Private school market shares are rising steadily in many developing countries, but we have a limited understanding of how private schools set prices, how parents respond, and how this affects enrollment and performance in equilibrium. To shed light on demand behavior and supply response, I present a model of school pricing that incorporates an unusual feature of schooling compared to other goods - a potential preference by parents for small classes, and hence low school enrollment - that interacts with schools having market power. I show that, for a relatively broad range of parameter values, these two features can lead to the surprising result that an increase in aggregate household income, and hence an increase in willingness to pay for private schooling, can actually cause equilibrium private school enrollment to decrease. To investigate how private school enrollment responds to rising household income in practice, I exploit aggregate community-level income shocks in Chile, which has had a nationwide school voucher system since 1981. These shocks are caused by different responses to the price of copper in different municipalities. I show that private school prices rise by 0.9% in response to a shock that causes a 1% rise in income while private school enrollment falls by 2.0%. I find that falling private school enrollment is primarily caused by the middle-income students at the top schools. Those middle-income students induced to downgrade by rising prices do not experience the test score gains from the income shock experienced by students in the rest of the income distribution. I structurally estimate an extended version of the model and find that both market power and parental preferences for reduced class size are contributing to the observed declines in enrollment.