Large cities are less green
We study how urban quality evolves as a result of carbon dioxide emissions as urban agglomerations grow. We employ a bottom-up approach combining two unprecedented microscopic data on population and carbon dioxide emissions in the continental US. We first aggregate settlements that are close to each other into cities using the City Clustering Algorithm (CCA) defining cities beyond the administrative boundaries. Then, we use data on emissions at a fine geographic scale to determine the total emissions of each city. We find a superlinear scaling behavior, expressed by a power-law, between emissions and city population with average allometric exponent across all cities in the US. This result suggests that the high productivity of large cities is done at the expense of a proportionally larger amount of emissions compared to small cities. Furthermore, our results are substantially different from those obtained by the standard administrative definition of cities, i.e. Metropolitan Statistical Area (MSA). Specifically, MSAs display isometric scaling emissions and we argue that this discrepancy is due to the overestimation of MSA areas. The results suggest that allometric studies based on administrative boundaries to define cities may suffer from endogeneity bias.
Allometry was originally introduced in the context of evolutionary theory huxley1936 () to describe the correlation between relative dimensions of parts of body size, for instance brain size in mammals, with changes in overall body size. In a classical result, Kleiber showed that metabolic rate, , and the body mass, , of a large range of mammal’s are related by an allometric power-law , where is the allometric exponent and is a constant kleiber1961 ().
In analogy with biological systems, Bettencourt et al. bettencourt2007 () showed that cities across US obey allometric relations with population size. Indeed, a large class of human activities can be grouped into three categories according to the value of the allometric exponent: (a) Isometric behavior (linear, non-allometric or extensive, ) typically reflects the scaling with population size of individual human needs, like the number of jobs, houses, and water consumption. (b) Allometric sublinear behavior (hipoallometric, non-extensive, ) implies an economy of scale in the quantity of interest because its per capita measurement decreases with population size. Hipoallometry is found, for example, in the number of gasoline stations, length of electrical cables, and road surfaces (material and infrastructure). (c) Superlinear behavior (hyperallometric, non-extensive, ) emerges whenever the pattern of social activity has significant influence in the urban indicator. Wages, income, growth domestic product, bank deposits, as well as rates of invention measured by patents and employment in creative sectors, display a superlinear increase with population size. These superlinear scaling laws indicate that larger cities are associated with optimal levels of human productivity and quality of life; doubling the city size leads to a larger-than-double increment in productivity and life standards bettencourt2007 (); bettencourt2010a (); bettencourt2010b ().
The optimal productivity of large cities raises the question of the consequences of urban growth to environmental quality. Indeed, it is intensely debated whether large cities can be considered environmentally “green”, implying that their productivity is associated with lower than expected greenhouse gases (GHG) and pollutant emissions bento2006 (); kahn2006 (); brownstone2009 (); dodman2009 (); puga2010 (); glaeser2010a (). For instance, some of these studies report that the level of commuting has a major contributing to the relation between GHG emissions and city size bento2006 (); kahn2006 (); brownstone2009 (); glaeser2010a (). As a consequence, compact cities would be more green due to the attenuation of the average commuting length. More recently, however, Gaigné et al. gaigne2012 () suggested that compact cities might not be as environmentally friendly as it was thought, mainly because increasing-density policies obligate firms and households to change place. This relocation of the urban system then generates a higher level of pollution. In this context, here we study the allometric laws associated with a particular type of GHG emissions from human activity by studying the relation between CO emissions of cities as a function of population size. We employ a bottom-up approach combining two unprecedented microscopic data on population and carbon dioxide emissions in the continental US. We first define the boundaries of cities using the City Clustering Algorithm (CCA) makse1995 (); makse1998 (); rozenfeld2008 (); giesen2010 (); rozenfeld2011 (); duranton2012 (); gallos2012 (); duranton2013 (); ioannides2009-2013 () which are then used to calculate the CO emissions. We find a superlinear allometric scaling law between emissions and city size. We also explore different sectors and activities of the economy finding superlinear behavior in most of the sectors. Our results pertain only emissions of CO. It will be desirable to extend it to the rest of GHGs. These results indicate that large cities may not provide as many environmental advantages as previously thought kahn2006 (); dodman2009 (); puga2010 (); glaeser2010a ().
