In 2001 Lew Rockwell penned an article, in which he mentioned the cost per student in public vs. private schools. At that time, according to his research, it was $6,000 vs. $3,100 with many public schools being in awful conditions, so it makes sense to speculate that the higher prices did not buy better quality. (Quality in a socialized industry? Say what?)
The numbers need to be updated: in 2004-05 the government schools cost $10,892 per student per year. The government does not publicize the relevant recent data on private schools for obvious reasons (e.g., their Characteristics of Private Schools in the United States: Results from the 2005-2006 Private School Universe Survey has no finance information); but I was able to obtaine a private schools tuition datasheet by request, and my calculations show that the average per student cost in 2003-04 was $5,410.
So, still government schooling costs at least twice as much as private education. I am not saying that this is the only reason to privatize all government-run schools, but it is surely a potent one.