M E A S U R E M E N T A N D O T H E R I S S U E S

ON INEQUALITY:

Montclair State University; Research Methods: Spring 2015

WHAT IS INEQUALITY?

“If there were in the world today any large number of people who desired their own happiness more than they desired the unhappiness of others, we could have a paradise in a few years” Bertrand Russell

WHAT IS INEQUALITY?

•  Intellectual and Emotional Rationalization…the justification for inequality

•  Scarcity: Real or Imposed

•  Mathematically: Naming and collecting groups that number greater than apportioned resources. The pigeonhole principle

WHAT IS INEQUALITY?

Sociologically: The separation and classification of people into categories. •  Inter-racial, Intra-racial, Familial •  Economically via pricing (Education, Health Care) •  Socially (Labor and Housing Markets (covenants),

Salaries and Wages)

MEASURING INEQUALITY

•  Normative Inequality: Social Welfare

•  Objective Inequality: Measure of difference in distribution

MEASURING INEQUALITY

•  Normative Inequality: Social Welfare

•  Objective Inequality: Measure of difference in distribution

Numeric quantification of the distributional apportionment of a resource or ‘good’.

MEASURING INEQUALITY

The Empirical Cumulative Distribution Function: ECDF

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ECDF OF 2008 INCOME

AllWhiteBlackEqual

0 50000 100000 150000

0.0

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1.0

Empirical Distribution (Function) on CPI-U-RS Money Income, 2008

Figure 1: Graph of empirical cumulative distribution function (ECDF) of Money Incomeof Households — Consumer Price Index Research Series Using Current Methods, CPI-U-RS. The solid green curve is an ECDF for a hypothetical uniform distribution where everyhousehold in the population has the weighted (by CPI binning) mean household income.The ECDF completely specifies the distribution; the fraction of households having less thanor equal to the income on the x�axis is the height of the curve on the y-axis. Generally, thefraction of black households is increasingly lesser than white households as income increases:the ten percentage point di↵erence at $10000 of income or less increased to a twenty pointdi↵erence at $50000 of household income. The CPI-U-RS right censors — here via aggre-gating — distributional information above $100000. The $150000 cuto↵ used in the graph isarbitrary. A steeper ECDF, as a rule of thumb, suggests more households at lower incomelevels. Notice the ECDF curve for blacks is steeper than that for whites. Data from [16].

Notice the ECDF, and thus measurements of inequality, are statistics or functions of data:here incomes of members of a population. It is desirable that a measurement of inequalityyield a compelling and useful illustration of a income distribution. Figure 1 illustrates a

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MEASURING INEQUALITY: THE LORENZ CURVE

The Lorenz Curve:

Encyclopedia of Race and Racism, 2nd Edition, Volume 2 – 4th/ 11/30/2012 09:36 Page 416

distribution function (CDF): the CDF yields the propor-tion of the population that has income less than or equalto each value. The Lorenz curve is a function that takes theunit interval, [0, 1] from the proportion of total income tothe proportion of the population. The CDF, in this setup,is a function on the real numbers, [0, 1], to the unitinterval—from amount of income to the proportion of thepopulation.

The Lorenz curve is:

(3)L(p) ! (N " x )#1 x(i )

!Np"!

i!1

(4)! (N " x )#1 (i $N)FN

!Np" !i!1

#1

on a ordered sample of incomes, x(), with samplemean, x ! 1

npi!1! x(i ). The value of the curve L(p) at

percent p is the total income held by p percent of thepopulation. The total income at p percent can be calcu-lated directly from the inverse of the empirical CDF;since the empirical CDF holds the fraction of the pop-ulation at each income (i.e., the quantiles), its inverse isjust the quantile associated with each fraction. This isillustrated in equation 3.

