NCEE 0

NATIONAL CENTER FOR

ENVIRONMENTAL ECONOMICS

Quantifying the Distribution of Environmental Outcomes for
Regulatory Environmental Justice Analysis

Kelly Maguire and Glenn Sheriff

Working Paper Series

Working Paper # 11 -02
April, 2011

^e.0 sr^ U.S. Environmental Protection Agency
g	ra National Center for Environmental Economics

s	z 1200 Pennsylvania Avenue, NW (MC 1809)

^ Washington, DC 20460
sP http://www.epa.gov/economics

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Quantifying the Distribution of Environmental Outcomes for
Regulatory Environmental Justice Analysis

Kelly Maguire and Glenn Sheriff

NCEE Working Paper Series
Working Paper # 11-02
April, 2011

DISCLAIMER

The views expressed in this paper are those of the author(s) and do not necessarily represent those
of the U.S. Environmental Protection Agency. In addition, although the research described in this
paper may have been funded entirely or in part by the U.S. Environmental Protection Agency, it
has not been subjected to the Agency's required peer and policy review. No official Agency
endorsement should be inferred.

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Quantifying the distribution of environmental outcomes

for regulatory environmental justice analysis

Kelly Maguirea, Glenn Sheriff13

Abstract: Economists have long been interested in measuring distributional
impacts of policy interventions. As environmental justice (EJ) emerged as an
ethical issue in the 1970s, the academic literature has provided statistical
analyses of the incidence and causes of various environmental outcomes as
they relate to race, income and other demographic variables. In the context of
regulatory impacts, however, there is a lack of consensus regarding what
information is relevant for EJ analysis, and how best to present it. This paper
helps frame the discussion by suggesting a set of questions fundamental to
regulatory EJ analysis, reviewing past approaches to quantifying distributional
equity, and discussing the potential for adapting existing tools to the regulatory
context.

Keywords: environmental justice, regulatory impact analysis, distributional
analysis, equity, inequality index

Subject Area: Distributional effects
JEL Codes: C43, D63, Q52

The views in this paper are those of the authors and do not reflect official U.S.
EPA policy. No endorsement should be inferred. The authors thank Ann
Wolverton for valuable comments and suggestions, and Mark Corrales and
Adam Wagstaff for helpful discussions.

a U.S. EPA National Center for Environmental Economics

b (corresponding author) U.S. EPA National Center for Environmental Economics, 1200
Pennsylvania Ave., NW (MC 1809T), Washington, DC 20460, 202-566-2265,
sheriff. glenn@epa. gov.

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1. Introduction

Economists have been interested in analyzing the distribution of environmental benefits
for almost as long as they have been calculating the benefits themselves. While the tools
for conducting benefits analysis are well developed, those for examining equity, or
distributional effects, are less so.

Most OECD countries routinely perform a regulatory impact analysis of significant new
environmental rules.1 These analyses typically contain an estimate of monetized benefits
and costs of options under consideration. They may also discuss how these benefits and
costs are distributed across various subgroups, economic sectors, or regions. In the U.S.,
various Executive Orders (EO) require some distributional analysis (e.g., EO 13045
addresses children's health, E.O. 13211 addresses energy issues). Relevant to this
discussion, EO 12898, Federal Actions to Address Environmental Justice in Minority
Population and Low-Income Populations, requires federal agencies to address
"disproportionately high and adverse human health or environmental effects...on
minority populations and low-income populations" [2], To date, however,
implementation of EO 12898 has been slow and inconsistent (see [3, 4] for critiques of
U.S. Environmental Protection Agency (EPA) implementation).

To be useful in the policy-making process, distributional analysis should facilitate the
ranking of alternative regulatory outcomes. Such rankings are inherently normative, and
thus should reflect the views of society as expressed through the political process as

1 As of 2000, half of OECD countries used regulatory impact analysis across the board,
with an additional six using it for specific types of regulation [1],

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opposed the views of the technical staff preparing the analysis. There is a tradeoff. Purely
descriptive analysis such as pollution exposure rates by subgroup may be difficult to
digest and interpret in a consistent manner. However, methods for aggregating the data
into easily presented rankings have the potential for implicitly reflecting the staffs value
judgments.

In addition, for the purposes of both decision-making and environmental justice there is a
need for consistency and transparency. These concepts are related. Consistency implies
that the decision-maker use a similar framework to make decisions across rules. If a
certain distribution of outcomes is preferred to another for one pollutant, then a similar
ordering should be preserved for others. For the purposes of EJ, defined by the U.S. EPA
to include "fair treatment and meaningful involvement," transparency in decision-making
is essential [5], Interested parties should be able to identify the information and
methodology used to make a decision is a way that is clear and accessible. In identifying
methods for use in EJ analysis for regulatory policy we are cognizant of the need for both
consistency and transparency.

Here, we present various methods used in the (mostly) economics literature to quantify
the distribution of environmental impacts, and evaluate their usefulness through the prism
of how the results can be used to guide the environmental regulatory process. We begin
Section 2 with a discussion of three fundamental questions that a distributional analysis
of environmental policy options needs to address. In Section 3 we discuss efforts to
describe environmental or health outcomes for different subgroups. In Section 4 we

2 The few examples discussed here are by no means comprehensive. For recent reviews
of the EJ literature see [6, 7, 8],

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describe methods to aggregate this information in a way that allows society to rank
policies in a transparent and consistent manner using inequality indices. In Section 5 we
offer concluding thoughts and some potential steps forward.

2. Three Fundamental Questions for Regulatory EJ Analysis

Environmental justice is a concern that certain subgroups, typically defined by race or

"3

income, have historically borne a disproportionate share of environmental burden. In the
context of new regulations it is important to understand the questions a distributional
analysis of environmental policy should address.

With regulatory impact analysis the primary concern is with the distributional effects
associated with the options under consideration, as opposed to the causes of inequities
typically investigated by the academic literature. The goal is to provide the decision-
maker and public with information regarding the degree to which regulatory options
under consideration remove or worsen previous disparities in environmental outcomes for
vulnerable communities, or create new disparities where none existed.

Before turning to the questions, it is important to identify the outcome to be measured.
Options include pollution (e.g., parts per million of ozone), health effects (e.g., number of
cases of asthma), and monetized benefits (e.g., willingness to pay for reductions in
asthma cases). Here, we adopt the position dominant in the environmental justice
community (if not the economic literature) that the distribution of physical outcomes

"3

Distributional effects could be assessed across many potential socio-economic variables.
Although environmental justice tends to focus on the distribution across race, ethnic
groups, and income levels, one could easily apply the tools to other subgroups based on
age, education, geographic location, etc.

