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The Cascade Identity: 2SLS as a Policy Parameter in Capacity-Constrained Settings
\noindentJEL codes: C36, D61, I23, I26, I28, D64 \\ Keywords: Instrumental variables, multiple treatments, heterogeneous treatment effects, capacity constraints, cascade effects, university admissions, charitable giving, gender quotas
Governments routinely adjust capacity in programs such as university fields of study, medical training, and public services. In such settings, admitting one additional individual typically displaces someone from another program, whose vacated slot is then filled from its own queue, triggering a chain of reallocations across the system. The policy-relevant effect of such an expansion is therefore not just the effect on the marginal entrant, but the total effect of this reallocation chain.
We show that a standard econometric object---the multi-treatment two-stage least squares (2SLS) coefficient---identifies exactly this total effect. The result is an algebraic identity: under instrument relevance and the condition that the instrument and the policy operate on the same allocation margin, the 2SLS coefficient equals the general-equilibrium shadow value of relaxing a capacity constraint, including all downstream reallocations through the system. No monotonicity condition is required, no restriction on the pattern of cross-program substitution is imposed, and the identity extends beyond queue-based allocation to any fixed-supply setting, including competitive markets with price instruments.
The margin-alignment condition holds by construction in centralized admissions systems, where capacity expansions shift the admission cutoff and the instrumental variable strategy exploits local randomness around that same threshold (e.g., ockert2010, kirkeboen2016, abdulkadiroglu2017, altmejd2021, bleemer2022), and more generally in any rationed setting where slots are filled from queues, including public housing lotteries jacob2012, military draft lotteries angrist1990, and oversubscribed public programs kline2016, gelber2016effects.
The mechanism is straightforward. Expanding a program draws individuals from other programs, which in turn refill their vacancies from their own queues, propagating the shock through the system. The first-stage matrix already encodes these reallocations---how shocks to one treatment shift enrollment across all others---and 2SLS aggregates them into their total outcome effect by inverting this matrix. The off-diagonal elements, often viewed as a complication for individual causal interpretation, are precisely the vacancy-creation rates through which the policy operates.
The result provides a different way of interpreting multi-treatment IV. With multiple treatments and heterogeneous effects, 2SLS coefficients are typically viewed as weighted averages of individual treatment effects, where the weights may be negative and depend on complex substitution patterns, making the estimand difficult to interpret mogstad2024handbook. The standard response has been to impose additional structure on choice behavior, such as latent index models heckman2008multiple, observed next-best alternatives kirkeboen2016, additive separability heckman2018, or marginal treatment response approaches mogstad2024policy. More recently, bhuller2024 show that positive weighting requires ruling out cross-treatment spillovers.
We instead study environments in which such spillovers are intrinsic: capacity-constrained systems in which individuals are allocated across competing programs. In such settings, the relevant object for welfare analysis is the total societal effect of a marginal capacity expansion, and we show that this is what the 2SLS coefficient identifies. It is not an average of individual treatment effects but a general-equilibrium policy parameter: the marginal societal value of expanding a treatment in a capacity-constrained system. Under homogeneous effects, the downstream cascade is inert and this parameter reduces to the standard individual treatment effect; under heterogeneity, they diverge.
haavelmo1943 showed that recovering the effect of a policy intervention requires inverting the full simultaneous system, not reading off the reduced form. The modern literature has developed this theme in capacity-constrained settings specifically. kline2016 show that in Head Start, the policy-relevant effect of expanding a program includes the downstream reallocations it triggers through competing alternatives: an equilibrium object that the standard Wald ratio misses and that recovering requires a structural model of the allocation mechanism. gandil2025trickle and arkhangelsky2025 develop methods to recover such equilibrium effects; the former by simulating counterfactual equilibria through a re-engineered version of the Danish admissions mechanism, the latter by constructing an adjusted outcome that appends each agent's equilibrium externality, estimated from LATEs at admission cutoffs. Both approaches require the researcher to build new objects beyond standard regression output.
Our result complements these contributions by showing that in any fixed-supply system, the standard multi-treatment 2SLS coefficient already is the equilibrium-adjusted estimand; no transformation of the outcome, no simulation of the mechanism, and no assumptions on individual choice behavior are needed. The result connects to carneiro2010, who show that marginal policy effects can be identified under weaker conditions than average effects, without extrapolating beyond the support of the instrument. The cascade identity is the multi-treatment analogue of this result in capacity-constrained systems, where the relevant margin is defined by the admission cutoff and the policy effect propagates through competing programs rather than operating on a single treatment. A researcher needs only the 2SLS coefficient $\beta_k$ and the marginal cost of capacity to conduct cost-benefit analysis. The Wald ratio $W_k$ provides the partial-equilibrium benchmark; the cascade correction $\beta_k - W_k$ is the general-equilibrium adjustment. Both ingredients are standard regression outputs.
We illustrate the framework with two applications using Swedish university admissions, where marginal applicants are allocated across fields through a centralized lottery mechanism. The first revisits the long-standing question of whether economics and business education erodes prosocial values. We find that the direct effect of expanding business on charitable giving is precisely zero: the marginal business student gives no more or less to charity than she would have in her next-best alternative. But the total policy effect is positive and significant, driven entirely by the cascade: opening a business seat draws a student away from a field that would have increased her giving, and refilling that vacancy draws someone in from outside higher education. The apparent prosociality gap between business students and others is not caused by business education; it reflects the prosocial effects of the fields that business students would otherwise have attended.
The second application analyzes gender-targeted STEM policies. Quotas and targeted expansions are among the most common instruments for increasing female representation in competitive fields, yet their system-level consequences are difficult to evaluate: admitting a woman to a competitive program displaces someone from a fallback program, and the vacancy left behind is filled from a mixed-gender queue. The cascade framework is informative about exactly this type of intervention, since it traces the full chain of reallocations triggered when the composition of the marginal entrant changes. We find that admitting one additional woman to competitive STEM generates 0.25 STEM degrees, of which roughly one-fifth accrues to men: the woman vacates a mid-tier STEM slot that is refilled from a predominantly male queue. A replacement policy---admitting a woman in place of the marginal man---redistributes STEM degrees across genders but barely changes the total. Both effects are invisible to single-instrument methods and emerge only through the cascade.
The remainder of the paper proceeds as follows. Section (ref) presents the policy parameter, econometric specification, and assumptions, and derives the cascade identity. Section (ref) discusses the assumptions and their scope. Section (ref) examines heterogeneity and replacement policies. The following sections apply the framework to field-of-study choice and prosocial behavior (Section (ref)) and to gender-targeted STEM policies (Section (ref)). Section (ref) concludes.
