Segregation and Tiebout Sorting:
Investigating the Link Between Public Goods and Demographic Composition*
H. Spencer Banzhaf† and Randall P. Walsh‡
March 2009
Support for this research was provided by the National Science Foundation, NSF SES-03-21566.
Additional support for Banzhaf was provided by the Property and Environment Research Center (PERC).
We thank Antonio Bento and participants in a 2008 ASSA Session and in seminars at Brown, Duke,
Georgetown, PERC, the University of Wyoming, and Yale for comments on earlier versions of this paper.
*
†
‡
Associate Professor, Department of Economics, Georgia State University.
Associate Professor, Department of Economics, University of Pittsburgh and NBER.
Segregation and Tiebout Sorting:
Investigating the Link Between Public Goods and Demographic Composition
In the white community, the path to a more perfect union means acknowledging that what ails the
African-American community does not just exist in the minds of black people; that the legacy of
discrimination - and current incidents of discrimination, while less overt than in the past - are real
and must be addressed. Not just with words, but with deeds—by investing in our schools and our
communities….
—Barack Obama
Introduction
Racial segregation of one form or another has been a recurring social problem throughout human
history. In the United States, substantial progress has been made since Brown vs. Board of
Education of Topeka and the Civil Rights Act of 1964 ended de jure racial segregation. But
despite the improvements in black educational achievement, the narrowing of black-white
income gaps, and even the election of the first African-American President, de facto racial
segregation continues to be one of the country’s most prevalent social issues.
Commensurate with the topic’s social and political relevance, recently several prominent
papers have modeled group segregation. Card, Mas, and Rothstein (2008) have shown that white
households still flee neighborhoods once they become 5-20% minority. Cutler, Glaeser, and
Vigdor (1999) and Kiel and Zabel (1996) show that white communities command a price
premium in part precisely because of their whiteness. Becker and Murphy (2000) show that
differences in willingness to pay for amenities can help drive group segregation. Hoff and Sen
(2005) and Sethi and Somanathan (2004) show how positive externalities from the activities of
richer households can drive such segregation. And Bayer, Fang, and McMillan (2005) and Sethi
and Somanathan (2004) show that reducing income inequality between groups can actually
increase neighborhood segregation, because richer minorities need no longer join whites to
obtain high levels of public goods.
These papers have made significant contributions to our understanding of the equilibrium
properties of segregation. However, none of these papers has considered the impact of placebased policy shocks. This is unfortunate because, as the above epigraph illustrates, many of the
policy remedies for reducing group inequity focus on investments in minority communities.
1
Investments in education are only one example. In addition, the U.S. Department of Housing
and Urban Development (HUD) provides Community Development Block Grants (totaling
$120B since 1974); the US EPA's Superfund and Brownfields Programs clean up contaminated
sites and encourage redevelopment; enterprise zones lower effect tax rates; and the 1992 Federal
Housing Enterprises Financial Safety and Soundness Act steers Fannie and Freddie investments
to low-income and minority communities. Such place-based policies are some of our primary
policy tools for reducing or compensating for segregation. Yet at the same time, some activists
express concern that they may trigger gentrification that ultimately harms incumbent residents
and benefits absentee landlords or gentrifying households (see e.g. the recent report by NEJAC
2006). Regrettably, the segregation literature to date is unable to speak to such effects.
Additionally, while recent work with models of public goods has shown how exogenous
improvements in public goods can trigger such gentrification effects (e.g. Sieg et al. 2004,
Banzhaf and Walsh 2008) this analysis has ignored the role of group-based segregation.
In this paper, we advance the literature by introducing an exogenous location-specific
public good (to be manipulated by public policy) into the segregation-based model. In this
sense, we combine features of the segregation and public goods literatures.1 First, following the
tradition of Schelling (1969, 1971), we include preferences for the endogenous demographic
make-up of the community.2 Working in this tradition, Schelling, Pancs and Vriend (2006), and
Card, Mas, and Rothstein (2008) show how such preferences and the resulting neighborhood
dynamics can create a “tipping point” that causes communities to become segregated (even
when everybody prefers some degree of integration). Second, following the tradition of Tiebout
(1956), we include heterogeneity in willingness to pay for public goods. Working in this
tradition McGuire (1974) and Epple, Filimon, and Romer (1984) have shown how households
can segregate themselves by groups or income strata according to their demand for local public
goods.3 In particular, as with other recent papers (Banzhaf and Walsh 2008, Sethi and
1
Glaeser and Scheinkman (2003) show how these two models can be nested in a more general social interaction
model.
2
Throughout our analysis a fundamental assumption is that individuals of a given type tend toward living together
due to preferences to live with individuals of their own type. We note, though, that identical results would be
obtained if the model assumed that concentrations of individuals of given type lead to social spillovers (i.e.
concentrations of businesses, community networks, restaurants, non-profit services, etc.) that disproportionately
benefit that type.
3
Bond and Coulson (1989) present a similar argument in the context of a “filtering” model of neighborhood
housing stocks.
2
Somanathan 2004), we adapt Epple et al.'s model of vertically differentiated communities.
In this paper, we combine both aspects into a general equilibrium model of sorting on
demographics and an exogenous public good. We then characterize the equilibria of such a
model and derive the comparative statics of policy shocks to the local public goods. The model
generates several results regarding the importance of place-based amenities. First, we establish
that, even when community sorting is driven by tastes for the exogenous place-based public
good and not by demographic tastes, some racial segregation will result, with the poorer group
enjoying lower levels of the public good. Second, we show that introducing tastes for an
endogenous demographic composition can drive further segregation, as suggested by Schelling's
“tipping model.” As a pragmatic matter, this is likely to further the differences in the average
public good enjoyed by the two groups.
Most importantly, we show that place-based interventions that improve the public good
in a low-quality high-minority community may actually increase group segregation, as richer
minorities are more likely to migrate into the community following the improvement.
Essentially, when differences in public goods become less important, group-based sorting begins
to dominate income-based sorting vis-à-vis the public good. In the final section of the paper, we
use large changes in the distribution of air pollution from industrial facilities that occurred in
California between 1990 and 2000 to test the predictions of the model. Consistent with our
model’s predictions, we find that large scale improvements in the dirtiest sites are associated
with increased racial sorting and decreased income sorting relative to exposure to toxic air
pollution.
Theory Model
In this section, we develop an equilibrium model of the link between race, demographic
composition, public goods and location choice. In the recent literature, the model is most similar
to work by Becker and Murphy (2000), Sethi and Somanathan (2004) and Banzhaf and Walsh
(2008), but differs from these papers in several important respects. It resembles Becker and
Murphy (2000) in combining preferences for an exogenous public good with the endogenous
demographic community, but differs in incorporating income distributions and heterogeneity in
willingness to pay for amenities. Generalizing their model in this way is crucial for motivating
income-based segregation. On the other hand, in adapting the vertically differentiated
3
framework of Epple et al. (1984), our model resembles Sethi and Somanathan (2004) and
Banzhaf and Walsh (2008). But it differs from Banzhaf and Walsh (2008), who only consider
the exogenous good, by including racial groups and demographic preferences. And it differs
from Sethi and Somanathan (2004), probably the most general model to date, by including an
exogenous public good.4 Including this public good is obviously essential to analyzing the kinds
of policy shocks that motivate our paper.