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(e.g., pollution or health effects), rather than their monetized value is most appropriate for
regulatory analysis. We also focus exclusively on the distribution of environmental
outcomes, not the distribution of economic costs (higher prices, reduced employment,
etc.) associated with a particular regulatory option.4 Whether to use pollution or health
effects depends on data availability. Since they most directly affect human well-being,
health effects are the most relevant outcome. When this information is unavailable,
pollution exposure levels may be a useful proxy, followed by ambient pollution
concentrations, plant emissions, and proximity to a source [9, 10],

Methods for attributing monetary value to environmental outcomes often employ
measures of individuals' willingness to pay for an improvement in environmental quality.
Willingness to pay for environmental quality, like any normal good, is typically
increasing in an individual's income; all else equal wealthier individuals are able, and
usually willing, to pay more for the same good. Since social inequities that spur an
interest in environmental justice are likely to be correlated with the distribution of
income, using willingness-to-pay-based monetized valuations of the distribution of
environmental benefits can be problematic.5

The analysis should begin with an understanding of the baseline distribution of the
environmental outcome of concern:

1. What is the baseline distribution of the environmental outcome?

4	For a recent survey of the economic literature analyzing the incidence of the costs of
environmental regulation (primarily by income group), see [7],

5	For more discussion on the difference between the "rights-based" (i.e., physical
outcomes) versus "preference-based" (monetized outcomes) approaches, see [10],

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Establishing a proper baseline distribution is crucial for two reasons. First, identification
of a pre-existing disparity presents an opportunity to tailor policy options to address the
disproportionate impact directly. Second, the baseline establishes a marker for
determining distributional impacts of the policy itself.

Once the baseline has been established, the analysis should predict the ex-post
distributional effects of the regulatory options under consideration.

2. What is the distribution of the environmental outcome for each regulatory

option?

While the options under consideration may be implemented uniformly (e.g., the same
standard would apply to all individuals, geographic locations, or types of facilities), the
distribution of the pollutant in the predicted post-regulatory scenarios may differ for
several reasons. First, the type of regulation may affect the post-regulatory distribution.
A uniform rate-based standard (e.g., per unit of output) means that facilities with higher
output will generally have higher post-regulatory emissions. Second, to the extent that
different types of individuals (e.g., low-income) have different sensitivities to a given
pollutant or different exposure pathways, some individuals will experience a different
post-regulatory scenario than others. Answering this question for prospective options
requires the capacity to model alternative outcomes.

Finally, it is important to assess the degree to which various policy options create or
remove disproportionate impacts.

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3. How do the policy options being considered improve or worsen the distribution
of the environmental outcome with respect to vulnerable subgroups?

Answering this question requires a methodology for comparing the answers to the first
two questions in order to determine at least whether a regulation represents an
improvement to the status quo and other considered options, and ideally an indication as
to how much.

Responses to these three questions can be presented in conjunction with aggregate net
benefits arising from the policy options. This combination of information would enable
policy makers to understand the possible tradeoffs between environmental justice and
overall economic efficiency implicit in the decision-making process. Even if there are
limited opportunities within the policy design itself to address any post-regulatory
distributional effects, clear documentation and acknowledgment of those effects is
informative to the decision-maker and the public, and may help guide future policy.

These three questions provide a basic framework to inform the distributional analysis for
environmental regulatory policy. This framework also enables analysts to identify if and
how existing disparities may be addressed through the regulatory context, recognizing
that legal, political, and enforceability constraints may prevent any action in this regard.

Note that such an analysis may not always be feasible. Data constraints may prevent the
identification of existing or post-regulatory disparities. The geographic distribution of the
pollutant may be unknown, for example. While advances in air monitoring and modeling
allow for more detailed assessments of how pollutants are dispersed, such analytical

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efforts require significant time and resource allocations. Some water pollutants are even
more problematic as little is known about the fate of a pollutant after discharge.

Moreover, answering these three questions is by no means sufficient for addressing all EJ
issues. For example, analysis that focuses on a single pollutant typically will not account
for the contribution of cumulative effects from other pollutants or multiple exposures
from sources outside the scope of the proposed rule. Disproportionately affected
communities may suffer from multiple stressors that have accumulated over decades. One
specific pollutant may show little impact or may even be distributed fairly evenly. In an
area with multiple waste sites or polluting facilities, however, the marginal effect of a
particular pollutant may be greater than in a community without such stressors.

Related to this point, analysis focusing on pollution concentrations or exposure levels,
rather than health outcomes may also fail to account for baseline differences in health
risks across racial and ethnic groups and income categories. Such differences may exist
due to genetic, cultural, or other un-accounted for factors. Increasingly, scientists are able
to document that the same exposure affects people differently, and those affects can vary
along racial and ethnic lines. Thus, the same exposure may result in significantly
different health effects depending on an individual's race, ethnicity, or income. In
addition, individuals with low incomes have less access to averting behaviors and
resources, like medical care, alternative water sources or housing options that allow them
to avoid exposures. Thus assuming that exposure affects everyone in the same manner
may be misleading.

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With these caveats in mind we now discuss ways to present information in a way that is
helpful for addressing these three questions.

3. Describing distributions

A tradeoff exists between providing information in a way that is useful to policy makers
and imposing ethical assumptions on the part of the analyst. This section describes
quantitative methods that have been used to describe the distributional effects of various
environmental outcomes with a minimum of ethical input.

Distributional effects are quantified in a variety of ways in the published academic
literature. While a consensus has not been reached on how best to analyze, quantify, and
present the results of an environmental justice analysis, a suite of methods has emerged
over the last few decades that can be categorized as visual displays, summary statistics,
and regression results. The variation in methods both within and across these categorizes
can be attributed to author preference or expertise, as well as the research question at
hand. In this section we survey key methods for quantifying distributional effects and
evaluate their effectiveness in addressing the policy questions outlined above.

3.1. Visual Displays

The use of charts, graphs, and maps can also be useful to provide an overview of the data
and results used in analysis. Beginning with the earliest study in our review,
Dorfman [11] examines the distribution of benefits and costs of environmental programs.
Results are shown graphically as a percent of household income. Shadbegian et al. [12] is
one of the few distributional analyses of a specific rule. They show the distribution of

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monetized benefits and costs from the Sulfur Dioxide trading program across U.S.
regions.

The graphical displays, as well as those that use maps to present information (e.g., [12,
13, 14, 15]) are a useful complement to other quantifiable information. Geographic
Information System generated maps are useful for suggesting trends, showing the general
location of where pollution is greatest or disparities are most pronounced. However, in
terms of analyzing the baseline or ex-post distribution of pollution, such displays are
suggestive at best, and lack the level of detail that required in a decision-making context.
In particular, they can be effective at conveying differences between baselines and policy
options if the differences are stark. For more subtle changes, they are not useful.

