Designing Around Selection: Identification and Inference Under Multi-Dimensional Unobserved Heterogeneity

We study identification and optimal policy design in a broad class of principal-agent models. We show that the most common empirical framework within the literature is equivalent to an unstructured potential-outcomes model augmented with three specific assumptions: the Law of Demand (LoD), or treatment-effect monotonicity; extrapolative model structure (EMS), which rules out lumpy agent responses to price changes; and rank invariance (RI), which restricts unobserved heterogeneity (UH) to be one-dimensional. This decomposition isolates the identifying content of each assumption and clarifies its economic role. The LoD and MS are empirically testable using exogenous price variation; RI, on the other hand, is a strong assumption ruling out many economically plausible behaviors, and also not empirically testable. We derive sharp bounds on counterfactual outcomes when RI is relaxed. The conventional 1-dimensional model delivers an upper bound on planner objectives, while the lower bound, which allows for arbitrary multi-dimensional UH, has an adversarial interpretation for policy design. We estimate empirical bounds and apply them to nonlinear pricing of rideshare services. The resulting robust pricing policy fully insures against worst-case latent selection while preserving most of the profit and consumer-surplus gains predicted by the conventional model. Our framework provides a tractable approach to robust policy design in adverse-selection settings including Mirrleesian taxation, regulation, labor supply, and procurement.