We consider a model with two communities, j  {C1, C2}, each comprised of an
identical set of fixed size housing stock with measure 0.5. There are also two demographic
types, r  {b, w}. Type-b has measure β < 0.5 and type-w has measure (1-β). There is
heterogeneity in income within type which is described by the continuous distribution functions
Fr. We impose the following assumption on the income distributions of type b and w
individuals:
𝑌𝑏Min = 𝑌𝑤Min = 0 < 𝐹𝑤−1 [
. 5 ∗ (1 − 2𝛽)
] < 𝑌𝑏Max ≤ 𝑌𝑤Max
1−𝛽
(1)
Essentially, this condition says that the groups’ income distributions cannot be “too different.”
In particular, this condition states that when individuals are stratified solely by income, there will
be some mixing of groups. One obvious interpretation of these types is as racial groups (think w
for white and b for blacks). However, in principle they could represent any pair of groups with a
poorer minority and richer majority (e.g. high school drop-outs vs. graduates).
Each individual is assumed to consume one unit of housing and to have preferences over
a numeraire (price set to unity), an exogenous community public good level Gj and endogenous
𝑗
community demographic composition 𝐷𝑟 . Individuals of each type experience the same service
flow of the public good. However, as we discuss below, they view the demographic composition
differently, so that D is indexed by type r as well as by location j.
Conditional on choosing location j, utility for an individual of type r and income y is
given by equation (2):
4
We also provide a more general characterization of the equilibria of our model, including integrated and segregated
equilibria. Sethi and Somanathan (2004) ignore segregated equilibria.
4
𝑗
𝑗
𝑈𝑟 = 𝑈[𝑌 − 𝑃𝑗 , 𝑉(𝐺 𝑗 , 𝐷𝑟 )] = U[x, Vrj]
(2)
The function U(.) is increasing at a decreasing rate in both of its arguments and V(.) is increasing
in both arguments. More compactly, let x be consumption and Vrj = V(Gj, Drj). We also require
the Inada condition that Ux→∞ as x→0.
Demographic preferences are captured by Drj = Dr(PCTrj), where PCTrj is the proportion
of type r in community j. The only restriction on Dr( ) is that Dr(1) > Dr(α) for all α<1-β.
Essentially, ceteris paribus, group-w individuals prefer an all-w community to filling out the
remaining housing supply in the community in which all type b individuals reside. This
assumption is consistent with a wide range of preferences including a general “bliss point”
specification, Drj = g[abs(PCTrj-ρr)], where g’<0, 1-β ≤ ρw ≤ 1, and ρb > 2β.
Additionally, without loss of generality, we assume that when public good levels differ
that the level of the public good is higher in Community 2 than in Community 1: GC2≥GC1.
Further, in order to close the model, we assume that the price level in Community 1 is equal to 0:
PC1 = 0. This normalization will allow us to work with households' willingness to pay to live in
C2, given Vrj and yi. Denote this willingness to pay as Bidry.
Equilibrium in the model is characterized by a sorting of individuals across the two
communities and a price level in Community 2, PC2, such that:
E1. In each community, for each type, Drj arises from the sorting of individuals:
𝑃𝐶𝑇𝑟 𝑗 =
∫𝑖∈𝑟,𝑗 1𝑑𝑖
∫𝑖∈𝑗 1𝑑𝑖
E2. Each individual resides in his preferred community. That is, for all individuals of
type r choosing to live in community j:
U[Yir – Pj, V(Gj, Drj] ≥ U[Yir – P-j, V(G-j, Dr-j]
E3. The measure of individuals choosing each community is equal to 0.5.
The structure of the model allows us to simplify the characterization of the sorting of
individuals across communities. Specifically, the concavity of U( ) in the numeraire and V( )
combined with the assumption that each household consumes an identical and fixed quantity of
5
housing implies that equilibria in the model will exhibit single crossing in income within each
type.5 In other words, in equilibrium, for each type there exists a boundary income, 𝑦𝑟 , such that
̅𝑟 , choose to live in the community that type
all individuals of type r with incomes greater than 𝑌
views as more desirable (i.e. the community with the higher Vrj).6
In will be particularly useful to consider the willingness to pay of these boundary
individuals. Define 𝐵𝑖𝑑𝑌̅𝑟 to be the price level in Community 2 that makes these individuals
indifferent. Recalling that PC1 is normalized to 0, 𝐵𝑖𝑑𝑌̅𝑟 is implicitly defined by equation (3):
𝑈[𝑌̅𝑟 , 𝑉(𝐺 𝐶1 , 𝐷𝑟𝐶1 )] = 𝑈[𝑌̅𝑟 − 𝐵𝑖𝑑𝑌̅𝑟 , 𝑉 (𝐺 𝐶2 , 𝐷𝑟𝐶2 )].
(3)
By definition, the bid makes these individuals indifferent between communities.
We begin our analysis with a discussion of what demographic compositions are feasible
given only condition (E3)—ignoring for the time being whether or not these can actually be
supported as an equilibrium. We first note that the entire demographic profile of a given sorting
of households can be expressed as a function of PCTwC1, the proportion of Community 1 that is
white:
PCTwC2 = 2 - 2β - PCTwC1,
PCTbC1 = 1 - PCTwC1, and
PCTwC2 = PCTwC1 + 2β - 1.
Consequently, the demographic qualities can also be expressed as a function of PCTwC1, Drj =
Drj(PCTwC1).
Further, given that single crossing holds within each type, we can also identify the
boundary incomes as a function of 𝑃𝐶𝑇𝑤𝐶1 , conditional on the relative levels of V(.):
5
See Epple et al. (1984) for a discussion of the single crossing property in locational equilibrium models.
Formally, the boundary income is defined as that income at which (PC2-Bidy) is at a minimum. Often, this will be
for an indifferent household, in which case (P2-Bidy) =0. However, there are equilibria where all type-b individuals
locate in the same community. In this case 𝑌̅𝑏 = 𝑌𝑏𝑀𝑖𝑛 when Vbj is higher in this community and 𝑌̅𝑏 = 𝑌𝑏𝑀𝑎𝑥 when
Vbj is higher in the other community. Because β<0.5, there must always be type-w individuals living in both
communities.
6
6
. 5 ∗ (𝑃𝐶𝑇𝑤 𝐶1 + 2𝛽 − 1)
]
𝛽
. 5 ∗ (1 − 𝑃𝐶𝑇𝑤 𝐶1 )
𝐹𝑏−1 [
]
𝛽
𝐹𝑏−1 [
𝑌̅𝑏 =
{
−1
𝐹𝑊
[
𝑌̅𝑤 =
{
𝑖𝑓
𝑉𝑏𝐶1 > 𝑉𝑏𝐶2 ,
𝑖𝑓
𝑉𝑏𝐶2 > 𝑉𝑏𝐶1 ,
. 5 ∗ (2 − 2𝛽 − 𝑃𝐶𝑇𝑤 𝐶1 )
] 𝑖𝑓
1−𝛽
. 5 ∗ 𝑃𝐶𝑇𝑤 𝐶1
−1
𝐹𝑊
[
]
𝑖𝑓
1−𝛽
𝑉𝑤𝐶1 > 𝑉𝑤𝐶2 ,
𝑉𝑤𝐶2 > 𝑉𝑤𝐶1 .