3.2. Summary Statistics

Summary statistics are a key component of any empirical analysis, providing the reader
with an important overview of the data used in the study. These statistics typically
include information on the number of observations associated with a particular variable,
some measure of central tendency, such as the mean or median, and a measure of
dispersion, such as the standard deviation. Although they are quite simple, these statistics
can provide useful insights into the patterns of disparities regarding environmental
outcomes, and require no ethical assumptions on the part of the analyst. In addition,
summary statistics can be applied consistently across regulatory scenarios and are
typically transparent to the reader. Information on the quantity of a particular pollutant
across income quintiles or racial groups, for example, gives insight into whether or not
the pollutant is evenly distributed, and this may be accompanied by some measure of

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statistical significance. With respect to the questions outlined above, these statistics are
useful for establishing baseline incidence of environmental burdens, and can be used to
measure both post-regulatory incidence and changes in incidence.

Asch and Seneca [16] and Harrison and Rubenfield [17] are two early studies of the
distribution of pollution in the U.S. Both articles examine the distribution of air pollution
across various demographic variables, including income and race. Relevant for the policy
questions we pose, the authors analyze both the baseline and the changes in air pollution
due to current regulations. Asch and Seneca [16] find that the baseline distribution of
particulate matter was regressive. Using the correlation between seven categories of
income and particulates in 284 U.S. cities they find that z-statistics show a positive
correlation for the lower income groups and that regulations helped ameliorate these
effects.

Harrison and Rubenfield [17] show baseline and control scenario exposure to NOx
concentrations for seven income groups in Boston. They show the concentration levels
across the income groups for the baseline and control scenarios and make some
qualitative statements about the results (e.g., the distribution of baseline concentrations is
fairly even across income groups, but the poor receive more benefits from reductions).

More recently, Brajer and Hall [18] examine changes in ozone and particulate matter with
respect to various demographic variables for the Los Angeles basin for 1990-1999. The
data are presented as "population weighted pollution levels" by county, race and income.
A Spearman rank correlation analysis shows correlation between pollution and socio-

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economic variables. They find that pollution has fallen over the decade in the region, but
the air quality gains are not evenly distributed.

While this brief review is not comprehensive, it provides a sense of the type of
information summary statistics convey in the literature. The methods are straightforward
and easily understood, and are useful for answering the first two questions in Section 2.
They provide useful baseline information regarding outcomes across subgroups, as well
as the correlation between group characteristics and environmental outcomes. When
combined with models that predict pollutant responses, they should be able to provide
similar information for alternative regulatory options.

Summary statistics are unlikely to contain sufficient information regarding the third
question, however. Focusing on averages or correlations can be misleading since a low
average exposure may mask very high exposure for a subset of individuals within a
group. There may be an undetected EJ problem if such hotspots occur primarily in
vulnerable subgroups, for example.

In addition, these statistics do not provide a clear, systematic ranking of alternatives.
Different policy options may involve tradeoffs between total improvements across all
groups and reducing the disparities among groups. Simple averages or correlations
provide no guidance regarding a transparent way to resolve these conflicts within one
regulatory analysis, much less consistently across rules.

3.3. Regression Analysis

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Regression analysis is a cornerstone of empirical economic analysis. It allows the
researcher to use the data in a way that can provide internally consistent, unbiased
hypothesis testing. In terms of environmental justice, regression analysis is frequently
used to identify the existence and causes of various environmental outcomes across
subgroups. By controlling for confounding factors, the researcher can identify the impact
of key independent variables on the measure of interest. There are numerous ways to
conduct regression analysis; here we highlight a few.

A common modeling framework is to use a probability-based model to account for the
fact that not all locations will experience a particular outcome (e.g., toxic release, facility
siting), and there may be systematic differences between areas with and without the
release. Baden et al. [19] conduct an analysis of Superfund sites using a logit model and
control for location characteristics, such as the population density, population size, and
state fixed effects. Results show a significant and positive relationship between the
percent Black and Hispanic and the probability of having a Superfund site, and that the
higher the income the less likely the area will have a site.

Downey et al. [20] examine toxicity-weighted U.S. air pollution Risk-Screen
Environmental Indicators data and its distribution across race and ethnicities. The authors
assign each of six race and ethnic groups within metropolitan areas a score based on their
exposure to air pollution. They use a logit model to examine how income affects the
probability of receiving a high score, controlling for community characteristics, such as
density, employment, region, etc. They find a strong link between income and disparities
in releases across 329 metropolitan areas, but the link with race is less significant.

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Wolverton [21] examines plant siting decisions with a conditional logit model, using
community characteristics at time of siting, rather than after construction. This distinction
is important since a rule can cause housing prices or wages to change in affected areas,
leading in turn to migration that alters demographic characteristics. Controlling for
several variables including property values, wage rates, education, employment, etc., she
finds that income, but not race, affects location decisions.

Arora and Cason [22] use a tobit model to examine the effect of neighborhood
characteristics on Toxics Release Inventory emissions by zip code for 1990. They first
estimate the probability that a geographic area has a facility with releases, and estimate
the size of the release in a second stage. The authors find that there is a significant
coefficient on race variables in the Southeast. The coefficients suggest that areas with
more non-white residents are more likely to have higher emissions. Income follows an
inverted U-pattern; i.e., emissions initially increase with income until reaching a point
after which emissions fall as income rises.

Fowlie et al. [15] use a matching approach to examine the relationship between emissions
of facilities participating in the California Regional Clean Air Incentives Market and
demographic variables. Their model allows them to examine emissions before and after
implementation of the emission trading, controlling for county attainment status,
community, and demographic variables. They compare the effects of the trading policy
with the counterfactual of traditional command and control regulation. They find that
neighborhood demographic characteristics are not a statistically significant predictor of
changes in emission levels.

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In general, regression analysis is useful for teasing out causal factors behind the
relationship between socio-economic variables and environmental outcomes. However,
for the purposes of an EJ regulatory analysis most (with the exception of [15]) do little to
inform the question of baseline and post-regulatory scenarios. Conducting a careful
regression analysis is highly time and data intensive. Consequently, it is likely to be
beyond the resources available for a regulatory impact analysis. Moreover, while studies
such as [15] indicate the effectiveness of race or income as a predictor of emissions for
different policy alternatives, they are not designed to rank alternatives.

4. Ranking distributions

While the methods described in the previous section are useful for addressing many
important questions, they are not able to rank outcomes in a way that provides answers to
our third question in a transparent manner. Fortunately, a set of tools for ranking
distributions is relatively well developed in the context of income and health outcomes.
The literature on applying these methods to rank environmental policy outcomes by their
distributional impacts is still in its infancy, however.

