Segregation and Tiebout Sorting:

Investigating the Link Between Public Goods and Demographic Composition*

H. Spencer Banzhaf† and Randall P. Walsh‡

July 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, Cornell, 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, and NBER.

‡ Associate Professor, Department of Economics, University of Pittsburgh, and NBER.

1

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 importance, 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 segregation. And Bayer, Fang, and McMillan (2005) and Sethi and

Somanathan (2004) show that reducing income inequality between groups can actually increase

group 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 place-

based policy shocks. This is unfortunate because, as the above epigraph illustrates, many of the

2

policy remedies for reducing group inequity focus on investments in minority communities.

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 effective 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. At the same time, 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 a model of group segregation. 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

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 a given type lead to social spillovers (i.e. concentrations of businesses, community networks, restaurants, non-profit services, etc.) that disproportionately benefit that type.

3

can segregate themselves by groups or income strata according to their demand for local public

goods. In particular, we adapt Epple et al.'s model of vertically differentiated communities, a

model used in other recent work as well (Banzhaf and Walsh 2008, Sethi and Somanathan 2004).

We combine both traditions into a general equilibrium model in which households choose

a community based on both its demographics and its 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 sorting is driven by tastes for the exogenous

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 on 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 on 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

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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

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) by including an exogenous public good.3 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, the minority, has measure β < 0.5 and type-w has measure (1-β). One

obvious interpretation of these types is as racial groups (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).

There is heterogeneity in income within type, which is described by the continuous

distribution functions Fr. We impose the following two assumptions on the income distributions

of type b and w individuals:

Fw(y) ≤ Fb(y) for all y (1)

����� � ����� � � ��� � � � �� � ���� � � � ����� � ����� (2)

Condition (1) says that type w is richer than type b in the sense of first-order stochastic

dominance. Condition (2) says that the groups’ income distributions cannot be “too different.”

In particular, when individuals are stratified solely by income, there will be some mixing of

3 We also provide a more general characterization of the equilibria of our model, including integrated and segregated equilibria. Sethi and Somanathan (2004) focus on integrated equilibria.

5

groups.

Each individual is assumed to consume one unit of housing and to have preferences over

a numeraire good (with a price set to unity), an exogenous community public good level Gj and

endogenous community demographic composition ���. 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 (3):

��� � ��� � � ! "#$� ! ���%& = U[x, Vrj] (3)

The function U(.) is increasing at a decreasing rate in both of its arguments and V(.) is increasing

in both arguments. More compactly, let Vrj = V(Gj, Dr

j). Consumption x is simply net income

after subtracting housing costs. We impose the Inada condition that Ux→∞ as x→0.

Demographic preferences are captured by Drj = Dr(PCTr

j), 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.4

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(PCTr

j-ρr)], where g’<0, 1-β ≤ ρw ≤ 1, and ρb > 2β.

Additionally, without loss of generality, we assume that when public good levels differ

the level of the public good is higher in Community 2 than in Community 1: GC2≥G

C1. Further,

in order to close the model, we assume that the price level in Community 1 is equal to 0: PC1 = 0.

This normalization allows 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

4 Our model is entirely capable of relaxing this assumption. However, in this paper we restrict our characterization of this case, which we view as most relevant.

6

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:

'()� *� * + �,-./�!�+ �,-./�

*

E2. Each individual resides in his preferred community. That is, for all type r

individuals choosing to live in community j:

U[Yi – Pj, V(Gj, Drj] ≥ U[Yi – P-j, V(G-j, Dr

-j]

E3. Housing markets clear, so 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 in V( ),

combined with the assumption that each household consumes an identical and fixed quantity of

housing, implies that preferences exhibit the “single crossing” property. This in turn implies that

the equilibria in the model will exhibit stratification in income within each type.5 In other words,

in equilibrium, for each type there exists a boundary income, �0�, such that all individuals of type

r with incomes greater than �0�, choose to live in the community that that type 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 1-,234 to be the price level in Community 2 that makes these individuals

indifferent. Recalling that PC1 is normalized to 0, 1-,234 is implicitly defined by equation (4):

�5�3�! "�$6�! ��6��7 � �5�3� � 1-,234 ! "�$68! ��68�7. (4)

5 See Epple et al. (1984) for discussion of single crossing and stratification in equilibrium sorting models.

6 Formally, the boundary income is defined as that income at which for all Y > �3�, �9!�� : �9!�; if "�� : "�;. Often,

this will be for an indifferent household. However, there are equilibria where all type-b individuals locate in the

same community. In this case �3� � ��<.= when Vbj is higher in this community and �3� � ��<>? when Vb

j is higher in the other community. Because β<0.5, there must always be type-w individuals living in both communities.

