Using a household production model of educational choices, we characterise a free-market situation in which some agents (high wagers) fully educate their children and spend a sizable amount of resources on them, while others (low wagers) educate them only partially. The free-market equilibrium is iniquitous, both because the households have different resources and because the children have different access to education. Public policy is thus called for, for vertical as well as horizontal equity purposes. Conventional wisdom has it that both objectives could be achieved using price control instruments, i.e. income taxes and price subsidies. We find instead that income taxes reduce equality of opportunity and that price subsidies cannot remedy this. Quantity controls become necessary: a compulsory education package, financed by a redistributive tax system, achieves both types of equity. Redistributive taxation and compulsory education are therefore best seen as complementary policies.