In this section, we outline how this literature has been adapted to address environmental
justice questions, identifying some shortcomings and suggesting some steps forward. We
begin with a set of visual ranking tools, Lorenz and concentration curves, that allow one
to determine easily if one distribution of outcomes is more "equitable" than another.
These tools are only applicable, however, for a small set of possible distributional
comparisons.

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We then discuss several inequality indices, the Gini coefficient, the concentration index,
the Atkinson index and the Kolm-Pollak index. Unlike the visual ranking tools, these
indices permit the analyst to rank any set of distributions. This universal applicability
comes at the expense of imposing additional normative assumptions, however.6

4.1. Visual ranking tools

We begin with two visual ranking tools, the Lorenz curve and the concentration curve.
These have the advantage of imposing relatively few ethical standards on an ordering;
however, they are unable to provide a complete ranking of distributions. In addition, they
do not provide much useful information regarding distribution of environmental
outcomes across subgroups, limiting their applicability to EJ analysis.

Lorenz Curves. Lorenz curves provide a means of ranking policy outcomes only if one
accepts the ethical premise that it is always desirable to transfer a unit the outcome
variable (good or bad) away from a highly exposed individual to one who is less exposed.
Some hypothetical Lorenz curves for distribution of a pollutant are depicted in Figure 1.
The horizontal axis of the graph indicates percentiles of the population ranked by
pollution exposure: 10 corresponds to the ten percent of the population least exposed to
the pollutant, 50 corresponds to the half of the population least exposed to pollution, etc.
The vertical axis represents the percent of pollution exposed by each percentile. The

6 This tradeoff can be most easily seen with the Gini coefficient and concentration index.
Although these two indices are derived respectively from the Lorenz and concentration
curves, they do not provide identical information as the curves. The indices can rank
distributions that the curves cannot, but they require the analyst to impose stronger ethical
restrictions.

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black diagonal line depicts a perfectly equal distribution of exposure: the lowest 10
percent of the population experience 10 percent of the exposure the lowest 50 percent of
the population experience half the exposure, etc.

Curves A, B, and C represent three hypothetical Lorenz curves in which pollution is not
distributed equally. In curve A, for example, the least exposed half of the population is
exposed to 30 percent of the pollution, while in curve B the least exposed half
experiences only 10 percent of the pollution.

Lorenz curves have the useful feature that the farther away the curve is from the diagonal,
the less equal is the distribution. This property can form the basis of a ranking system.
Suppose A and B represent the predicted distributions of two regulatory options. For
now, let us suppose that the two policies result in the same amount of pollution per
capita. Option A results in a more equitable distribution than Option B. The only value
judgment that needs to be imposed to make a preference ranking is that one care at all
about distributional equity. It does not matter how much one cares about exposure at the
top or bottom of the distribution. As long as one prefers a more equal distribution to a
less equal one, a curve that is closer to the diagonal (such as A) is preferable to a curve
that is farther (such as B).

Although Lorenz curve analysis imposes minimal value judgments on the part of the
analyst, it has several drawbacks that limit its practical usefulness. First, it is only a
partial ordering, meaning that it can only draw meaningful comparisons for options
whose Lorenz curves do not cross. A policy generating curve C, for example, cannot be
compared with curves A and B since it is closer to the diagonal for some range of the

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population, but farther for others. This property is problematic with several options since
the more curves being analyzed the more likely that some will cross.

Figure 1. Lorenz curves.

Cumulative percent of population, ranked by exposure

Second, Lorenz curve analysis is ordinal; one can say that A is preferred to B, but not by
how much. This ordinal property is related to a third issue. Lorenz curve analysis ignores
differences in average exposure levels. For example, if we abandon the assumption that
each distribution has the same average pollution level, the exposure levels of the most
highly exposed individual in distribution B may be lower than the least exposed in
distribution A. It may be undesirable to conclude that A is preferred to B simply because
the exposure is more equitably distributed. One cannot evaluate a tradeoff between lower
average exposure levels and a less equitable distribution using standard Lorenz curves.

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Finally, for purposes of environmental justice analysis, Lorenz curves have the
shortcoming that they are not easily disaggregated by population subgroups. It is
straightforward to use Lorenz curves to compare distributions of pollutants within a sub-
group (e.g., define the population and exposure percentiles in terms of individuals below
a poverty threshold). It is not so easy to use Lorenz curves to evaluate distributions across
subgroups (e.g., to make statements to the effect that a regulation causes pollution to be

n

more equitably distributed across racial groups).

Concentration Curves. Like the Lorenz curve, the vertical axis of the concentration
curve displays the share of an outcome variable experienced by a population. The
horizontal axis displays the cumulative percent of the population ranked by socio-

o

economic status (typically income). The height of the concentration curve indicates the
share of the outcome experienced by a given cumulative proportion of the population.
Figure 2 displays hypothetical concentration curves. A perfectly equal distribution of
outcomes corresponds to a concentration curve along the 45° line.9

Unlike Lorenz curves, concentration curves can cross the 45° line, and even lie
completely above it if lower income is correlated with higher outcomes. Concentration
curves can rank distributions in a manner similar to Lorenz curves; for a good outcome, a
higher curve is socially more desirable. Concentration curve rankings implicitly employ
social preferences such that it is always desirable to transfer a good environmental

n

Although Lorenz curves can be decomposed by subgroup [23], this decomposition does
not allow one to rank distributions as in the aggregate Lorenz curve analysis.

o

In contrast, a Lorenz curve would display the population ranked by exposure.

9 Kakwani [24] first developed this analysis to study income tax progressivity. Wagstaff
et al. [25] proposed its use in measuring the equity of health outcomes.

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outcome away from a relatively rich individual towards a poorer one, even if the poorer
individual is slightly poorer and significantly healthier [26], Note that this normative
judgment may be more controversial than the corresponding assumption used for Lorenz
curve analysis (that it is desirable to shift good health outcomes to the relatively ill).

Figure 2. Concentration curves.

Cumulative percent of population, ranked by income

Concentration curve analysis suffers from the same shortcomings as Lorenz curve
analysis. It is unable to rank distributions whose curves cross, thus providing only a
partial ordering. It is ordinal, and ignores differences in average exposure levels. It is also
unable to evaluate changes in distributions between subgroups (other than those based on
income).

