Scaling of Urban Income Inequality in the United States
Elisa Heinrich Mora, Jacob J. Jackson, Cate Heine, Geoffrey B. West, Vicky Chuqiao Yang, Christopher P. Kempes
SScaling of Urban Income Inequality in theUnited States
Elisa Heinrich Mora
1, 2 , Jacob J. Jackson
3, 2 , Cate Heine
4, 2 ,Geoffrey B. West , Vicky Chuqiao Yang , and Christopher P.Kempes Minerva Schools at KGI, San Francisco, CA 94103, USA Santa Fe institute, Santa Fe, NM 87501, USA Brown University, Providence, RI 02912, USA Massachusetts Institute of Technology, Cambridge, MA 02139,USAMarch 1, 2021
Abstract
Urban scaling analysis, the study of how aggregated urban fea-tures vary with the population of an urban area, provides a promisingframework for discovering commonalities across cities and uncoveringdynamics shared by cities across time and space. Here, we use the ur-ban scaling framework to study an important, but under-explored fea-ture in this community—income inequality. We propose a new methodto study the scaling of income distributions by analyzing total incomescaling in population percentiles. We show that income in the leastwealthy decile (10%) scales close to linearly with city population, whileincome in the most wealthy decile scale with a significantly superlin-ear exponent. In contrast to the superlinear scaling of total incomewith city population, this decile scaling illustrates that the benefitsof larger cities are increasingly unequally distributed. For the poorestincome deciles, cities have no positive effect over the null expectationof a linear increase. We repeat our analysis after adjusting income byhousing cost, and find similar results. We then further analyze the a r X i v : . [ phy s i c s . s o c - ph ] F e b hapes of income distributions. First, we find that mean, variance,skewness, and kurtosis of income distributions all increase with citysize. Second, the Kullback-Leibler divergence between a city’s incomedistribution and that of the largest city decreases with city popula-tion, suggesting the overall shape of income distribution shifts withcity population. As most urban scaling theories consider densifyinginteractions within cities as the fundamental process leading to thesuperlinear increase of many features, our results suggest this effect isonly seen in the upper deciles of the cities. Our finding encouragesfuture work to consider heterogeneous models of interactions to forma more coherent understanding of urban scaling. Throughout human history, the global urban population has grown continu-ously. More than half of the global population is currently urbanized, placingcities at the center of human development [1]. It is estimated that by 2030,the number of megacities, cities with more than 10 million inhabitants, willincrease from 10 to approximately 40 [1]. Thus, there is an urgent need fora quantitative and predictive theory for how larger urban areas affect a widevariety of city features, dynamics, and outcomes [2]. Perhaps most critically,we need this theory to address how larger cities positively and negativelyaffect socioeconomic outcomes and the quality of life of individuals.Previous research has demonstrated power-law-like relationships between ur-ban population (also referred to as size later in the text) and many urbanfeatures such as GDP, patents, crime, and contagious diseases that persistglobally [3–7]. These relationships can often be described by Y = Y N β , (1)where Y is an urban feature, such as GDP or number of crime instances, N is the population of the city, Y is a constant, and β is the scaling exponent.For many urban outputs, the scaling exponent β is greater than 1, suggestinggreater rates of productivity (in both the positive and negative sense) inmore populated cities. These observations, known as urban scaling, suggestthat a small set of mechanisms significantly influence a variety of urbanfeatures across diverse cities [8, 9]. Understanding these mechanisms hasimportant implications for developing more prosperous and safer cities. In2his framework, desirable aspects with β > have positive returns to scale,while desirable aspects with β < have a less than linear return to scale,demonstrating a diseconomy of scale. Similarly, for undesirable features β > shows a diseconomy of scale since the associated per-capita costs would beincreasing with city size.One important aspect of urban features that remains under-explored in theurban scaling framework is economic inequality. Inequality has fundamentalimplications for individuals’ quality of life and the productivity and stabil-ity of societies [10]. Past research has heightened debate about economicinequality and its relationship with economic growth and general welfare[11–17]. Many have raised concern of its negative effects on political stabil-ity [11, 18], crime [19] and corruption [20]. It has been shown that moreunequal places have higher murder rates, grow more slowly, and the corre-lation between area-level inequality and population growth is positive [21].Economic inequality is usually measured in terms of the dispersion in thedistribution of