Datasets. We use two geo-referenced datasets on population and CO emissions in the continental US defined in a fine geometrical grid. The population dataset is obtained from the Global Rural-Urban Mapping Project (GRUMPv1) grump2011 (). These data are a combination of gridded census and satellite data for population of urban and rural areas in the United States in year (Fig. 1a and Sec. 3). The GRUMPv1 data provides a high resolution gridded population data at arc-second, equivalent to a grid of at the Equator line.
The emissions dataset is obtained from the Vulcan Project (VP) compiled at Arizona State University vp2013 (). The VP provides fossil fuel CO emissions in the continental US at a spatial resolution of ( grid) from to . The data are separated according to economic sectors and activities (see Sec. 3 for details): Commercial, Industrial, and Residential sectors (obtained from country-level aggregation of non-geocoded sources and non-electricity producing sources from geocoded location), Electricity Production (geolocated sources associated with the production of electricity such as thermal power stations), Onroad Vehicles (mobile transport using designated roadways such as automobiles, buses, and motorcycles), Nonroad Vehicles (mobile surface sources that do not travel on roadways such as boats, trains, snowmobiles), Aircraft (Airports, geolocated sources associated with taxi, takeoff, and landing cycles associated with air travel, and Aircraft, gridded sources associated with the airborne component of air travel), and Cement Industry.
We analyze the annual average of emissions in for the total of all sectors combined (see Fig. 1b) and each sector separately (Fig. 2). The choice of data (rather than as in population) reflects the constraint that it is the only year for which the quantification of CO emissions has been achieved at the scale of individual factories, powerplants, roadways and neighborhoods and on an hourly basis vp2013 ().
To define the boundary of cities, we use the notion of spatial continuity by aggregating settlements that are close to each other into cities rozenfeld2008 (); giesen2010 (); rozenfeld2011 (); duranton2012 (); duranton2013 (); ioannides2009-2013 (). Such a procedure, called the City Clustering Algorithm (CCA), considers cities as constituted of contiguous commercial and residential areas for which we know also the emissions of CO from the Vulcan Project dataset. By using two microscopically defined datasets, we are able to match precisely the population of each agglomeration to its rate of CO emissions by constructing the urban agglomerations from the bottom up without resorting to predefined administrative boundaries.
We also use the US income dataset available in ASCII format by US Census Bureau censusbureau2013a () for the year . This dataset provides the mean household income per capita for the US counties. For each county, we combined the income data and the administrative boundaries censusbureau2013b () in order to relate them with the geolocated datasets (Fig. 1c and Sec. 3).
We first apply the CCA to construct cities aggregating population sites at site . The procedure depends on a population threshold and a distance threshold . If , the site is populated. The length represents a cutoff distance between the sites to consider them as spatially contiguous, i.e. we aggregate all nearest-neighbor sites which are at distances smaller than . Thus a CCA cluster or city is defined by populated sites within a distance smaller than as seen schematically in Fig. 3. Starting from an arbitrary seed, we add all populated neighbors at distances to the cluster smaller than until no more sites can be added to the cluster. The scaling laws produced by the CCA depend weakly on and . and we are interested in a region of the parameters where the scaling laws are independent of these parameters.
This aggregation criterion based on the geographical continuity of development was shown to provide strong evidence of Zipf’s law in the US and UK rozenfeld2008 (); giesen2010 (); rozenfeld2011 (); duranton2012 (); duranton2013 (); ioannides2009-2013 () in agreement with established results in urban sciences gabaix1999 (); ioannides2003 (); giesen2011 (); giesen2012 (). For cut-off lengths above , it was shown that CCA clusters verify the Zipf’s law and the Zipf’s exponent is independent of . Next, we first present results for aggregated clusters at , and then show the robustness of the scaling laws over a larger range of parameter space.
