School Choice and the Housing Market

Grigoryan (2021) develops a unified general-equilibrium framework that jointly models school assignment mechanisms and the housing market to evaluate the welfare and distributional consequences of replacing traditional neighborhood assignment (NA) with the Deferred Acceptance (DA) mechanism. The paper fills a gap in the matching theory literature, where preferences and priorities are typically treated as exogenous, by making residential choices endogenous: families first observe which school assignment mechanism the district announces, then optimally select a neighborhood given market-clearing prices and other families’ choices, and finally children are assigned to schools through the announced mechanism.

The model features a continuum of families, each with a type defined by valuations over all neighborhood–school pairs, a finite set of neighborhoods and schools (one school per neighborhood), and competitive equilibrium prices. Three mechanisms are compared: NA (each child attends the neighborhood school), DA without neighborhood priority (DA), and DA with neighborhood priority (DN), where neighborhood residents receive priority at their local school.

The paper’s first major result (Theorem 3) is that DN unambiguously generates weakly higher aggregate welfare than NA. The proof exploits the fact that DN preserves NA’s option — families can still guarantee admission to the neighborhood school by living there — while additionally allowing families to access seats at other schools that go unclaimed by neighborhood residents. Although price effects under DN can make some individual families worse off relative to NA, aggregate welfare (inclusive of house sellers) is always weakly higher under DN. In simulations with 1,000 students, 10 neighborhoods, and 10 schools, DN yields average aggregate welfare gains of 2.40% relative to NA across the 18 parameter configurations studied.

The welfare comparison between DA (without neighborhood priority) and NA is ambiguous in the general model: simulations show DA producing gains as large as +5.65% and losses as large as −18.26% relative to NA, depending on the degree of preference alignment across families (parameter α) and the variance in school capacities (parameter γ). DN also dominates DA in aggregate welfare under two sufficient conditions — identical ordinal preference rankings over neighborhoods and schools (Assumption 1 or 2) — though counterexamples exist when these assumptions fail.

The second major result (Theorem 5, Corollaries 1–2) concerns the welfare of lowest-income families, defined as those with budget (maximum willingness to pay for housing) equal to zero or sufficiently close to zero. Under two jointly sufficient conditions — (1) neighborhoods that are underdemanded (zero-priced) under NA remain underdemanded under DA/DN, and (2) the schools in those underdemanded neighborhoods are themselves underdemanded — both DA and DN generate weakly higher welfare for the lowest-income families than NA. These conditions hold whenever families share common ordinal preference rankings (Corollary 1) and in the uniform economy where each valuation profile is equally likely (Corollary 2). The conditions are shown to be approximately necessary in a robustness sense (Theorem 6): for any economy violating them, an arbitrarily close economy exists in which a positive measure of zero-income families prefer NA. In simulations, DN raises lowest-income welfare by an average of 26.51% and DA by an average of 38.25% relative to NA.

The paper also proves existence of a competitive equilibrium for the continuum economy under DA and DN via the Schauder-Tychonoff fixed-point theorem (Theorem 2), exploiting the continuity of school assignment probabilities in families’ neighborhood choices. In discrete economies, assignment externalities can preclude equilibrium existence, but approximate equilibria exist in sufficiently large discrete markets and all welfare comparisons carry over approximately. The existence proof technique applies to general assignment games with externalities including peer preferences and complementarities.

Scope conditions: results are derived for a model without direct peer externalities or endogenous school quality; a supplementary extension to local public financing finds that the aggregate welfare superiority of DA over NA may not survive when school spending is capitalized into housing prices, though the lowest-income welfare sufficiency conditions of Theorem 5 do extend to that environment.

Q: What is the core research question and why does the housing market matter for evaluating school choice?

A: The paper asks how replacing neighborhood assignment with the Deferred Acceptance mechanism affects aggregate welfare and the welfare of the lowest-income families, accounting for the fact that families choose where to live in response to the school assignment mechanism. The housing market matters because under neighborhood assignment families can guarantee enrollment at a preferred school by purchasing a house in that school’s neighborhood; switching to DA changes these strategic incentives, alters equilibrium prices, and therefore changes who ends up in which neighborhood before any school assignment takes place. Ignoring residential choices would miss this feedback loop between assignment rules and housing demand.