The Gini Index The empirical Gini index is a functionfrom an observed distribution to a scalar on the unitinterval. The Gini coefficient returns the scaled ‘‘concen-tration’’ of a distribution defined as the ratio of observeddistance from equality to the maximum distance fromequality. This distance is just the area between the 45! lineLorenz curve for a uniform distribution and the observedLorenz curve divided by 1/2—the area between a uniformLorenz curve and a singular Lorenz curve—on the spaceof the Lorenz curve, the unit square [0, 1] x [0, 1]. TheGini coefficient is one, its maximum, on a singular

0.0 0.2 0.4 0.6 0.8 1.0

1.0

0.8

0.6

0.4

0.2

0.0

Lorenz Curves on CPI-U-RS Money Income, 2008

All, 0.37 White, 0.36 Black, 0.42 Equal, 0

Figure 2. The solid diagonal line is on a hypothetical uniform distribution where every household inthe population has the weighted (by CPI binning) mean household income. The Lorenz curve iscompletely specified by the (empirical) distribution function; the fraction of total income held by pproportion on the x-axis is the height of the curve on the y-axis. The greater the area between the 45!

line and the Lorenz curve, the greater the income concentration. The maximum possibleconcentration—a population where the income is fully concentrated in one person—is 1/2: the area inLorenz space under the 45! line. A maximally concentrated population has a Gini of one; the Lorenzcurve is the dotted black line. Estimated Gini indexes by ‘‘white’’ and ‘‘black’’ racial identification onthe binned CPI-U-RS in the legend.

Inequality: Overview

416 E NCYCLO PEDI A O F RA CE AND RA CIS M, 2ND EDIT IO N

MEASURING INEQUALITY: THE LORENZ CURVE

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Lorenz Curves on CPI−U−RS Money Income, 2008

All, 0.37White, 0.36Black, 0.42Equal, 0

Figure 2: Graph of empirical Lorenz curves on Money Income of Households — ConsumerPrice Index Research Series Using Current Methods, (CPI-U-RS). The solid green curveis on a hypothetical uniform distribution where every household in the population has theweighted (by CPI binning) mean household income. The Lorenz curve is completely specifiedby the (empirical) distribution function; the fraction of total income is held by p proportionon the x-axis is the height of the curve on the y-axis. The greater the area between the45� line and the Lorenz curve the greater the income concentration. The maximum possibleconcentration — a population where the income is fully concentrated in one person — is1/2: the area in Lorenz space under the 45� line. A maximally concentrated population hasa Gini of 1; the Lorenz curve is the dotted black line. Estimated Gini indexes by “white”and “black” racial identification on the binned CPI-U-RS in the legend. Data from [16].

The Theil Index

Theil’s index as a measure on an observed population of N total individuals, is:

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MEASURING INEQUALITY: THE GINI COEFFICIENT

•  The scaled ‘‘concentration’’ of a distribution •  Defined as the ratio of observed distance from

equality to the maximum distance from equality.

•  This distance is just the area between the 45 degree line Lorenz curve for a uniform distribution and the observed Lorenz curve divided by 1/2

MEASURING INEQUALITY: THE GINI COEFFICIENT

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distribution—one where all of the ‘‘good’’ in a populationis held by one person. The minimum, zero, is returned on auniform distribution—one where everyone in a populationholds an equal amount of the good.

There are many ways to calculate Gini’s index on asample x; the coefficient is also defined as a function of themean deviation, for example. It is illustrative to write it asa function of the Lorenz curve, showing the connectionbetween the univariate Gini, the Lorenz curve, and theobserved distribution function as a measure of inequality.

with FN-1 the inverse of the observed distribution

function: the function that returns the pth quantileof the observed data. In this way (colloquially speaking),the ordered observed data x()—which is the list 1/N,

2/N, . . ., joined with the smallest, next largest, . . .,largest amounts of the good—generates all of theinequality information for these measures.

For a uniformly distributed population, where allpersons have equivalent income:

• the observed distribution function is a 45! line fromthe origin.

• the Lorenz curve is a 45! line from the origin to theright-hand corner of the unit square.

• the Gini coefficient is zero.

For a singularly distributed population, where oneperson holds all the income:

• the observed distribution function is a step function:zero everywhere but at the total income where itreaches 1.

• the Lorenz curve is the lower and right sides of theunit square.

• the Gini coefficient is 1.