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In general, both visual ranking tools have some advantages over the visual displays
discussed in the previous section. In some cases, both Lorenz and concentration curves
allow comparisons across policy alternatives. In addition, concentration curves are able to
provide information regarding the equity of an environmental outcome with respect to
one demographic variable of interest, income. However, both curves share the main
shortcomings of the other visual displays; they are only effective at comparing
distributions if there are sufficiently stark differences. If the curves for different policy
options cross, this analysis provides no effective ranking methodology.

4.2. Inequality Indices

An inequality index is a mathematical tool for converting distributions of goods (e.g.,
income) or bads (e.g., pollution) into a single number. That number can then be used to
generate an ordering for any set of outcomes, thus addressing the partial ordering issue
inherent in the Lorenz and concentration curve analyses. For example, a distribution with
a higher inequality index number is less equal, and hence less preferred than one with a
lower number. Moreover, some inequality indices can be decomposed in a manner that
allows one to evaluate inequality both within and between subgroups of interest. An
index value can also have cardinal (rather than just ordinal) significance, i.e., the
magnitudes, not just the rankings, contain useful information. However, these useful
features come at the cost of imposing subjective value judgments. In addition, their
usefulness for evaluating distributions of bads can be problematic.

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Here, we focus on four families of inequality indices: the Gini coefficient, the
concentration index, the Atkinson index, and the Kolm-Pollak index.10 These indices can
be divided into the categories of relative (Gini coefficient, concentration index, and
Atkinson index) and absolute (Kolm-Pollak index) indices. Relative indices are
unaffected by proportionate changes in the outcome variable. They are therefore
convenient for analysis of variables using different units of measurement (e.g., currencies
for income analysis). In contrast, absolute indices are unaffected by a uniform shift in the
outcome variable (i.e., the addition of a constant to every individual's outcome). These
properties are mutually exclusive, and there is no unambiguous reason to choose one
category of index over another. As argued by [28], however, relative indexes can be
misleading. Suppose the income of both members of a population of two individuals
doubles. If prices do not change the difference in purchasing power between the two
would also double, suggesting that the new distribution is less equal. An absolute
inequality index would increase to reflect this change, while relative index would not.

Blackorby and Donaldson [29, 30] show that relative and absolute indices that depend
only on one variable have an associated ordinal social evaluation function.11 The equally
distributed equivalent (EDE) value of a distribution is the amount of the outcome variable
that, if given equally to every individual in the population, would leave society just as
well off as the actual, unequal distribution. The EDE thus embodies a set of social
preferences and is a measure of social welfare that enables rankings of distributions with

10For a discussion of other index numbers in the context of income distribution, see [27];
in the context of environmental outcomes, see [9],

uThe proofs do not apply to the concentration index since it depends on two variables,
environment and income.

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23

different means. The Gini coefficient, Atkinson index, and Kolm-Pollak index can all be
expressed as functions of their associated EDEs.

Choosing a specific type of index with which to rank policies is thus equivalent to
choosing a particular social evaluation function on which to base the policy decision.
Since the values of the associated social evaluation function do depend on the average
value of the outcome variable (not just the distribution), they provide an additional tool
with which the analyst can compare policy outcomes that differ in both mean and
distribution in a logically consistent manner.

Although the social evaluation functions are ordinal, the associated inequality indices are
cardinal. A relative index answers the question, "What percent of the average amount of
the good would society be willing to sacrifice if the remainder were allocated evenly
across the population?" An absolute index answers the question, "What is the amount of
the good per capita society would be willing to sacrifice if the remainder were allocated
evenly across the population?" Thus, magnitudes, not just ranking of the indexes are
significant.

Gini coefficient. The Gini coefficient is the most widely used inequality index. Its
popularity is likely due more to the fact that it is easily understood as an increasing
function of the area between a Lorenz curve and the diagonal line representing perfect
equality than to desirable theoretical properties. It can therefore be used to rank
distributions whose Lorenz curves cross. To be able to do this, the Gini coefficient
requires additional ethical assumptions.

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24

Specifically, it has the feature that the effect of a transfer on the index number depends
on the individuals' rank, not the difference in outcomes. In contrast to the widely
accepted principle that an inequality index should place greater weight on transfers
among the relatively worse off, for a typical bell-shaped distribution a transfer between
individuals in the middle of the distribution will have a higher effect on the Gini
coefficient than a transfer between two similarly distanced individuals at either

12

tail [31], Similarly, the Gini coefficient has the undesirable property that the effect of a
transfer on the index depends on the endowment of a third individual; if that individual is
ranked between the first two, the transfer will have a greater impact than if not (since
there will be a greater rank difference between the first two individuals in the former
case). Finally, and particularly troublesome for EJ analysis, the Gini coefficient cannot
generally be used to decompose aggregate inequality into within and between group

13

components in an internally consistent manner [30],

Although it is a simple matter to compute a Gini coefficient if the outcome of concern is
a bad (rather than a good), the resulting measure does not have a sensible associated
social evaluation function (since it would be increasing in the bad). It is an ordinal
ranking of dispersion, but loses the cardinal interpretation of a relative inequality measure
since the EDE is smaller than the mean (for a bad it should be larger). Thus, it does not
indicate the percent increase in average pollution that could be tolerated in exchange for a

12

There are ways of modifying the Gini coefficient to introduce flexibility in the weights
placed on different segments of the population [32, 33], These techniques are rarely used
in practice, however.

13

Specifically, constructing an EDE for each subpopulation and then using these to
construct an aggregate EDE for the entire population does not yield the same result as
calculating the aggregate EDE directly.

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25

perfectly equal distribution. Consequently, the Gini coefficient can provide useful
comparisons for distributions with the same mean level of a bad, but cannot be used in
conjunction with a social evaluation function to rank distributions with different means.
Moreover, using the Gini coefficient in this way can be misleading since it can generate
different policy rankings if one uses a bad as the outcome variable versus its
complementary good. Calculating the Gini coefficient for ambient concentrations of parts
per billion of an air pollutant, for example, yields a different ranking of policy outcomes
than using the same data to calculate a Gini coefficient for parts per billion of "clean" air.

There are several examples of applications using the Gini coefficient to analyze
distributions of health and environmental outcomes. Among the first were [34], who used
a Gini coefficient to track evolution in age at death (a good) over time in Great Britain.
Heil and Wodon [35] use a Gini coefficient to examine the distribution of predicted C02
emissions across countries grouped by income. Millimet and Slottje [36] use the Gini
coefficient to compare distributions of pollution across states grouped by income class.14
Millimet and Slottje [37] use the Gini coefficient to evaluate the effect of regulatory
compliance costs on the distribution of toxics reported in the U.S. Toxic Release
Inventory across U.S. states and counties. They combine regression results with
Spearman correlations between demographic characteristics and emissions to argue that
policies that increase inequality as measured by the Gini coefficient increase racial
disparities. In these studies, the Gini coefficient has been used primarily as an ordinal
measure of dispersion, without attendant welfare implications.

