An improved pollution index

Table 2 1997 ambient concentrations and Pollutant Standards Index PSI values selected counties SO 2 24 h O 3 1 h, 2nd Max PM 10 24 h, 99th p’ile d State County CO 8 h NO 2 1 h, AM c ppm ppm mgm 3 ppm mgm 3 Ambient concentrations in 1997 a 2.2 0.0193 0.131 79 PA 65.5 Montgomery 10.48 129 0.097 0.0178 4.4 AZ Pima 107.42 0.113 99 IL Cook 5.3 0.0336 PSI 6alues for each gas b 100–200 50–100 PA Montgomery 50–100 50–100 AZ Pima 50–100 50–100 50–100 IL Cook 50–100 50–100 PSI 6alue for county b 100–200 PA Montgomery AZ Pima 50–100 IL Cook 50–100 a Source: EPA 1998, Table A-12, pp. 122–139. The data for SO 2 are reported in ppm. They were converted to mgm 3 using the EPA’s conversion factor of 1 ppm = 2620 mgm 3 D. Mintz, personal communication, March 29, 1999. b Own calculations based on Table 1. c Arithmetic mean. d 99th percentile. Consider the three counties shown in Table 2. The data in this table represent the composite average of 1997 readings for these counties at various monitoring sites. They are interpreted to represent the air quality on a typical day in each of these counties. 2 PSI values are calculated on the basis of Table 1. According to this index, Cook and Pima Counties are equally polluted whereas Mont- gomery County has worse air quality. However, note that the PSI is in the 50 – 100 range for all five gases in the case of Cook County, but only for two gases in case of Pima. Furthermore, the ambient concentrations of all gases except O 3 are lower in Montgomery County as compared to the other two counties. Yet, according to the PSI, Montgomery is the most polluted of the three counties since the index value is determined entirely by its O 3 concentration, which puts it in the 100 – 200 range regardless of the ambient concentration of the other gases. The aim of the paper is to develop an index that is based on the level of each pollutant, their individual physical impacts, and the consequent welfare losses while building upon the framework established under the PSI. The welfare losses provide the common metric in terms of which of the ambient concentrations of different environ- mental indicators may be aggregated into an over- all pollution index.