We can similarly characterize the derivatives of these boundary incomes with respect to
𝑃𝐶𝑇𝑤 𝐶1:
𝑑𝑌̅𝑏
=
𝑑𝑃𝐶𝑇𝑤 𝐶1
𝑑𝑌̅𝑤
𝑑𝑃𝐶𝑇𝑤 𝐶1
0.5
>0
𝛽𝑓𝑏 (𝑌̅𝑏 )
0.5
−
<0
{ 𝛽𝑓𝑏 (𝑌̅𝑏 )
𝑖𝑓
𝑉𝑏𝐶1 > 𝑉𝑏𝐶2 ,
𝑖𝑓
𝑉𝑏𝐶2 > 𝑉𝑏𝐶1 ,
0.5
< 0 𝑖𝑓
(1 − 𝛽)𝑓𝑤 (𝑌̅𝑤 )
=
0.5
>0
𝑖𝑓
{ (1 − 𝛽)𝑓𝑤 (𝑌̅𝑤 )
−
𝑉𝑤𝐶1 > 𝑉𝑤𝐶2 ,
𝑉𝑤𝐶2 > 𝑉𝑤𝐶1 .
To build additional intuition with the model, consider momentarily the case where the
public goods G are identical across communities. Panels 1 through 3 of Figure 1 illustrate the
relationship between PCTwC1, demographic composition, V( ) and 𝑌̅. The figure illustrates
results for the following specification: β=0.25, Drj = (PCTrj-ρr)2, ρb=0.5, ρw=0.9, and Vrj = Drj +
Gj. That is, V( ) is separable in D and G and D is defined by a quadratic function around a bliss
point ρ, with type w preferring more segregation.7 The utility function U( ) is Cobb-Douglas in
consumption and V, with an expenditure share of 0.75 on consumption. Incomes are assumed to
be uniformly distributed with Yrmin=0, Ybmax=1, and Ywmax=1.1. Finally, GC1=GC2=1.
In their recent work on tipping points, Card, Mas and Rothstein (2008) that ρw is around 0.9, which is also
consistent with the survey literature. This literature also suggests that an appropriate level for ρb is around 0.5
Farley et al. (1978) and Farley and Krysan (2002).
7
7
Panel 1 shows the racial compositions in each community as a function of PCTwC1.
(These are defined by an identity.) The figure shows that racial compositions are identical when
PCTwC1 = PCTwC2 = 1-β = 0.75. Panel 2 shows the corresponding values of V( ). It
demonstrates that our mild condition that Dr(1) > Dr(α) for all α<1-β, when combined with
condition (E3), implies that—in equilibrium—both types always prefer the community with the
larger percentage of their own type.8 For type-w individuals, V( ) is maximized at their bliss
point of 0.9. Beyond this bliss point, VwC1 is decreasing in PCTwC1, but still remains higher than
VwC2. Type-b individuals never achieve their bliss point and Vbj is always increasing in PCTbj.
Because public good levels are identical across the two communities, for both types, the level of
V( ) is equated across communities at the point where the communities have identical
demographic compositions (PCTwC1 = 1-β). Panel 2 also shows that when the levels of G are
identical across the two communities, the two types will have opposite rankings of the two
communities: whenever PCTwC1 < 1-β, VbC1 > VbC2 but VwC2 > VwC1, whereas when PCTwC1 > 1β, the opposite holds.
Panel 3 shows the link between PCTwC1 and the boundary incomes. Below 1-β, type-w
individuals prefer Community 2. As a result, single-crossing implies that they will sort such that
the richest type-w individuals locate in C2. Over this range, the boundary income is equal to that
of the poorest type-w individual in C2 (alternatively the richest type-w individual in C1). As
PCTwC1 increases, the number of rich type-w households living in C2 must decrease. Thus, the
boundary income must increase. But when PCTwC1 increases above 1-β, the relative ranking of
the communities changes and rich type-w individuals now locate in C1. Thus, the boundary
income is now decreasing in PCTwC1. Because in this example the community switch occurs
where type-w individuals are equally distributed between the two communities, there is only a
kink in boundary income at the switch. As we shall see, in cases where public good levels differ
and indifference between the two communities occur where PCTwC1 ≠ 1-β, there will be a
discontinuity in the boundary income function.
While the boundary income profile for type b looks similar to that of the type w, the
underlying process is different. Below 1-β, type-b individuals prefer Community 1. Thus, by
8
We note that while the assumption that both types strictly prefer the community with more of their own type is
sufficient for the findings that follow, it is by no means necessary. However, imposing this condition greatly
simplifies the analysis.
8
single crossing, the type-b boundary income will be that of the poorest type-b individual in C1
(alternatively the richest type-b individual in C2). When PCTwC1 = 1-2β = 0.5, all type-b
individuals are in C1 and 𝑌̅𝑏 = YbMin = 0.
From the foregoing analytical framework, we make the following observations about the
model.
Observation 1. Some type-w individuals always live in both communities. This must be
the case because the measure of type-w individuals is greater than 0.5. More specifically,
the share of type w in any community can never fall below 1-2β. In contrast, it is
possible that there will be no type-b individuals in one of the two communities.
Observation 2. When VwC1 ≠ VwC2, there will be some 𝑌̅𝑤 at which type-w individuals are
indifferent. All type-w individuals with income higher than 𝑌̅𝑤 will reside in the more
desirable (to w) community and poorer type-w individuals will live in the less desirable.
Moreover, 𝑌̅𝑤 can never drop below 𝐹𝑤−1 (1 − 2𝛽). In contrast, it is possible that b's
boundary income may fall to zero.
Observation 3. The community that type w perceives as more desirable will be more
expensive: VwC2 > VwC1 ↔ PC2 > 0. Moreover, PC2 = 𝐵𝑖𝑑𝑌̅𝑤 . This must be so, for if a
community was more desirable to type w and less expensive, all type-w individuals
would choose to live in it, and the demand for housing would exceed supply.
These observations are general to the model and do not depend on the assumption of equal public
goods or the other features of the example associated with Figure 1.
The discussion to this point demonstrates how all of the feasible allocations of
individuals and associated boundary incomes are uniquely determined by PCTwC1. We now turn
to an analysis of which allocations can be supported as equilibria. As noted in Observation 1,
because β<0.5, it is possible for all type-b individuals to live in the same community. We will
refer to equilibria where all type-b individuals live in the same communities as “segregated” and
equilibria where some type-b individuals live in each community as “integrated.” We make the
following observations about each of these cases.
Observation 4. For segregation of types with all type-b individuals locating in C1 to be
supportable as an equilibria, it must be the case that 𝐵𝑖𝑑𝑌̅𝑏 ≤ 𝐵𝑖𝑑𝑌̅𝑤 . (Otherwise, if BidY̅b
> BidY̅w , then the boundary type-b individual would outbid the boundary type-w
individual for housing in C2.) In particular, if VbC2 > VbC1 and VwC2 > VwC1 then 𝐵𝑖𝑑𝑌̅𝑤 ≥
𝐵𝑖𝑑𝑌̅𝑏 > 0. If VbC1 > VbC2 and VwC2 > VwC1 then 𝐵𝑖𝑑𝑌̅𝑤 > 0 > 𝐵𝑖𝑑𝑌̅𝑏 . And if VbC1 > VbC2
and VwC1 > VwC2 then 0 > 𝐵𝑖𝑑𝑌̅𝑤 ≥ 𝐵𝑖𝑑𝑌̅𝑏 . Note also that if VbC1 > VbC2then 𝑌̅𝑏 = YbMin = 0.