14Since the Gini coefficient does not satisfy the consistency in aggregation both of these
studies required a group overlap term in addition to between and within group terms.

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26

Concentration index. The concentration index is similar to the Gini coefficient, being an
increasing function of the difference between the 45° line and the concentration (rather
than Lorenz) curve.15 It ranges from -1 (the entire outcome is borne by the poorest
individual) to 1 (the entire outcome is borne by the wealthiest individual). Since the
concentration curve can cross the 45° line, zero either indicates perfect equality or that
the area above the curve is exactly equal to the area below it. As with the Gini
coefficient, the effect of allocating a unit of the outcome variable to an individual is
weighted by the individual's rank. With the concentration index, the relevant rank is
income, rather than the outcome variable.

The concentration index can provide a complete ordering in the sense that lower values
are always more "pro-poor" (for distribution of a good) than higher values. The cardinal
relationship between magnitudes of concentration index numbers lacks the clear intuition
of the other three indices considered here, however.16

Like the Gini coefficient, the concentration index value depends on individuals' ranks,
not absolute differences. It also shares the trait that ordering based on the concentration
index can be sensitive to whether the outcome variable is expressed as a good or its "bad"
complement [40], It inherits from the concentration curves the questionable normative
assumption that transfers of a good environmental outcome from rich to poor is always
desirable [41],

15For details on the practical use of the concentration index, see [38],

16 This is not to say that there is no intuitive interpretation. Koolman and van
Doorslaer [39] provide a link between the index value and the proportionate amount of
the outcome variable that would need to be redistributed from the richest to the poorest
half of the population in order to attain an index value of zero (not necessarily equality).

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27

Atkinson Index. The Atkinson index satisfies a several desirable theoretical properties
lacking in other relative indices [28, 29, 31], Among these are that it is a function of
individual allocations rather than rank, and it can be disaggregated into subgroups in a
consistent manner (see also [42]).

In its formula, the Atkinson index explicitly incorporates ethical considerations with an
inequality aversion parameter that ranges from zero to infinity. This parameter introduces
some flexibility, allowing the analyst to specify the amount society is willing to trade a
reduction in the outcome variable for one individual for an increase for another. A value
of zero implies that society is indifferent between transfers among any two individuals.
The higher the parameter's value, the more weight society places on transfers to
individuals with lower outcomes. Since the choice of a parameter value is entirely
normative, it is common to calculate Atkinson indexes for several values to determine
how sensitive rankings are to the choice.

Although the Atkinson index has many desirable properties when used to analyze
distributions of goods, it is not so convenient for analyzing bad outcomes. As with the
Gini coefficient, imputing a bad into the Atkinson formula removes any cardinal welfare
significance since the associated social evaluation function would be increasing in the
bad. It also causes the index to place more weight upon the most well-off individuals
(those with low outcomes), rather than the worst off. The Atkinson index is generally not

17

defined for negative numbers, thus precluding a simple redefinition of bads in that way.
Transforming a bad into a good by replacing it with its complement (e.g., parts per billion

17

Even for examples in which negative values are defined the Atkinson Index generates
the perverse result that a progressive redistribution reduces social welfare [43],

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28

of a pollutant to parts per billion of "clean" air, or the probability of not dying from
cancer) may have the undesirable result of rendering an index value so small as to be

18

within the rounding error of a computer.

Although the Atkinson index is commonly used in income distribution analysis, it has
rarely been used to measure environmental or health outcomes. Waters [43] used an
Atkinson index to analyze distribution of access to health care (a good) in Ecuador. Levy
et al. [14] use the Atkinson Index to evaluate the distribution of mortality risk resulting
from alternative power plant air pollution control strategies in the United States. Levy et
al. [44] use the Atkinson index to analyze reduction in mortality risk from particulate
matter reductions from regulating transportation. Each of these studies used the Atkinson
index as a measure of dispersion without welfare significance.

Kolm-Pollak index. The Kolm-Pollak index shares the desirable theoretical properties of
the Atkinson index [28, 30, 42], Like the Atkinson, it uses an inequality aversion
parameter to specify the relative importance of allocations to different segments of the
population. Higher values correspond to greater weight being placed on the worse off and
zero indicating complete indifference to the allocation.

18

To put this in perspective, consider the relative income distribution of a society of
billionaires who differed in wealth by only a few dollars. It would be almost perfectly
equal, with the value of the corresponding Atkinson index being extremely close to zero.
Note that this does not mean that the distributional effects are insignificant. If the good
were clean air or probability of not dying from cancer the percent reduction society
would be willing to give up for an equal distribution might be quite small, but the value
of that reduction might be significant. Nonetheless, presenting the results in a manner
such that a regulation changes the Atkinson Index from 9.59 x 10 6 to 9.51 x 10 6, may
not be easy to interpret.

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29

In contrast with the other indices examined here, however, the Kolm-Pollak index readily
accommodates bad outcomes. It is inappropriate to input bad values directly into the
index. However, one can simply multiply them by minus one and subtract the result from
some arbitrary benchmark. This operation preserves the appropriate social evaluation
function ranking and is equivalent to measuring the distribution of a complementary
"good." The property of an absolute index that adding the same amount to everyone in
the population does not change its value helps in this regard; the value of the index is
independent of the benchmark level. Intuitively, a difference of a few dollars among the
incomes of a population of billionaires (in parts per billion of clean air) has as much
impact on the Kolm-Pollak index as it does among a population of paupers (since the
index is unchanged by adding or subtracting the same value from everyone in the
population). To date, the Kolm-Pollak index has not been used in the analysis of
environment or health outcomes, and there are few examples of its application in income
analysis ( [45] is an exception).

In general, the Atkinson and Kolm-Pollak inequality indices have the potential to inform
all three questions posed in Section 2. They can provide a concise snapshot the dispersion
of environmental outcomes for baseline and policy scenarios, both within and across
population subgroups. In terms of ranking outcomes, they can be used to determine
whether policy alternatives improve the dispersion of outcomes, holding the total amount
of the outcome constant. For good outcomes the social evaluation functions associated
with both indices can also be used to rank alternatives for which both the dispersion and
total amount of pollution vary. Only the Kolm-Pollak index appears suitable for
evaluation of bad outcomes, however.