3. An improved pollution index

To construct the index, environmental indicators are first aggregated into the appropriate attribute; attributes are then aggregated into the overall pollution index. Thus, if X i n X i n R + denotes the ambient concentration of each indicator i i = 1, 2, . . ., I in region n n = 1, 2, . . ., N, the index for region n is defined as I n = fA s n , where A s n = gX n i . Here s denotes the set of environmental attributes s = 1, 2, . . ., S, A s n refers to a particular attribute for region n, and f· and g· are contin- uous functions with well defined first and second order partial derivatives. Note that I n may be defined directly in terms of the indicators as I n = fgX i n = hX i n , I n R + . 2 EPA cautions that these data should not be used to rank counties according to their air quality since they represent the air quality in the immediate vicinity of the monitoring stations and may not be representative of the county-wide situation EPA, 1998, p. 139. The data is used for illustrative purposes. Fig. 1. A hypothetical damage function. When is a region considered polluted? Follow- ing the rule implied by the PSI, a region is pol- luted when any environmental indicator exceeds the threshold below which damages from expo- sure to it are not significant. This is indicated by X i min in Fig. 1. An environmental indicator is a pollutant when X i n \ X i min . A region is polluted if X i n \ X i min for any i. This implies that the index value is independent of ambient concentration of indicator i if and only if X i n 5 X i min . When all environmental indicators are at or below their respective minimum levels, the region is not pol- luted and the index value is zero. That is, I n = Á Ã Í Ã Ä fgX i n , X j n \0 if X i n \ X i min , X j n \ X j min , ij fgX j n \0 if X i n 5 X i min , X j n \ X j min , ij if X i n 5 X i min Öi . To determine the severity of pollution, the physical impacts due to exposure are combined with their welfare consequences. Panel A of Fig. 1 depicts an epidemiological dose-response func- tion. 3 It relates the probability of damage to ambient concentrations. The figure suggests that at the lower range of ambient concentrations probability of damage rises rapidly. Eventually, as the probability of damage approaches 1, the growth rate of probability declines. Now consider panel B of the same figure. This shows the loss in welfare due to increased expo- sure to an environmental indicator. EPA 1997 see in particular figure 18 found some evidence of rapidly rising welfare losses due to increased exposure to the criteria pollutants under the CAA. Under the formulation, welfare losses are modeled as a smoothly convex function of ambi- ent concentrations over the interval X i min , X i max . 4 At low ambient concentrations welfare losses in- crease at an increasing rate following the rapid rise in the probability of damage. At higher con- centrations, even though the marginal probability of damage begins to taper off, welfare continues to decline rapidly since the absolute probability of damage is high. However, the function is discon- tinuous at X i max . Once damage is near certain, marginal welfare losses taper off. This might be because almost all the populations or materials in the region are afflicted by the damage. In this case, the index may be set at some arbitrarily high level indicating severe pollution. The index does not particularly distinguish between these regions since the marginal welfare loss due to increased ambient concentrations is minimal. 3 EPA 1997 used a similar functional form to estimate the change in probability of adverse health effects due to increased exposure to the criteria pollutants under the CAA. An exam- ple is the rising probability of shortness of breath or chest tightness as ambient SO 2 concentrations increase. 4 Since the attributes are themselves sub-indices of environ- mental quality, it is assumed that each attribute is also a monotonically increasing and convex function of a subset of environmental indicators. That is, f ’ \ 0, g’ \ 0, h´’ \ 0, and the appropriate second order conditions hold. Fig. 2. Concave isopollution lines. Once ambient concentrations of environmental indicators are translated into their welfare im- pacts, the standard micro-economic concept of substitutability can be invoked Bourguignon and Chakravarty, 1998. This allows one to use the construct of isopollution lines and surfaces in order to compare environmental quality over time and space. An isopollution surface is the hyper- plane defined by all regions with identical envi- ronmental quality. Its two-dimensional analog is an isopollution line. Consider a region that is polluted with respect to one indicator only. That is, X i n \ X i min and X j n 5 X j min , for all i j, i, j = 1, 2, . . ., I. Then the isopollution lines and surfaces will be parallel to the X i axis: the environmental quality in this region is determined by the ambient concentration of indicator i only. In the more general case, suppose there are two pollutants, X 1 and X 2 . 5 Suppose also that X 1 min B X 1 m B X 1 n and X 2 m \ X 2 n \ X 2 min . In other words, region n is more polluted in terms of indicator 1 while the opposite is true for region m. Finally, assume that the environmental quality in these two regions is the same so that the numerical value of the environmental pollution index for these regions is identical. Then the isopollution line connecting these regions will be concave in the X 1 , X 2 pollutant space. The concavity of the isopollution lines arises from the strictly convex damage function shown in panel B of Fig. 1, and the decreasing marginal rate of substitution between pollutants. First note that isopollution lines must be downward sloping. If pollutant 1 decreases, then pollutant 2 must increase in order for the environmental quality to remain unchanged. Next, suppose that X 1 is at some high level whereas X 2 is at some relatively low level, as shown by point A in Fig. 2. At this point, the welfare impact due to a marginal change in the concentration of indicator 1 is high, whereas that due to a change in the concentration of indicator 2 is low. Thus, a marginal decrease in X 1 may be ‘substituted’ by a relatively larger increase in X 2 , while maintaining the level of environmental quality. The opposite will be true in the case of point B. In the limiting case, the isopollution lines may be negatively sloped straight lines implying perfect substitution, or in- verted L-shaped lines implying perfect comple- mentarity. A final property of the pollution index relates to scale effects. It is assumed that the index displays non-decreasing returns to scale: a propor- tional increase in the level all pollutants results in a proportional, or more than proportional, in- crease in the index value. The properties outlined above are intuitive and general. A function that satisfies these properties is the constant elasticity of substitution CES function with appropriate sign restrictions. There- fore, this function is used to aggregate the indica- 5 This assumption is not restrictive since the results can be easily extended to higher dimensions. The two dimension case is used here as it is easy to depict graphically. tors into attributes and further into the overall index as shown in the following equation. I n = Á Ã Í Ã 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 Ã Í Ã 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