7

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) (clearing in the housing markets), 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 - PCTw

C1, (5)

PCTbC1 = 1 - PCTw

C1, and (6)

PCTwC2 = PCTw

C1 + 2β - 1. (7)

Consequently, the demographic qualities can also be expressed as a function of PCTwC1, Dr

j =

Drj(PCTw

C1).

Further, given that single crossing holds within each type, we can also identify the

boundary incomes as a function of '(�6�, conditional on the relative levels of V(·):

�3� �@ABAC��� D� � � � '(E6� F �� � ��� G -H "�6� : "�68!

��� D� � � �� � '(E6��� G -H "�68 : "�6�!I (8)

�3� �@ABACJ�� D� � � �� � �� � '(E6��� � � G -H "�6� : "�68!

J�� D� � � '(E6�� � � G -H "�68 : "�6��

I (9)

We can similarly characterize the derivatives of these boundary incomes with respect to

'(E6�:

8

,�3�, '(E6� �@ABAC * ����H���3�� *: � -H "�6� : "�68!� ����H���3�� * � -H "�68 : "�6�!

I

,�3�, '(E6� �@ABAC�* ����� � ��H���3�� * � -H "�6� : "�68!

����� � ��H���3�� *: � -H "�68 : "�6��I

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 �3. The figure illustrates

results for the following specification: β=0.25, Drj = (PCTr

j-ρr)2, ρb=0.5, ρw=0.9, and Vr

j = Dr

j +

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, Yb

max=1, and Ywmax=1.1. Finally, GC1=G

C2=1.

Panel 1 of Figure 1 shows the racial compositions in each community as a function of

PCTwC1. These are defined by the market clearing identities established in equations (5)-(7)

above.) The figure shows that racial compositions are identical when PCTwC1 = PCTw

C2 = 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 PCTw

C1, but still remains higher than VwC2. Type-b individuals never

7 In their recent work on tipping points, Card, Mas and Rothstein (2008) find that ρw is around 0.9, which is also consistent with the survey literature. This survey literature also suggests that an appropriate level for ρb is around 0.5 Farley et al. (1978) and Farley and Krysan (2002).

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.

9

achieve their bliss point and Vbj is always increasing in PCTb

j. Because in this example 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, Vb

C1 > VbC2 but Vw

C2 > VwC1, whereas when PCTw

C1 > 1-β, 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

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 �3� = 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.

10

Observation 2. When VwC1 ≠ Vw

C2, there will be some �3� at which type-w individuals are

indifferent. All type-w individuals with income higher than �3� will reside in the more desirable (to w) community and poorer type-w individuals will live in the less desirable.

Moreover, �3� can never drop below ���5���� � �� �� � ��K 7. 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 ≥/≤ Vw

C1 ↔ PC2 ≥/≤ 0. Moreover, PC2 = 1-,23L . 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 depicted in 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 1-,23M ≤ 1-,23L. (Otherwise, if 1-,23M

> 1-,23L, then the boundary type-b individual would outbid the boundary type-w

individual for housing in C2.) In particular, if VbC2 > Vb

C1 and VwC2 > Vw

C1 then 1-,23L ≥ 1-,23M > 0. If VbC1 > Vb

C2 and Vw

C2 > VwC1 then 1-,23L > 0 > 1-,23M . And if Vb

C1 > VbC2

and VwC1 > Vw

C2 then 0 > 1-,23L ≥ 1-,23M . Note also that if VbC1 > Vb

C2 then �3� = YbMin =

0. Otherwise, �3� = 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 1-,23L = 1-,23M . 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

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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 (10):