There are $K$ programs, indexed by $j \in \{1,\dots,K\}$, and an outside option $j = 0$. Each program has fixed and binding capacity. Individual $i$ assigned to alternative $j$ generates a specific societal value. A planner considering whether to expand program $k$ by one slot needs the total expected societal effect of this expansion, including all downstream reallocations. Denote this object $T_k$.
The marginal entrant to program $k$ generates a direct benefit $W_k$ in expectation. But she may have come from another program $j$, creating a vacancy there. When program $j$ refills that vacancy from its ranking list, the new entrant to $j$ may herself have come from program $m$, creating a further vacancy. The cascade continues until a slot is filled from the outside option.
Let $r_{kj}$ denote the vacancy creation rate: the number of vacancies created in program $j$ per new admission to program $k$. Then $T_k$ satisfies the recursion
Each reallocation in program $j$ carries societal value $T_j$---the same object we are defining---so the equation is a fixed-point condition: the value of expanding program $k$ equals the direct effect on the marginal entrant plus the value of all induced reallocations, each of which is itself an equilibrium object defined by the same primitives.
Equation (ref) is not an identifying assumption. It is a consequence of three primitives: (i) the definition of $T_k$ as the total societal effect of a one-slot expansion, (ii) fixed total supply of every program other than $k$, and (iii) the allocation mechanism fills vacancies from ranked queues.
It is useful to express ((ref)) in matrix form. Collect the vacancy creation rates into a $K \times K$ matrix $M$ with entries $m_{kj} = r_{kj}$ and zeros on the diagonal. Let $\mathbf{W} = (W_1, \dots, W_K)^T$ denote the vector of direct effects. The cascade recursion becomes
with solution $\mathbf{T} = (I - M)^{-1}\,\mathbf{W}$ whenever the spectral radius of $M$ is less than one.
The Neumann series expansion
has a direct economic interpretation. Each power of $M$ corresponds to one round of the cascade:
This is Walrasian t\^atonnement: a supply shock propagates through successive rounds of reallocation until all vacancies are filled. Convergence of the series, given spectral radius of $M$ less than one, is stability of the equilibrium. Equation (ref) computes the infinite t\^atonnement in closed form. Appendix (ref) works through an instructive two-program case in detail.
The policy parameter $T_k$ is therefore the general-equilibrium shadow value of relaxing the capacity constraint in program $k$. The direct effect $W_k$ is the partial-equilibrium shadow value; the cascade correction $T_k - W_k$ is the general-equilibrium adjustment. The question is whether standard econometric tools can recover $T_k$.
Consider the linear model
estimated by two-stage least squares using instruments $(Z_{i1},\dots,Z_{iK})$. $A_{ij}$ indicates admission of individual $i$ to programme $j$. Predetermined covariates and fixed effects are suppressed; all results extend immediately.
Define the first-stage matrix $\Pi$ with entries
so that $\pi_{jk}$ measures how instrument $k$ shifts enrollment in program $j$.
The reduced form satisfies
or in matrix form
The 2SLS estimand is $\boldsymbol{\beta} = (\Pi^T)^{-1}\mathbf{RF}$: the reduced form, passed through the inverse of the full first-stage matrix.
We require two conditions linking the econometric objects to the planner's problem.
Assumption (ref) is an institutional condition satisfied by construction in centralized admission systems (Sweden, Norway, Denmark, Chile), where both the instrument and capacity expansions shift the cutoff along the same ranked queue. It applies equally to school choice mechanisms with waitlists and other settings where programs fill to capacity from ranked lists. In less centralized settings, the condition need not hold by construction but remains a plausible interpretation of the instrument whenever the researcher treats the IV estimate as informative about capacity policy.
Assumption (ref) is thus not a new requirement imposed by the cascade framework. It is the same condition that makes any LATE estimate in these settings interpretable as a policy-relevant parameter. When kirkeboen2016 argue that their estimates are “informative about policy that (marginally) changes the supply of slots in different fields,” they are asserting precisely that the instrument and the policy operate on the same margin, i.e. that the person who would be admitted under a marginal expansion is the same type of person whose admission is shifted by the instrument. Without this alignment, the LATE identifies the effect on lottery compliers but says nothing about the effect of expanding capacity, because the marginal expansion might admit a different person through a different channel. Any researcher who interprets a lottery- or cutoff-based IV estimate as informative about capacity policy is implicitly invoking Assumption (ref). We merely formalize what the literature already assumes.
A prominent view in applied microeconomics holds that the reduced form is more policy-relevant than the IV coefficient, because it captures the effect of the instrument itself and requires fewer assumptions angristpischke2009, imbens2014. Assumption (ref) accepts the premise of this argument, i.e. that the instrument is a relevant policy, but overturns the conclusion. When the policy is a capacity expansion that propagates through the system, the reduced form captures only the effect of one lottery draw on the lottery winner: a partial-equilibrium object that ignores all downstream reallocations. It is the 2SLS coefficient---obtained by inverting the full first-stage matrix---that traces the cascade and recovers the general-equilibrium policy effect. The distinction between the reduced form as a prediction tool and the structural equation as a policy tool dates to haavelmo1943: when the government intervenes by changing a constraint, the structural parameters govern the response---not the conditional expectations from single equations.
The key implication of Assumption (ref) is that the instrument-based first stage identifies the vacancy creation rates and direct effects from the planner's problem:
The logic is simple. Per unit increase in $Z_k$, program $k$ gains $\pi_{kk}$ students and program $j$ loses $|\pi_{jk}|$ students. Rescaling to one new admission to $k$, the number of vacancies created in $j$ is $-\pi_{jk}/\pi_{kk}$. Similarly, the reduced form $\mathrm{RF}_k$ is the total outcome change per unit increase in $Z_k$, so the outcome change per new admission---the direct effect---is the Wald ratio $\mathrm{RF}_k / \pi_{kk}$. These are the vacancy creation rates and direct effects that enter the cascade recursion of Section (ref); Assumption (ref) ensures they are identified by the instrument variation. The same marginal responses that govern the effect of the instruments also govern the effect of a marginal capacity expansion. This bridges the policy parameter defined in Section (ref) and the econometric objects defined in Section (ref).
Assumption (ref) does impose a stability requirement: the ranking list that governs who fills a vacancy is not itself affected by the expansion. For a marginal change, this is innocuous -- no applicant or program responds to a single-slot shift. For larger reforms, however, it rules out endogenous responses on both sides of the market: applicants must not adjust their application or ranking decisions, and programs must not alter admission criteria or capacity in other dimensions. gandil2025trickle illustrates the importance of this condition: when applicants are allowed to adjust their application lists, simulated ripple effects increase substantially. The cascade identity therefore characterizes the marginal expansion at current capacity levels; applying it to non-marginal reforms requires the first-stage matrix to remain stable, a substantively stronger assumption.
