Homework Proposal #3: Due: Wednesday, April 11 Level 3: (50 pts. max – about 3-4 pages) Topic:Choose an issue in the debate over setting standards in K-12 curriculum, and discuss your issue in the context of the Avery and Pathak paper. The Relevance to Current Events:Avery and Pathak target their paper specifically at the school choice and school vouchers literature, which is a topic high on our current Secretary of Educations agenda. Directions:Read the introduction, the discussion, the extensions and the conclusion of Avery and Pathak.Write up a discussion, in your own words, about how the one town model is different from the two town model.Fun Stuff:If you would like more context for the vouchers literature, there are some additional references provided.
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NBER WORKING PAPER SERIES
THE DISTRIBUTIONAL CONSEQUENCES OF PUBLIC SCHOOL CHOICE
Parag A. Pathak
Working Paper 21525
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
We are grateful to Arda Gitmez, Ed Glaeser, Richard Romano, and Tim van Zandt for superb comments.
Pathak thanks the National Science Foundation for financial support under award SES-1056325.
Avery thanks INSEAD for hospitality, as much of this paper was written while he was a visiting scholar
at INSEAD. The views expressed herein are those of the authors and do not necessarily reflect the
views of the National Bureau of Economic Research.
At least one co-author has disclosed a financial relationship of potential relevance for this research.
Further information is available online at http://www.nber.org/papers/w21525.ack
NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official
© 2015 by Christopher Avery and Parag A. Pathak. All rights reserved. Short sections of text, not
to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including
© notice, is given to the source.
The Distributional Consequences of Public School Choice
Christopher Avery and Parag A. Pathak
NBER Working Paper No. 21525
JEL No. H44,I20
School choice systems aspire to delink residential location and school assignments by allowing children
to apply to schools outside of their neighborhood. However, the introduction of choice programs affect
incentives to live in certain neighborhoods, which may undermine the goals of choice programs. We
investigate this possibility by developing a model of public school and residential choice. We consider
two variants, one with an exogenous outside option and one endogenizing the outside option by considering
interactions between two adjacent towns. In both cases, school choice rules narrow the range between
the highest and lowest quality schools compared to neighborhood assignment rules, and these changes
in school quality are capitalized into equilibrium housing prices. This compressed distribution generates
incentives for both the highest and lowest types to move out of cities with school choice, typically
producing worse outcomes for low types than neighborhood assignment rules. Paradoxically, even
when choice results in improvement in the worst performing schools, the lowest type residents may
Harvard Kennedy School of Government
79 JFK Street
Cambridge, MA 02138
Parag A. Pathak
Department of Economics, E17-240
77 Massachusetts Avenue
Cambridge, MA 02139
In 1974, Judge W. Arthur Garrity Jr. ruled that if a Boston school is more than 50% non-white,
then it would be subject to racial balancing. Garritys ruling ignited a fierce debate between
school choice proponents and neighborhood assignment advocates that continues more than forty
years later. Though Boston stands out, courts were involved with student assignment in numerous
districts, before many of these districts adopted some form of school choice. In choice plans,
pupils can apply to schools outside of their neighborhood, and the district uses this information for
centralized placement. Choice plan proponents argue that they would result in a more equitable
distribution of school access and lead to improvements in school productivity.1 Despite these
ambitious intentions, however, choice plans remain controversial, and there have been many recent
calls to return to neighborhood assignment across several districts.2
The aim of this paper is to provide a simple model to explore how the link between school
assignment rules, house prices, and the residential choices of families affect the distributional consequences of public school choice plans. Our model is motivated by empirical evidence showing how
the housing market and residential choices reflect school assignment rules (see, e.g., Black (1999),
Kane, Riegg, and Staiger (2006), Reback (2006), and Bayer, Ferreira, and McMillan (2007)). By
contrast to other recent work that emphasizes the connection between assignment rules and the
incentives for schools to improve their quality (see, e.g., Hoxby (2003), MacLeod and Urquiola
(2009), Barseghyan, Clark, and Coate (2015), and Hatfield, Kojima and Narita (2015)), we focus
on the effect of outside options in nearby towns on locational decisions of families living in a town
that adopts a school choice assignment rule.