In order to assign the total CO emissions to a given CCA cluster, we superimpose the obtained cluster to the CO emissions dataset. If a populated site composing a CCA cluster falls inside a CO site, we assign to the populated site the corresponding CO emissions proportional to its area , considering that the emissions density is constant across the CO site of . For a given CCA cluster, we then calculate the population (POP) and CO emissions by adding the values of the constitutive sites of the cluster.
Scaling of emissions with city size. Figure 4 shows the correlation between the total annual CO emissions and POP for each CCA cluster for and (). We perform a non-parametric regression with bootstrapped confidence bands silverman1986 (); hardle1990 () (see Sec. 3). We find that the emissions grow with the size of the cities, on average, faster than the expected linear behavior. The result can be approximated over many orders of magnitudes by a power-law yielding the following allometric scaling law:
where and () is the allometric scaling exponent obtained from Ordinary Least Squares (OLS) analysis montgomery2006 () for this particular set of parameters and (see Sec. 3 for details on OLS and on the estimation of the exponent error, all emissions are measured in log base of metric tonnes of carbon per year).
In addition, we investigate the robustness of the allometric exponent as a function of the thresholds and . Figure 5a shows as a function of the cut-off length for different values of population threshold (, , , and ). We observe that increases with until a saturation value which is relatively independent of . Performing an average of the exponent in the plateau region with over , we obtain . Thus, we find superlinear allometry indicating an inefficient emissions law for cities: doubling the city population results in an average increment of in CO emissions, rather than the expected isometric . This positive non-extensivity suggests that the high productivity found in larger cities bettencourt2007 (); bettencourt2010a () is done at the expense of a disproportionally larger amount of emissions compared to small cities.
Figure 5b investigates the emissions of cities as deconstructed by different sectors and activities of the economy. We perform non-parametric regression with bootstrapped confidence bands of (see Fig. 6 for and by each sector) versus and we find that the exponents for different sectors saturate to an approximate constant value for . We assign an average exponent, over the plateau per sector as seen in Table 1. The sectors with higher exponents (less efficient) are Residential, Industrial, Commercial and Electric Production with , above the average for the total emissions. Onroad vehicles contribute with a superlinear exponent , yet, below the total average. The exponent for Nonroad vehicles is also below the average at , while Aircraft sector displays approximate isometric scaling with . Cement Production displays sublinear scaling , although the reported data is less significant than the rest with only datapoints of cities available.
We further investigate the dependence of the allometric exponent on the income per capita of cities by aggregating the CCA clusters by their income (INC) and plotting the obtained in Fig. 7 (see also Fig. 8). We find an inverted U-shape relationship, which is analogous to the so-called environmental Kuznets curve (EKC) kuznets1955 (); grossman1995 (); kahn2006 (). We observe that initially increases for cities with low income per capita until an income turning point located at $ per capita (in US dollar). After the turning point, decreases indicating an environmental improvement for large-income cities. However, the allometric exponent remains always larger than one regardless of the income level (except for the lowest income) indicating that almost all large cities are less efficient than small ones, no matter their income.
Comparison with MSA. A further important issue in the scaling of cities is the dependence on the way they are defined krugman1996 (); rozenfeld2008 (); giesen2010 (); rozenfeld2011 (); duranton2012 (); duranton2013 (); ioannides2009-2013 (). Thus, it is of interest to compare our results with definitions based on administrative boundaries such as the commonly used Metropolitan Statistical Areas (MSA) fragkias2013 () provided by the US Census Bureau censusbureau2013c (). MSAs are constructed from administrative boundaries aggregating neighboring counties which are related socioeconomically via, for instance, large commuting patterns. A drawback is that MSAs are available only for a subset ( cites) of the most populated cities in the US, and therefore can represent only the upper tail of the distribution krugman1996 (); rozenfeld2011 (); ioannides2009-2013 () (see Sec. 3 for details).
