum rates of research students you must have graduate under your supervision.)