The response to your "let their salaries increase" pointed out that there are districts (and counties and states) where this simply does not and can not happen given how government policies are structured. Much of that the result of quite deliberate "starve-the-beast" anti-tax policies, but also frequently a simply quite inadequate economic base.
In the deep south of the US, say, the state of Mississippi, you have a de-facto segregated school system. Whites, even poor whites, attend private schools, largely religious, known as "Segregation Academies". Public schools are almost wholly the domain of blacks.
http://www.theatlantic.com/national/archive/2012/12/in-south...
https://en.wikipedia.org/wiki/Educational_segregation_in_Sun...
https://en.wikipedia.org/wiki/Segregation_academy
Markets don't exist independent of power constructs, a fact explicitly recognised by Adam Smith: "Wealth, as Mr Hobbes says, is power." Much of Smith's Wealth of Nations is about those dynamics.
This thread reads as if two ideological agendas are being pushed on the other, rather than some seeking of greater truth. I'm finding your line of argument the less pesuasive and less sincere.
Look, teacher pay is in control of the government, not the market. And nurse pay and nursing demand has already had demonstrable impact on students' selection of what to study. (You can observe this by looking at the real world.)
Is there a particular hypothesis that you're arguing against? Is it that increasing pay won't attract students to the line of study? Do you disagree with the equation (Teaching Budget) = (Teacher Pay) * (Number Of Teachers) in a material way?
In fact I ... wait a second.
Me: > I proposed increasing their pay,
You: > No, you didn't.
Me (upthread): > let their salaries increase.
Well now, looks like I did. (You can observe this by looking at the real world.)