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Econ-416 · economics · Economics

Returns to expanding college access

LLM status Open unverified

Statement

Original subject group: Education, families and demography

Audit classification: Background; proposed extension. Distinguished capacity-wide applicant net returns from the already estimated local admission margin and national welfare.

Definitions and precise research target

Editorial formalization. Consider applicants near the admission threshold of one public-university system. Let $k\in\{0,1,\ldots,\bar k\}$ be an administratively feasible integer capacity expansion, where $\bar k$ is a declared maximum measured in additional seats, and let $E_i(k)$ be applicant $i$'s enrollment destination after admissions, financial aid, and peer composition adjust. Define discounted lifetime earnings $Y_i(k)$, education resource cost $C_i(k)$, and a prespecified monetary nonpecuniary benefit $N_i(k)$. For the baseline applicant cohort, define \[V(k)=E\!\left[\sum_i\{Y_i(k)+N_i(k)-C_i(k)\}\right]-K(k),\] where $K(k)$ is additional infrastructure cost not already included in $C_i(k)$. Determine the discrete marginal net value $V(k+1)-V(k)$ for $k<\bar k$ and distinguish it from a local admissions-cutoff earnings effect. The identification strategy must address alternative colleges, displacement of other applicants, completion, peer effects, and declining instructional resources per pupil. Lifetime quantities beyond observed follow-up require explicit extrapolation and uncertainty intervals. Report separately outcomes for marginal admits and changes for existing students.

Fix the full baseline applicant set, including existing admits, rejected applicants and substitutes at other institutions. $k$ adds exactly $k$ seats under a stated admissions/aid rule, with capacity and peer adjustments declared. $Y_i(k)=\sum_{t=1}^{T}\beta^ty_{it}(k)+\operatorname{terminal}_i(k)$; the terminal term represents a stated lifetime extrapolation, not observed earnings. $N_i(k)$ is a monetary nonpecuniary benefit under a common reference convention, and $C_i,K$ partition direct educational and infrastructure opportunity costs. Earnings are a private-income measure, not automatically national output: specify whether price/general-equilibrium wage changes are excluded or incorporate all taxpayers and affected workers for a social welfare extension. The stated $V$ is an applicant-cohort net-return objective, rather than a universal utilitarian welfare functional. All expectations use a stated baseline population and probability law. Identification means every admissible data-generating model with the same observed-data law yields the same displayed target; otherwise report the identified set. Exact equations, horizons and assumptions are editorial, rather than source-given canonical conjectures.

Variants and assumption changes

  1. Editorial alternative-college variant. Stratify marginal admits by their no-expansion destination using explicit principal-stratum assumptions or bounds. Gains over no college and over another college can differ.
  2. Editorial national-welfare variant. Include affected workers, taxpayers and owners with endogenous wages and baseline-money welfare. Compare with applicant earnings under the same financing rule.

References and source audit

1. Existing local admissions-cutoff result; capacity-wide marginal welfare remains a different editorial target. Locator: Working-paper abstract; revision December 2025. Primary indexed abstract inspected; direct page retrieval failed. URL: https://www.nber.org/papers/w32296.
  1. Jack Mountjoy. 2024. Marginal Returns to Public Universities. Working-paper abstract; revision December 2025. https://www.nber.org/papers/w32296. DOI: https://doi.org/10.3386/w32296.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-416-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-416

Capacity-wide college returns and the limits of admission discontinuities

Problem ID: I26-412. Status: Unresolved. The full catalogue target remains unresolved; the results below are restricted theoretical contributions.

A target that includes existing students

The question asks for $\Delta V_k=V(k+1)-V(k)$ for an integer seat expansion, with a fixed baseline applicant population, alternative colleges and instructional responses. An admissions-cutoff effect is useful evidence, but it is generally only one component. The working-paper source studies local admission margins [mountjoy]; the calculation below does not reinterpret its estimates as an expansion experiment or supply new empirical estimates.

Let $\mathcal I$ contain $N$ applicants, including those already attending and those displaced at substitute institutions. Define $D_i(k)\in\{0,1\}$ as attendance at the focal system. At a given $k$, impose the following restricted, transparent decomposition of applicant net value: \[ Z_i(k)=Y_i(k)+N_i(k)-C_i(k) =a_i+\tau_iD_i(k)+\eta_i(k). \tag{1} \] The direct gain $\tau_i$ is relative to applicant $i$'s declared outside destination, with its resource costs and completion risk included. The residual $\eta_i(k)$ contains peer, instructional-resource and outside-institution effects. We normalize $\eta_i(0)=0$ for convenience, not by identification. This is an accounting model whose direct gains have policy meaning only if their reference destinations and held-fixed inputs are specified. Without such restrictions any chosen $\tau_i$ can be offset by $\eta_i(k)$.

Proposition (Marginal decomposition): Under integrability, writing $\Delta D_i=D_i(k+1)-D_i(k)$ and $\Delta\eta_i=\eta_i(k+1)-\eta_i(k)$, \[ \Delta V_k={\mathbb{E}}\sum_i\tau_i\Delta D_i +{\mathbb{E}}\sum_i\Delta\eta_i-\Delta K_k. \tag{2} \] If a deterministic rule enrolls exactly one additional applicant $j_k$, retains all previous enrollees and changes no other direct destination indicator, the first term is ${\mathbb{E}}\tau_{j_k}$.
Proof: Subtract (1) at adjacent capacities, sum over the fixed population and subtract infrastructure costs. Finite sums commute with expectations. Under the additional enrollment rule every indicator difference is zero except $\Delta D_{j_k}=1$. ∎

This elementary identity identifies the missing margins precisely. More seats need not produce exactly one additional enrollee if take-up changes, and an enrolled student may displace someone elsewhere. Those changes belong to the fixed cohort's other destination effects, not to an omitted population. Tuition and grants are transfers between parties; (1) uses educational opportunity costs for $C_i$, with the relevant fiscal boundary separately declared. The target is applicant net returns, so it need not equal national welfare.

