The Cascade Identity: 2SLS as a Policy Parameter in Capacity-Constrained Settings

A growing literature shows that two-stage least squares (2SLS) with multiple treatments yields coefficients that are difficult to interpret under heterogeneous treatment effects and cross-effects in the first stage. We show that in capacity-constrained allocation systems, these cross-effects are not a nuisance but the source of a clean policy interpretation. When treatments are rationed and the instrument operates on the same margin as the policy of interest, the 2SLS coefficient $β_k$ equals the total societal effect of expanding treatment $k$ by one slot, including all cascading reallocations through the system. The mechanism is general: it applies whenever fixed supply constrains allocation, whether through ranked queues, waitlists, or market-clearing prices. This cascade identity $\mathbf{T} = \mathbfβ$ holds for any first-stage matrix, under arbitrary treatment effect heterogeneity, and requires only instrument relevance and that the instrument operates on the same margin as the policy. The result applies to university admissions, school choice, medical residency matching, public housing, and other rationed allocation settings. We provide an empirical application using lottery-based admission to Swedish university programs and charitable giving as the outcome.