Furthermore, we find that the MSA construction violates the expected extensivity bettencourt2007 (); rozenfeld2011 () between the land area occupied by the MSA and their population since MSA overestimates the area of the small agglomerations rozenfeld2011 (). This is indicated in Fig. 9, where we find the regression:
with and (). This approximate square-root law implies that the density is not constant across the MSAs:
On the contrary, CCA clusters capture precisely the occupied area of the agglomeration leading to the expected extensive relation between land area and population as seen also in Fig. 9:
with and , with small dispersion , implying that the density of population of CCA clusters is well-defined (extensive), i.e. it is constant across population sizes,
In summary, while the CCA displays almost isometric relation between population and area, the MSA shows a sublinear scaling between these two measures. As a consequence, the emission of CCA is independent of the population density, as expected. On the other hand, from Eq. 2 and Eq. 6, the MSA leads to a superlinear scaling between them, .
The non-extensive character of the MSA areas is due to the fact that many MSAs are constituted by aggregating small disconnected clusters resulting in large unpopulated areas inside the MSA. This is exemplified in some typical MSAs plotted in Fig. 10, such as Las Vegas, Albuquerque, Flagstaff and others. The plots show that a large MSA area is associated to a series of disconnected small counties, like it is seen, for instance, in the region near Las Vegas. This clustering of disconnected small cities inside a MSA results into an overestimation of the emissions associated with the Las Vegas MSA, for instance. The same pattern is verified for many small cities, specially in the mid-west of US, as seen in the other panels. For some large cities, like NY, the agglomeration captures similar shapes as in the occupied areas obtained with CCA, although it is also clearly seen that the area of the NY MSA contains many unoccupied regions. Therefore, the occupied area of a typical MSA is overestimated in comparison to the area that is actually populated as captured by the CCA, the bias is larger for small cities than larger ones. This endogeneity bias leads to an overestimation of the CO emissions of the small cities as compared to large cities. Consequently, we find a smaller allometric exponent for MSA than CCA with an almost extensive relation:
with and ( = 0.71, see Fig. 11). This result is consistent with previous studies of scaling emissions of MSA by Fragkias et al. fragkias2013 (), who used MSAs and found a linear scaling between emissions and size of the cities, and also Rybski et al. rybski2013 (), who used administrative boundaries to define cities in countries. Table 2 and 3 summarize the results of CCA and MSA cities.
Thus, the measurement bias in the MSAs leads to smaller found for MSA as compared with CCA, since low-density MSAs have relatively large areas. Hence, the CCA results, which are not subject to that endogeneity bias, should be considered the main source of information on emissions. They show a positive link between emissions and population size as well as the expected extensive behavior of the occupied land. This analysis calls the attention to use the proper definition of cities when the scaling behavior of small cities needs to be accurately represented. Indeed, this issue arises in the controversy regarding the distribution of city size for small cities since the distribution of administrative cities (such as US Places) are found broadly lognormal (that is, a power law in the tail that deviates into a log-Gaussian for small cities) eeckhout2004 (); levy2009 (); eeckhout2009 (); glaeser2010b (); ioannides2009-2013 (), while the distribution of geography-based agglomerations like CCA is found to be Zipf distributed along all cities (power-law for all cities) makse1995 (); makse1998 (); rozenfeld2008 (); giesen2010 (); rozenfeld2011 (); duranton2012 (); duranton2013 ().
In general, we expect that when the scaling obtained by CCA is extensive, then any agglomeration of CCA such as MSA, should give rise to extensive scaling too. However, when there are intrinsic long-range spatial correlations in the data (like in non-extensive systems with ), agglomerating populated clusters (as done with MSA) may give different allometric exponents depending on the particular administrative boundary used to define cities. It is of interest to note that, beyond MSA fragkias2013 (), there are other administrative boundaries used in the literature to define cities, like for instance US-Places studied in eeckhout2004 (); levy2009 (); eeckhout2009 (). This measurement bias is a generic property of any non-extensive system, such as a physical system at a critical point. Thus, scaling laws obtained using administrative boundaries to define cities which cluster data in a somehow arbitrary manner may need to be taken with caution.
