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In this paper, we provide a suite of tools for empirical market design, including optimal nonlinear pricing in intensive-margin consumer demand, as well as a broad class of related adverse selection models. Despite significant data limitations, we are able to derive informative bounds on demand under counterfactual price changes. These bounds arise because empirically plausible DGPs must respect the Law of Demand and the observed shift(s) in aggregate demand resulting from a known exogenous price change(s). These bounds facilitate robust policy prescriptions using rich, internal data sources similar to those available in many real-world applications. Our partial identification approach enables viable nonlinear pricing design while achieving robustness against worst-case deviations from baseline model assumptions. As a side benefit, our identification results also provide useful, novel insights into optimal experimental design for pricing RCTs.