For simplicity, we assume that each family has one child and consider a world of (primarily)
one-dimensional types, which could be interpreted either as wealth or status (of the family) or
ability (of the child) or some combination of them. We assume that the quality level of a school is
determined by the average of the types of families/children who enroll in that school. With utility
The first US school choice plan was in Cambridge, Massachusetts, where the district decided in 1981 to introduce
a choice plan to empower parents with choice, to include and treat fairly all students, to promote diversity, and to
promote school improvement through the competitive mechanism (CPS 1981).
For instance, Theodore Landsmark, a well-known advocate of Bostons busing plan in the 1970s, called for
a return to neighborhood assignment (Landsmark 2009). Former Boston Mayor Thomas Menino encouraged the
Boston school committee to adopt a plan that assigns pupils closer to home, and a plan restricting the amount of
choice outside of neighborhoods was adopted in 2014 (for more details, see Pathak and Shi 2014). Other districts
have also severely scaled back their choice plans such as Seattle (see Pathak and So¨nmez (2013) for details).
functions that provide incentives for assortative matching, students segregate by type.
town with multiple school districts uses a neighborhood assignment rule, endogenous differentiation
of housing prices and school qualities emerge in self-confirming fashion in equilibrium.
extreme, a neighborhood known for highest quality schools will have the highest housing prices
and will attract only highest types, and thus will continue to have high quality schools.
at the other extreme, lowest types will locate in neighborhoods with low quality schools.
consequence of these market forces, lowest types are relegated by self-selection and equilibrium
pricing to subpar schools, and thus, the educational system can be expected to widen rather than
narrow the inequality between initially high and low types.3
Our primary question is whether a town can improve outcomes for low types by adopting a
school choice rule, whereby all families have equal access to all schools in that town. In practice,
school rosters still tend to be somewhat differentiated by neighborhood within a town that adopts
school choice for several reasons: some towns allow for residential preferences in school assignment
(Abdulkadirog?lu and So¨nmez 2003; Dur, Kominers, Pathak, and So¨nmez 2013; Calsimiglia and
Guell, 2014); families may have preferences for schools near them (Hastings, Kane, Staiger 2009;
Abdulkadirog?lu, Agarwal, and Pathak 2015); and wealthier families tend to use more sophisticated
strategies in school assignment lotteries (Pathak and So¨nmez 2008). There is even some evidence
that the process of defining school boundaries can be captured by wealthy families – in the spirit of
gerrymandering – with the consequence that school choice rules can even reinforce the incentives for
school segregation by wealth within a particular town (Tannenbaum 2014). Even when a school
lottery is scrupulously designed to eliminate residential preferences and other features that may
favor wealthy families, segregated sorting may still result in an asymmetric equilibrium (Calsimiglia,
Martinez-Mora, and Miralles 2014), depending on the specific algorithm used for the assignment
To make the strongest possible case for school choice, we abstract away from these practical
These ideas have their roots in Tiebout (1956) and Schelling (1971, 1978), and have been explored extensively
by (among many others) Benabou (1993, 1996), Durlauf (1996), and Loury (1977) in studies of intergenerational
mobility, by Fernandez and Rogerson (1996) and Nechyba (2003b) in studies of the effects of different tax regimes
for funding public schools, and by Epple and Romano (1998, 2003) and Nechyba (2000, 2003a) in studies of school
vouchers. Epple and Sieg (1999) empirically examine the relationship between locational equilibrium and community
income distribution, while Rothstein (2006) provides empirical evidence of the relationship between neighborhood
sorting and school quality. Epple and Romano (2015) analyze efficient allocations in a multi-community model with
details and assume that, in fact, all schools in a town that adopts school choice assignment rule have
exactly the same quality equal to the average of types who locate in that town in equilibrium.
We then ask how the adoption of a school choice rule by a particular town affects the locational
choices of families in the resulting housing market equilibrium, with some families choosing to move
to that town and others choosing to leave it.