Q: What are the three mechanisms compared, and how do they differ?

A: Neighborhood assignment (NA) assigns each child to the school in their neighborhood with certainty. DA without neighborhood priority allocates seats by student preference rankings and lottery numbers, with market-clearing cutoffs determined iteratively; no residential location confers a priority advantage. DN (DA with neighborhood priority) works like DA but grants neighborhood residents a priority of 1 at their local school and 0 at all other schools, effectively guaranteeing neighborhood families a seat at their local school while filling remaining seats by lottery among non-neighborhood applicants.

Q: What does Theorem 3 establish, and what is the intuition for why DN dominates NA in aggregate welfare?

A: Theorem 3 establishes that for any competitive equilibrium under DN and any competitive equilibrium under NA, aggregate welfare is weakly higher under DN. The intuition is that DN preserves all options available under NA — a family can always choose the neighborhood corresponding to its most-valued school and be guaranteed admission there — while additionally providing access to seats at other schools not claimed by their own neighborhood residents. The proof maps DN’s CE onto a Walrasian equilibrium of a continuum assignment game and invokes the welfare-maximization property of such equilibria from Gretsky, Ostroy, and Zame (1992).

Q: Why is the welfare comparison between DA and NA ambiguous?

A: Under NA, families with the highest cardinal valuations for a particular school can guarantee admission by purchasing a house in that neighborhood, and this targeted sorting can raise aggregate welfare when preferences over schools are strongly aligned. Under DA (without neighborhood priority), no location guarantees school admission, so families lose this signaling device; but DA allows families to live in preferred neighborhoods without sacrificing school quality, which raises welfare when preferences are heterogeneous. Neither effect dominates in general: in simulations, DA ranges from −18.26% to +5.65% relative to NA across the parameter space.

Q: What role do neighborhood priorities play as a “signaling device,” and when does DN dominate DA?

A: Neighborhood priorities allow families to credibly signal high valuations for a school by choosing to live in that school’s neighborhood, analogously to signaling devices in matching markets without money. When families have identical ordinal preference rankings over neighborhoods and schools (Assumptions 1 or 2), DN generates weakly higher aggregate welfare than DA because any DA assignment probability can be replicated under DN by mixing over neighborhoods, but the converse is not true. Counterexamples exist when preference rankings differ across families, so the DN-over-DA dominance is not universal.

Q: What are the sufficient conditions for lowest-income families to prefer DA/DN to NA, and how tight are they?

A: The two joint conditions are: (1) neighborhoods that have zero price (are underdemanded) under NA also have zero price under DA or DN after the mechanism switch; and (2) the schools located in those underdemanded neighborhoods are themselves underdemanded (have zero admission cutoffs) under DA/DN. Condition (1) reflects that the poorest neighborhoods are unlikely to become highly sought-after merely because the assignment mechanism changed. Condition (2) is consistent with the empirical finding of Owens and Candipan (2019) that in large US metropolitan areas the poorest neighborhoods typically have underperforming schools. Theorem 6 shows these conditions are approximately necessary: any economy violating them is arbitrarily close to one where a positive measure of zero-budget families prefer NA, so robustness requires them.

Q: What do the simulations show about the magnitude of welfare effects for lowest-income families?

A: In simulations with 10 lowest-income families (budgets of 0.05) among 1,000 total, DN raises lowest-income welfare by an average of 26.51% relative to NA and DA raises it by an average of 38.25% relative to NA, across the 18 parameter configurations. The gains are larger when preferences for neighborhoods and schools are less correlated (lower α) and when school capacities are more uniform (higher γ). DA consistently outperforms DN for lowest-income families in the simulations, even though DN dominates NA in aggregate welfare more reliably.

Q: How does the paper handle equilibrium existence given the externalities created by residential choices?

A: Because a family’s expected utility from a neighborhood depends on other families’ neighborhood choices (through their effect on school assignment probabilities), standard existence results for assignment games do not directly apply. For the continuum economy, the author proves that school assignment probabilities under DA/DN are equicontinuous in families’ neighborhood choices, which enables application of the Schauder-Tychonoff fixed-point theorem to guarantee the existence of a competitive equilibrium (Theorem 2). In finite discrete economies, assignment externalities can prevent equilibrium existence (illustrated by an example in Appendix B), but approximate equilibria exist for sufficiently large discrete markets, and all welfare comparisons hold approximately.

