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

We study identification and optimal policy design within principal-agent models. The modal empirical specification is equivalent to a potential-outcomes framework augmented with three assumptions: treatment-effect monotonicity, extrapolative structure, and rank invariance (RI), which restricts unobserved heterogeneity to be one-dimensional. This decomposition isolates identifying content of each assumption. The first two assumptions are interpretable and testable; RI is untestable and rules out economically plausible behaviors. We derive sharp counterfactual bounds when RI is relaxed. We estimate empirical bounds and apply them to optimal nonlinear pricing. The robust policy fully insures against worst-case latent selection while still enabling profit and consumer-surplus gains.