The incentive for flight of high types from a town that adopts school choice has been discussed
in the literature on the residential consequences of school desegregation or busing. For instance,
Baum-Snow and Lutz (2011) attribute the decline in white public school enrollment in urban centers
to court-ordered desegregation decrees, finding that migration to other districts plays a larger role
than private school enrollment. In the context of our model, withholding the option of paying for
a high quality school will drive high types to other towns that offer that option. But this same
logic applies inexorably as well to predict flight of low types when a town adopts school choice.
In fact, any model that predicts that school choice results in a narrowing of the range between
highest quality and lowest quality schools in a town and allows for changes in school qualities to
be capitalized into housing prices will generate a prediction that the adoption of school choice will
produce incentives for types at both extremes to move. Yet to our knowledge, ours is the first paper
to model how narrowing the gap between highest and lowest quality schools provides equilibrium
incentives for flight of low types (in addition to high types) from the public schools in that town.
Our approach is also inspired by past studies of the effects of private school vouchers, especially
Epple and Romano (1998) and Nechyba (2000). These papers develop ambitious models that
include multi dimensional student types, define school quality as a function of tax funding and
average peer quality, and allow for tax regimes, housing prices, and residential choices of families
to be determined endogenously in equilibrium, then typically use computational methods to assess
the welfare implications of different voucher plans. Subsequent papers by these authors, Epple and
Romano (2003) and Nechyba (2003a), consider the effects of public school choice in this framework.
Epple and Romano (2003) provide an example in their concluding remarks (p. 273-274) where a
public school choice rule induces exit by either low or high-income households, but do not conduct
a formal analysis along those lines as the framework of that example is quite distinct from the
models they analyze in the main section of the paper.
While we make a conscious decision to exclude many features in this earlier literature, our model
is not a special case of any of these models for two important reasons. First, the models in the
voucher literature typically assume that each family must purchase a house in a given town, where
private schools provides the sole channel for flight from the public schools. Then private schools
only attract high types, as enrollment in a private school then effectively requires a family to pay
twice for schooling: first, paying for a public school in the form of housing costs and then paying a
separate tuition to switch to private school. Second, some of the models, particularly Epple and
Romano (2003), assume that there is a fixed price for houses attached to the lowest quality school
in a town. But this is not an innocuous assumption, as it implies that changes in the quality of
the worst school in the town are not capitalized into market prices, and thus improvements in the
quality of the worst school are necessarily beneficial to low types. In sum, although our model is
superficially simpler than these earlier models, it allows for important effects that are excluded by
the modeling choices in that literature.
Our results are also related to the literature on gentrification and the displacement hypothesis,
which conjectures that neighborhood revitalization will result in higher prices that in turn cause lowincome and minority residents to move. The empirical evidence on the existence and magnitude
of displacement effects of gentrification is mixed (Vigdor, 2002; Atkinson, 2004; Freeman, 2005;
McKinnish, Walsh, and White, 2010; Autor, Palmer, and Pathak 2014), perhaps because there is
considerable endogenous selection in the location (Guerrieri, Hartley, and Hurst, 2013) and racial
composition of neighborhoods where gentrification occurs (Card, Mas, and Rothstein, 2008; Hwang
and Sampson, 2014).
The paper is organized as follows.
Section 2 describes and analyzes the partial equilibrium
model of the effects of a school choice assignment rule in a single town when school qualities are
driven by peer effects and residential choices, while outside options in other towns are fixed exogenously. Section 3 extends the model to a general equilibrium in two towns where the school qualities
and residential housing prices in each town (and thus outside options for all participants) are determined endogenously in equilibrium. Section 4 discusses empirical implications and extensions of
the model. Section 5 concludes. Proofs not in the main text are in the appendix.
The One Town Model
We focus on the locational equilibrium associated with school assignment rules in a particular town
t. Each family i is assumed to have one child who will enroll in school as a student, where each
family/student has a two dimensional type. The first dimension is binary, identifying partisans
who have a particular interest in living in town t.