The framework imposes little additional structure beyond these two assumptions. Treatment indicators need not be mutually exclusive: a student can be “ever admitted” to multiple programs, and the cascade operates on net enrollment flows. No monotonicity condition is required. Cross-effects are allowed: $\pi_{jk} \neq 0$ for $j \neq k$, as they generically are in capacity-constrained systems. Flows need not balance: $\sum_j \pi_{jk} \neq 0$, so instruments may draw individuals both from other programs and from the outside option. Indeed, at least some instruments must draw entrants from the outside option: if all column sums $\sum_j \pi_{jk}$ were zero, the first-stage matrix would be singular and the 2SLS coefficient would not exist.
The cascade recursion (ref) is derived from the primitives fixed supply and queue-based reallocation. It is not assumed to match the econometric objects. To verify that the identity is not an artifact of the algebraic setup, Appendix (ref) computes $T_k$ independently by brute force: expanding a program by one slot in a fully simulated multi-round admissions system with retakes, programme switching, new cohorts, noncompliance, and defiers. It then sums up the resulting outcome changes across all individuals. The 2SLS coefficient $\hat{\beta}_k$, estimated on the same simulated data, agrees with this experimentally computed $T_k$ to within sampling error (paired $t$-statistics of 1.35 and 0.60 across 1{,}000 replications).
Under homogeneous treatment effects, every individual gains $\Delta_j = Y(j) - Y(0)$ from enrolling in program $j$, regardless of type. Since the average treatment effect and the marginal policy effect coincide when effects are constant, the homogeneous case provides a useful benchmark for interpreting the cascade identity.
Consider expanding program $k$ by one slot. The new entrant may have come from another program $j$. Her net gain is $\Delta_k - \Delta_j$: she gains $\Delta_k$ from entering $k$ but loses $\Delta_j$ from leaving $j$. Her vacated seat in $j$ is filled by the next person on $j$'s ranking list, who may in turn have come from program $m$, gaining $\Delta_j - \Delta_m$. This person's departure from $m$ is again filled from $m$'s ranking list, and so on. The cascade terminates when a seat is filled by someone from the outside option, who gains $\Delta_\ell$ for whatever program $\ell$ she enters.
Under homogeneity, the intermediate terms telescope. Summing along the chain: \[ (\Delta_k - \Delta_j) + (\Delta_j - \Delta_m) + \cdots + \Delta_\ell = \Delta_k. \] Every intermediate program cancels: it appears once as a gain (for the person entering) and once as a loss (for the person leaving). The only term that survives is $\Delta_k$, the effect at the origin of the chain. The cascade is a zero-sum reallocation at every intermediate step, and the sole net effect is that one additional person, wherever she sits at the end of the chain, has been drawn into the system through program $k$'s expansion.\footnote{When treatment indicators are not mutually exclusive, the terminal entrant in the cascade need not come from the outside option; she may already be enrolled in other programs. However, the second-stage equation is linear and additive in the treatment indicators, so under homogeneous effects the incremental gain from adding program $k$ is $\Delta_k$ regardless of which other programs the individual is already enrolled in. The telescoping argument goes through unchanged.} Hence $\beta_k = \Delta_k = Y(k) - Y(0)$, the standard textbook interpretation.
This makes precise what heterogeneous treatment effects add. When effects differ across individuals, the person who enters program $j$ at one link of the cascade may gain more or less than the person who left $j$ at the previous link. The intermediate terms no longer cancel. The 2SLS coefficient $\beta_k$ captures the full chain of non-cancelling gains and losses. The cascade identity $\mathbf{T} = \boldsymbol{\beta}$ says that 2SLS performs this accounting automatically: it aggregates the direct effect and all downstream spillovers into a single policy-relevant number.
The distinction between the Wald ratio $W_k$ and the 2SLS coefficient $\beta_k$ is most naturally understood as a distinction between partial and general equilibrium effects. The Wald ratio is the effect on the lottery winner; the cascade correction $\beta_k - W_k$ is the effect on everyone else. In this sense, $\beta_k$ is the general-equilibrium shadow value of relaxing the capacity constraint in programme $k$, while $W_k$ is the partial-equilibrium shadow value.
A demand-system analogy makes this transparent. Suppose $K$ goods are supplied in fixed quantities and prices $p = (p_1,\dots,p_K)$ clear the markets. The first-stage matrix $\Pi$ corresponds to the matrix of demand derivatives, with entries $\pi_{jk} = \partial E[Q_j]/\partial p_k$. See Appendix (ref) for details.
A marginal increase in the supply of good $k$ requires a reduction in its price $p_k$ to clear the market. The welfare gain from this price change, holding all other prices fixed, is exactly the Wald ratio $W_k$. This is a partial-equilibrium object: it ignores that the change in $p_k$ shifts demand in all other markets through the cross-price effects $\pi_{jk}$.
These cross-effects require further price adjustments in every other market. Each adjustment has its own welfare effect and feeds back into the system through additional cross-price responses. The economy converges to a new equilibrium through a sequence of such adjustments.
This is Walrasian t\^atonnement in its standard market economy form. The cascade identity $\mathbf{T} = \boldsymbol{\beta}$ then says: in a fixed-supply economy, the 2SLS coefficient is the general-equilibrium shadow value of relaxing the supply constraint. The Wald ratio is the partial-equilibrium shadow value; the cascade correction is the GE adjustment.
The cascade identity targets a general-equilibrium policy effect, whereas the existing literature on multi-treatment IV targets partial-equilibrium individual treatment effects. The distinction maps directly into the PE/GE interpretation of Section (ref).
kirkeboen2016 and heinesen2024instrumental identify local average treatment effects by conditioning on each individual's next-best alternative and imposing an irrelevance condition: if instrument $k$ does not shift an individual into treatment $k$, it does not shift her into any other treatment either. Together, these assumptions imply that within each next-best stratum, the off-diagonal elements $\pi_{jk}$ are zero.
With no cross-program spillovers, the policy expansion does not propagate the shock beyond programme $k$, and the distinction between partial and general equilibrium disappears. In this case, the Wald ratio and the 2SLS coefficient coincide, \[ \beta_k = W_k, \] and both recover the same partial-equilibrium effect.
The cascade identity instead allows for non-zero cross-terms and explicitly incorporates the resulting spillovers. The off-diagonal elements that kirkeboen2016 and heinesen2024instrumental eliminate are exactly the cross-program flows that transmit the shock through the system. Their approach shuts down general-equilibrium adjustment; ours traces it out by inverting the full matrix $\Pi$.
The two approaches therefore answer different questions. For individual-level pairwise LATEs, the irrelevance and next-best assumptions are required and our result offers no shortcut. For the system-level effect of a capacity expansion, these assumptions are not needed: the first-stage matrix already contains the information required to recover the general-equilibrium response.
