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

Governments routinely adjust capacity in rationed programs such as university fields, medical training and public housing, where admitting one individual displaces others and triggers chains of reallocation. We show that in such settings, the standard multi-treatment two-stage least squares (2SLS) coefficient identifies exactly the total societal effect of a marginal expansion, including all downstream reallocations. The result is an algebraic identity: under instrument relevance and a single alignment condition, satisfied in centralized admissions systems, the 2SLS coefficient equals the general-equilibrium shadow value of relaxing a capacity constraint, while the single-instrument Wald ratio captures only the direct effect. Their difference recovers the full equilibrium adjustment without additional structure. Monotonicity is not required. The identity extends beyond queue-based allocation to any fixed-supply setting, including competitive markets with price instruments. We apply the framework to two policy questions in Swedish university admissions, where marginal students are allocated across fields through a centralized lottery mechanism. First, revisiting the debate on whether economics and business education erodes prosocial values, we find that the direct effect of expanding business on charitable giving is precisely zero, but expanding the less competitive fields that business students are displaced from has large prosocial effects. Second, analyzing gender-targeted STEM policies, we find that admitting four women to competitive STEM generates one additional male STEM degree through downstream vacancies. Both are general-equilibrium effects invisible to single-instrument methods.