The second dimension is student type,
which is independent and identically distributed according to distribution f (x) on [0, 1], where f
is continuous and differentiable and there is a positive constant ? such that f (x) > ? for each x.
We assume that there is a unitary actor for each household and refer interchangeably to families
and students as decision makers. To ease exposition, we frequently refer to the value of x as the
one-dimensional type of a student, neglecting partisanship.
Each family has a separable utility function that takes as arguments the type, xi , the quality of
school j chosen by the family, yj , and the price of attending that school, pj . Since we study rules
for assigning students to public schools which are freely provided, pj is simply the cost of housing
associated with school j (and quality yj ). We write this utility function as
u(xi , yj , pj ) = ?ij + v(xi , yj ) – pj ,
where ?ij = ? > 0 if family i is partisan to town t and school j is in town t, and ?ij = 0 otherwise.
The choice of a separable utility function of this form facilitates interpretation of marginal utility
and marginal cost of changes in school quality at equilibrium prices, while still producing results
that are qualitatively consistent with the prior literature.
A critical assumption of the model involves properties of v, the value function for schooling.
Assumption 1 v is continuous, differentiable, strictly increasing in each argument, v(0, 0) = 0,
and there is a positive constant ? > 0 such that
= ? for each (xi , yj ).
Assumption 1 implies that v satisfies the property of strictly increasing differences in (xi , yj ).4
That is, if xH
i > xi and yj > yj , then
i , yj ) – v(xi , yj ) > v(xi , yj ) – v(xi , yj ).
This assumption induces a motivation for assortative matching of students to schools, as high
types are willing to pay more for an increase in school quality than low types.5 The assumption
that v(0, 0) = 0 simply normalizes the boundary values for v.
See, for example, Van Zandt (2002).
If the one-dimensional type in the model is initial wealth, then it is natural to use a slightly different formulation
of utility, as is standard in the prior literature, namely u(xi , yj , pj ) = h(xi – pj , yj ) for some function h. Then, so
long as pj , the price for attending school j, is an increasing function of the quality of that school, h11 < 0 and h12 > 0
are jointly sufficient for u to exhibit strictly increasing differences in (x, y). Since hij refers to the second derivative
of h with respect to i and j, these sufficient conditions correspond to assumptions of decreasing marginal utility in
net wealth in combination with higher marginal utility for school quality as net wealth increases.
We assume that measure mt of families are town-t partisans and that the measure of houses
available in town t is Mt = mt , so that it is possible for all of these families to live in town t. We
also assume a competitive market for schools outside of town t such that schools of quality y are
available at competitive price p(y) for each y, which we identify below. Further, we assume a large
number of non-partisans of each type x who would be willing to locate in town t under sufficiently
In a rational expectations equilibrium, the full set of prices p(y) induces enrollment choices by
each student so that a school of quality y has associated housing price p(y), and enrolls students
with average type y.
Then if schools of every quality level y are available in equilibrium, there
must be perfect assortative matching in equilibrium, with all students of type x enrolling at schools
with quality y = x.6
Lemma 1 The competitive pricing rule p(y) =
(z, z)dz induces a (non-partisan) student of
type x to choose a school of quality x.
Lemma 1 identifies a unique pricing rule for self-sorting of all types into homogeneous schools.
In the One Town Model, we assume that schools of every quality level y are available outside
town t at associated (housing) price p(y). Thus, we denote the (outside option) value available in
equilibrium to a partisan of town t with type x as
p(x) = v(x, x) – p(x).
With these outside options in place for schools and housing outside of town t, we can now study the
effect of different school assignment rules on equilibrium outcomes in town t. For a neighborhood
assignment rule, the houses in town t are exogenously partitioned into separate districts 1, 2, …, D,
The competitive market for public schools outside the given town is quite similar to the nature of private schools
in Nechyba (2000, 2003a), where in equilibrium, each private school enrolls students of a single ability level, much
as a school of quality y outside town t is chosen only by students of type y in our model. One important distinction
is that students who opt for an outside option in our model do not also have to pay for a house in town t, whereas
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