The paper studies the optimal education policy of a budget-constrained utilitarian government. Households differ in their income and in the intellectual ability of their children; income is observed by the government, but ability is private information. Households can choose to use private education, but cannot borrow to finance it. The results we obtain are striking. The optimal education policy is elitist: it increases the spread between the education achievement of the bright and the less bright children, compared to both private provision and the first-best policy. It is also such that the education received by a child depends positively on their parental income, unless they are bright. Finally, the optimal education policy is input regressive, in the sense of Arrow (1971): households with higher income and brighter children contribute less towards the cost of the education system than households with lower income and less bright children.