A fully explicit nonidentification construction

Suppose baseline observations include all earnings and destinations at $k=0$, plus a valid local randomized-admission or discontinuity experiment holding total instructional capacity fixed. Assume it identifies a direct net gain $\tau=10$ for the next applicant. There are $m$ existing enrollees. Construct two models with identical baseline data and identical local experiments. In model A, capacity expansion changes only the next applicant's destination and costs $\Delta K=2$. Thus $\Delta V_0=8$. In model B, the same expansion reduces each existing enrollee's net lifetime outcome by $d>0$, for example because the added seat comes with no instructional resources. Existing-student losses occur only when the administrative capacity regime changes; individual local admission assignments at fixed capacity produce no such loss. Both models therefore reproduce every stipulated observable law, but \[ \Delta V_0^{B}=8-md. \tag{3} \] At $m=100,d=0.1$, the two values are $8$ and $-2$ in the same arbitrary monetary units. All outcomes can be made nonnegative by giving existing students sufficiently high baseline earnings. The intervention still adds exactly one seat, so negative value does not rely on violating feasibility.

The construction is also valid when the local experiment identifies separate no-college and alternative-college gains: keep those direct potential outcomes unchanged and alter only the capacity-regime term. Thus improved identification of individual enrollment returns does not by itself identify the spillover sum in (2). The example establishes nonidentification on a class allowing these resource responses; it does not claim such losses actually occur.

A sharp sensitivity interval

Assume exactly one new enrollee, whose direct gain is bounded by $\tau_{j_k}\in[\ell,u]$ almost surely or whose expected gain is in that interval. Suppose capacity effects are partitioned into $m$ existing students with expected changes in $[-d,0]$, and $r$ other cohort members with expected changes in $[-e,e]$. The entrant's own quality change is already in its direct-gain interval. Infrastructure increment $\kappa$ is known. No cross-person restrictions are imposed.
Theorem (Sharp rectangular-model bounds): The identified interval under these restrictions and no further information is \[ \Delta V_k\in[\ell-md-re-\kappa,\ u+re-\kappa]. \tag{4} \] Every point of this interval is compatible with the stipulated baseline and local-admission observations if the unobserved expansion responses may vary independently within their intervals.
Proof: Summing the component bounds in (2) proves containment. For the lower endpoint set the direct mean to $\ell$, every existing student's change to $-d$, and every other person's change to $-e$. For the upper endpoint choose $u$, zero and $e$. These assignments concern unobserved capacity regimes and can leave all stipulated observed potential outcomes unchanged. Convexly interpolate the component values to attain every intermediate total. Deterministic responses suffice, so no unobserved cross-world correlation is required. This proves sharpness for this explicitly rectangular class. ∎

If direct marginal returns are estimated only over a nearby score interval, extend them using a declared Lipschitz bound. For score distance $h$ and slope envelope $L$, a known cutoff mean $\tau(c)$ implies $\ell=\tau(c)-Lh$ and $u=\tau(c)+Lh$. This follows directly from $|\tau(x)-\tau(c)|\le L|x-c|$. It is an assumption about transport, not a consequence of continuity at the original cutoff. Capacity choices can then be robustly ranked only when their intervals separate; otherwise the bounds correctly refuse a sign conclusion.

Lifetime extrapolation and a direct design

Let follow-up end at $T$, with annual earnings differences after $T$ bounded by $L_t\le\Delta y_{it}\le U_t$. For discount factor $\beta\in(0,1)$, the unobserved tail lies in \[ \sum_i\sum_{t>T}\beta^tL_t \ \le\ \text{tail}\ \le\ \sum_i\sum_{t>T}\beta^tU_t, \tag{5} \] assuming the series converge absolutely. A constant symmetric bound $|\Delta y_{it}|\le M$ gives half-width $NM\beta^{T+1}/(1-\beta)$. Existing-student tails must be included as well as entrants' tails. A terminal value estimated from earnings growth is not an additional observed outcome; its uncertainty must enter the same accounting.

A direct identification route is to randomize complete capacity rules $k$ and $k+1$ across comparable independent systems or cohorts, with no cross-system spillovers, and measure the same baseline-population total $S=\sum_iZ_i-K$. Random assignment $A$ with known propensity $p$ identifies \[ {\mathbb{E}}[S(k+1)-S(k)] ={\mathbb{E}}\left[\frac{AS}{p}-\frac{(1-A)S}{1-p}\right]. \tag{6} \] Conditioning on assignment proves (6). It allows arbitrary interference within the system and directly includes existing students, but requires observing the total and implementing the stated aid, resource and admission rules. Following only students who enroll would change the population and invalidate the equality for $V$.

The unresolved work is empirical: estimating the resource and displacement responses, collecting long follow-up, and supporting transport across feasible capacities. Equations (2)--(6) supply an auditable decomposition, sharp sensitivity analysis and a design sufficient for identification in a restricted setting. They neither deliver the desired system-specific marginal values without data nor resolve general-equilibrium national welfare.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt decomposes capacity-wide college value, proves sharp rectangular sensitivity bounds, and supplies lifetime tail controls and a system-level randomized identification design. Resource dilution, alternative-college displacement, long-run returns, and transport across capacities still need evidence to determine actual marginal expansion values.

References

  • [mountjoy] J. Mountjoy, Marginal Returns to Public Universities, NBER Working Paper 32296 (2024), revised December 2025. Primary indexed abstract checked; cited here for the local-admission research setting only. https://www.nber.org/papers/w32296.

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