,1-,234, '(E6� � D ,�3�, '(E6� ��?68 � �?6�� F N�O68"P68 ,��68, '(E6� � �O6�"P6� ,��6�, '(E6�QG ��?68

(10)

where Uxj is the derivative of the utility function with respect to the numeraire, evaluated in

Community j at the appropriate income (i.e. �3�) 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 necessitated to change PCTwC1. As is shown in Panel 3, for

type w the boundary income increases over the range 0.5 to 0.75 and, because the positive value

of the bid function implies ��?68 � �?6�� >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 0.5 to 0.75 in the figure, reducing the

bid function. In the case of type-w individuals, the effect of closing the gap 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 PCTw

C1 > 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.

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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 wY switches communities with a type-b individual arbitrarily close to

bY , 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.

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=G

C2, there will be three equilibria: two stable segregated equilibria with PCTw

C1 = 1 and PCTwC1 = 1-2β respectively and one unstable integrated equilibrium

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 �3� is equal to 0 or YbMax and the marginal individual

will be confined to incomes above or below these levels respectively.

13

with PCTwC1 = 1-β.

Proof: See Appendix

We now turn to an 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=PCTw

C2. 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.

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.

10 We do note that the boundary income functions are now non-differentiable at the value of PCTw

C1 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).

14

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 10, 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

polar 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 $R6� < GC2

such that for all GC2

> GC1 > $R6�

there will exist exactly three equilibria, two symmetric stable segregated equilibria and

one unstable integrated equilibrium. Further, there will exist $RR6� < GC2

satisfying

V[GC2

, Db(1)] > V[$RR6�, Db(1-2β)] such that for all GC2

> GC1 > $RR6� 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

public goods component. In particular, individuals will be stratified by income instead of

segregated by race.

15

To operationalize this intuition in our model, assume that, for given levels of GC2, DrC1,

and DrC2, BidY

r→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 (Condition 2), 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, Dr

C1, and DrC2, STUVWXYZ[ 1-,234 � �3�. Then, there

exists $\6� such that for all GC1

<$\6�<GC2

there is a single equilibrium which is

integrated and 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.

However, C2, the high public good community, will still have a higher proportion of

type-w residents than C1:

Proposition IV Under the conditions of Proposition III, for sufficiently low G

C1, PCTW2>PCTW1.

Proof: see the appendix.

The proof of Proposition IV follows intuitively from Condition (1), first-order stochastic

dominance in the income distributions. Essentially, since type-w is richer on average, when

there is perfect stratification by income then the richer community also has a higher share of

type w. In this way, our model captures as a special case the earlier insights from McGuire

(1974), who showed how segregation can be driven by group differences in the willingness to

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.

16

pay for public goods. It also speaks to important policy issues. For example, the “environmental

justice” movement has shown that minorities are disproportionately exposed to pollution (e.g.

Bullard 2000). This model suggests that discrimination (at least in pollution patterns) is not

necessary to drive this result, as it may be driven by differences in incomes and sorting on

amenities (Been 1994).

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

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 shows how the equilibria evolve as GC1 improves. On the vertical axis it shows

17

levels of PCTwC1 that are sustainable as equilibria for different levels of GC1, shown on the

horizontal axis. The figure continues to illustrate the example shown in Figure 3. At the far left

of the figure, at very low levels of GC1, there is a single integrated equilibrium. As GC1

improves, there is a slight but steady decrease in PCTwC1 as richer minorities migrate into C1.

Then a “tipping point” is reached, with whites “fleeing” C1 to the full extent possible in

equilibrium, and the communities becoming completely segregated. Eventually, as GC1

improves to the point that it is better than GC2, rich whites begin to move in and the process

slowly reverses, until another tipping point is reached and “gentrification” leads to the

community become all-w. Figure 5 similarly 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, Bullard 2000, 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 5. 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 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

12 This for instance is the case in the segregated equilibria supported in regions C and C’ in Figure 5.

18

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 5).13 On the surface, this result resembles that of Bayer et al.

(2005) and Sethi and Somanathan (2006), who find that increasing the number of high socio-

economic status minorities 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, Cameron and McConnaha (2006), Greenstone

and Gallagher (2008), and Vigdor (2009) 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 6 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.

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 (11) and (12).