The cascade interpretation gives the 2SLS coefficient a direct welfare meaning. The marginal value of public funds for expanding program $k$ is
where $c_k$ is the marginal cost of a slot. The numerator $\beta_k$ is the 2SLS coefficient from the main specification. This object is the multi-treatment analog of the marginal policy effect in carneiro2010 and connects directly to the MVPF framework of hendren2020: the welfare consequence of an infinitesimal expansion of program $k$, expressed per unit cost, identified from local variation at the admission margin without extrapolation across the full distribution of treatment effects. In the spirit of chetty2009, the 2SLS coefficient serves as a sufficient statistic for welfare analysis in capacity-constrained systems; the equilibrium adjustment is already embedded in the estimator, requiring no additional structural modeling.
This contrasts with approaches such as kline2016 and gandil2025trickle, where the Wald ratio identifies only the direct effect on the lottery winner, and the analyst must model or simulate the downstream re-allocations, and their fiscal consequences, to recover the policy-relevant effect. In the cascade framework, these equilibrium adjustments are already embedded in the 2SLS coefficient. The researcher needs only the marginal cost of capacity to conduct cost-benefit analysis; no additional structural modeling of reassignment flows or costs in other programs is required.
The result highlights a key advantage of the cascade interpretation: while individual-level LATE parameters are often difficult to map into policy-relevant welfare objects, the multi-treatment 2SLS coefficient directly identifies the marginal policy effect needed for cost-benefit analysis in capacity-constrained systems.
A natural question in any treatment evaluation is whether effects differ across subgroups, for example, by gender, age, or family background. In the standard LATE framework with a single uncapped treatment, the approach is straightforward: estimate the model separately on each subgroup. The resulting coefficient identifies the LATE for compliers within that subgroup. The cascade framework can address the analogous question: what is the societal effect of a group-targeted expansion, such as reserving an additional slot for women? But the system-level nature of the estimand requires a decomposition that accounts for how the cascade propagates through mixed-group queues.
Consider estimating $\beta_k$ on women only, i.e. dropping all men from the sample and running 2SLS with the female subsample. The resulting coefficient $\beta_k^f$ inverts the women-only first-stage matrix $\Pi^f$ and uses the women-only reduced form. By the cascade identity applied to this subsample, $\beta_k^f$ equals the societal effect of expanding programme $k$ by one women-only slot, where the entire downstream cascade also operates exclusively through women.
This is a fictional policy. The system does not have a gender. In the real admission system, when a woman vacates a slot in programme $j$, the next person on $j$'s ranking list may be a man. The cascade is gender-blind from the second step onwards. The women-only estimate imposes a single-gender system that does not exist, producing a parameter that does not correspond to any implementable policy.
Two well-defined heterogeneity questions can be asked instead. We describe each in turn.
The first question is: when programme $k$ expands by one slot, how much of the total societal effect accrues to women versus men?
This is answered by running full-sample 2SLS with group-specific outcomes as dependent variables. Define $Y_i^f = f_i \cdot Y_i$ and $Y_i^m = (1-f_i) \cdot Y_i$, where $f_i$ is a female indicator. Estimate
by 2SLS on the full sample, using the same instruments and endogenous variables as the main specification. Define $\beta_j^{m*}$ analogously from the regression with $Y_i^m$ as the dependent variable.
Since $Y_i^f + Y_i^m = Y_i$, linearity of 2SLS implies
for each $k$. The decomposition is exact and additive. Both $\beta_k^{f*}$ and $\beta_k^{m*}$ inherit the cascade interpretation from the full-sample specification: the first-stage matrix $\Pi$ is the actual mixed-gender system, and the cascade operates through mixed-gender queues at every step. We are simply partitioning the outcome changes at each step into those experienced by women and those experienced by men.
The coefficient $\beta_k^{f*}$ answers: “of the total societal effect of expanding programme $k$ by one slot, how much is due to changes in women's outcomes?” This includes women who enter programme $k$ directly, women who enter other programmes through the cascade, and women who are displaced. It is a well-defined policy parameter that requires no fictional single-gender system.
This approach generalises immediately to any partition of the population: age groups, parental income quartiles, prior education levels. For any exhaustive set of group indicators $g \in \{1,\dots,G\}$ with $\sum_g \mathbf{1}[g_i = g] = 1$, define $Y_i^{(g)} = \mathbf{1}[g_i = g] \cdot Y_i$ and estimate full-sample 2SLS for each group outcome. The coefficients sum to the total: $\sum_g \beta_k^{(g)*} = \beta_k$.
The second question is: what is the total societal effect of admitting one more woman (specifically) to programme $k$, rather than one more person of unspecified gender?
This question asks whether it matters for society who the marginal entrant is. It is the closest analogue to the standard heterogeneous LATE and requires a different construction.
When programme $k$ admits one more woman:
The total effect of admitting one more woman to programme $k$ is therefore
where $\mathrm{RF}_k^f$ and $\pi_{jk}^f$ are the reduced form and first stage estimated on women only, and $\beta_j$ is the full-sample 2SLS coefficient.
The first term is female-specific: the direct effect on the marginal woman that fills the new slot. The cascade correction uses female-specific vacancy creation rates (since it is a woman who potentially departs programme $j$) but full-population cascade effects (since the refill draws from the mixed-gender queue). All ingredients are standard output from two sets of 2SLS regressions: the women-only first stage and reduced form, and the full-sample $\beta_j$.
Comparing $T_k^{|f}$ with $T_k^{|m}$ (defined analogously for men) reveals whether the gender of the marginal entrant matters for the total societal effect. The two can differ for two reasons. First, the direct effect may differ: the marginal woman admitted to programme $k$ may gain more or less than the marginal man would ($\mathrm{RF}_k^f / \pi_{kk}^f \neq \mathrm{RF}_k^m / \pi_{kk}^m$). Second, the cascade triggered by a woman may differ from the cascade triggered by a man, because women and men tend to vacate different programmes when admitted to $k$ ($\pi_{jk}^f \neq \pi_{jk}^m$). For example, if women admitted to a STEM programme disproportionately vacate slots in health sciences while men disproportionately vacate slots in business, the downstream reallocation differs by gender even though the refill at each step draws from the same mixed-gender queue. If neither channel is operative ($T_k^{|f} = T_k^{|m}$), the policy maker expanding programme $k$ need not be concerned with the gender composition of the marginal entrants.
The cascade framework extends naturally from capacity expansions to replacement policies: interventions that hold total capacity fixed but change the composition of entrants. A quota, an affirmative action rule, or a reallocation of slots across applicant types are all examples. The key observation is that the marginal policy effect $T_k$ is defined as a derivative, so it is symmetric: expanding program $k$ by one slot has effect $+T_k$, and contracting by one slot has effect $-T_k$. A policy that simultaneously admits one type and removes another is the difference of two conditional policy effects.