Q: How does the paper’s model relate to and extend prior theoretical work on school choice and welfare?

A: Prior theoretical work (e.g., Calsamiglia et al. 2015; Xu 2019; Avery and Pathak 2020) uses stylized models with single-parameter family types, identical ordinal school rankings, supermodular valuations, and no preferences over neighborhoods. This paper allows an unrestricted preference domain — families have arbitrary valuations over all neighborhood–school pairs — which generates novel findings: in the general model, lowest-income families do not necessarily benefit from DA (contrary to Calsamiglia et al. and Xu), aggregate welfare comparisons between DA and NA are ambiguous (whereas they are trivially resolved in the special cases of prior work), and neighborhood priorities can be welfare-improving even relative to DA without priorities.

Q: Does the paper address the extension to endogenous school quality or local public financing?

A: In Supplementary Appendix B, the model is extended to allow school spending to be financed by local property taxes, making school quality endogenous to neighborhood housing values. In that environment, the aggregate welfare superiority of DA/DN over NA may not hold: DA attracts non-neighborhood applicants to high-priced neighborhoods, and if those schools are a poor match for those applicants absent the spending, social welfare may fall — a result analogous to Barseghyan et al. (2013). However, the paper reports that the sufficiency conditions for lowest-income family welfare comparisons (Theorem 5) do extend to the local public financing environment, preserving the distributional results.

Q: What does the paper say about alternative mechanisms such as Immediate Acceptance (Boston mechanism) and Top Trading Cycles?

A: The Supplementary Appendix studies these alternatives. For Immediate Acceptance (IA), the paper shows that when there are neighborhood priorities, lowest-income families may prefer DA to IA, echoing the finding that IA is not strategyproof and may disproportionately hurt low-income families who are worse at gaming the system or have worse outside options (Pathak and Sonmez 2008; Calsamiglia et al. 2015). Top Trading Cycles and further extensions are also analyzed in the Supplementary Appendix, though detailed results are not developed in the main text.

Neighborhood Assignment (NA): The baseline mechanism under which each family’s child is automatically enrolled in the school located in their chosen residential neighborhood, with no option to attend schools outside that neighborhood.

Deferred Acceptance without Neighborhood Priority (DA): A strategyproof centralized assignment mechanism in which seats are allocated by families’ stated preference rankings and lottery numbers via market-clearing admission cutoffs; residential location confers no priority advantage at any school.

Deferred Acceptance with Neighborhood Priority (DN): A version of DA in which families residing in a neighborhood receive priority 1 at their neighborhood school and priority 0 at all other schools, guaranteeing neighborhood residents a seat at their local school before remaining seats are allocated by lottery to non-neighborhood applicants.

Competitive Equilibrium (CE): A pair of neighborhood choices and a price vector such that (1) each family optimally selects the neighborhood maximizing expected utility net of price (subject to budget), (2) neighborhood capacities are not exceeded, and (3) neighborhoods with excess capacity are priced at zero.

Underdemanded Neighborhood/School: A neighborhood whose equilibrium price is zero (excess housing supply) or a school whose admission cutoff is zero (excess capacity), meaning any applicant who lists it can gain admission.

Assignment Externality: The indirect dependence of a family’s expected utility on other families’ neighborhood choices, which operates through the effect of the population distribution across neighborhoods on the family’s school assignment probabilities under DA or DN. This externality can preclude competitive equilibrium existence in discrete economies.

Aggregate Welfare: The utilitarian sum of all families’ expected utilities from their neighborhood–school assignments, not netting out neighborhood prices (so it includes the welfare of house sellers as passive agents); the comparison criterion for Theorems 3 and 4.

Signaling Device (neighborhood priority as): The interpretation that neighborhood priorities allow families to credibly reveal high valuations for a school by choosing to live in that school’s neighborhood, analogously to signaling instruments in matching markets without monetary transfers; the mechanism through which DN can improve welfare relative to DA.