13 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.

19

'(]^_`a^(�.b � cb F d. F �b `ee�(^`_.b F f.b (11)

'(]^_`a^(�.� � c� F d. F �� `ee�(^`_.� F f.�! (12)

with �� : �b : �, then first differencing the equations leads to:

g '(]^_`a^(�. � gc F ��g `ee�(^`_. F ��� � �b� `ee�(^`_.b F h. (13)

If `ee�(^`_.b is omitted from equation (13), the estimate of β1 will be biased downward

when g `ee�(^`_ and `ee�(^`_b 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, the role of public goods 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.

20

Empirical Illustration

In a nutshell, our model predicts that an exogenous improvement in public goods that closes the

public goods gap should change locational equilibria in such a way as to put more emphasis on

sorting by race.14 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.15 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

14 It might seem equally true that it should put less emphasis on sorting by income. However, this is not necessarily so. The reason can be seen in the logic of Figure 6. As GC1 rises and PCTw

C1 falls, both effects lead to a higher absolute value of the slope in the cross-sectional relationship between race and G. In contrast, as GC1 rises and mean income rises in C1, there are two offsetting effects on the cross-sectional relationship between income and G.

15 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).

21

emissions of all chemicals reported since 1988.16 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. These block-level data are aggregated back up to the circle-

communities assuming uniform distributions within each block. 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 5). Ten percent of our communities were exposed in the baseline period (1988-

1990), with 4 percent of our communities (or approximately 40% of those exposed in 1988-

1990) 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 7

shows the link between baseline TRI exposure and the 10-year change in emissions. The figure

clearly demonstrates that the largest improvements occurred in the communities with the highest

baseline emissions.

Using these data, we test for changes in the structural relationships between race and

pollution over time. In particular, we estimate

ic]-jk)-cl�m � hbm F hnm �̂m F homp�m F hqe� F r�m (14)

where �̂m 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 p�m is the continuous level of emissions from nearby

facilities allocated to community j. The variable I captures the extensive margin and any non-

pollution related disamenity of the polluting facilities (visual, noise, smell), while e captures

16 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.

22

emissions weighted by EPA’s toxicity index.

Finally, to help control for confounding influences, we include controls for location

effects, Lj, including distance to the coast and degrees latitude. However, our main approach to

controlling for unobserved spatial amenities is to employ successively more stringent time-

invariant neighborhood dummies, including school district dummies, zip code dummies, and

community fixed effects. These effects should capture unobserved public goods. For example,

school districts have the advantage of mapping directly into an important but difficult-to-measure

local public good. When community fixed effects are employed, the model is essentially that of

equation (13), a difference-in-differences model correcting for changes in the parameters over

time.

Table 1 displays the results. It shows the estimated effects, in each year, of a typical level

of exposure in 1990 (for exposed communities). That is, for each year, it shows

t

I

j

j

j j

t

e ee

µµ +

> ∑∑ |

1990

1990 )0(1

1.17

The table clearly shows that as the public goods gap shrinks from 1990 to 2000, sorting on race

increased—precisely the predictions in the model. As the time-invariant locational controls get

more local, the estimated levels of the cross-sectional sorting parameter shrink, perhaps because

of unobservables or perhaps simply because less variation remains for estimating the model. But

in all cases, the estimated change in the relationship is consistently in the range of 2-3 percentage

points and highly significant. That is, the effect of a “typical” polluting plant was associated

with more minorities in both years, but in 2000 the typical plant was associated with an increase

in nearby minorities 2-3 percentage points more than the same sized plant in 1990.

These results assume that the same reduced form relationship between race and pollution

holds throughout California. But if pollution reductions are spread unevenly across the state,

then changes in the equilibrium sorting patterns may also be uneven. To capture this possibility,

we allow the pollution parameter to change differently in each county. In particular, we replicate

17 Similar results are obtained by using the average of 2000 emissions as an alternative normalization.

23

the above model, interacting the 2000 pollution coefficients with the average level of pollution

reductions in the community’s county. Controlling for zip code dummies, we find that in the

average county, a typical plant continued to be associated with more nearby minorities in 2000

than 1990, by 3.7 percentage points, quite close to the results shown in Table 1. However, this

masks some heterogeneity. A county experiencing now change in pollution saw only a 2.2

percentage point change, while the county experiencing the greatest improvement in pollution

saw a 5.5 percentage point change in such sorting. These differences are statistically significant.