Suppose the planner replaces the marginal type-$m$ entrant in program $k$ with a type-$f$ entrant, holding total capacity fixed. This is equivalent to two simultaneous operations: admit one more type-$f$ individual (effect $T_k^{|f}$) and remove one type-$m$ individual (effect $-T_k^{|m}$). The net effect of the replacement is
Each term decomposes using equation (ref):
The replacement effect $R_k^{f \leftarrow m}$ therefore has two sources. First, the direct effects differ: the type-$f$ entrant may gain more or less from program $k$ than the type-$m$ entrant she replaces ($\mathrm{RF}_k^f / \pi_{kk}^f \neq \mathrm{RF}_k^m / \pi_{kk}^m$). Second, the two types trigger different cascades: type-$f$ and type-$m$ individuals vacate different fallback programs when admitted to $k$ ($\pi_{jk}^f \neq \pi_{jk}^m$), so the downstream reallocations differ even though both cascades draw from the same mixed-type queues at every subsequent step.
The replacement effect can be further decomposed by who benefits. For any partition of the population into groups $g \in \{1,\dots,G\}$, define group-specific outcomes $Y_i^{(g)} = \mathbf{1}[g_i = g] \cdot Y_i$ and let $\beta_j^{(g)*}$ denote the full-sample 2SLS coefficient from regressing $Y_i^{(g)}$ on the treatment indicators. Then
where \[ T_k^{|f,(g)} = \frac{\mathrm{RF}_k^{f,(g)}}{\pi_{kk}^f} + \sum_{j \neq k} \frac{-\pi_{jk}^f}{\pi_{kk}^f} \, \beta_j^{(g)*} \] measures how admitting one more type-$f$ individual to program $k$ affects group $g$'s outcomes, and $T_k^{|m,(g)}$ is defined analogously. By linearity, the group-specific replacement effects sum to the total: $\sum_g R_k^{f \leftarrow m,(g)} = R_k^{f \leftarrow m}$.
The replacement formula (ref) answers a question that arises naturally in any capacity-constrained system: what happens when the planner changes who fills a slot rather than how many slots exist? The expansion effect $T_k^{|f}$ and the contraction effect $-T_k^{|m}$ each include a direct component and a cascade component. The direct components measure the difference in treatment effects between the two types at program $k$'s margin. The cascade components measure the difference in the downstream reallocations that each type triggers: which programs they vacate, who fills those vacancies, and what those individuals would otherwise have done.
The decomposition by beneficiary (ref) makes the distributional consequences of replacement policies transparent. A gender quota in program $k$, for instance, replaces the marginal man with a woman. The group-specific replacement effects reveal how this swap affects women's and men's outcomes separately, including all effects working through the cascade. The direct effect of the quota operates on the two individuals directly involved (the woman who enters and the man who is displaced). The cascade effect operates on everyone else in the system who is reallocated as a consequence. If the woman and the man who are swapped would have attended different fallback programs, the cascade triggered by the quota differs from the cascade that would have occurred without it, and the distributional consequences extend well beyond the two individuals at program $k$'s margin.
All ingredients are standard regression output. The type-specific first stages and reduced forms come from subsample IV regressions; the full-population $\beta_j$ and $\beta_j^{(g)*}$ come from full-sample 2SLS with the overall and group-specific outcomes. No additional structural modeling is required. The replacement effect is identified under the same assumptions as the expansion effect: instrument relevance and ranking-list refill.
The cascade estimator is particularly well-suited to field-of-study effects because the counterfactual for a student marginally admitted to, say, economics, STEM or medicine is not “no higher education” but rather enrollment in whatever program they would have attended instead. A single-instrument Wald ratio that ignores this substitution confounds the effect of attending economics with the effect of not attending a program that may itself increase prosocial behavior. The cascade identity resolves this by estimating the full societal effect of expanding each field by one slot, including the downstream reallocation of students across substitute programs that such an expansion triggers.
Our application uses Swedish data. In the Swedish university admissions system, applicants submit a single ranked list of programs and are admitted through a national clearing process administered by UHR (Universitets och högskolerådet). Admission is determined by a merit score (meritvärde) based on upper secondary grades, results from the Swedish Scholastic Aptitude Test (Högskoleprovet), or prior undergraduate credits. Each program allocates seats across parallel tracks corresponding to these criteria. When qualified applicants exceed available seats, the admission cutoff falls at the lowest merit score among admitted students within each track. Ties at the cutoff are resolved by lottery, conducted independently within each program and admission round. The lottery produces a complete ranking of applicants within each merit-score bracket, with position determined by chance alone.
The lottery tie-breaking rule defines a natural set of pivotal groups: for each program and admission round, the pivotal group consists of all applicants whose merit score equals the admission cutoff. Within this group, admission is determined by the lottery draw alone, while applicants above the cutoff are admitted with certainty and those below are rejected with certainty. We construct a luck variable equal to the applicant's normalized rank within the pivotal group, which is uniformly distributed between 0 and 1 and independent of all pre-determined characteristics conditional on group membership. Crucially, the lottery ranking also determines the marginal applicant in the natural policy experiment of expanding a program by one seat: it is the highest-ranked rejected applicant, i.e. the next in the lottery queue who would be admitted under a marginal capacity expansion. This alignment between the instrument and the policy margin is what gives the 2SLS estimator its institutional interpretation.
This design is closely related to the regression discontinuity approach used in related work on Scandinavian higher education altmejd2021, kirkeboen2016, which identifies causal effects by comparing applicants just above and just below the admission cutoff. Our setting differs in that applicants within the same merit-score bracket are identical on the running variable by construction, and admission among them is determined by lottery. Continuity assumptions and bandwidth choices are therefore not required, and the instrument can be treated as the outcome of a randomization within each pivotal group.
Not all programs use pure lottery tie-breaking. Until 2012, a small number of programs applied gender quotas at the admission margin. Other programs resolve ties using Högskoleprovet scores rather than random draws, generating a deterministic rather than stochastic margin. Both deviations are flagged in the administrative data. We exclude all affected programs from the analysis, retaining only programs for which the within-pivotal-group luck variable is consistent with random assignment. The final analytical sample comprises 11,604 pivotal groups across 28 admission rounds between 2008 and 2021, with an average pivotal group size of approximately 12 applicants.
We group the 11,604 pivotal groups in our sample into seven mutually exclusive fields using the standard Swedish SUN classification (Standard f\"{o}r svensk utbildningsnomenklatur). The fields are: business including economics (SUN 34 and 314), social science (SUN 2--3 excluding business), pedagogy (SUN 1), medicine (SUN 721, 724, 727), health (remaining SUN 7), STEM (SUN 4--5), and a residual other category. Appendix (ref) discusses the aggregation weights and the conditions under which field-level estimates inherit the cascade interpretation from the underlying program-level effects.
