This paper shows that the design of education policy involves a potential conflict between welfare and social mobility. We consider a setting in which social mobility is maximized under the least elitist public education system, whereas welfare maximization calls for the most elitist system. 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. The ranking between the welfare- and mobility-maximizing degrees of elitism may even be reversed. Utilitarian welfare is always higher when private supplementary education is available, but social mobility may be reduced.