0.1

0.2

0.3

0.4

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0.6

2000 2002 2004 2006 2008

2000 2002 2004 2006 2008

!1.0

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!0.2

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0.6

2000 2002 2004 2006 2008

2000 2002 2004 2006 2008

50

0

100

150

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I a

10111213141516

I w

Theil’s Index

Figure 3. Illustration of Theil’s index calculated on wealth (left-hand column) and income (right-hand column) using the University ofMichigan’s Health and Retirement Survey (HRS) data: 2000, 2002, 2004, 2006, and 2008. The upper row is the across term; thelower row is the within term. Both terms are fixed by log base b = min( !xg/ !x), the ratio of the poorer (black) group sample mean to theoverall mean. The small number of observations in 2008 (n = 724) inflate the (estimate of) standard error—via ordinary bootstrap.Across-group inequality appears to be stable or decreasing from 2000 to 2008; within-group inequality appears to be increasing from2000 to 2006. Confidence bars are at 95 percent significance.

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Inequality: Overview

E NCYCLO PEDI A O F RA CE AND RA CIS M, 2ND EDIT IO N 417

MEASURING INEQUALITY

For a uniformly distributed population, where all persons have equivalent income: •  the ECDF is a 45 degree line from the origin. •  the Lorenz curve is a 45 degree line from the origin

to the right-hand corner of the unit square. •  the Gini coefficient is zero.

MEASURING INEQUALITY

For a singularly distributed population, where one person holds all the income: •  the ECDF is a step function: zero everywhere but at

the total income where it reaches 1. •  the Lorenz curve is the lower and right sides of the

unit square. •  the Gini coefficient is 1.

INEQUALITY WORLDWIDE

and East to those of the North and West. There has been theoretical and empirical studyof this historical and ongoing inequality between the North and South. The tableau hasbecome more varied recently — some nations of the South have experienced recent gains inwealth and others in the North have begun what may be a prolonged descent. Some of thismay be interpreted by the most elementary market theories; perhaps more can be explainedby how ‘trade’ is realized among the developed, developing and underdeveloped worlds.26

While there are strong historical antecedents for present-day international inequality it isgenerally regarded as fact that the state of cross-country economics fosters ongoing inequityin income, wealth and resources.27

Figure 5: Estimated Lorenz curve and Gini coe�cient on 1.5� gridded GNP (from world bankHotspots Report) superimposed over heat map of data. The world distribution of GNP isstaggeringly unequal; most of the world is very poor relative to few places of extreme wealth.The green line is the Lorenz curve at equality; the dotted black line is the curve at perfectinequality. The ratio of the area between the green line and unbroken black line and thetotal area below the green line is the Gini coe�cient. Data from [21].

The state of international inequality is undeniably striking. Using no other distinction thanlocation above or below the equator, the nations of the North have an average GDP 270percent greater than that in the South.28 International inequality at individual levels is per-haps even more severe. In Figures 5 and 6 the Gini concentration coe�cients are calculated

12

INEQUALITY WORLDWIDE

over regions aggregated by geographical and administrative boundary — the Gini coe�cientwhen calculated over countries vs. area is six percent greater. Similarly, and tellingly, theP90/P10 ratio — the ratio of incomes at the 90th and 10th percentiles — calculated acrosscountries is nearly twice as large as when calculated across gridded areas.29

Figure 6: Estimated Lorenz curve and Gini coe�cient on country level GNP for 2011 su-perimposed over heat map of by country data. Compare with Figure 5. The concentrationof income (.849 vs. .799) appears greater over aggregation by country and the fraction ofincome to the poorer half of the population appears lesser (.014 vs .031). The green line isthe Lorenz curve at equality; the dotted black line is the curve at perfect inequality. Theratio of the area between the green line and unbroken black line and the total area belowthe green line is the Gini coe�cient. Data from [31].

This imbalance has been called an economic gradient towards the comparatively industriallyproductive and resource-poor nations of the North from the resource-rich but underdevelopedcountries of the South. Alternatively, it could be characterized as the worldwide tendencyfor the international distribution of goods to move away from peoples of color.30 The in-equalities measured here via coe�cients for concentration are proxies for real and miserablehuman su↵ering.31 Prospectively these gross levels of international inequality are more thandisconcerting, they are dangerous.32

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INEQUALITY WORLDWIDE Cross-national inequality in resources, income and wealth, generate vulnerability via reducedcapacity to mitigate health and economic disasters. Figure 7 is a prospective illustration oflocations that may be particularly vulnerable to future hazards.

