There is no such thing as 'free education fully financed by the government'. A zero-tuition college education simply means that instead of the students bearing the cost of attaining their degrees the taxpayers bear it. Students and parents misperceive the price of education, considering it to be free, even though it comes out of their pockets in taxes.
And why should low-income taxpayers finance the education of wealthier students? Proponents of state-financed education argue that
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need to end the government subsidies so that college costs will be borne only by willing parties. We don't need to try to make college attendance universal; we need to allow people to choose for themselves the type and extent of education that best suits them. Making college education fully government-financed goes against the basic objective of higher education - to allow the intellectual cream of society to specialize in their chosen field of interest.