The structural equation is
where $Y_i$ are application-specific outcomes in the years following the admission decision, $A_{ij}$ indicates enrollment in field $j$, and $\mathbf{X}_i$ contains field of application, prior giving (application 1), gender, age, year of admission, and application priority. We estimate the system by 2SLS on the sample of pivotal-group members, using field-specific lottery instruments $Z_{ij} = L_i \times \mathbf{1}(\text{applied to field } j)$ as excluded instruments, where
is the applicant's normalized lottery rank within pivotal group $g$ of size $n_g$. By construction, $L_i$ is symmetric and uniformly distributed on $(0,1)$ within each pivotal group and therefore orthogonal to all variables fixed within groups, including field of application, year of admission, and merit score. Pivotal group fixed effects are consequently not required for identification. Section (ref) verifies balance on predetermined individual characteristics and demonstrates robustness to alternative control specifications (including pivotal group fixed effects).
By the cascade identity of Section (ref), the coefficient $\hat{\beta}_j$ recovers the full societal effect of expanding field $j$ by one seat, inclusive of all downstream student reallocations across substitute fields. A central feature of the cascade framework is that the cascade effect itself requires no additional estimation beyond what is already standard practice. The difference between the full 2SLS coefficient $T_k$ and the single-instrument Wald ratio $W_k = \mathrm{RF}_k / \pi_{kk}$ recovers exactly the downstream reallocation effect:
Both objects are standard regression output: $T_k$ is the coefficient from the full 2SLS system and $W_k$ is the coefficient from a single-instrument IV regression restricted to field $k$'s own lottery. Their difference, reported in column (5) of Table (ref), measures the societal effect of the reallocation triggered by a marginal expansion of field $k$; the effect on everyone else in the system, net of the direct effect on the lottery winner.
Universities have long claimed that their purpose extends beyond the transmission of technical skills to the cultivation of prosocial values, civic responsibility, and moral character.\footnote{Classical and progressive educational theory emphasizes education as character formation and preparation for democratic citizenship; see, e.g., aristotle_politics and dewey1916. A large empirical literature studies the causal impact of education in general on prosociality and civic engagement, typically exploiting compulsory schooling reforms or other sources of exogenous variation in years of schooling dee2004, milligan2004, glaeser2002, helliwellputnam2007, persson2015, almen2025. dee2020 and willeck2022 provide recent surveys.} The content of these commitments differs systematically across fields. Programs in health care, social work, and education frame their mission around public service and human development, while programs in economics, business, and engineering emphasize competition, innovation, and market-based problem solving. Whether these differences in mission translate into differences in the prosocial behavior of graduates is an open empirical question with direct implications for education policy.
A long-standing literature documents that economics and business students exhibit lower levels of cooperative behavior, charitable giving, and trust relative to students in other fields frank1993, frey2003, bauman2011, carter1991, sundemo2025. Two competing explanations have proved difficult to disentangle. The first is treatment: economics education itself shapes values by reinforcing a model of human behavior premised on narrow self-interest---the homo economicus assumption---or by legitimizing self-interested action through the language of incentives and efficiency marwell1981, frank1993. The second is selection: students who choose economics are already less prosocially oriented before enrollment, and the field attracts rather than creates self-interested individuals bauman2011, girardi2024. Fully resolving the selection-versus-treatment debate requires exogenous variation in field assignment itself, i.e., precisely what the Swedish admissions lottery provides.
We measure prosociality using annual data on charitable giving drawn from Swedish tax registers, which record all donations qualifying for tax deductions,\footnote{The tax deduction for charitable donations was available during 2012--2015 and reintroduced from 2019 onward. Giving data are therefore observed for these years only. Since our admission sample covers 2008--2021, nearly all applicants have at least one post-admission observation of giving. However, for applicants admitted before 2012, we lack a pre-admission measure of giving. To maintain a common sample across all specifications, we set prior giving equal to zero for these applicants. Section (ref) shows that the main estimates are robust to alternative treatments of this variable.} for the full population of university applicants over 2008--2021. Giving is observed both before and after the admission decision, allowing us to control for pre-existing levels and trends and isolate the causal effect of field attendance. As a revealed-preference measure that is private, voluntary, and financially costly, it is free from the social desirability bias that plagues survey-based attitude measures.
Figure (ref) displays the estimated first-stage matrix. Each cell reports the effect of a high lottery score in one field (column) on the probability of eventual admission (i.e.\ ever admitted during 2008--2021) to the same or another field (row), measured over a two-to-twelve year window following the lottery draw.\footnote{An applicant is only coded as ever admitted if she was admitted to the field through competition. This means that all admissions that did not have applicants in the final queue, i.e. that were not capacity constrained, are counted towards the outside option instead. Technically, the outside option thus includes a few students that were admitted to university programmes that had surplus capacity and could not fill all its slots.} The diagonal elements are large, positive, and uniformly significant: a high lottery score for a given field substantially raises the probability of eventual admission to that same field, confirming strong compliance. The off-diagonal elements are negative throughout, consistent with the capacity-constraint mechanism underlying the cascade identity: a student who secures admission to one field is displaced from others.
The pattern of cross-effects reveals the substitution structure of the Swedish higher education market. Business, social science, STEM, and medicine exhibit substantial mutual substitution, indicating that applicants to these competitive fields hold overlapping preferences and treat them as close alternatives. Teaching and health, by contrast, show negligible cross-effects with other fields. Their applicant pools are largely distinct, and the counterfactual for a rejected applicant is more likely to be no higher education at all rather than enrollment in a substitute field. This reflects the lower average merit-score cutoffs for teaching and non-specialist health programs, which draw from a different part of the applicant distribution than the more competitive fields. Appendix (ref) shows a more granular table at the 3-digit SUN-level, revealing that there is also some substitution within fields.
These cross-effects are the structural inputs to the cascade estimator. The off-diagonal elements of the first-stage matrix determine the cascade correction that separates the direct effect of enrolling in a field from the indirect effect of displacing students into substitute fields. The heatmap thus summarizes the identification structure of the entire analysis: strong diagonals confirm that each field's own lottery instrument provides credible variation, while the pattern of off-diagonal substitution maps the reallocation mechanism through which a marginal capacity expansion propagates across the system.
Table (ref) reports the main results. Column (1) reports the full 2SLS estimate $T_k$, interpreted as the total societal effect of admitting one additional student to field $k$, including both the direct effect on the marginal entrant and all downstream reallocation effects through the capacity-constrained system. Column (2) reports the own-instrument Wald ratio $W_k$, which captures only the direct effect on the lottery winner, ignoring what happens to displaced students. Column (5) reports the cascade $T_k - W_k$.