In summary, we find that CCA urban clusters in the US have sub-optimal CO emissions as measured by a superlinear allometric exponent . The exponent decreases for cities with low and high income per capita in agreement with an EKC hypothesis kahn2006 (). From the point of view of allometry, larger cities may not represent an improvement of CO emissions as compared with smaller cities.
Population dataset. The United States population dataset for the year
is a part of the Global Rural-Urban Mapping Project (GRUMPv1). The
GRUMPv1 is available in shapefile format on the Latitude-Longitude projection
(Fig. 12a) and it was developed by the International Earth Science
Information Network (CIESIN) in collaboration with the International Food
Policy Research Institute (IFPRI), the World Bank, and the Centro
Internacional de Agricultura Tropical (CIAT) grump2011 () (Fig.
1a). The GRUMPv1 combines data from administrative units and urban
areas by applying a mass-conserving algorithm named Global Rural Urban
Mapping Programme (GRUMPe) that reallocates people into urban areas, within
each administrative unit, while reflecting the United Nations (UN) national
rural-urban percentage estimates as closely as possible grump2011 (). The
administrative units (more than units with population
inhabitants) are based on population census data and their administrative
boundaries. The urban areas (more than areas with population
inhabitants) are based on night-time lights data from the National Oceanic
and Atmospheric Administration (NOAA) and buffered settlement centroids (in the
cases where night lights are not sufficiently bright). In order to provide a
higher resolution gridded population data ( arc-second, equivalent to a grid
of at the Equator line), the GRUMPv1 assumes that
the population density of the administrative units are constant and the
population of each site is proportional to the administrative unit areas located
inside of that site. We exported the original data to the ASCII format on
Lambert Conformal Conic projection (Fig. 12b), available to download at
http://jamlab.org. Both projections parameters are defined as follow:
Projection name: Latitude–Longitude (LL)
Horizontal datum name: WGS84
Ellipsoid name: WGS84
Denominator of flattening ratio:
Projection name: Lambert Conformal Conic (LCC)
Standard parallels: ,
Latitude of projection origin:
Geographic coordinate system: NAD83
Emissions dataset. The second dataset used in this study is the annual mean of the United States fossil fuel carbon dioxide emissions with the grid of for the year . Full documentation is available at http://vulcan.project.asu.edu/pdf/Vulcan.documentation.v2.0.online.pdf. This dataset was compiled by the Vulcan Project (VP) and it is already available in binary format on the Lambert Conformal Conic projection defined above. The VP was developed by the School of Life Science at Arizona State University in collaboration with investigators at Colorado State University and Lawrence Berkeley National Laboratory vp2013 (). The VP dataset is created from five primary datasets, constituting eight data types: The National Emissions Inventory (NEI) containing the Non-road data (county-level aggregation of mobile surface sources that do not travel on roadways such as boats, trains, ATVs, snowmobiles, etc), the Non-point data (county-level aggregation of non-geocoded sources), the Point data (non electricity-producing sources identified as a specific geocoded location) and the Airport data (geolocated sources associated with taxi, takeoff, and landing cycles associated with air travel); The Emissions Tracking System/Continuous Emissions Monitoring (ETS/CEM) containing the Electricity production data (geolocated sources associated with the production of electricity); The National Mobile Inventory Model (NMIM) containing the On-road data (county-level aggregation of mobile road-based sources such as automobiles, buses, and motorcycles); The Aero2k containing the Aircraft data (gridded sources associated with the airborne component of air travel), and finally, the Portland Cement containing the cement production data (geolocated sources associated with cement production).