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5. Conclusions

For at least the past thirty years, the academic literature has used a variety of methods for
quantifying the relationship between environmental quality and vulnerable sub-
populations. In general, methods have been chosen with respect to their usefulness in
answering questions posed by a particular study. As a result, there has been little attempt
to develop a consistent framework to be used across studies, much less one suitable for
the questions likely to be important for regulatory analysis. While use of a common
environmental justice metric would be convenient for making comparisons and drawing
conclusions across academic studies, it is essential for undertaking regulatory impact
analysis in a consistent and transparent manner across different rules. In this section we
discuss how well the tools presented in Sections 3 and 4 answer the questions for
regulatory EJ analysis posed in Section 2.

Visual displays, whether GIS maps, Lorenz curves, or concentration curves have the
advantage of illuminating sharp disparities. Maps, for example, can be very effective at
indicating situations in which pollution levels are highly concentrated in locations with
large numbers of residents belonging to vulnerable subpopulations. They are less useful
for analysis of alternatives in which differences are less pronounced and obvious to the
naked eye. Nor do they suggest a means of ranking tradeoffs between total pollution
reductions and reductions in disparities. Similarly, Lorenz and concentration curves are
most helpful when there are sharp differences in policy options. They are not as
informative if policy alternatives generate curves that cross. In general, visual displays
suffer the disadvantage that they are not easily comparable across many alternatives,

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31

whether for an analysis of several options for implementing a given rule, or a
comprehensive analysis across rules.

Subgroup summary statistics such as mean exposure rates have the advantage of being
simple to calculate and easily understood. They provide useful information regarding
baseline conditions, potentially providing a signal if vulnerable subgroups are more
highly exposed.

These statistics have two important shortcomings, however. First, they do not provide
detailed information regarding distribution of outcomes within a group. This information
can be important since the impact of a pollutant may be more of a concern if it is
concentrated in a hotspot among a relatively small group of individuals, than if it is
evenly spread across the sub-population. Second, they do not provide a clear ranking of
alternatives in a systematic way. Different policy options may involve tradeoffs between
total improvements across all groups and reducing the disparities among some groups.
Simple averages do not provide a transparent way to resolve these conflicts.

Regression analysis can be effective in determining causality (e.g., if race a determining
factor in pollution exposure). This approach can be useful for identifying existing
baseline disparities and for conducting retrospective studies. It does not appear to be well
suited, however, for ranking impacts of hypothetical regulatory options.

Inequality indices seem to be a promising tool for addressing all three questions posed in
Section 2. They provide a means of evaluating the distribution of environmental
outcomes both within and across subgroups at baseline. Inequality indices can use model
simulation results to predict distributional effects of various regulatory alternatives.

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Moreover, due to their associated social evaluation functions, they provide a transparent
and consistent means of ranking alternatives for which both total pollution levels and
their relative distributions vary. They do so at the cost of imposing restrictive value
judgments on the analysis, especially with respect to the level of inequality aversion.
Sensitivity analysis over a range of inequality aversion parameter values can moderate
this normative influence.

Inequality indices have the advantage of a robust theoretical literature describing their
properties as well as many practical applications in the context of income distribution
analysis. Two of the most commonly used indices in that context, the Gini coefficient and
the Atkinson index, have undesirable theoretical properties if used to measure the
distributions of a "bad" like pollution, rather than a "good" like income. Specifically, the
corresponding social evaluation functions are not well-behaved, thus invalidating their
potential for ranking options that have different tradeoffs between total improvements
and reducing disparities. The concentration index, commonly used to evaluate health
outcomes by income levels, has a relatively weak theoretical foundation; the
corresponding social evaluation function is not well understood. Perhaps more
importantly for EJ analysis, however, is its inability to evaluate distributions across
subpopulations that are not defined by income.

In contrast, the Kolm-Pollak index shares the desirable theoretical traits of the Atkinson
index while being able to accommodate evaluation of distributions of bads. In contrast
with the other indices, however, it has a very thin record of empirical applications in the

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context of income distribution and, to our knowledge, no published applications in the
context of environmental outcomes.

Where does this leave the analyst in terms of determining a consistent and transparent
method for evaluating distributional effects in regulatory analysis? Inequality indices
show potential for meeting the needs of consistency in a regulatory analysis. Data are
likely to be available across regulatory settings to estimate a Kolm-Pollak index, which
shows the most promise for evaluating adverse environmental outcomes. This index
could thus enable the decision maker to evaluate EJ consistently for a variety of rules. In
addition, visual displays, summary statistics, and regression analysis provide useful
supplementary information that can contribute to a richer understanding of potential EJ
issues than a set of index numbers alone.

The two main impediments to using a Kolm-Pollak index in an EJ component of
regulatory analysis are the lack of peer-reviewed applications and its lack of familiarity
among policy-makers and the public. For it to become a useful policy tool, both of these
issues need to be addressed by further academic research and pilot applications. Research
regarding an appropriate range of values for the inequality aversion parameter is
particularly important. Such research may involve initial costs associated with both
mastering practical techniques involved in its calculation, as well as costs to the user in
terms of understanding the output. Such costs are likely to be small, however, compared
to the relative advantage of a better understanding the distributional effects of
environmental policy.

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References

1.	Organisation for Economic Cooperation and Development (OECD). Tools to
improve regulatory design. In Regulatory Policies in OECD Countries: From
Interventionism to Regulatory Governance; OECD: Paris, France, 2002.

2.	Federal Register. Executive Order 12898: Federal Actions to Address
Envrionmental Justice in Minority Populations and Low Income Populations.
Available online: http ://www. gpo. gov: 80/fdsvs/pkg/FR-1994-02-16/html/94-
3685.htm

3.	Government Accountability Office (GAO). EPA Should Devote More Attention to
Environmental Justice When Developing Clean Air rules. A Report to the Ranking
Member, Subcommittee on Environment and Hazardous Materials, Committee on
Energy and Commerce, House of Representatives, Washington, DC, US, 2005.

4.	GAO, Environmental Justice: Measurable Benchmarks Needed to Gauge EPA
Progress in Correcting Past Problems. Testimony Before the Subcommittee on
Superfund and Environmental Health, Committee on Environment and Public
Works, U.S. Senate. In GAO: Washington, DC, US, 2007.

5.	Environmental Protection Agency (EPA). Interim Guidance on Considering
Environmental justice During the Development of an Action; US EPA:
Washington, DC, US, 2010.

6.	Ringquist, E.J. Assessing Evidence of Environmental Inequities: A Meta-
Analysis. Journal of Policy Analysis and Management 2005, 24, 223-247.

7.	Parry, I.W.H.; Sigman, H.; Walls, M.; Ill, R.C.W. The Incidence of Pollution
Control Policies. In The International Yearbook of Environmental and Resource
Economics', Tietenberg, T.; Folmer, H., Eds.; Edward Elgar: Northampton MA,
2006.

8.	Bowen, W. An Analytical review of environmental justice research: What do we
really know? Environmental Management 2002, 29, 3-15.