9
Otherwise, 𝑌̅𝑏 = YbMax. The opposite and symmetrical case holds for equilibria where all
type-b individuals locate in C2.
Observation 5. In all integrated equilibria, both types agree on which community is more
desirable and 𝐵𝑖𝑑𝑌̅𝑤 = 𝐵𝑖𝑑𝑌̅𝑏 . For integration, some members of each type must be
willing to pay to live in the more desirable community, while others must be unwilling to
pay the price and so live in the less desirable community.
The upshot of these observations is that equilibria can be identified by an analysis of the
bid functions of the boundary incomes for the two types. To illustrate this point, Panel 4 of
Figure 1 plots the boundary income bid functions associated with the previously defined
example. Type-w individuals prefer Community 2 when PCTwC1 < 0.75 and so the bid function
is positive over this range. The situation reverses above PCTwC1 = 0.75 and the bid function
becomes negative. When the demographic compositions are identical in the two communities,
type-w individuals are indifferent and the bid function is 0 (recall that we have equal public good
levels in this example).
As PCTwC1 increases, the change in the boundary income bid function is represented by
Equation (4):
𝑑𝐵𝑖𝑑𝑌̅𝑟
𝑑𝑌̅𝑟
𝑑𝐷𝑟𝐶2
𝑑𝐷𝑟𝐶1
1
𝐶2
𝐶2
𝐶1
1𝐶1
𝐶2
𝐶1
(𝑈
)
=
[
−
𝑈
+
(𝑈
𝑉
−
𝑈
𝑉
)]
𝑥
𝑥
𝑉
𝐷
𝑉
𝐷
𝑑𝑃𝐶𝑇𝑤 𝐶1
𝑑𝑃𝐶𝑇𝑤 𝐶1
𝑑𝑃𝐶𝑇𝑤 𝐶1
𝑑𝑃𝐶𝑇𝑤 𝐶1 𝑈𝑥𝐶2
(4)
where Uxj is the derivative of the utility function with respect to the numeraire, evaluated in
Community j at the appropriate income (i.e. ̅
Yw in the case of type w) and all other derivatives
are similarly defined. The expression shows two effects. The first term in the brackets captures
the effects of the change in the boundary income associated with the change in PCTwC1. As is
shown in Panel 3, for type w the boundary income increases over the range .5 to .75 and, because
the positive value of the bid function implies (UxC2 -UxC1 ) >0, the effect of the change in the
boundary income acts to increase the bid function. The second term in the brackets captures the
effect of changes in the V( ) function as PCTwC1 increases. Given the functional form
assumptions in this specification, the difference is decreasing over the range .5 to .75 in the
figure, reducing the bid function. In the case of type-w individuals, the effect of closing the gap
10
in V( ) dominates the income effect over the entire range and thus the bid function is always
negatively sloped.
For type-b individuals, preferences are reversed. The bid function is always negative for
PCTwC1 < 0.75 and always positive for PCTwC1 > 0.75. In terms of the slope of the bid function,
the basic dynamics are the same as with type-w individuals, except that for very low or very high
levels of PCTwC1 the change associated with the boundary income dominates the change
associated with closing the gap in V( ). This result is driven by the steeper slope of the boundary
income function, which arises from the fact that there are fewer type-b individuals.
Evaluating the figure, we can see that there are three equilibria in this example. At the
far left, type-w individuals are willing to pay to live in Community 2, type-b individuals are not,
and C2 is all-w. At the far right is the opposite and symmetric segregated equilibrium. In the
middle, there is an integrated equilibrium in which the two communities have the same
composition and everybody is indifferent.
In addition to the existence of such equilibria, we are also concerned with stability of
equilibria. We adopt the following definition of local stability.
Definition. An equilibrium is locally stable if, when a type-w individual with an income
arbitrarily close to Yw switches communities with a type-b individual arbitrarily close to
Yb , the relative levels of their bid functions are such that the marginal individual that was
relocated from C2 to C1 now has a higher bid for C2 than does the marginal individual
that was moved from C1 to C2.9
In this case, the individual moved from Community 2 will outbid the other individual and move
back into Community 2, restoring the initial equilibrium.
As discussed above, equilibria occur either at corners (the case of segregation) or in the
interior (the case of integration). Whenever corner equilibria exist they will be locally stable.
For interior solutions, it is straightforward to demonstrate that the local stability conditions are
satisfied whenever the type-b boundary income bid function crosses the type-w boundary income
bid function from above.
9
Note that in general we must consider two possibilities: first, moving a type-w individual from C2 to C1 and a
type-b individual from C1 to C2; and second, moving a type-w individual from C1 to C2 and a type-b individual
from C2 to C1. When either C1 or C2 is completely comprised of type w individuals, there will only be one possible
perturbation. Note also that by Observation 4, with segregation 𝑌̅𝑏 is equal to 0 or YbMax and the marginal individual
will be confined to incomes above or below these levels respectively.
11
Thus, from Panel 4 in Figure 1, we see that while there are three equilibria, only the two
symmetric segregated equilibria are stable. This result is not limited to the special case presented
here. For any specification satisfying our general assumption, the following proposition holds.
Proposition I.
Whenever GC1=GC2, there will be three equilibria: two stable segregated equilibria with
PCTwC1 = 1 and PCTwC1 = 1-2β and one unstable integrated equilibrium with PCTwC1 =
1-β.
Proof: See Appendix
We now turn to analysis of cases where public good levels differ across communities. To
build intuition for how relationships change when public goods differ, Figure 2 replicates
Figure 1 for a case with unequal public good levels (GC1=0.9 and GC2=1). Because the set of
feasible racial compositions is independent of public good levels, the first panel of Figure 2 is
identical to Figure 1. Panel 2 shows that, when GC1 falls, VrC1 shifts down for both types. As a
result, type-b individuals are now indifferent between the two communities when PCTwC1 =
0.65. For type-w individuals the indifference point is now at PCTwC1 = 0.92.
Panel 3 shows the impact of the differential public good levels on boundary income.
When public goods were equated across communities, both types were indifferent at the point
where PCTwC1=PCTwC2. As a result, even though relative preferences for communities switch
at this point, reversing the income sorting, the boundary income functions were continuous
(though kinked) at the indifference point. Differential public good levels separate the
indifference point for the two types moving them away from equal sorting. As a result, there is
now a discontinuity in the boundary income functions for each type at their indifference point.
Finally, Panel 4 of Figure 2 presents the bid functions under the new public good levels.
In this example, the situation remains qualitatively similar to the case where the public good
levels were equal.10 There are still two stable segregated equilibria and one unstable integrated
equilibrium, with the location of the unstable integrated equilibrium now shifted slightly to the
right to where PCTwC1 = 0.76. Thus, for small differences in public good levels, it is possible to
support a segregated equilibrium with all type-b individuals in the high public good community.
10
We do note that the boundary income functions are now non-differentiable at the value of PCTwC1 where the two
communities are equally desirable. But the bid function approaches zero from both sides at this point, so the
boundary income bid functions are still continuous (though kinked).