income or wealth, such as in the Gini Coefficient. Some pastresearch has noted larger cities are correlated with increasing Gini coefficientin income distribution [22–24] , but it remains unclear if there are systematicrelationships between other features of the income distributions and urbanarea size. Furthermore, characterizing distributions by a single metric maylose important information [11] – for example, does being poor in biggercities correspond to a higher or lower standard of living than being poor ina smaller city?A few recent studies [25, 26] have investigated the scaling of total incomein various income brackets in Australia. These studies find that the totalincome in lower income brackets scales sublinearly or linearly, while higherbrackets scale superlinearly, suggesting greater income agglomeration in thehigher income categories in more populated cities. While these studies areinformative and provide a new measure for inequality in terms of absoluteincome (instead of relative income, as in the Gini Coefficient), a limitation isthat this measure confounds inequality with average income, which increaseswith city population. In particular, the “equal” situation in this new measureof inequality is when the total income for all income brackets scales linearly.However, given that total income scales superlinearly in cities globally [3, 4],this “equal” situation is unlikely to occur. For example, even if the shapesof income distributions remain identical, income bracket aggregations followdistinct scaling relationships as a result of differences in mean. Figure 13 roup by income brackets Group by deciles Mean increases with city size, variance does notIncomeMean and variance both increase with city sizeIncomeLarger citySmaller city METHOD S C E N A R I O (A) (B)(C) (D) Figure 1: Illustration comparing two methodologies—scaling obtained fromgrouping by income bracket (A and C) and that by decile (B and D). Us-ing simulated log-normal income distributions in two scenarios—log-meanincreases with city size while log-variance remains the same (A and B), andlog-mean and log-variance both increases with city size (C and D). The in-come distributions are illustrated on a log-scale. The income-bracket group-ing (A and C) leads to differences in the groups’ income scaling for bothscenarios, and fails to distinguish whether larger cities have more dispersionin their income distributions. The decile grouping (B and D) leads to thedifferences in the groups’ scaling observed only when the dispersion increaseswith the population. The insets show how scaling exponent ( β ) varies withincome groups (bracket or decile). 4 and C illustrates this behavior using simulated log-normal distributions.While the measure of inequality proposed in [25, 26] can be valuable for someapplications, it would be useful to untangle the increase in mean from thegreater dispersion in income.In this manuscript, we address a few keys questions: (1) How does incomeinequality (adjusted for shifting average income) systematically change withcity size? (2) How different is the income of rich and poor people (measuredby percentiles of the population) in small and large cities, and how does thisdifference scale with city size? (3) Are poor people in a larger city better offthan poor people in a small city, after adjusting by the cost of living? Howabout the same for rich people?Here, we propose a new method to study the scaling of inequality by analyz-ing total income scaling in population percentiles. We show that income inthe least wealthy decile (10%) scales almost linearly with city size, while thatin the most wealthy decile scales with a significantly superlinear exponent.This illustrates that the benefits of larger cities are increasingly unequallydistributed, and for the poorest income deciles, city growth has no positive ef-fect on income growth over the null expectation of a linear increase. We thenintroduce systematic considerations of the entire distribution of income toshow which income distribution features are changing with city size. We findthat the mean, variance, skewness, and kurtosis of the income distribution allscale systematically with city size. We introduce a KL-divergence procedureto systematically compare all moments and find that comparisons with thelargest cities also demonstrate a systematic scaling with city size, indicatingthat the overall shape of income distribution is radically shifting with citysize. We then attempt to identify actual changes in purchasing power withcity size by normalizing income by housing costs, which also grow superlin-early with city population. Finally, we discuss how these observations canbe connected with the proposed mechanisms underlying urban scaling. We propose a new method to investigate the scaling of income aggregatedby deciles in each city (i.e., the bottom 10%, the next 10%, and so on). The5umber of individuals in decile n of city i is, N ( n ) i = N i / , where N i is thepopulation of city i . The total income in decile n of city i , Y ( n ) i is, Y ( n ) i = (cid:88) j ∈ D ( n ) y i,j , (2)where