9.	Levy, J.I.; Chemerynski, S.M.; Tuchmann, J.L. Incorporating concepts of
inequality and inequity into health benefits analysis. International Journal for
Equity in Health 2006, 5, Art. 2.

10.	Pearce, D. Conceptual Framework for Analyzing the Distributive Impacts of
Environmental Policies, OECD: Paris, 2003.

11.	Dorfman, R. Incidence of the Benefits and Costs of Environmental Programs.
American Economic Review Papers and Proceedings 1977, 67, 333-340.

12.	Shadbegian, R.; Gray, W.; Morgan, C. Benefits and Costs from sulfur dioxide
trading: A distributional analysis. In Acid in the Environment: Lessons Learned
and Future Prospects, Visgilio, G. R.; Whitelaw, D. M., Eds.; Springer: 2007.

13.	Perlin, S.A.; Setzer, R.W.; Creason, J.; Sexton, K. Distribution of Industrial Air
Emissions by Income and Race in The United States: An Approach Using the
Toxic Release Inventory. Environmental Science & Technology 1995, 29, 69-80.

14.	Levy, J.I.; Wilson, A.M.; Zwack, L.M. Quantifying the efficiency and equity
implications of power plant air pollution control strategies in the United States.
Environmental Health Perspectives 2007, 115, 743-750.

-------
35

15.	Fowlie, M.; Holland, S.; Mansur, E. What Did the Market Deliver, and to Whom?
An Evaluation of Southern California's RECLAIM Program. American Economic
Review Forthcoming.

16.	Asch, P.; Seneca, J.J. Some Evidence on the Distribution of Air Quality. Land
Economics 1978, 54, 278-297.

17.	Harrison, D.; Rubinfeld, D.L. The Distribution of Benefits from Improvements in
Urban Air Quality. Journal of Environmental Economics and Management 1978,
5, 313-332.

18.	Brajer, V.; Hall, J.V. Changes in the distribution of air pollution exposure in the
Los Angeles basin from 1990 to 1999. Contemporary Economic Policy 2005, 23,
50-58.

19.	Baden, B.M.; Noonan, D.S.; Turaga, R.M.R. Scales of Justice: Is there a
Geographic Bias in Environmental Equity Analysis? Journal of Environmental
Planning and Management 2007,50, 163-185.

20.	Downey, L.; Dubois, S.; Hawkins, B.; Walker, M. Environmental Inequality in
Metropolitan America. Organization & Environment 2008, 27, 270-294.

21.	Wolverton, A. Effects of Socio-Economic and Input-Related Factors on Polluting
Plants' Location Decisions. The B.E. Journal of Economic Analysis and Policy
2009, 9, Article 14.

22.	Arora, S.; Cason, T.N. Do Community Characteristics Influence Environmental
Outcomes? Evidence from the Toxics Release Inventory. Southern Economic
Journal 1999, 65, 691-716.

23.	Bishop, J.A.; Chow, K.V.; Zeager, L.A. Decomposing Lorenz and Concentration
Curves. International Economic Review 2003, 44, 965-978.

24.	Kakwani, N.C. Measurement of Tax Progressivity: An International Comparison.
Economic Journal 1977, 87, 71-80.

25.	Wagstaff, A.; van Doorslaer, E.; Paci, P. Equity in the finance and delivery of
health care: Some tentative cross-country comparisons. Oxford Review of
Economic Policy 1989, 5, 89-112.

26.	Fleurbaey, M. Health, Equity and Social Welfare. Annales d'Economie et de
Statistique 2006, 21-59.

27.	Chakravarty, S.R. Ethical Social Index Numbers; Springer-Verlag: New York,
1990.

28.	Kolm, S.-C. Unequal Inequalities I. Journal of Economic Theory 1976, 72, 416-
442.

29.	Blackorby, C.; Donaldson, D. Measures of relative equality and their meaning in
terms of social welfare. Journal of Economic Theory 1978, 18, 59-80.

30.	Blackorby, C.; Donaldson, D. A Theoretical Treatment of Indices of Absolute
Inequality. International Economic Review 1980, 27, 107-136.

31.	Atkinson, A.B. On the measurement of inequality. Journal of Economic Theory
1970, 2, 244-263.

32.	Weymark, J. A. Generalized Gini inequality indices. Mathematical Social Sciences
1981, 7, 409-430.

33.	Yitzhaki, S. On an Extension of the Gini Inequality Index. International
Economic Review 1983, 24, 617-628.

-------
36

34.	Illsley, R.; Le Grand, J. Measurement of Inequality in Health. In Economics and
Health; Williams, A., Ed. Macmillan: London, 1987.

35.	Heil, M.T.; Wodon, Q.T. Future Inequality in CO2 Emissions and the Impact of
Abatement Proposals. Environmental and Resource Economics 2000, 77, 163-
181.

36.	Millimet, D.L.; Slottje, D. Environmental Compliance costs and the distribution
of emissions in the U.S. Journal of Regional Science 2002, 42, 87-105.

37.	Millimet, D.L.; Slottje, D. An environmental Paglin-Gini. Applied Economics
Letters 2002, 9, 271-274.

38.	O'Donnell, O.; van Doorslaer, E.; Wagstaff, A.; Lindelow, M. Analyzing Health
Equity Using Household Survey Data: A Guide to Techniques and Their
Implementation; The World Bank: Washington, DC, 2008.

39.	Koolman, X.; van Doorslaer, E. On the interpretation of a concentration index of
inequality. Health Economics 2004, 13, 649-656.

40.	Clarke, P.M.; Gerdtham, U.-G.; Johannesson, M.; Bingefors, K.; Smith, L. On the
measurement of relative and absolute income-related health inequality. Social
Science and Medicine 2002, 55, 1923-1928.

41.	Bleichrodt, H.; van Doorslaer, E. A welfare economics foundation for health
inequality measurement. Journal of Health Economics 2006, 25, 945-957.

42.	Blackorby, C.; Donaldson, D.; Auersperg, M. A new procedure for the
measurement of inequality within and among population subgroups. Canadian
Journal of Economics 1981, 14, 665-685.

43.	Waters, H.R. Measuring equity in access to health care. Social Science and
Medicine 2000, 57, 599-612.

44.	Levy, J.I.; Greco, S.L.; Melly, S.J.; Mukhi, N. Evaluating Efficiency-Equality
Tradeoffs for Mobile Source Control Strategies in an Urban Area. Risk Analysis
2009, 29, 34-47.

45.	Atkinson, A.B.; Brandolini, A. On Analyzing the World Distribution of Income.
The World Bank Economic Review 2010, 24, 1-37.

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