12
However, comparison of the bid functions in Figures 1 and 2 suggest that, at least for this
example, as the public good gap increases this equilibrium will no longer be supported. As we
shall see below, this result holds in general.
As the public good gap increases, it becomes difficult to make general statements
regarding the character of equilibria. This is because, as is clear from an examination of
Equation 4, the slopes of the bid functions are highly sensitive to local variations in the density
of the income functions and the relative curvatures of the utility functions. For instance, it is
relatively straightforward to generate models with multiple segregated and multiple stable
equilibria when the public good differentials are moderate in size.
Nevertheless, the model does provide sharp predictions regarding equilibria for cases
when there are “small” or “large” differences in public goods, and a comparison of these two
cases provides important policy insights. We begin by considering the case where there are
small differences in the public goods levels. As stated in Proposition 1, when the level of public
goods in the two communities are identical, there will always be one unstable integrated
equilibrium and two symmetric stable segregated equilibria. We can also extend this observation
to a measure of Gj as stated in Proposition II.
Proposition II
For any given level of GC2, there will exist 𝐺̃ 𝐶1 < GC2 such that for all GC2 > GC1 > 𝐺̃ 𝐶1
there will exist exactly three equilibria, two symmetric stable segregated equilibria and
one unstable integrated equilibrium. Further, there will exist 𝐺̃ 𝐶1 < GC2 satisfying
V[GC2, Db(1)] > V[𝐺̃ 𝐶1 , Db(1-2β)] such that for all GC2 > GC1 > 𝐺̃ 𝐶1 there will exist a
stable segregated equilibria with all type-b individuals locating in C1.
Proof: See Appendix
Proposition II addresses the case where differences between the public good levels are
small enough that demographic preferences dominate in the determination of equilibria. It states
that in this range, the stable equilibria are characterized by segregation of the types. Moreover,
segregation with type-b in the low-public good community is guaranteed to exist over a wider
range of G's.
Consider now the opposite extreme, where differences in public goods are very large and
effectively drive the sorting behavior, swamping any effect of the demographics. Intuitively, this
case will resemble earlier results from Epple et al. (1984) and other related papers with only a
13
public goods component. In particular, individuals will be stratified by income instead of
segregated by race.
To operationalize this intuition in our model, assume that, for given levels of GC2, DrC1,
and DrC2, BidYr→Y as GC1→-∞.11 Then, at the limit, the boundary income bid functions are
equal to the boundary incomes themselves and both types view Community 2 as more desirable
regardless of demographic sorting. By construction, the richest type-b individual is richer than
the boundary type-w individual when PCTwC1 =1-2β and the poorest type-b individual is poorer
than the boundary type-w individual when PCTwC1 =1. As a result, the boundary income of
type-b individuals crosses the boundary income of type-w individuals exactly once from above.
In this situation, there is a single equilibrium. The equilibrium is stable and integrated with all
individuals with income above the population median income locating in Community 2.
Proposition III formalizes this result.
Proposition III
Assume that, for given levels of GC2, DrC1, and DrC2, 𝐶1→−∞
lim 𝐵𝑖𝑑𝑌̅𝑟 = 𝑌̅𝑟 . Then, there
𝐺
̂ C1 such that for all GC1<𝐺̂ 𝐶1 there is a single equilibrium which is integrated and
exists G
stable.
Proof: see the appendix.
That is, when public good differences are sufficiently large, the communities will be integrated
by group but stratified by income.
To summarize the discussion so far, when public good levels in the two communities are
relatively similar the only stable equilibria will be segregated. At intermediate differences in
public good levels it is difficult to make general statements about equilibria. However, the stable
segregated equilibria with all type-b individuals in Community 1 will exist over at least part of
this range. Finally, when differences in public good levels are high there will be a single stable
equilibrium with integration.
To further illustrate the implications of Propositions II and III, Figure 3 displays the bid
functions for the model of Figures 1 and 2—fixing the level of GC2 at 1 and varying the level of
GC1 from 0.25 to 1. When GC1 =0.25 the public goods difference is large enough that the
11
This condition is less restrictive than it might first seem. It is for example satisfied by Cobb-Douglass or CES
preferences. We would also note that while this is a sufficient condition for the emergence of a single integrated
equilibrium as public good differences increase, it is by no means necessary.
14
outcome resembles the limiting notion of proposition III where the bid functions equal the
boundary income functions. Here there is a single stable integrated. As GC1 increases to 0.5, the
bid functions no longer track the boundary income functions as closely, but there is still a single
stable and integrated equilibrium. When GC1=0.7, the bid functions no longer cross and the only
equilibria is a stable segregated equilibria with all type-b individuals located in the low public
good community. In other words, closing the gap in public goods by improving the public good
level in Community 1 causes a change from integrated equilibrium to segregated equilibrium.
This stable segregated equilibria exists for all 1 ≥ GC1 > 0.6. When GC1=0.8, this stable
segregated equilibrium continues to exist. In addition, two new equilibria appear, an unstable
and a stable integrated equilibrium. Finally, once GC1=0.9, we are in the realm of proposition II
with two stable segregated equilibria and one unstable integrated equilibrium.
The figure illustrates the results of Propositions II and III. Namely, when public good
differences are large, integrated equilibria (with income stratification) are especially salient.
When public good differences shrink, segregated equilibria are especially salient. Figure 4 offers
a schematic of these results, showing the stable equilibria at different qualitative values of GC1.
Below, we confirm these patterns using data on pollution changes in California from 1990 to
2000.
These results speak to three important policy and empirical issues in the literature. The
first and most central to our application are policy concerns centered on the correlation between
low-income and/or minority populations and the levels of local public goods like public school
quality, public safety, parks and green space, and environmental quality. The “environmental
justice movement,” for example, has highlighted such correlations with air pollution and local
toxic facilities (see e.g. Been 1994, Ringquist 2006, Noonan 2008). Our model confirms the
intuition that such sorting can be produced by income differentials. For example, given the
correlation that exists between race and income in today’s society, this will be the case in regions
A and A’ of Figure 4. Moreover, racial preferences can strengthen this result by driving even
fairly rich minorities into the minority district, as they are not willing to pay the extra price for
15
the high public good community’s whiteness (a theme raised in the law literature by, e.g., Ford
1994).12
However, both advocates and analysts have raised concerns that improving public goods
in low-quality neighborhoods may drive forms of gentrification (Sieg et al. 2004, Banzhaf and
McCormick 2006, NEJAC 2006). While there may well be price effects, our model suggests that
it is unlikely that public good improvements will lead to large turnovers in racial or other group
compositions. To the contrary, Propositions II and III together imply that improvements in
public goods may increase segregation (for example an improvement that leads to a move from
region A to regions D in Figure 4).13 On the surface, this result resembles that of Bayer et al.
(2005) and Sethi and Somanathan (2006), who find that improving the minority income
distribution can increase segregation. They find that such people-based policies have this effect
because richer minorities now have enough mass to form their own high public-good
communities. Our results suggest that place-based policies pose the same dilemma, but for a
different reason: group-based sorting becomes more salient when there is less reason to sort on
the exogenous public good. It appears that segregation is a deep-rooted social problem.