D ( n ) are the individuals in income decile n , and y i,j is the income ofindividual j in city i . See Supplemental Materials for more details on thedecile assignment in our computational implementation.Figure 1 C and D illustrate this method on simulated log-normal incomedistributions. Panel C represents the situation in which cities shift in log-mean with city size, but do not shift in log-standard deviation, and panelD represents the situation in which cities increase both log-mean and log-standard deviations with city size. We consider the former case an exampleof the “equal” situation, and this method should lead to no variation in scalingexponents across deciles. Variations in scaling exponent only occur for thelatter case. We also contrast the results of our method with that of thegrouping by income bracket method in Figure 1 A and C, where variationsin scaling exponents occur for both scenarios.Figure 2: Examples of the estimated income distributions using censustract data. Income is measured in US dollars. The three metropoli-tan areas shown are: New York-Newark-Jersey City, NY-NJ-PA, popula-tion 20,316,622; Minneapolis-St. Paul-Bloomington, MN-WI, population3,670,397; Santa Fe, NM, population 204,396.6 .2 Data and income distribution estimation The primary dataset used in our analysis is the 2015 American CommunitySurvey conducted by the US Census Bureau (see Supplementary Materials formore detail). We use the income data reported on the level of census tracts,small local areas of on average 4500 people, of which on average 2300 reportedincome. We infer the individual-level income distribution in MetropolitanStatistical Areas (MSAs) by applying the Gaussian kernel density estima-tor with a widened Silverman bandwidth function on the census-tract-leveldata. This method assumes income in each census tract is distributed asa Gaussian. The mean equals the average income of the census tract, andthe standard deviation is calculated as a function of the number of datapoints. Aggregating the Gaussian probability density functions (PDFs) foreach census tract in the MSA produces an estimated income PDF for theMSA. Examples of the estimated individual-level income distribution for afew MSAs are shown in Figure 2.
The estimated income distributions for US cities are grouped into deciles:the 10% of the population which reports the lowest income is grouped intothe first decile (decile β , and corresponding confidence intervals, by per-forming an ordinary least square regression of the log-transformed variables, log( Y ( n ) i ) = β log( N ( n ) i ) + c , and β and c are the fitted parameters. Thismethodology is consistent with previous research such as [3]. We further analyze how the shapes of the income distributions vary withcity population. We first compute the first four statistical moments, mean,variance, skewness, and kurtosis, for income distributions of each city, andanalyze how they vary with population. We then compute the Kullback-Leibler (KL) divergence between each city’s income distribution and that ofthe largest city (New York-Newark-Jersey City MSA). The KL divergencemeasures how different one distribution is from another, while the zero value7ndicates the two distributions are identical, and a greater value indicatesmore divergence. Mathematically, the KL divergence between two discretedistributions of random variable x , P ( x ) and Q ( x ) is, KL ( P || Q ) = (cid:88) x P ( x ) log( P ( x ) /Q ( x )) . (3) In order to normalize income by the cost of living, we calculate total housingcost in a census tract as cost = 12 ( u rent r + u own o ) , where the averagemonthly rent r , the average monthly owner costs o , and the number of unitsof each type u rent and u own are all taken from the 2015 American CommunitySurvey (see Supplementary Materials for more detail and access information).We then repeat the decile-grouped analysis on income adjusted for housingcost, as well as analyze how the proportion of income spent on housing varieswith city size in each decile. The results for scaling of income aggregated in deciles are summarized inFigure 3. For the lowest two deciles, the scaling exponent β is linear orslightly sublinear ( . ). For upper deciles, β is consistently superlinear,as high as . as compared to the scaling exponent of total income in ourdataset, β = 1 . . This shows that scaling effects are not equivalent for allsegments of the population. The poorest two deciles in bigger cities makeabout the same income as their counterparts in smaller cities, while thewealthiest eight deciles in bigger cities make more than their counterparts insmaller cities, where the difference increases with the decile. We further analyze how income distributions vary with urban area popula-tion by studying the statistical moments of the income distributions. Wefirst examine the first four moments: mean, variance, skewness, and kurto-sis. 8
A) (B)
