We show that, when school quality is measured by the educational standard and attaining the standard requires costly effort, secondary education needs not be a hierarchy with private schools offering better quality than public schools, as in Epple and Romano, 1998. An alternative configuration, with public schools offering a higher educational standard than private schools, is also possible, in spite of the fact that tuition levied by private schools is strictly positive. In our model, private schools can offer a lower educational standard at a positive price because they attract students with a relatively high cost of effort, who would find the high standards of the public school excessively demanding. With the key parameters calibrated on the available micro-econometric evidence from the US, our model predicts that majority voting in the US supports a system with high quality private schools and low quality public schools, as assumed by Epple and Romano, 1998. This system, however, is not the one that would be selected by the social planner, who prefers high quality public schools combined with low quality private schools.