Second, our results may help explain some recent empirical puzzles. As noted above, the
environmental justice literature shows that minority households are correlated with undesirable
facilities like hazardous waste sites. But recently, both Cameron and McConnaha (2006) and
Greenstone and Gallagher (2008) have found in a difference-in-difference context that
improvements to such sites do not appear to reduce these correlations. They suggest this may be
because of long-lasting “stigma” of the sites or the ineffectualness of cleanup. Our results
suggest another explanation: the reduced form relationship may change after a cleanup, so that
the correlation on race becomes even stronger. Figure 5 plots the relationship between Gj and
PCTbj for our two communities for each value of GC1 used in Figure 3. The bottom panel shows
the cross-sectional relationships. The top panel shows difference-in-differences for each
successive improvement in GC1. It clearly shows that while the cross-sectional differences have
the expected negative slope, the difference-in-differences have the opposite slope.
This for instance is the case in the segregated equilibria supported in regions C and C’ in Figure 4.
This of course depends on which of the two stable equilibria obtain. However, A move to the stable segregated
equilibria with all type b individuals in the high public good community would require a much larger shift in
populations than is required for a move to the stable segregated equilibria with all type b individuals in the low
public good community.
12
13
16
One way to think about this problem is in terms of a mis-specification of the standard
difference-in-differences regression. Our model suggests that the correlation between race and
public goods increases as the low public good community sees increases in its public good level.
If we have a reduced form relationship as in Equations (5) and (6).
𝑃𝐶𝑇𝑀𝐼𝑁𝑂𝑅𝐼𝑇𝑌𝑖0 = 𝑡0 + 𝛼𝑖 + 𝛽0 𝑃𝑂𝐿𝐿𝑈𝑇𝐼𝑂𝑁𝑖0 + 𝜖𝑖0
(5)
𝑃𝐶𝑇𝑀𝐼𝑁𝑂𝑅𝐼𝑇𝑌𝑖1 = 𝑡1 + 𝛼𝑖 + 𝛽1 𝑃𝑂𝐿𝐿𝑈𝑇𝐼𝑂𝑁𝑖1 + 𝜖𝑖1 ,
(6)
with 𝛽1 > 𝛽0 > 0, then first differencing the equations leads to:
Δ𝑃𝐶𝑇𝑀𝐼𝑁𝑂𝑅𝐼𝑇𝑌𝑖 = Δ𝑡 + 𝛽1 Δ𝑃𝑂𝐿𝐿𝑈𝑇𝐼𝑂𝑁𝑖 + (𝛽1 − 𝛽0 )𝑃𝑂𝐿𝐿𝑈𝑇𝐼𝑂𝑁𝑖0 + 𝜇𝑖
(7)
If 𝑃𝑂𝐿𝐿𝑈𝑇𝐼𝑂𝑁𝑖0 is omitted from equation (7), the estimate of β1 will be biased downward when
Δ𝑃𝑂𝐿𝐿𝑈𝑇𝐼𝑂𝑁 and 𝑃𝑂𝐿𝐿𝑈𝑇𝐼𝑂𝑁0 are negatively correlated (i.e. if the dirtiest areas are being
cleaned up). In this case, the problem may simply be one of omitted variables. Including
baseline pollution levels is suggested as a control.
A third application of our model is to the recent revival in the interest in “tipping models”
of racial segregation (e.g. Pancs and Vriend, 2006, and Card, Mas, and Rothstein, 2008).
However, although Schelling (1969, 1971) noted the link between public goods and demographic
sorting in his early work, their role has generally been under appreciated in recent models of
segregation. For example, Card, Mas, and Rothstein (2008) recently have conducted a study of
"tipping" behavior in US Cities. They identify tipping points using tract level data, looking for
break points in the change in the white population as a flexible function of the baseline minority
composition of the tract. They assume tipping points are identical for all tracts within a
metropolitan area. However, our model suggests this is unlikely to be the case. When two tracts
have large differences in locational amenities, integration is supportable even with large
proportions of minorities. When the tracts have small differences in amenities, however, more
segregation results. In other words, one community with, say, 20% minorities and high public
goods may continue to attract whites, while another even with 15% minorities and low public
goods may not. This may be one reason why there appears to be more noise in their predictions
around their estimated tipping point (see their Figure 4). Our model suggests more precise
results can be obtained by adjustment for differences in public goods using multiple regression or
other methods.
17
Empirical Illustration
In a nutshell, our model predicts that an exogenous improvement in public goods that closes the
public goods gap should in general change locational equilibria in such a way as to put more
emphasis on sorting by race and less emphasis on sorting by income. We illustrate these effects
with an application to 1990-2000 changes in demographics surrounding the dirtiest polluters in
California. In particular, we use a data set previously assembled by Banzhaf and Walsh (2008).
At the core of the data are 25,166 “communities” defined by a set of tangent half-mile
diameter circles evenly distributed across the urbanized portion of California. This approach has
the virtue of establishing equally sized communities with randomly drawn boundaries, in
contrast to political and census boundaries which may be endogenous.
These neighborhoods are matched to the presence of large industrial emitters of air
pollution and their emissions levels, as recorded in the Toxic Release Inventory (TRI), in each
year from 1990 to 2000. This is a commonly used metric in the literature on the relationship
between environmental quality and demographic composition.14 Firms handling more than
10,000 pounds each year of certain hazardous chemicals have been required to report these
emissions since 1987. This censoring at the reporting threshold gives rise to a kind of errors-in
variables problem. As in the usual case, this is likely to have a “conservative” effect on our
results, biasing them to zero, as some exposed communities are included in the control group.
See de Marchi and Hamilton (2006) for further discussion of the data. We use a toxicity
weighted index of all emissions in the TRI, developed by the US Environmental Protection
Agency and available in its Risk Screen Environmental Indicators model (RSEI). We use a
three-year lagged average, anchored respectively on 1990 and 2000, of the toxicity-weighted
emissions of all chemicals reported since 1988.15 Emissions from each plant are assumed to
disperse uniformly over a half-mile buffer zone, and are allocated to each residential community
accordingly.
The data are also matched to 1990 and 2000 block-level census data on the total
populations of each racial group. They are similarly matched to block-group-level data including
14
E.g., Arora and Cason (1996), Brooks and Sethi (1997), Kriesel et al. (1996), Morello-Frosch et al. (2001),
Rinquist (1997), Sadd et al. (1999).
15
The list of reporting chemicals greatly expanded in 1994. To maintain a consistent comparison of TRI emissions
over time, we have limited the data to the common set of chemicals used since 1988.
18
average incomes. Block-group-level data are allocated down to blocks by population weights,
and block-level data are aggregated back up to the circle-communities. See Banzhaf and Walsh
(2008) for additional details including summary statistics.
These data will provide a good test of our model if a) reductions in TRI emissions in
California between 1990 and 2000 are “large” enough to lead to changes in the reduced form
relationship between TRI emissions and demographic compositions, and b) reductions in
emissions are targeted to the most polluted locations (i.e. a move from region A to regions C or
D in Figure 4). Ten percent of our communities were exposed in the baseline period (19881990), with 4 percent of our communities (or approximately 40% of those exposed in 19881990) losing their exposure by 1998-2000. While it is hard to quantify how large a change is
“large” enough to shift the reduced form relationship, it seems reasonable to assume that this
level of improvement is at least a candidate for shifting the reduced form relationship. Figure 6
shows the link between baseline TRI exposure and the 10-year change in emissions. The figure
clearly demonstrates that the largest improvements were targeted to the communities with the
highest emissions.
