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Econ-419 · economics · Economics

Scaling housing mobility programs

LLM status Open unverified

Statement

Original subject group: Urban, housing and spatial economics

Audit classification: Published research agenda. Removed double counting of children earnings within family monetary welfare and specified market saturation.

Definitions and precise research target

Editorial formalization. Consider eligible and noneligible baseline households in one metropolitan market. At prespecified saturation $s$, an explicitly randomized share of eligible families receives search and moving assistance; housing, rents, schools and neighborhoods respond. For a fixed baseline household population including relevant owners and taxpayers, define monetary benefits $b_i$ and weights $\omega_i>0$ as follows. The proposed empirical target is \[W(s)-W(0)=E\sum_i\omega_i b_i(s).\] Identify or bound it using credible intervention variation and a declared model of price, selection and capacity responses. Financing, ownership income, resources and public services enter once. Local trial effects do not automatically identify full equilibrium responses; report selection, transport and distributional assumptions.

Saturation $s$ is a market-level allocation probability under a fully specified lottery of offers, not an individual's treatment. Let $\mathcal U_i^s$ be expected lifetime utility of every baseline household, eligible or otherwise, under that saturation, averaging over the lottery and equilibrium responses. Include all ownership income, child outcomes, financing and program resources in budgets or utility, using a common declared horizon/terminal value. Set $b_i(s)$ by $\mathcal U_i^0(b_i(s))=\mathcal U_i^s(0)$ and $W(s)=E\sum_i\omega_i b_i(s)$, assuming unique integrable benefits. Do not add children's earnings to a family money metric that already includes the resulting consumption. Fixed saturation points are estimable under randomized market saturation and a stated between-market interference restriction; individual offers in a single low-saturation trial identify a different contrast. 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 capacity variant. Compare fixed housing/school supply with endogenous supply under the same saturation lottery. Congestion and rents change $W(s)$.
  2. Editorial distributional variant. Change announced welfare weights and report eligible, noneligible and owner benefits separately. Positive aggregate benefits need not benefit every group.

References and source audit

1. Explicit agenda on mechanism weights, scaling mobility programs, and incumbent residents of revitalized areas. Locator: Journal of Economic Perspectives 35(4), 197–222; Discussion and Conclusion, pp.216–217. Full text or primary publication page inspected. URL: https://lkatz.scholars.harvard.edu/sites/g/files/omnuum5961/files/lkatz/files/chyn_katz_jep_2021_all.pdf.

2. Opportunity-market review exists; older review supplies explicit agenda, not this abstract. Locator: NBER Working Paper 35419, July 2026; abstract. Primary indexed metadata/abstract inspected; direct record failed. URL: https://www.nber.org/papers/w35419.

  1. Eric Chyn; Lawrence F. Katz. 2021. Neighborhoods Matter: Assessing the Evidence for Place Effects. Journal of Economic Perspectives 35(4), 197–222; Discussion and Conclusion, pp.216–217. https://lkatz.scholars.harvard.edu/sites/g/files/omnuum5961/files/lkatz/files/chyn_katz_jep_2021_all.pdf. DOI: https://doi.org/10.1257/jep.35.4.197.
  2. Lawrence F. Katz. 2026. Neighborhood Effects and Missing Markets for Opportunity. NBER Working Paper 35419, July 2026; abstract. https://www.nber.org/papers/w35419. DOI: https://doi.org/10.3386/w35419.

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-419-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-419

Housing saturation, rent transfers and a sharp scaling obstruction

Problem ID: R28-415. Status: Unresolved. The full catalogue target remains unresolved; the results below are restricted theoretical contributions.

A market model with a fixed welfare boundary

The target is the change in baseline households' monetary welfare when a market-level share of eligible families is offered housing mobility assistance. The local treatment effect of an offer is distinct from the welfare of a saturated market. We solve an illustrative quasilinear housing model in which these differences can be calculated exactly. The review by Chyn and Katz [source] motivates the scaling question; the functions and numerical values below are declared assumptions, not fitted evidence.

There are $Q_0>0$ incumbent renter households occupying a desirable location, owners of its existing housing, and a mass $E$ of eligible outside families. All belong to the baseline welfare population, as do taxpayers financing the program. Each household has money-linear utility, so the catalogue's baseline-money equivalent benefit equals its utility change. Welfare weights are initially one. At saturation $s$, assume every offered family moves and no unoffered family does, so $q=sE$ new units are demanded. The lottery selects a random set of $q$ eligible families; each has common expected gross location benefit $v$, already including all valued child outcomes. No child-earnings term is added separately.

Incremental inverse housing supply is $p(q)=p_0+\kappa q$, $\kappa\ge0$, interpreted as marginal real construction cost and competitive rent. Baseline incumbents retain their units. Incremental construction costs are \[ C(q)=\int_0^q p(x)\,dx=p_0q+\frac\kappa2q^2. \] Moving assistance consumes $cq$ real resources, $c\ge0$, paid for by lump-sum taxes. Its budgetary transfer to providers is not an extra benefit; providers' real costs have been included. Assume fixed baseline ownership and no outside parties receiving rents. These assumptions deliberately exclude endogenous take-up, school congestion and migration by incumbents.