Figure 3: Scaling of income (in US Dollars) by population for deciles ofUS MSAs. (A) Scaling of total income in deciles (B) Scaling exponents( β ) of each decile and corresponding 95% confidence intervals. The dashedline is β = 1 to help guide the eye. Higher-income deciles exhibit greaterscaling exponents than lower income deciles, and the lowest deciles exhibitnear-linear scaling. The scaling exponents for aggregated income in city,combining all deciles, is 1.07.The scaling of the four moments of the estimated individual income distri-bution for all cities in our data is shown in Figure 4. The first moment, themean, shows the well-characterized urban agglomeration effect: per-capitaincome increases with city size [3]. The second, and third moments bothincrease similarly with city population, suggesting a widening of the distribu-tion and increasing asymmetry with greater urban population. This can alsobe qualitatively observed in the example distributions in Figure 2. Lastly, thekurtosis also increases with population size, showing an increasingly heavytail with greater urban population.We find a stronger relationship for higher statistical moments, indicatingthat for larger American cities, there is a more evident increase in the thirdand fourth moments. This means that there is a stronger increase in thegrowing tail of the distribution, in comparison to the first two statistical mo-ments. This gives us an interesting indication of the distribution of economicbenefits.Another useful perspective on the scaling of the income distributions is tocompare large and small cities using measures that consider the entire dis-9 A) (B) (C) (D)
Figure 4: First four statistical moments of the estimated income distributionsas a function of city size. The texts in each panel display the scaling exponent, β , and in the bracket, corresponding 95% confidence intervals.tribution through the KL divergence. Figure 5 shows the KL divergence be-tween each US city and the largest city, as a function of the log-transformedcity population. The KL divergence, in general, decreases with city popula-tion, and approaches zero as the population approaches that of the largestcity. This behavior suggests that as cities get smaller, their income distri-butions are increasingly dissimilar to that of the largest city. The Pearsoncorrelation between the two variables in Figure 5 is − . , while the Spear-man correlation is − . . The Pearson correlation measures the linear cor-relation between two variables, while the Spearman correlation measures therank correlation, and assesses how well relationship between two variablescan be described by a monotonic function, regardless of linearity. This find-ing suggests that population and the KL divergence tend to change together,but not necessarily at a constant rate. While we can identify a general scalingtrend, our data also exhibit frequent outliers and deviations. While the differences in income scaling that we have identified are impor-tant, they are not necessarily grounded in differences in the experiences ofurban residents—cost of living can vary drastically across and within UScities, and if cost of living is changing in the exact same way as income,differences in income scaling between groups begin to lose meaning. In or-der to understand whether the differences in income scaling we see betweendeciles create differences in affordability and purchasing power, we look atchanges in housing cost with city size. By analyzing, in combination, ag-10igure 5: Kullback-Leibler divergence between the estimated income distri-butions and that of the largest city, as a function of log population. TheSpearman correlation is -0.718. .gregate household income and aggregate housing cost for each census tract,we find that aggregate housing cost scales faster than aggregate income for every decile, implying that while income per person increases with city size,larger cities may still be overall less affordable. This difference is more dra-matic for the poorer deciles—in the bottom decile, housing cost scales with β = 1 . while income scales with β = 1 . ; in the top decile, housing costscales with β = 1 . while income scales with β = 1 . . This is visualized inFigure 6A—income exponents begin to catch up to housing cost exponentsin richer deciles, but never as high as housing cost. Perhaps more intuitively,in Figure 6B, we can see that the ratio between total housing cost and to-tal income grows with city size for every decile, but more dramatically forpoorer deciles. Together, these results imply a widening gap between richerand poorer residents in affordability of cities with city size. Here we proposed a new method to study the scaling of income distribu-tions and income inequality in urban areas. The aggregated income in in-come deciles scale systematically with city size. The bottom decile scaleswith an exponent slightly below and the top decile with an exponent of β = 1 . . This result suggests that the benefits of larger cities are increas-11igure 6: Comparing the scaling of housing cost and income. (A) The scalingof total income, total housing cost, and the difference between total incomeand housing cost, for each decile. Housing cost scales with greater exponentsthan income for all deciles. The