These data types supply the CO emissions sectors: Aircraft, Cement, Commercial, Industrial, Non-road, On-road, Residential, and Electricity. In order to represent all the sectors in a grid, the VP assumes that the CO emissions of each site is given by the contributions of the geocoded and non-geocoded (via area-weighted proportions) sources located inside of that site. We exported the original data to the ASCII format, available to download at http://lev.ccny.cuny.edu/~hmakse/soft_data (Fig. 1b and Fig. 2).
Income per capita dataset. We also use the US income dataset available in ASCII format by US Census Bureau censusbureau2013a () for the year . This dataset provides the mean household income per capita for the US counties. For each county, we combined the income data and the administrative boundaries (Fig. 1c) in order to relate them with the geolocated datasets. The US county boundaries are also available to download in ASCII format by the US Census Bureau censusbureau2013b (). However, we already joined these datasets and provided them to download at http://lev.ccny.cuny.edu/~hmakse/soft_data.
Superimposing the datasets. We superimposed the population and CO datasets on the Lambert Conformal Conic projection in order to estimate the CO emissions on a higher grid level (). We checked if each population site is inside of a CO site. If so, we assigned the CO value as proportional to its area (), considering that the CO density is constant in each CO site. For the population and income datasets, we checked if each population site (actually, the center of mass) is inside of some US county boundary. If so, we assigned the income value for that site equal to the income value for the county. We performed this test taking into account that a horizontal line (in the polygon direction), starting in a point that is inside of a polygon, hits on it an odd number of times, while a point that is outside of the polygon, hits on it an even number of times.
MSA. The definitions of Metropolitan Statistical Area (MSA), Primary Metropolitan Statistical Area (PMSA) and Consolidated Metropolitan Statistical Area (CMSA) are provided by the US Census Bureau censusbureau2013c (). The MSAs are geographic entities defined by some counties socioeconomically related with population larger than . The PMSAs are analogous to MSAs, however they are defined by just one or two counties also socioeconomically related with population larger than . Finally, the CMSA are large metropolitan region defined by some PMSAs close to each other. In order to set a relation between the definition of MSA/CMSA cities and CCA cities, we show the most populated MSA/CMSA cities and the largest CCA cities associated to them in Table 1 and 2. The largest CCA city associated to a given MSA/CMSA is defined by the most populated CCA city whose center of mass is inside of that MSA/CMSA boundary. All datasets are available to download from censusbureau2013c (), including the population and administrative boundaries of MSA/CMSA. Additionally, we make them available at http://lev.ccny.cuny.edu/~hmakse/soft_data.
Nadaraya-Watson Method. In order to calculate the allometric scaling exponents, we performed well-known statistic methods hardle1990 (). For one data distribution , we apply the Nadaraya-Watson method nadaraya1964 (); watson1964 () to construct the kernel smoother function,
where is the number of points and is a Gaussian kernel of the form,
where the is the bandwidth estimated by least squares cross-validation method racine2004 (); li2004 (). We compute the () confidence interval (CI) by the so-called quantile function over random bootstrapping samples with replacement.
For our case, the distribution is the set of values , where is from to the number of CCA cities . Furthermore, we calculate the exponents by the ordinary least square (OLS) method nr2007 (). Let us to consider the terms,
The regression exponents ( and in the equation ) are given by,
If the errors are normally and independently distributed, the standard error of each exponent is given by montgomery2006 (),
where is the Student-t distribution with of CI and degrees of freedom, and the variances and are given by,
Finally, we show the value of the regression exponents as,
-squared. The is the coefficient of determination or -squared and is calculated as following:
The by emission sector and the average are in Table 1.
We gratefully acknowledge funding by NSF, CNPq, CAPES and FUNCAP. We thank X. Gabaix and S. Alarcon for helpful discussions.
EAO, JSA, and HAM designed research, performed research, and wrote the manuscripts.
Competing financial interests: The authors declare no competing financial interests.
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