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：越来越多的文献表明，在使用多个处理变量的两阶段最小二乘法（2SLS）中，若存在处理效应异质性和第一阶段交叉效应，其估计系数将难以解释。我们证明，在容量受限的分配体系中，这些交叉效应并非干扰因素，而是赋予政策参数清晰解释的根源。当处理变量受到配额限制，且工具变量与待研究政策作用于同一边际时，2SLS系数$β_k$等于将处理$k$扩展一个配额所产生的总社会效应，其中包括通过系统产生的所有级联再分配效应。该机制具有普遍性：只要固定供给制约着资源配置（无论是通过排序队列、等待名单还是市场出清价格），这一结论均成立。这一级联恒等式$\mathbf{T} = \mathbfβ$适用于任意第一阶段矩阵，可容纳任意形式的处理效应异质性，且仅需满足工具变量相关性和工具变量与政策作用于同一边际这两个条件。该结果可应用于大学招生、学校选择、住院医师匹配、公共住房及其他配额分配场景。我们以瑞典大学课程的抽签录取作为实证案例，以慈善捐赠行为作为结果变量。