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.

翻译：日益增长的文献表明，在存在异质性处理效应及第一阶段交叉效应的情况下，采用双重工具变量法（2SLS）处理多种干预措施时所得的系数难以解释。本文证明，在容量约束型分配系统中，这些交叉效应并非干扰因素，而是构成清晰政策解释的根源。当干预措施受到配给且工具变量与待研究政策作用于相同边际时，2SLS系数$β_k$等于将干预措施k扩展一个名额所产生的社会总效应——包含经由系统的所有级联再分配过程。该机制具有普适性：只要固定供给对分配形成约束（无论是通过排序队列、等待名单还是市场出清价格），均可适用。此级联恒等式$\mathbf{T} = \mathbfβ$对任意一阶段矩阵均成立，可应对任意处理效应异质性，仅需满足工具变量相关性和作用于与政策相同边际这两个条件。该结果适用于大学招生、择校、住院医师匹配、公共住房及其他配给型分配场景。我们基于瑞典大学项目的抽签录取制度，以慈善捐赠作为结果变量进行了实证应用分析。