Figure 7: Estimated 99th percentile of hazard distribution on multivariate data (GDP, Pop-ulation, Peak Ground Acceleration, Floods, Cyclones, Drought, Volcanoes, Landslides) col-lected in 2003 or earlier. While there are a few locations identified in the developed world,the majority are in developing or underdeveloped regions; in particular notice Central andSouthern America, the Caribbean and South-East Asia. The western share of Hispaniola(Haiti) is identified as a 99th percentile vulnerable location. See [6].

The observed di↵erence in inequality when GDP is collected over country boundaries versusover gridded locations — these are the maps in Figures 5 and 6 — reflects the gulf inliving standards among countries, in particular between the developed and developing world,and suggest that across country inequality remains strong even in the presence of withincountry inequalities. Inequality is realized transversely as well as lengthwise; across countryinequality means that equivalent quintiles are unequal in magnitude.33 The story here is thathuman lives are often miserably di↵erent depending upon country of residence.34 Whetheror not the covariates of inequality are causal or secondary, cross country inequalities areassociated with very di↵erent within country experiences.

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INEQUALITY WITHIN COUNTRIES

To some degree charity is a racket in a capitalist system, a way of making our obligations to one another feel optional and of keeping poor people feeling a sense of indebtedness to the rich.

INEQUALITY WITHIN COUNTRIES

To some degree charity is a racket in a capitalist system, a way of making our obligations to one another feel optional and of keeping poor people feeling a sense of indebtedness to the rich. -Shawn Carter

INEQUALITY WITHIN COUNTRIES

Inequality within countries

To some degree charity is a racket in a capitalist system, a way of making ourobligations to one another feel optional and of keeping poor people feeling a senseof indebtedness to the rich.

— Shawn Carter.35

Figure 8: Gini coe�cients by country overlaid with scatter plot of log GDP vs. Gini’s(x100) with Lowess ([23]) curve. Higher coe�cients — darker shading on the map — meanhigher distributional inequality within country. GDP is on log scale; the normal range ofGDP distorts the relationship between the Gini and GDP. Overall, the Gini coe�cient, i.e.distributional inequality, tends to decrease with GDP. On the map, contrast Sweden andGermany with China, Brazil, the United States and South Africa. Points above the curveare countries with more than average inequality at level of GDP. See [59] or [31] for data.

Figure 8 illustrates Gini coe�cients at thecountry level with the relationship between ob-

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INEQUALITY WITHIN COUNTRIES

0.5 0.6 0.7 0.8 0.9 1.0

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HDI2010

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Chile

Czech Republic

Egypt

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Israel

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United States

Uzbekistan

(a) HDI

0 200 400 600 800

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# Incarcerated / 100k

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Colombia

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(b) Rate of Jailing

10 20 30 40 50 60 70

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% Share of Economy `Unofficial'

Gini

Bolivia

Brazil

Egypt

Japan

Mexico

Malaysia

Nigeria

Sweden

United States

Uzbekistan

(c) ‘Uno�cial’ Economy

Figure 9: Plots of Human Development Index (HDI), Number Incarcerated per 100 thousand,and Percent Share of Economy due to ’Uno�cial’ Activity vs. Gini coe�cient. Data from[59], [27] and [29], in order.

served Gini and observed GDP. A comparison of Figures 6 and Figure 8: many countriesat relatively high income levels have high levels of within country inequality. For exampleBrazil, China and the United States have relatively high GDPs and relatively high inequal-ity. In contrast, Japan, Germany and Sweden have relatively high GDPs and lower levelsin within country income inequality. This suggests the relationship between inequality andcountrywide wealth is not straightforward: GDP alone cannot explain the within countrydi↵erences in income distribution.

