This paper examines the degree of elitism of public education under two different social objectives. It illustrates a potential conflict between welfare and social mobility. In the absence of private supplementary education, utilitarian welfare increases with the degree of elitism of the public education system. On the other hand, elitism decreases the steady state proportion of heterogenous dynasties (those comprised of a skilled parent and an unskilled child, or vice versa) which is our measure of social mobility. Consequently, social mobility is maximized under the least elitist public education system. We then open up the possibility for skilled parents to invest in private supplementary education for their child. We show that when private education is available, the degree of elitism that maximizes social mobility increases, while the welfare-maximizing degree of elitism decreases, provided that the inequality in productivity between the two types of agents is large enough. We provide a numerical example where the ranking between the welfare-and mobility-maximizing degree of elitism is reversed when private education is allowed — i.e., where the public education system that maximizes social mobility is more elitist than the one that maximizes welfare. Finally, we show that utilitarian welfare is always (weakly) higher when private supplementary education is available. However, to maximize social mobility it may be preferable to ban private supplements.