The results reveal substantial heterogeneity across fields. Teaching, medicine, health, and STEM all generate large and statistically significant total effects on charitable giving, with point estimates ranging from 0.045 to 0.077. For teaching, virtually the entire effect is direct: the cascade is small and insignificant (0.002), indicating that vacancies created by an additional teaching seat do not create prosocial effects. This is consistent with the first-stage evidence that teaching draws from a distinct applicant pool with few close substitutes. The counterfactual for a rejected teaching applicant is more likely no higher education than enrollment in another prosocial field. For medicine, health, and STEM, by contrast, the cascade accounts for a meaningful share of the total effect, ranging from 0.01 to 0.02, all statistically significant. Expanding these competitive fields displaces applicants who would otherwise have enrolled in other programs with positive prosocial effects, and these programs recruit, in turn, individuals from outside higher education. This reallocation adds to the total societal benefit of expanding these disciplines.
The results for business and social science are particularly instructive for the debate on economics and prosociality. The own-instrument Wald ratio for business ($W_k = 0.0042$) is precisely estimated and statistically indistinguishable from zero. This is a clean estimate of what attending business does to a student's prosocial behavior relative to their counterfactual: it finds no effect whatsoever. Students who are marginally admitted to business programs give no more or less to charity than they would have had they attended their next-best alternative. The total policy effect of expanding business with one seat is, however, positive ($T_k = 0.0196$), and the difference is the cascade ($T_k - W_k = 0.0154$, $p < 0.01$). Opening one additional seat in business draws a student away from their next-best alternative---social science, medicine, or another field with a positive prosocial effect---and filling that vacancy generates a societal gain that the own-instrument estimate misses entirely.
Social science presents a similar but empirically sharper pattern. The total policy effect is positive and significant ($T_k = 0.037$, $p < 0.05$), yet the own-instrument Wald ratio is half as large and statistically insignificant ($W_k = 0.019$). The significant cascade ($T_k - W_k = 0.018$, $p < 0.01$) accounts for the difference. A researcher reporting only $T_k$ would conclude that expanding social science raises charitable giving; a researcher reporting only $W_k$ would conclude it has no effect. Both conclusions are correct since they answer different questions. The cascade decomposition reveals that the policy effect of expanding social science operates substantially through the reallocation of displaced students into fields with stronger prosocial effects, rather than through the direct effect of social science attendance itself. This is precisely the confound that a single-instrument Wald estimate cannot disentangle, and that the cascade framework resolves.
A broader implication for the literature is that estimates of the prosocial “cost” of economics or business education may conflate two distinct effects: the impact of studying business and the impact of not studying the alternative. Our results indicate that the latter drives much of the observed differences. In particular, fields such as pedagogy, health, and medicine appear to generate substantial increases in prosocial behavior. From this perspective, economics and business do not so much erode an underlying baseline of prosociality; rather, they fall short of the prosocial gains induced by competing fields.
The field-level estimates in Table (ref) are uniformly positive (except for “other”, a small group of diverse programs), suggesting that higher education broadly increases prosocial behavior. The average $\hat{\beta}_k$ of 0.0377 is statistically significant, providing causal evidence in favor of the long-standing view that expanding university education cultivates moral character beyond the transmission of technical skills (in this sample, the average charitable giving was about 0.1 in 2021; see Table (ref) in Appendix (ref) for field-specific averages). A natural question is how this average, which reflects the mean effect of expanding each field by one slot, relates to what a researcher would typically estimate when asking whether higher education in general increases prosocial behavior: regressing the outcome on a single indicator for any enrollment, instrumented by $L_i$ or a set of field-specific interactions. This is the standard single-regressor local average treatment effect (LATE). Appendix (ref) discusses the relationship between the average of $\beta_1, \dots, \beta_K$ and this pooled 2SLS estimate, and Table (ref) reports the results.
A central objective of higher education policy in many countries is to increase female representation in STEM fields: science, technology, engineering, and mathematics blickenstaff2005, ceci2011. Interventions range from targeted recruitment and mentoring programs to explicit gender quotas or preferential admission rules. Two distinct policy instruments are commonly discussed. The first is expansion: creating an additional slot in a STEM program and reserving it for a woman. The second is replacement: holding capacity fixed and admitting a woman in place of the marginal man. Both are standard tools of affirmative action.
The cascade framework developed in Sections (ref)--(ref) provides a unified analysis of both policy instruments. The expansion effect is the gender-conditional policy effect $T_k^{|f}$; the replacement effect is $R_k^{f \leftarrow m} = T_k^{|f} - T_k^{|m}$. Each can be decomposed by beneficiary: how much of the effect accrues to women's STEM degrees versus men's? The cascade matters because the woman who enters and the man who is displaced vacate different fallback programs, triggering different downstream reallocations through the mixed-gender admission queues. A policy that appears to move one woman into STEM may, through the cascade, also move men into or out of STEM at other points in the system.
We apply this decomposition to Swedish admissions data. Among applicants to STEM programs in the pivotal groups, approximately 57.7 percent ultimately obtain a STEM degree, while only 27.2 percent of applicants are female (see Table (ref) in Appendix (ref)).
We classify STEM programs into three tiers based on admission selectivity: competitive STEM (top quartile), mid-tier STEM, and non-STEM. The outcome is a binary indicator for holding a STEM degree in 2022. This three-tier classification allows cascade effects to operate within STEM, which would be obscured by a single STEM indicator. For example, admitting a woman to a competitive STEM program may free up a spot in a mid-tier STEM program, which is then filled from a predominantly male applicant pool.
Figure (ref) displays the gender-specific first-stage matrices. The diagonal elements confirm strong compliance for both genders, with women showing somewhat larger own effects in competitive STEM (0.549 vs.\ 0.443) and mid-tier STEM (0.394 vs.\ 0.359), reflecting that women in pivotal groups for STEM programs are more strongly affected by the lottery, which is consistent with men having more fallback options within STEM. The off-diagonal elements reveal the substitution patterns that drive the cascade. The key asymmetry is in the competitive STEM column: when women win the competitive STEM lottery, they vacate mid-tier STEM at a rate of 0.134; about 24% of the own effect. For men, the corresponding rate is 0.229, or 52% of the own effect. Men admitted to competitive STEM are thus twice as likely as women to have come from mid-tier STEM rather than from outside STEM. Conversely, women admitted to competitive STEM vacate non-STEM slots at a higher rate than men (0.088 vs.\ 0.068), indicating that a larger share of marginal women in competitive STEM would otherwise have left STEM entirely. This asymmetry is the mechanism behind the gendered cascade: when a woman enters competitive STEM, the mid-tier vacancy she creates is smaller, but the direct effect on the marginal women is bigger. When a man enters, the direct effect on the marginal man is smaller, but the cascade is bigger since his admission frees more slots in mid-tier STEM.
