I believe private schools on average pay their teachers less than do public schools. However, private schools do better academically than their public school counterparts. More money for better education is just so much BS and the proof is in the pudding.
Take a look at socio-economic status of students and student teacher ratio, these two are prime indicators in any school be it private or public.
The issue of more money for teachers is to attract more teachers. I think the certification rate of teachers in the classroom was around 30% just a few years ago. There are so many students and not enough teachers that most states are forced to use teachers aides and substitutes as teachers. Right now, the limit is set to 30 students per teacher mainly because there is not enough teachers to do the work.
Now, moron got into office and pushed for NCLB, which basically put an enormous amount of paper work on teachers and holds them responsible for learning growth(even in SPED students) up to 100% proficiency by 2013. Now, who would want to do this job for 45K a year? It is very easy to post data about a teacher making $36 an hour to argue a point, but in reality the money does not equal the responsibility. In many other "professions" a manager with this many people under him and this type of responsibility would be making 6 figures.
If you want better education, you need to pay for the teachers. Many are the first to blame teachers, yet the last to want to pay for them. Teachers have been going "beyond the call of duty" for so long that most are accustomed to that and expect that. Yet if you ask these people will they work 10 hours overtime a week without pay, and be responsible for the growth of 150 people a week, they will tell you where to stuff it.