housing-adjusted income exhibits similarvariation across deciles as total income. (B) Ratio between housing cost andhousehold income as a function of city population. In the poorest deciles(dark brown), the proportion of income spent on housing increases sharplywith city size; in the wealthiest deciles (orange), this proportion remainsstagnant.ingly unequally distributed, and for the poorest income deciles, cities haveno positive effect over the null expectation of a linear increase. Much hasbeen written about the apparent increasing gains of large cities [3, 4], such asgreater GDP, higher wages, and more patents per capita. Our results showthat the increasing benefits of city size are not evenly distributed to peoplewithin those cities. We further show systematic variations in distributioncharacteristics. Besides greater mean, distributions of bigger cities also ex-hibit greater spread, greater asymmetry, and heavier tail. These perspectivescan be explicitly connected to traditional measures of income inequality, suchas the Gini coefficient. Like the Gini Coefficient, our method characterizesthe overall dispersion of income distributions (see Figure 7), but it also pro-vides more detailed information that is not characterized by Gini, such ashow the urban agglomeration effect alters the incomes of relatively poor or12ich people differently.Figure 7: Changes in the Gini coefficient with urban population in simulatedlog-normal distributions. For a scenario of parallel decile scaling (Figure 1B)and for a scenario where the deciles have divergent scaling (Figure 1D). Asexpected, the divergent scaling observation corresponds to increasing Ginicoefficient with population.Although our results appear to closely align with those of Sarkar et al.[25, 26],which analyze Australian income data, the difference in methodology (aggre-gation by income brackets vs. by deciles) should lead to different interpre-tations of the scaling exponents derived. In particular, the baseline “equal”situation is different in the two methods—in Sarkar et al., when total incomein all income brackets scale with the linear exponent, and in our methods,when total income in all deciles scale with the same exponent (either linearor nonlinear).Our paper offers new contributions to the literature. First, we develop a newmethod to study income inequality in the urban scaling framework, whichuntangles the systematic shift in mean from the study of income inequality.This method enables us to study how income agglomeration effects vary be-tween relatively rich and poor people, after accounting for the systematicallyincreasing mean with population size. Second, our analysis including housingcost demonstrates that despite agglomeration effects on income, bigger citiesare less affordable for people of all deciles in the sense that they spend pro-portionally more of their income on housing; this is especially true for lower-13ncome people. Third, our analysis extends beyond the single-parametercharacterization of income inequality. We analyze more complex propertiesof income distributions through analyzing statistical moments and KL di-vergence, and reveal systematic variations with city size. Fourth, our resultssuggest new directions for understanding mechanisms of urban agglomerationeffects—it is important to extend beyond theories considering homogeneousdensifying interactions to those which account for heterogeneity.Understanding the underlying mechanisms of why inequality is systematicallyscaling with city size is of great future interest with many potential implica-tions. Urban scaling theory in general proposes densifying interactions withincities as the fundamental process leading to the superlinear increase of manyfeatures [8, 9, 27, 28]. Our analysis shows that the superlinear scaling is notseen within all subsections of the city. The superlinear scaling of total wealthis driven by the top income deciles, and is not matched proportionally by thelowest deciles. This adds another dimension to considerations of the under-lying mechanisms of urban scaling theory: what processes are leading to theincreasingly unequal distribution of wealth in larger cities? We explored theidea of city heterogeneity as an indirect proxy for heterogeneous interactionrates. One hypothesis of the mechanism driving superlinear scaling of in-come with city size is that larger cities foster more and more diverse socialand economic interactions, creating opportunities for the exchange of ideasand resources. Existing literature credits superlinear growth of income incities to more opportunities for social contacts and interactions in large cities[3, 8]. Increased social contact with city size has been empirically confirmed[29], and ties between individual’s exposure to diverse social connections andeconomic outcomes have been shown empirically as well [30]. Together, thisseems to suggest that cities that are better mixed, allowing diverse partsof the population to be exposed to one another, should be overperformingwith respect to urban scaling. 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