Using these data, we test for changes in the structural relationships between income and
pollution and race and pollution over time. In particular, we estimate
𝑀𝑒𝑎𝑛𝐼𝑛𝑐𝑜𝑚𝑒𝑗𝑡 = 𝛿0𝑡 + 𝛿𝐼𝑡 𝐼𝑗𝑡 + 𝛿𝑒𝑡 𝑒𝑗𝑡 + 𝛿𝐿 𝐿𝑗 + 𝜖𝑗𝑡
𝑃𝑐𝑡𝑀𝑖𝑛𝑜𝑟𝑖𝑡𝑦𝑗𝑡 = 𝜇0𝑡 + 𝜇𝐼𝑡 𝐼𝑗𝑡 + 𝜇𝑒𝑡 𝑒𝑗𝑡 + 𝜇𝐿 𝐿𝑗 + 𝜖𝑗𝑡
where 𝐼𝑗𝑡 is an indicator for whether community j was “exposed” to TRI pollution (i.e. was within
a half-mile of such facilities) in year t and 𝑒𝑗𝑡 is the continuous level of emissions from nearby
facilities allocated to community j. The variable I captures the extensive margin and any nonpollution related disamenity of the polluting facilities (visual, noise, smell), while e captures
emissions weighted by EPA’s toxicity index.
Finally, to help control for confounding influences, we include controls for location
effects, including distance to the coast and degrees latitude. However, our main approach to
controlling for unobserved spatial amenities is to employ successively more stringent
neighborhood dummies, including school district dummies, zip code dummies, and community
fixed effects. These effects should capture unobserved public goods. School districts have the
19
advantage of mapping directly into an important but difficult-to-measure local public good.
Table 1 displays the results, with income in the top panel and percent minority in the
bottom panel. The table shows the estimated effects, in each year, of a plant with emissions

1
equal to the 1990 average. That is, for each year, it shows ˆet 
  1(e1990
 0)
 j j

1990 
e
 ˆIt .16

j

j|ej 0

The table clearly shows that as the public goods gap shrinks from 1990 to 2000, sorting on
income diminished and sorting on race increased—precisely the predictions in the model. In the
top panel, we see that an average plant is generally associated with poorer populations, but the
effect is $1,000 smaller (in constant dollars) in 2000 than in was in 1990. This difference is
highly significant and quite robust to different spatial controls, although the level effects in each
year are quite different. As we use more local fixed effects for spatial controls, the sorting on
income appears to disappear entirely in 2000.
The lower panel similarly shows that an average plant is generally associated with more
minorities, but the effect is about 12 percentage points higher in 2000 than 1990. Again, this
difference is highly significant and quite robust to the level of spatial controls.
These results are surprisingly consistent with the models predictions, suggesting that
sorting on environmental amenities does drive correlations between those amenities and
demographics. And yet, changes in those amenities can trigger feedbacks with the sorting on
demographics, so that the demographic changes are counter-intuitive and do not reflect the
structural (i.e. cross sectional) patterns.
Conclusions
Our model suggests that any analysis of the distribution of spatially delineated public goods
across demographic groups must account for endogenous sorting on those demographics, while
at the same time studies of spatial patterns in demographics must account for public goods. We
show that when sorting includes demographics, changes in public goods can lead to counterintuitive results. In particular, improving public good levels in disadvantaged communities can
actually increase segregation. This result has two important implications. For the empirical and
econometric literature, it suggests that structural parameters can vary greatly over time, implying
16
Similar results are obtained by using the average of 2000 emissions.
20
caution is required when using difference-in-differences estimators. More importantly, it
suggests that segregation may be a deeply ingrained feature that is difficult to shake.
21
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24
Figure 1:
, Community Sorting and Bid Functions (
Panel 1: Racial Composition
).
Panel 2: V Function
1
2
Pct
Pct
Pct
Pct
0.9
0.8
W1
W2
B1
B2
V1W
V2W
1.95
V1B
V2B
0.7
0.6
1.9
0.5
0.4
1.85
0.3
0.2
1.8
0.1
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
1.75
0.5
0.55
Panel 3: Y-Bar
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
Panel 4: Bid Functions
0.7
0.01
ybarw
ybarb
0.6
Bid ybarw
0.008
Bid ybarb
0.006
0.5
0.004
0.002
0.4
0
0.3
-0.002
-0.004
0.2
-0.006
0.1
-0.008
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
-0.01
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
25
Figure 2:
, Community Sorting and Bid Functions (
).
Panel 1: Racial Composition
Panel 2: V Function
1
2
Pct
Pct
Pct
Pct
0.9
0.8
W1
W2
B1
B2
1.95
1.9
0.7
1.85
0.6
0.5
1.8
0.4
1.75
V1W
0.3
V2W
1.7
0.2
V1B
1.65
0.1
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
V2B
1.6
0.5
0.55
Panel 3: Y-Bar
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
Panel 4: Bid Functions
-3
0.7
20
x 10
Bid ybarw
ybarw
0.6
Bid ybarb
ybarb
15
0.5
10
0.4
0.3
5
0.2
0
0.1
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
-5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
26
Figure 3: Impact of Changes in G on Bid Functions
0.12
0.06
Bid ybarw
Bid ybarw
Bid ybarb
0.1
0.08
0.04
0.06
0.03
0.04
0.02
0.02
0.01
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
Bid ybarb
0.05
1
0.035
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
0.025
Bid ybarw
Bid ybarw
Bid ybarb
0.03
Bid ybarb
0.02
0.025
0.015
0.02
0.01
0.015
0.005
0.01
0
0.005
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
-0.005
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
-3
20
x 10
0.01
Bid ybarw
Bid ybarw
0.008
Bid ybarb
15
Bid ybarb
0.006
0.004
10
0.002
5
-0.002
0
-0.004
0
-0.006
-0.008
-5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
-0.01
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Proportion White Community 1
0.9
0.95
1
27
Figure 4: Schematic of Stable Equilibria (Propositions III and IV)
28
Figure 5: Difference in Difference and Cross Sectional Relationships
Difference in Differences
0.25
Change in G
0.2
.25 to .5
.5 to .7
.7 to .8
.8 to .9
.9 to 1
Change in Percent Black
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Change in G
Cross Sectional Relationships
0.5
G1=.25
G1=.5
G1=.7
G1=.8
G1=.9
Percent Black
0.4
0.3
0.2
0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
Level of G
0.8
0.9
1
1.1
29
Figure 6. Improvements in TRI Emissions, 1990-2000 (3-year average)
1988-90 to 1998-2000 Change
3.1e+08
-3.1e+08
3.1e+08
0
1988-90 Average Hazard-Weighted TRI Exposure
30
Table 1. Estimated Effect of Average TRI Exposure in 1990 and 2000 and Difference
β2000
β1990
Δβ
Latitude, Dist Coast
-19,563***
(534)
-20,059***
(403)
496
(623)
School Dist dummies
-15,168***
()
-16,407***
(443)
1,239**
(631)
Zip Code dummies
-6,444***
(423)
-7,707***
(351)
1,263**
(510)
238
(480)
-1,259***
(368)
1,498***
(399)
Latitude, Dist Coast
29.22***
(0.82)
16.31***
(0.63)
12.91***
(1.01)
School Dist dummies
18.99***
(0.66)
6.71***
(0.55)
12.29***
(0.82)
Zip Code dummies
11.49***
(0.58)
-0.76*
(0.46)
11.49***
(0.58)
Community FE
2.95***
(0.67)
-9.26***
(0.46)
12.21***
(0.52)
Outcome/Spatial controls
Mean Income (1990$)
Community FE
Pct Minority (pct points)

Estimates are for the effect of
ˆet 
1
  1(e1990
 0)
 j j
e
j|ej 0
1990
j

  ˆ t . That is, they are the effect in year t of a plant
 I

with average 1990 emission levels. The effect accounts for the extensive margin of such a “typical plant” as well as
the intensive margin of emission levels.
