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Power Laws and Air Pollution

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Abstract

Many natural and anthropogenic quantities are well-characterized by heavy-tailed distributions. Earthquakes and cities are oft-cited examples. This paper tests for power law behavior in a different context: air pollution emissions and health damage. Pollutants covered include fine particulate matter, sulfur dioxide, nitrogen oxides, ammonia, and volatile organic compounds. Emissions data are processed through an integrated assessment model to generate marginal (per-ton) and gross pollution damages. The cross-sectional distribution of emissions and damages are then tested for power law behavior. Evidence of power law behavior is detected for all pollutants. The length of the power law tail varies according to pollutant. The estimated Pareto shape parameter for gross damage is close to unity which is evidence of Zipf’s Law. Finally, sources in the power law tail are in counties that are between four and seven times more likely to be non-compliant with federal air pollution standards than counties with sources in the body of the distributions. The fitted power law parameters may help to guide regulators in categorizing sources with the purpose of designing targeted policies to bring non-attainment counties into compliance.

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Notes

  1. In the present context, uniform policy refers to regulation that does not distinguish between various sources of pollution in the stringency of emission limits.

  2. The lowest rank in Fig. 5 is about 7500. While there are nearly 10,000 sources in the empirical model used to estimate marginal damage, some of these sources produce zero emissions in 2008. Thus, for the GED calculation, these sources generate zero GED. Because x min must occur at a non-zero value [31], such sources are dropped from the emission and GED distributions.

  3. The NH3 and VOC distributions are explored here because (1) space constraints preclude graphical analyses of all the pollutants, and (2) VOC and NH3 exhibit the most significant differences in the fitted power law parameters. The reader is encouraged to consult the appendix for a detailed description of the procedures used.

  4. The results in Figure 7 use the x min values estimated at the 0.01 level of significance.

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Authors and Affiliations

  1. Department of Economics, Middlebury College and NBER, Middlebury, VT, 05753, USA

    Nicholas Z. Muller

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Correspondence to Nicholas Z. Muller.

Appendix

Appendix

This appendix consists of two sections. First A.1 describes the procedures used to produce the QQ-plots shown in Figs. A3 and A4. Section A.2 contains supplemental figures cited in the main body of the article.

1.1 Methods for Producing QQ-Plots

First, using VOC as an example, observations in the power tail of the marginal damage distribution are dropped. Note that this relies on the estimated x min value produced by the UMPU method. So, all ranks above x min are dropped. Then, a two parameter Weibull distribution is fitted to the body of the distribution. Using the estimated scale and shape parameters from this fitting procedure, which uses maximum likelihood estimation, a simulated Weibull distribution is produced. Next, a lognormal distribution is fit to the body of the marginal damage distribution. And then a lognormal distributed random variable is simulated with the mean and variance from the lognormal fitting procedure. The observed data and two simulated variables are log-transformed and the QQ-plots are produced. This procedure is repeated for NH3 and for the emissions and GED distributions. The results are shown in Figs. 11 (NH3) and 12 (VOC) (Fig. 8).

Fig. 8
figure 8

Visual power law diagnostic: marginal damage distributions. Top left PM2.5, top right NH3, bottom left SO2, bottom right NO x

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Fig. 9
figure 9

Visual fit evaluation of Pareto (blue), exponential (green), and lognormal (red) distributions relative to GED data (black). Left panel CSN estimated power law tail above rank = 341. Right panel GI estimated power law tail above rank = 1202

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Fig. 10
figure 10

Share of GED contributed by different ranks of sources. Horizontal axis shows fraction of sources with highest GED sources at origin. Heavy black line GED. Lowest solid line fraction of damage predicted by CSN estimate of \( \left\{\widehat{\alpha}\right\} \) in: F(GED) = S (α − 2)/(α − 1) from [31]. Dot-dash line fraction of damage predicted by GI estimate of \( \left\{\widehat{\alpha}\right\} \) in: F(GED) = S (α − 2)/(α − 1). Dashed line fraction of damage predicted by UMPU estimate of \( \left\{\widehat{\alpha}\right\} \) in: F(GED) = S (α − 2)/(α − 1). Dotted line fraction of damage predicted by ME estimate of \( \left\{\widehat{\alpha}\right\} \) in: F(GED) = S (α − 2)/(α − 1)

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Fig. 11
figure 11

QQ-plots in log-transform space on (non-power law) body of NH3 distributions. Top panel marginal damage distribution. Middle panel emission distribution. Bottom panel GED distribution. Horizontal axes depict quantiles from theoretical distribution of best fit to marginal damage or emissions. Vertical axes depict quantiles from marginal damage, GED, or emission distributions

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Fig. 12
figure 12

QQ-plots in log-transform space on (non-power law) body of VOC distributions. Top panel marginal damage distribution. Middle panel emission distribution. Bottom panel GED distribution. Horizontal axes depict quantiles from theoretical distribution of best fit to marginal damage or emissions. Vertical axes depict quantiles from marginal damage, GED, or emission distributions

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Muller, N.Z. Power Laws and Air Pollution. Environ Model Assess 21, 31–52 (2016). https://doi.org/10.1007/s10666-015-9466-2

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