Exact welfare and rent cancellation

Theorem (The incidence decomposition): For $0\le q\le E$, aggregate baseline-money welfare is \[ W(q)-W(0)=(v-p_0-c)q-\frac\kappa2q^2. \tag{1} \] The same value follows by summing newcomers' benefit, incumbent renters' loss, owners' gain and program-resource costs exactly once.
Proof: Newcomers gain $(v-p(q))q$. Incumbent renters lose $Q_0\{p(q)-p_0\}=Q_0\kappa q$. Owners' incremental net revenue, subtracting new construction costs, is \[ Q_0\kappa q+p(q)q-C(q). \] Summing cancels all rent payments and receipts, leaving $vq-C(q)$. Subtract program resources $cq$ to obtain (1). Alternatively this is gross location benefit minus the two real resource costs. Both routes give the same expression, checking that owners and taxes are not double-counted. ∎

The newcomer effect alone is $v-p_0-\kappa q$ per mover. Multiplying it by the number of movers omits an owners' gain of $\kappa q^2/2$ net of incumbent renter losses. Conversely adding the entire rent increase as a social cost would count a transfer twice. These cancellations rely on including owners in the welfare boundary and equal weights. With owner weight $\omega_O$ and incumbent weight $\omega_I$, the weighted incidence includes $(\omega_O-\omega_I)Q_0\kappa q$; distribution then matters even for a pure transfer.

For $\kappa>0$, differentiating (1) gives marginal welfare $v-p_0-c-\kappa q$ and optimum \[ q^*=\min\{E,\max\{0,(v-p_0-c)/\kappa\}\}. \tag{2} \] Strict concavity proves uniqueness except for coincident boundary cases. When $\kappa=0$, welfare is linear and the sign of $v-p_0-c$ selects zero or full saturation, with all quantities optimal at equality. If assistance offers induce heterogeneous take-up, the identity $q=sE$ fails and (2) optimizes a different feasible object; one must solve the take-up and offer-lottery equilibrium first.

Why a low-saturation trial does not determine scaling

Suppose a trial identifies household outcomes and market prices for all quantities $0\le q\le q_0$. Consider two inverse supply functions \[ p_A(q)=p_0+\kappa q,\qquad p_B(q)=p_A(q)+L(q-q_0)_+,\quad L>0. \tag{3} \] Both are continuous, nondecreasing and agree exactly over every observed quantity. Use identical preferences, assignment, take-up and outcome shocks in the two models. Then all trial observable laws agree. At $q>q_0$, however, aggregate welfare differs by \[ W_B(q)-W_A(q)=-\frac L2(q-q_0)^2. \tag{4} \] This follows by integrating the extra marginal supply cost. The welfare gap can be arbitrarily negative if $L$ is unrestricted, despite perfect identification of the low-saturation experiment. Prices beyond support and their effect on construction, not individual selection bias, create this obstruction.

The same logic applies to a congestion function affecting schools: two functions can agree on the trial's enrollment support and diverge after its maximum. A random offer design at one saturation does not turn those unobserved values into identified quantities. Under literal fixed supply with no new construction, additional households must displace someone; the incumbent-retention assumption becomes infeasible rather than an innocuous zero-elasticity limit.

A sharp extrapolation envelope

Suppose $p(q_0)$ and welfare through $q_0$ are known, and additional supply obeys an absolutely continuous slope bound \[ 0\le p'(q)\le L\quad\text{for almost every }q>q_0. \] Keep the same known $v,c$ and the same resource-accounting model. For $d=q-q_0\ge0$, \[ (v-c-p(q_0))d-\frac L2d^2 \le W(q)-W(q_0)\le(v-c-p(q_0))d. \tag{5} \]
Proof: The derivative restriction implies $p(q_0)\le p(q_0+x)\le p(q_0)+Lx$ for $x\in[0,d]$. Integrate these inequalities and subtract from $(v-c)d$. Constant marginal cost attains the upper endpoint; slope $L$ throughout attains the lower endpoint. Intermediate constant slopes attain every intermediate value. All extensions agree with observed supply through $q_0$, so the interval is sharp in this extension class. ∎

For example if the net marginal gain at $q_0$ is $2$, $L=1$ and proposed extra quantity is $d=5$, the extension lies in $[-2.5,10]$. Its sign is unresolved despite a positive measured marginal gain at the pilot. With $d=1$, the interval $[1.5,2]$ certifies a gain under the declared restrictions. These are sensitivity calculations in arbitrary units, not estimates of a housing program.

Identification at supported saturations and the remaining gap

Randomizing whole markets between supported saturations identifies mean changes in an observed total money metric if markets are independent and the same baseline households, owners and taxpayers are tracked. Formally, if $A$ selects saturation $s$ with probability $p$, inverse-propensity weighting of the complete market total identifies ${\mathbb{E}}[W(s)-W(0)]$. This allows within-market equilibrium responses. It does not identify utility-equivalent benefits from earnings alone; the money metric requires a declared utility, income and price convention.

The general problem remains unresolved because housing and school supply, take-up, household substitution, owner residence, distributional weights and lifetime benefits must be estimated or bounded in the application. The concrete contributions here are a solved market subclass, an exact transfer audit, a pair of observationally equivalent pilot models and sharp scaling bounds. Their restrictions explain precisely which additional information would narrow the uncertainty.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves a quasilinear housing-saturation model with consistent rent incidence and proves sharp supply-slope extrapolation bounds beyond pilot support. Endogenous take-up, school capacity, substitution, heterogeneous welfare weights, and lifetime money-metric benefits still need identified market-level responses.

References

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