These results are surprisingly consistent with our model’s 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 simple differences-in-

differences do not reflect the true 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 counter-

intuitive 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

caution is required when using difference-in-differences estimators. More importantly, it

suggests that group segregation may be an inherent feature of household sorting that is difficult

to overcome.

24

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27

Fig

ure

1: PCTW1, C

om

mu

nit

y S

ort

ing a

nd

Bid

Fu

nct

ion

s (G

C1=

G C

2=

1)

Pan

el 1

: R

acia

l C

om

posi

tion

Pan

el 2

: V

Funct

ion

Pan

el 3

: Y

-Bar

Pan

el 4

: B

id F

unct

ions

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

Proportion White Community 1

Pct W1

Pct W2

Pct B1

Pct B2

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

11.75

1.8

1.85

1.9

1.952

Proportion White Community 1

V1W

V2W

V1B

V2B

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Proportion White Community 1

ybar w

ybar b

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1-0.01

-0.008

-0.006

-0.004

-0.0020

0.002

0.004

0.006

0.008

0.01

Proportion White Community 1

Bid ybar w

Bid ybar b

28

Fig

ure

2: PCTW1, C

om

mu

nit

y S

ort

ing a

nd

Bid

Fu

nct

ion

s (G

C1=

0.9

, G

C2=

1)

Pan

el 1

: R

acia

l C

om

posi

tion

Pan

el 2

: V

Funct

ion

Pan

el 3

: Y

-Bar

Pan

el 4

: B

id F

unct

ions

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

Proportion White Community 1

Pct W1

Pct W2

Pct B1

Pct B2

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

11.6

1.65

1.7

1.75

1.8

1.85

1.9

1.952

Proportion White Community 1

V1W

V2W

V1B

V2B

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Proportion White Community 1

ybar w

ybar b

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1-505

10

15

20x 10-3

Proportion White Community 1

Bid ybar w

Bid ybar b

Figure 3: Impact of Changes in G

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.850

0.02

0.04

0.06

0.08

0.1

0.12

Proportion White Community 1

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.850

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Proportion White Community 1

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85-5

0

5

10

15

20x 10

-3

Proportion White Community 1

Figure 3: Impact of Changes in GC1

on Bid Functions

0.85 0.9 0.95 1

Proportion White Community 1

Bid ybarw

Bid ybarb

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.850

0.01

0.02

0.03

0.04

0.05

0.06

Proportion White Community 1

0.85 0.9 0.95 1

Proportion White Community 1

Bid ybarw

Bid ybarb

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85-0.005

0

0.005

0.01

0.015

0.02

0.025

Proportion White Community 1

0.85 0.9 0.95 1

Proportion White Community 1

Bid ybarw

Bid ybarb

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Proportion White Community 1

29

0.9 0.95 1

Bid ybarw

Bid ybarb

0.9 0.95 1

Bid ybarw

Bid ybarb

0.9 0.95 1

Bid ybarw

Bid ybarb

30

Figure 4: Evolution of Equilibria as GC1

improves.

31

Figure 5: Schematic of Stable Equilibria (Propositions II and III)

32

Figure 6: Difference in Difference and Cross Sectional Relationships

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Change in G

Change in Percent Black

Difference in Differences

.25 to .5

.5 to .7

.7 to .8

.8 to .9

.9 to 1

Change in G

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0

0.1

0.2

0.3

0.4

0.5

Level of G

Percent Black

Cross Sectional Relationships

G1=.25

G1=.5

G1=.7

G1=.8

G1=.9

33

Figure 7. Improvements in TRI Emissions, 1990-2000 (3-year average)

1988-90 to 1998-2000 Change

1988-90 Average Hazard-Weighted TRI Exposure0 3.1e+08

-3.1e+08

3.1e+08

34

Table 1. Estimated Effect of Average TRI Exposure in 1990 and 2000 (and Difference) on

Percent Minority.

