Here is an article bemoaning how little teachers are paid in the USA, compared to orthodontists.
And here are the gratuitous pictures used in the article to help explain what else teachers do to make ends meet.

Answer (reprise): a free market in education.
Actually, that would probably result in quite a few fabulously well paid teachers, a lot of adequately paid teachers, and even more very, very badly paid teachers, desperate to get plum jobs but mostly never getting them. Like acting in other words.
The reason that teachers are "underpaid" and orthodontists better paid is that poking about in people's brains is a lot more appealing than poking about in their mouths.
Also, most of teaching is basically child-minding rather than actual teaching, and any old twat can learn to do that, whereas not any old twat can orthodont. Orthodonting even adequately is hard. If done incompetently, huge damage would routinely result after only a few hours of idiotic orthodonting. How often do idiot teachers do the kind of serious and irreparable damage to a pupil that an idiot orthodontist would do almost every time if he was an idiot? Therefore people are rationally willing to pay extra for properly qualified orthodonting.
Plus, nobody can agree what good teaching is, whereas there is widespread agreement about what good orthodonting is. Therefore, people are rationally willing to pay a lot for an actual agreed product, but they skimp on that which cannot be rationally decided on. Instead they (rationally) pick with a cheap pin. The more of a free market in teaching there is, the less this is true, but it would still remain somewhat true, I think, no matter how free the teaching market.
There is more widespread agreement about what a good school is (as opposed to "good teaching"), so people pay fortunes to buy, in fees or in mortgage payments to be in the right area. But this money doesn't find its way through to the mere teachers, on the whole, for the reasons stated above.
Nevertheless, a few free market teachers do already get paid a lot. I have already written here about Tony Buzan and Michel Thomas. They both get paid a lot. As do British TV's star history (two links here) teachers.
Well, most orthodontists (at least here in the U.S. -- I don't know how things work over in the land of the National Health) are private practioners. They are either in a solo private practice or they have banded together with other dental professionals into a partnership (and thus, presumably, enjoying savings in the costs because of being able to share receptionists, office equipment, computer software, etc.). Thus, there are some very successful orthodontists who enjoy high incomes, there is a vast middle group who are reasonably successful and reasonably prosperous, and there are some who are not very successful, either do to poor business practices, poor relationships with patients, lack of professional skill and competence, etc. and who either change occupations or scrape along earning less than other orthodontists, perhaps even in the income range of school teachers.
I suppose we could establish a system of public orthodonture. Yes, a system of Public Orhtodonture Facilities, paid for by taxes, where the public could come... No, wait, not just orthodonture... There woulld be Publich Oral Health Facilities, subdivided into General Dentistry and Orthodonture and Periodontal Care, etc. And dentists and periodontists and orthodontists and oral surgeons and dental hygenists would be salaried public employees. There would be a salary schedule, so much for starting pay, regular step increases with each year of service and additional amounts for those with additional academic training. There would be no merit pay raises or bonuses; everyone would be paid according to this schedule. It wouldn't matter if patients hated you or loved you, if you had a skilled but gental touch or if you inflicted pain regularly (and even, perhaps, came to relish inflicting a little pain now and then). If you had five years of service you got paid this amount; if you had eleven years of service you got paid that amount, and so on...
Would orthodontists then be well paid?
And don't forget the need for dental administrators. Each building would need a Principal Dentist (and an Assistant Principal Dentist and two Deputy Associate Principal Dentists) and a department of Oral Hygiene Counseling and a department of Dental Care Supplies Dispersement, etc. And, of course, the District Administrative Office and a Town Superintendent (and Deputy and Assistant and Associate Superintendents) and a State Division of Oral Hygiene Supervision and then a Federal Department of Dental Health (and, of course, vast new spending to enforce the new federal rules for the No Tooth Left Behind act).
So orthdontists would not be as well paid... Althoug the Dental Association (the oral health care union) local chapters would negotiate contracts that would get some good fringe benefits fo make up for the low salaries and the lack of merit-based pay, vacations, holidays, sick leave, personal leave, health insurance, and a good pension plan. The contract might cover working conditions, maximum number of patients to be seen per day, per week, etc.
There might not be enough orthodontists (nor enough general dentists and hygienists, etc.) so the government would have to encourage enrollment in new schools of dentistry. This might bring in plodding, unimaginative, uninspired drudges into the profession, people just putting in their time, collecting a steady paycheck, looking forward to retirement, enjoying their job security.
Okay, ignore the spelling errors... but let me fix my punctuation in that next to last paragraph lest it appear to say the opposite of what I meant: " local chapters would negotiate contracts that would get some good fringe benefits fo make up for the low salaries and the lack of merit-based pay -- vacations, holidays, sick leave, personal leave, health insurance, and a good pension plan. The contract might cover working conditions, maximum number of patients to be seen per day, per week, etc. " There... so that it is more obvious that the vacations and holidays, etc. are what is to make up for the lack of merit pay.