Table (ref) reports the full-sample cascade decomposition. Expanding competitive STEM by one slot generates 0.215 STEM degrees in total, of which the cascade accounts for 0.087 ($p < 0.01$)---roughly 40% of the total effect. The cascade is driven by substitution within the STEM hierarchy: the marginal competitive-STEM admit vacates a mid-tier slot, which is refilled from mid-tier's queue. For mid-tier STEM, the cascade is smaller (0.026, $p < 0.05$) because the marginal admit comes predominantly from outside STEM. The non-STEM row confirms the logic from the other direction: the Wald ratio is significantly negative ($-0.054$), but the positive cascade (0.034) partially offsets the loss as vacated STEM slots are refilled.
Table (ref) applies the gender-conditional formula of equation (ref) with STEM degree as the outcome, decomposed into women's and men's STEM degrees. The headline result is in the competitive STEM row in Panel A: admitting one more woman generates 0.250 total STEM degrees, of which 0.197 are women's and 0.053 are men's ($p < 0.01$), approximately one man's STEM degree for every four women's. The effect on men is entirely a cascade effect: the woman admitted to competitive STEM vacates a mid-tier slot that is refilled from a predominantly male queue.
The mid-tier row reveals a different pattern. Admitting one more woman to mid-tier STEM generates essentially zero additional women's STEM degrees ($-0.022$, insignificant): the marginal woman does not increase her chance of ending up with a STEM degree when admitted to mid-tier STEM. Mid-tier STEM is thus not the binding constraint for women's STEM attainment. The effect on men is positive (0.028) but imprecisely estimated.
The non-STEM row is instructive. Admitting one more woman to a non-STEM program has no net effect on total STEM degrees (0.001), but the decomposition shows why: it reduces women's STEM degrees by 0.016 ($p < 0.01$) and increases men's by the same amount ($p < 0.01$). Some women drawn into non-STEM would otherwise have been in STEM; these vacated STEM slots are mostly filled by men. The policy is almost exactly zero-sum for total STEM, but it redistributes degrees from women to men.
Panel B of Table (ref) reports the symmetric analysis for men. The contrast with Panel A is sharp. Admitting one more man to competitive STEM generates 0.205 STEM degrees, virtually all of which are men's (0.205) and essentially none women's (0.001). Unlike the female case, the male cascade does not benefit the other gender: men at the competitive STEM margin vacate programs whose queues do not contain women who would gain STEM degrees. The asymmetry traces directly to the first-stage matrices in Figure (ref): men admitted to competitive STEM vacate mid-tier STEM at a much higher rate than women do (0.229 vs.\ 0.134), but the mid-tier refill draws predominantly from men, so the cascade circulates within the male STEM pipeline without pulling women in.
Table (ref) reports the replacement effect $R_k^{f \leftarrow m} = T_k^{|f} - T_k^{|m}$: the net consequence of replacing the marginal man with a woman, holding capacity fixed. The point estimates follow mechanically from Table (ref); the contribution here is inference on the difference.
At competitive STEM, the net effect on total STEM degrees is small and insignificant (0.043): the quota barely changes total STEM production. But the decomposition reveals that this near-zero masks a significant redistribution: the quota generates 0.196 additional women's STEM degrees ($p < 0.01$) and destroys 0.153 men's ($p < 0.01$). The quota redistributes rather than creates.
The redistribution is not one-for-one. Each woman who replaces a man generates 0.196 women's STEM degrees but eliminates only 0.153 men's --- the net gain of 0.043 reflects the asymmetry in the two cascades documented above. Because the woman's fallback is more likely to be outside STEM while the man's is mid-tier STEM, removing the man destroys fewer downstream STEM degrees than admitting the woman creates. The quota is mildly STEM-expanding, but the effect is too small to distinguish from zero.
These results illustrate a general feature of affirmative action in capacity-constrained systems: the policy does not operate in a vacuum. Whether the instrument is expansion or replacement, the cascade propagates the intervention through the system, generating distributional consequences that extend beyond the individuals directly targeted. In our setting, expanding competitive STEM toward women generates one man's STEM degree for every four women's; replacing the marginal man with a woman redistributes degrees across genders but barely changes the total. The cascade framework makes these downstream effects visible and measurable using only standard IV output. The same logic applies to any rationed setting where slots are reallocated across types---including corporate board quotas matsa2013, ahern2012 and race-based university admissions policies, where banning or introducing affirmative action triggers cascading reallocations across tiers of the system bleemer2022.
The prevailing view in the econometric literature is that 2SLS with multiple treatments yields coefficients that are difficult to interpret under heterogeneous treatment effects. We show that in capacity-constrained allocation systems---a class that includes university admissions, school choice, medical residency matching, public housing, and other rationed settings---the 2SLS coefficient $\beta_k$ has a direct policy interpretation: the total societal effect of expanding treatment $k$ by one slot, including all cascading reallocations through the system. The result is an algebraic identity that holds for any first-stage matrix, requires only instrument relevance and that the allocation mechanism transmits supply expansions through the instruments, and imposes no restrictions on treatment effect heterogeneity or individual choice behavior. The identity applies not only to queue-based systems but to any allocation mechanism operating over goods with fixed supply, including competitive markets with price instruments.
For applied researchers working in capacity-constrained settings, the practical implication is straightforward. Multi-treatment 2SLS is not merely a convenience for combining instruments, it is the estimator that answers the policy question of whether to expand capacity. The cascade decomposition into own effects and downstream reallocation effects requires no additional estimation beyond what is already standard practice: the difference between the multi-treatment 2SLS coefficient and the single-instrument Wald ratio recovers the cascade directly.
The result has limitations. Assumption (ref) requires that the instrument and the policy operate on the same margin: a condition satisfied by construction in centralized systems but potentially violated in decentralized settings where vacancies are filled through different channels than those generating the instrument variation. The fixed-supply condition is essential: if competing programs can expand endogenously in response to the demand pressure created by the cascade, the identity breaks down. In practice, most publicly funded systems enforce fixed supply through administrative budget separation, regulatory constraints, or policy choice, but the assumption should be assessed case by case. Finally, the cascade identity characterizes a system-level policy effect that is in general difficult to decompose into individual-level treatment effects. Under homogeneous effects, the system-level and individual-level parameters coincide; under heterogeneity, they diverge, requiring the researcher to be clear about which parameter is being targeted.
Our empirical applications demonstrates the cascade framework using Swedish university admissions. The results show that the cascade correction is quantitatively important: for competitive fields like business and social science, the entire policy effect of expansion operates mainly through the downstream reallocation of displaced students rather than through the direct effect on the marginal entrant. This finding reframes the long-standing debate on whether economics education erodes prosocial values: the apparent prosociality gap between economics students and others is driven primarily by the prosocial effects of the fields that economics students would otherwise have attended, not by any negative effect of economics itself. The second application demonstrates that gender-targeted STEM admission policies generate distributional consequences that extend well beyond the individuals directly affected: approximately one-fifth of the STEM-degree effect of admitting a woman to competitive STEM accrues to men through the cascade.