as I said, teachers are overpaid, given their output.
By the way, the top of class people do not need to waste their time on students. It does not take a genius at higher education to teach algebra. It takes someone that knows algebra and how to communicate/teach.
The key is not in taking the creme de la creme and thrusting them into teaching jobs - we need them in jobs that produce economic output - not jobs that can be done by anyone who has a simple mastery of the subject being taught. Think about it - the answer is not in getting higher pay - the pay is too much already. The key is in getting people that know how to teach and holding them accountable for results.
Algebra is an example of an easy high school subject. What about Calculus? Or AP Chemistry?
Anyways, I'm not arguing that the current pool of teachers are being over or under paid. I think that most of them are incompetent and shouldn't be teaching in the first place.
However, if we lower teacher's pay even further, that would only further deteriorate the quality of teachers in the educational system.
As you said, we need teachers with mastery in a subject, good communication skills, and diligence/self-control (kids can be such annoying pricks). What I am saying is that people who fit this description of skillset can find better paying jobs in private industrial sectors. So what could possibly motivate them to become teachers? (Other than altruism, a good heart, etc.)
Teaching might not directly contribute much to economic output, but in the long run it can leverage a LOT of output.
If we sampled 10,000 children and sent them to some shitty, backwards public school system, how much economic output would they produce when they become adults?
On the other hand, if those same 10,000 children were sent to a different school system with superior teachers and resources, how much economic output would they produce as adults?
Obviously it's impossible to take the exact same kids and have them go through school twice, so we can't really do a controlled experiment. But I reckon they would do a lot better in the second scenario.
Now scale this to hundreds of thousands, or even millions of students. The difference in output would be massive. So my point is, finding higher caliber teachers and paying them more right now might seem like an economic inefficiency, but it's a long term investment that's supposed to pay off when the students graduate and start producing output. If we only looked at the short run, then why even bother sending kids to school when we can have them deliver newspapers or wash dishes full-time?