tors into attributes and further into the overall index as shown in the following equation.
I
n
= Á
à Í
à Ä
s:A
s
\ 0
d
s
{A
s n
− r
1
}
− n
1
r
1
\ Ö n
0, if A
s n
= 0 Ö s, n
and A
s n
= Á
à Í
à Ä
i: D
i
\ 0
v
i
{D
i
X
i n
}
− r
s
− n
2
r
s
\ Ö n
0, if D
i
= 0 Ö i
and D
i
X
i n
\ 0, if X
i n
\ X
i min
Ö i, n such that D´\0 D
i
X
i n
= 0, if X
i n
5 X
i min
Ö i, n where
s
d
s
= 1,
d
s
] 0,
n
1
] 1,
− B r
1
B −
1
i
v
i
= 1,
v
i
] 0,
n
2
] 1,
− B r
s
B −
1 Here D
i
· represents the society-wide dose-re- sponse function. The weights v
i
reflect society’s relative valuation of the marginal damage due to
pollutant i. Consider exposure to carbon monox- ide CO, which may cause angina and eventually
heart attacks. Then v
CO
represents the relative welfare change due to the change in the probabil-
ity of a heart attack caused by a unit change in the ambient concentrations of CO. By analogy,
the ds represent the relative change in welfare due to
the marginal
changes in
environmental attributes.
The nested CES functions reflect the double aggregation procedure suggested at the beginning
of this section. It is assumed that the elasticity of substitution between environmental indicators, r
s
, is the same for all indicators that make up at-
tribute A
s
, but varies across attributes. This is different from r
1
, which is the constant substitu- tion elasticity between the different attributes. If
desired, the individual indicators could be directly aggregated into the overall pollution index.
4. Air quality in the US
In this section, the proposed methodology is illustrated by computing an index of air pollution
using 1997 data for 74 counties and 61 Metropoli- tan Statistical Areas MSAs in the US. Data for
the five criteria pollutants included in the PSI, namely, PM
10
, CO, NO
2
, O
3
, and SO
2
, are used. EPA 1997 identified the dose-response functions
for these pollutants. However, to keep the analy- sis simple and directly comparable with the PSI,
the following normalization rule is assumed, where the threshold values for each gas are based
on the PSI:
D
i
X
i n
= Á
à Í
à Ä
5 + X
i n
− X
i min
X
i max
− X
i min
if X
i n
\ X
i min
Ö i, n if X
i n
5 X
i min
Ö i, n X
i min
corresponds to the ambient concentration below which there are no perceptible damages.
Therefore, X
i min
= 50 of NAAQS for each gas.
The health-effects descriptor for this range is ‘good’ indicating no general health effects for the
population EPA, 1994.
6
X
i max
for each gas is set equal to the concentration corresponding to a PSI
value of 500 and ‘hazardous’ health effects. PSI values above 400 may result in the premature
death of the ill and elderly people, and healthy people might experience adverse symptoms that
affect normal activity.
An advantage of the CES function is that the scale effects parameter and the elasticity of substi-
tution can vary over the domain of the function while retaining the general properties identified in
Section 3 Bourguignon and Chakravarty, 1998. The pollutant space is divided into five areas and
the elasticity of substitution varies by area.
7
The general rule followed is that as a region becomes
more polluted it gets harder to substitute de-
6
By using the PSI-based values we focus on short-term human health effects alone, ignoring any other damage to
materials and populations.
7
A clarification regarding the nomenclature. Regions de- noted by the letter n refer to geographical spaces — counties
and MSAs in this case. Areas refer to different parts of the pollutant space as delineated in Fig. 3.
Fig. 3. Defining areas of air pollution.
creases in the ambient concentration of one indi- cator for increases in the ambient concentration
of another, while maintaining the level of air quality. This follows from the rapidly increasing
welfare losses at higher ambient concentrations of each gas. Thus, the absolute value of r increases
as ambient concentrations increase.
This is depicted in Fig. 3.
8
To be consistent with the PSI methodology, a region belongs to the
area corresponding to the gas with the highest relative ambient concentration, regardless of all
other gases. Constant returns to scale are assumed for all areas. Also, due to lack of data equal
weights are assumed for all gases.
9
Parameter values are summarized in Appendix A.
Area 0 is not polluted since ambient concentra- tions of all indicators are below the level at which
damage occurs. By definition, the index value is zero in this area.
