Would you please refute this Libertarian argument about privatization of schools?

In handing out vouchers, you’re increasing the demand for more expensive schools. This is because more people are now able to afford more expensive education. Because the privatization of schools would render school prices subject to basic free market principles, as the demand increases so does the price. Basically schools would just increase their prices. If you hand out $10,000 vouchers, the least expensive education would become $10,000. The $10,000 vouchers would then just be a waste of money (for the wealthier tax payers and the government that then has to eat the opportunity cost ociated with distributing the vouchers). So now government is paying for absolutely nothing except the perpetuation of a system conducive to socio-economic inequity, and the average American would end up paying A LOT more for education then they do currently. You could institute a price ceiling, but then you’re using the same bureaucratic tactics you criticized earlier.