Spatial controls L β2000 β1990 ∆β R2

Latitude, Dist Coast

21.3*** 24.3*** 3.0*** 0.10

(0.6) (0.8) (1.0)

School Dist dummies

11.7*** 14.3*** 2.6*** 0.45

(0.5) (0.7) (0.8)

Zip Code dummies

4.2*** 7.0*** 2.8*** 0.69

(0.5) (0.6) (0.7)

Community fixed effects

-1.6*** 0.2 1.8*** 0.94

(0.5) (0.6) (0.5)

Estimates are for the effect oft

I

j

j

j j

t

e ee

µµ +

> ∑∑1990

1990 )0(1

1. That is, they are the effect in year t of the

typical exposure experienced in 1990 (if any) compared to no exposure. The effect accounts for the extensive margin of such “typical exposure” as well as the intensive margin of emission levels.

Robust standard errors are in parentheses.

***Significant at 1% level, ** 5%, * 10%.

35

Appendix.

Proof of Proposition I

We begin by noting that when public good levels are equal:

(a) 1-,23M 1-,23L when PCTwC1 < 1-β,

(b) 1-,23M : 1-,23L when PCTwC1 > 1-β, and

(c) 1-,23M � 1-,23L � � when PCTwC1 = 1-β.

(a) is true because Drj is highest in the community with more of type j. With GC1=G

C2, it follows that Vr

j is also higher in that community. (b) follows by the symmetric logic. (c) also follows by similar logic: in this case, Dr

C1=DrC2 so VC1=V

C2. The rest follows from Observations 4 and 5.

Proof of Proposition II21

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 are 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 > Vb

C1.

21 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.

36

Proof of Proposition III.

Part (i).

We begin by re-writing Equation 4 as follows:

,1-,234, '(E6� � ,�3�, '(E6� F DN�O68"P68 ,��68, '(E6� � �O6�"P�6� ,��6�, '(E6�Q � �?6�G ��?68

By assumption, Bidy → y as GC1→-∞. Thus, by the Inada conditions �?68→∞. Therefore, for

sufficiently low GC1, the entire term in brackets → 0 and we can focus on the first term s234st6u�WX.

As shown in the text,

,�3�, '(E6� � � ����H���3�� * �

,�3�, '(E6� � ����� � ��H���3�� *: �

whenever "�6� : "�68. Therefore,

,1-,23M, '(E6� �

and

,1-,23L, '(E6� : �� In other words, the boundary income bid function is monotonically increasing in PCTw

C1 for

type w and monotonically decreasing for type b.

Part (ii).

Next, note that at PCTwC1=1-2β, �3�=��<>? and �3�=��� vb�w�x��x y ����� � ��� given β<0.5.

Again, since Bidy → y, it follows that 1-,23M=��<>? and 1-,23L=����� � ���. Using Equation

(1), we then have 1-,23M > 1-,23L �** Note that by Observation 4, this cannot be an equilibrium.

37

Part (iii)

Similarly, note that at PCTwC1=1, �3�=��<.= =0 and �3�=��� v b�w��xy > 0. Thus 1-,23L > 1-,23M= 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, 1-,23M > 1-,23L .

By part (i), we know 1-,23M is monotonically decreasing in PCTwC1 and 1-,23L is monotonically

increasing. By part (iii) we have that at PCTwC1=1, 1-,23M < 1-,23L . Therefore, the two

boundary income bid functions cross once, with 1-,23M crossing 1-,23M 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.

Proof of Proposition IV.

By Observation (5), in an integrated equilibrium 1-,23L = 1-,23M. Since Bidy → y as GC1→-∞, it

follows that �3� � �3�=*�3. By first order stochastic dominance, F(�3��zF(�3�� ≥ 1.

Therefore,

��3����3�� {� � ��3��� � ��3��**

and

���3���� � ����3�� { *��� � ��3����� � ���� � ��3����

This shows that the ratio of type b to type w is higher in C1 than C2. Adding 1 written as

�� � ���� � ��3����� � ���� � ��3��� to both sides of the equation and inverting both sides leads to

38

�� � ����3���� � ����3�� F ���3�� �� � ���� � ��3����� � ���� � ��3��� F ��� � ��3���!

which is the desired result.