Area 1 is a low pollution area. Here the PSI value lies between 50 and 100 and the health
effects are ‘moderate’ with few or no health effects
8
The two-dimensional case is used to facilitate the graphical presentation. The logic is easily extended to the case of many
gases.
9
The VEQI study VCUCES, 1999 surveyed 28 experts to determine the weights for S0
2
, O
3
, and NO
2
. The values obtained ranged from 0.32 to 0.36.
for the general population EPA, 1994. In this area, r is set at its limiting value = − 1. This
implies perfect substitution between the environ- mental indicators, and negatively sloped straight
line isopollution contours. Of course, in the two gases-case, when the ambient concentration of
one gas is below 50 of the NAAQS, the index value is independent of that gas and the isopollu-
tion lines become parallel to the corresponding axis.
Area 2 is the area where PSI values fall between 100 and 300. In this range, the air quality is
considered ‘very unhealthy’ and there are wide- spread symptoms in the general population. There
is imperfect substitution between the gases and r = −
2. In area 3, air quality is ‘hazardous’. At least one
gas has an ambient concentration such that the PSI value is between 300 and 500, and no gas has
exceeded its X
i max
concentrations. r is set at B
− 2.5.
The most severely polluted area is area 4 where one or more of the gases records ambient concen-
trations beyond the level at which PSI = 500. For all practical purposes, welfare losses in this area
are determined entirely by this gas. Therefore, the isopollution lines become parallel to the relevant
X
i
axis as shown in Fig. 3 and r − . The computed Air Pollution Index API values
for each county and MSA in the data set are presented in Fig. 4 in ascending order of API
value. The least polluted regions are those where only one gas has ambient concentrations in excess
of 50 of its NAAQS. Generally, this gas is O
3
, though in Colorado Springs PM
10
exceeds 50 m
gm
3
. At the next level are counties and MSAs with index values around 2. Typically, ambient
concentrations of PM
10
and O
3
exceed the threshold defined by 50 of their respective
NAAQS. In regions with index values around 3, three gases exceed this threshold.
In each of the above cases, counties and MSAs fall in area 1 as defined in Fig. 3.
10
When index values are around 3.2, ambient O
3
concentrations exceed the corresponding NAAQS and regions fall
in area 2 of the pollution space. In addition, the ambient concentration of one other gas, typically
PM
10
, exceeds the corresponding X
i min
. The index value jumps to approximately 3.9 when the ambi-
ent concentration of a second gas exceeds 50 of its NAAQS value and O
3
remains above 0.12 ppm. Regions with four gases recording ambient con-
centrations above their respective X
i min
values have index values between 4 and 5. Regions in area 1
have index values closer to 4 and those in area 2, i.e. with ambient O
3
concentrations exceeding 0.12 ppm, have index values close to 4.5. Some of the
most polluted regions in the data set fall in area 1 and record ambient concentrations for all five
gases above 50 of their respective NAAQS. The API values for these regions are just above 5.
Fig. 4. Air Pollution Index API and Pollutant Standards Index PSI compared.
10
There is only one region, Honolulu County, that falls in area 0. All other counties and MSAs fall in area 1 or 2.
Fig. 4 compares the API with the PSI. A striking difference between the two indices is that unlike the
PSI, which does not distinguish between regions with index values within broad ranges, the API can
furnish a detailed ranking by index value. Such a ranking would affect the implementation of an
environmental policy that distributes resources to alleviate pollution. In the current data set there are
only three pairs of countiesMSAs with identical API values. These are Chittenden and Min-
neapolis-St. Paul rank 4, Milwaukee – Waukesha and Washington, DC rank 33, and Fort Worth-
Arlington and Jersey City rank 36.
In general, regions with PSI values between 50 and 100 have lower API values, indicating relatively
lower pollution, than regions with PSI values between 100 and 200. Recall, however, that PSI
values are determined by the highest relative ambi- ent concentration alone, ignoring all other gases.
Some cases, therefore, contradict this conclusion. Consider the regions with API values just above 4.
By the PSI, these counties and MSAs would be less polluted than those with API values between 3.2
and 4.
These contradictory results are reinforced by considering Pima, Montgomery, and Cook coun-
ties. Their API values are 2.04, 3.22, and 5.04, respectively. Even though Cook County is in area 1,
it is regarded as one of the most polluted regions in the data set, with ambient concentrations for all five
gases exceeding 50 of their respective NAAQS. The PSI would consider Cook County less polluted
than Montgomery, and equally polluted as Pima County, which has a much lower API value.
5. Conclusions and suggestions for further research