Robust standard errors are in parentheses.
***Significant at 1% level, ** 5%, * 10%.
31
Appendix.
Proof of Proposition I
We begin by noting that when public good levels are equal:
(a)
(b)
(c)
𝐵𝑖𝑑𝑌̅𝑏 < 𝐵𝑖𝑑𝑌̅𝑤 when PCTwC1 < 1-β,
𝐵𝑖𝑑𝑌̅𝑏 > 𝐵𝑖𝑑𝑌̅𝑤 when PCTwC1 > 1-β, and
𝐵𝑖𝑑𝑌̅𝑏 = 𝐵𝑖𝑑𝑌̅𝑤 = 0 when PCTwC1 = 1-β.
(a) can is true because Drj is highest in the community with more of type j. With GC1=GC2, it
follows that Vrj is also higher in that community. (b) follows by the symmetric logic. (c) also
follows by similar logic: in this case, DrC1=DrC2 so VC1=VC2. The rest follows from
Observations 4 and 5.
Proof of Proposition II20
The first part of the proof follows directly from the continuity of the bid functions and the
implicit function theorem. It states that the single unstable integrated equilibrium and the
segregated equilibrium with all type-b individuals in C2 that are present when public goods are
equal continues to hold and be the only possible equilibria for at least small perturbations in the
G’s around this point.
The second part of the proposition states that the stable segregated equilibrium with all
type-b individuals in C1 is more robust. When public good levels are higher in Community 2,
the boundary income bid function for type-w individuals will always be positive when PCTwC1 =
1-2β (the sorting with all type b individuals in C1), since type w perceives both the G and the D
components to be higher in C2. Conversely, type-b individuals always prefer the demographic
composition of C1 under this sorting. Thus, when PCTwC1 = 1-2β, their boundary income bid
function will remain negative until the difference in public goods becomes large enough to offset
these demographic preferences. As a result, the segregated equilibrium with all type-b
individuals in C1 will continue to exist at least until GC1 becomes low enough that VbC2 > VbC1.
20
Note: Still need to flesh this out and show that a) that this equilibria survives longer than the segregated equilibria
with type b in community 2, and b) that once this equilibirium disappears, it is gone for good.
32
Proof of Proposition III.
Part (i).
We begin by re-writing Equation 4 as follows:
𝑑𝐵𝑖𝑑𝑌̅𝑟
𝑑𝑌̅𝑟
𝑑𝐷𝑟𝐶2
𝑑𝐷𝑟𝐶1
1
𝐶2 𝐶2
𝐶1 1𝐶1
=
+
[(𝑈
𝑉
−
𝑈
𝑉
) − 𝑈𝑥𝐶1 ] 𝐶2
𝑉 𝐷
𝑉 𝐷
𝐶1
𝐶1
𝐶1
𝐶1
𝑑𝑃𝐶𝑇𝑤
𝑑𝑃𝐶𝑇𝑤
𝑑𝑃𝐶𝑇𝑤
𝑑𝑃𝐶𝑇𝑤
𝑈𝑥
By assumption, Bidy → y as GC1→-∞. Thus, by the Inada conditions 𝑈𝑥𝐶2 →∞. Therefore, for
𝑑𝑌̅
𝑟
sufficiently low GC1, the entire term in brackets → 0 and we can focus on the first term 𝑑𝑃𝐶𝑇𝑤
𝐶1 .
As shown in the text,
𝑑𝑌̅𝑏
0.5
=
−
<0
𝑑𝑃𝐶𝑇𝑤 𝐶1
𝛽𝑓𝑏 (𝑌̅𝑏 )
𝑑𝑌̅𝑤
0.5
=
>0
𝑑𝑃𝐶𝑇𝑤 𝐶1 (1 − 𝛽)𝑓𝑤 (𝑌̅𝑤 )
whenever 𝑉𝑟𝐶1 > 𝑉𝑟𝐶2 . Therefore,
𝑑𝐵𝑖𝑑𝑌̅𝑏
<0
𝑑𝑃𝐶𝑇𝑤 𝐶1
and
𝑑𝐵𝑖𝑑𝑌̅𝑤
> 0.
𝑑𝑃𝐶𝑇𝑤 𝐶1
In other words, the boundary income bid function is monotonically increasing for type w and
monotonically decreasing for type b.
Part (ii).
0.5−𝛽
Next, note that at PCTwC1=1-2β, 𝑌̅𝑏 =𝑌𝑏𝑀𝑎𝑥 and 𝑌̅𝑤 =𝐹𝑤−1 ( 1−𝛽 ) < 𝐹𝑤−1 (1 − 2𝛽) given β<0.5.
Again, since Bidy → y, it follows that 𝐵𝑖𝑑𝑌̅𝑏 =𝑌𝑏𝑀𝑎𝑥 and 𝐵𝑖𝑑𝑌̅𝑤 =𝐹𝑤−1 (1 − 2𝛽). Using Equation
(1), we then have 𝐵𝑖𝑑𝑌̅𝑏 > 𝐵𝑖𝑑𝑌̅𝑤 . Note that by Observation 4, this cannot be an equilibrium.
Part (iii)
33
0.5
Similarly, note that at PCTwC1=1, 𝑌̅𝑏 =𝑌𝑏𝑀𝑖𝑛 =0 and 𝑌̅𝑤 =𝐹𝑤−1 (1−𝛽) > 0. Thus 𝐵𝑖𝑑𝑌̅𝑤 > 𝐵𝑖𝑑𝑌̅𝑏 = 0.
Again, by Observation 4, this cannot be an equilibrium.
Part (iv)
We now combine parts (i) to (iii). By part (ii), we have that at PCTwC1=1-2β, 𝐵𝑖𝑑𝑌̅𝑏 > 𝐵𝑖𝑑𝑌̅𝑤 .
By part (i), we know 𝐵𝑖𝑑𝑌̅𝑏 is monotonically decreasing in PCTwC1 and 𝐵𝑖𝑑𝑌̅𝑤 is monotonically
increasing. By part (iii) we have that at PCTwC1=1, 𝐵𝑖𝑑𝑌̅𝑏 < 𝐵𝑖𝑑𝑌̅𝑤 . Therefore, the two
boundary income bid functions cross once, with 𝐵𝑖𝑑𝑌̅𝑏 crossing 𝐵𝑖𝑑𝑌̅𝑏 from above. This is a
stable integrated equilibrium. Moreover, as noted above the segregated equilibria are ruled out
by Observation 4.
Therefore, a single equilibrium exists, it is integrated, and it is stable.
34