Why do teachers at private schools in the United States get paid less than teachers at public schools?
In aggregate, this is likely true. However, there will be plenty of exceptions to the rule. In general, tuition-based schools will have more budgetary constraints than larger school districts and the employees are less likely to have union representation. My first job offer when seeking work as a teacher was from a small religious school in an impoverished inner-city neighborhood. Had I taken this position, I would have been the entire department for my subject area and responsible for teaching the religion classes. On the one hand, I would be able to pretty much do my own thing. On the other hand, I would probably still be living with my parents and still saving up to get married, buy a house, have children etc. now 15 years into my career. As you may guess, these types of jobs have a wild turnover rate and are often filled by teachers either looking for a starter job or who wish to supplement their pensions in retirement. It is also important to consider the working conditions of teachers in private schools. Granted, resources are always scarce in our profession; however, the classroom dynamic can be vastly different. Students in private school classrooms are either receiving a scholarship or have parents paying tuition. The solution to most serious discipline problems is "Thank you for your tuition money but you aren't a very good fit here." I have been the recipient of many ex-private and charter school students in my public school classrooms over the years. They are usually fun students with which to work; but have difficulty in overly strict environments. Because of the tuition at private schools, the parents of those students are also much more invested, which allows for greater homework expectations and opportunities for fundraising. The vast majority of private schools, and to an extent charter schools, are also selective in nature. Many also have admissions exams. Such rigorous admission standards coupled with increased parental involvement creates a very different, and often less stressful, classroom dynamic and teaching experience. Educators in general are not subject to traditional free-market rules due to the nature of their profession as a public service. Most would agree, that were teachers truly subject to the free market, that society would not be able to afford the best of us comparing our advanced degrees, certifications and responsibilities beyond the classroom to the earnings of other certified professionals. Unfortunately, the earnings of teachers across the profession are artificially low. Those lacking the leverage of union representation are then subject to even lower salary offers.
Teachers at a private school have salaries that are paid by the tuition of the students who attend. Teachers at a public school have salaries that are paid by taxes, because society has recognized that we all benefit from well-educated citizens. A little money from many people creates a lot more revenue than a lot of money from a few people.
Teachers' salaries in the public schools began to rise with the formation of the teachers' unions and the collective bargaining process. Very few private schools are unionized - there is no collective bargaining process. Some teachers are willing to accept the lower salaries paid by private schools as they prefer the environment of a private school to a public school.
The major reason is that in the United States the government and tax revenue can only support public schools. In most of the rest of the world that's not the case, so the U.S. can seem a bit strange.
The result is that private schools are supported only by tuition dollars paid by families, supplemented by voluntary donations from supporters. Since the large majority of school expenses are salaries, even the wealthiest private schools often pay their teachers less than typical public schools, and all private schools have fewer administrators and support personnel, requiring teachers to do more.
Public school teacher salaries also benefit from unionization, allowing the collective bargaining power of the teachers to pressure communities to raise taxes to pay teachers more. By contrast, unions have rarely been successful in private schools, in part because it's hard for teachers to demand that the children pay more, and in part because private schools tend to be more tight-knit communities.