We estimate the demand for health insurance in the California Affordable Care Act marketplace (Covered California) without using parametric assumptions about the unobserved components of utility. To do this, we develop a computational method for constructing sharp identified sets in a nonparametric discrete choice model. The model allows for endogeneity in prices (premiums) and for the use of instrumental variables to address this endogeneity. We use the method to estimate bounds on the effects of changing premium subsidies on coverage choices, consumer surplus, and government spending. We find that a $10 decrease in monthly premium subsidies would cause between a 1.6% and 7.0% decline in the proportion of low-income adults with coverage. The reduction in total annual consumer surplus would be between $63 and $78 million, while the savings in yearly subsidy outlays would be between $238 and $604 million. Comparable logit models yield price sensitivity estimates towards the lower end of the bounds.