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

Niklas Bengtsson, Per Engström

Abstract

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.

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

Abstract

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 equals the total societal effect of expanding treatment 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 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.
Paper Structure (40 sections, 2 theorems, 43 equations, 2 figures, 5 tables)

This paper contains 40 sections, 2 theorems, 43 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Model and Assumptions
  3. Setup
  4. Capacity-constrained allocation
  5. The Cascade Identity
  6. The policy experiment
  7. The cascade equation
  8. Main result
  9. Homogeneous treatment effects: the cascade as a zero-sum game
  10. Example using potential outcomes
  11. Solving the moment conditions.
  12. Expressing the coefficients in terms of complier types.
  13. Discussion
  14. Relation to bhuller2024
  15. Relation to kirkeboen2016
  16. ...and 25 more sections

Key Result

Proposition 1

Under Assumptions 1 and 2, the 2SLS coefficients equal the societal policy effects: That is, $\beta_k$ equals the total societal effect of expanding program $k$ by one slot, including all cascading reallocations through the system.

Figures (2)

  • Figure 1: First-stage matrix. Effect of the instrument (lottery admission) on being admitted to a specific field. Rows indicate fields admitted to ($D_k$) and columns indicate field applied to. Red = positive coefficient, blue = negative. Faded color = not significant ($|t|<1.96$). $t$-values in parentheses. 149579 observations, t-statistics computed using clustered standard errors on 11,604 pivotal groups.
  • Figure 2: First-stage matrix for more granular field definitions. Each cell depicts the sign and strength of the lottery instrument on being admitted to each program (first-stage coefficients). Red = positive, blue = negative. Faded = not significant ($|t|<1.96$).

Theorems & Definitions (6)

  • Proposition 1: Cascade Identity
  • proof
  • Remark 1: Intuition
  • Remark 2: Generality beyond queues
  • Remark 3: What heterogeneity does
  • Proposition 2: General Cascade Identity