Within country inequality is associated with societal distress, or more poetically, a ‘melan-choly of the soul.’36 These melancholies can be quantified: in general more unequal societieshave higher rates of incarceration and graft and relatively lower scores on human develop-ment. Figures 9a, 9b, and 9c are plots of the Human Development Index (HDI), incarcerationrate and percent share of economy devoted to ’uno�cial’ country vs. the country wide Ginicoe�cient. In panel 9a notice that Chile, in particular, and the US have relatively highlevels of inequality at relatively high HDI scores. In panel 9b, notice that the US has anincarceration rate almost 130% as large as its closest competitor.37 38

Inequality may not be a concomitant of economic development but an argument for thecontrapositive — that economic development is a predictor of greater equality — may bespecious as well.39 Any general statement about the relationship between economic inequalityand economic development must be conditioned on the local scenery: some countries haveenjoyed decreasing inequality with economic growth while other countries have remainedpersistently, or even increasingly unequal, as their economies have boomed. For instance,figure 10 illustrates the aggressive consolidation in fishing rights in New Zealand over thepast two decades contributing to greater inequality.40

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US INEQUALITY

The problem facing...people here in America is bigger than all other personal or organizational differences...we must stop worrying about the threat that we seem to think we pose to each other...and concentrate our united efforts toward solving the unending hurt.

US INEQUALITY

The problem facing...people here in America is bigger than all other personal or organizational differences...we must stop worrying about the threat that we seem to think we pose to each other...and concentrate our united efforts toward solving the unending hurt. -Malcolm X

US INEQUALITY

recent US data — these estimates suggest contemporary levels of income inequality in theUS are moderate.43 Over time, and especially during the last two decades, US inequalityhas dramatically increased to levels reminiscent of the late 1920s.44

Figure 11: Heatmap of US state by state Gini coe�cients with “White” only versus “Black”only state by state estimates overlaid. State level measurements of US inequality appearrelatively moderate yet with notable variation across state and under stratification by racialidentification. Dashed black line: Lowess smoothed curve; dotted grey line: 45� from origin.States above the dotted grey line have greater Gini coe�cients for “Blacks‘’ versus “Whites‘’.Generally, there appears to be a positive association between white and black inequality Ginicoe�cients. Black inequality is generally higher than white inequality. Of note: Hawaii hasvery few black residents and a relatively low Gini coe�cient for within ’Black’ inequality;Washington D.C. is a special case, it has the highest within “Black” inequality coupled withthe third lowest “White” Gini coe�cient. Data from [53].

There are multiple causes, or at least covariates of, widening American income and wealthinequality: from an increasingly regressive tax policy, declining American educational qualityat even the best institutions, and the post-industrial financialization of the US economy.45

Much of the contemporary increase in economic misery, as measured by recent increases inmeasured income and wealth inequality, can be attributed to the ‘Great Recession’ of 2007-2009. Recent research does suggest, though, that the United States is becoming persistentlyand uniquely unequal among the developed world.46

The ‘exceptional’ nature of American inequality is not without its accompaniments: inparticular the US has the highest rate of incarceration — of any country — and one ofthe highest rates of violent criminal activity in the developed world.47 The US places in

18

US INEQUALITY

the bottom quintile of wealthy countries in many measures of rule of law.48 The number ofpersons in US jails and prisons is particularly vulgar: an estimated 1 out every 100 persons isbehind bars; the observed probability of being in America and in jail is the world’s highest.49

The character and magnitude of these maladies tends to vary across US states, in somecases, with an apparent association to income inequality. Figure 12 plots estimated Ginicoe�cients by state versus violent crime rate, math proficiency scores for 8th graders andinfant mortality rates. Crime and infant mortality rates within a state appear to increasewith the Gini coe�cient, math proficiency scores tend to decrease.50

0.38 0.40 0.42 0.44 0.46 0.48

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Gini vs. Violent Crime Rate

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Gini vs. Infant Mortality

(d) Infant Mortality

Figure 12: Plots of Gini coe�cients, by state, versus: GDP, 2007 violent crime rate, per-centage of eighth graders scoring proficient in mathematics, and infant mortality.

US economic inequality is even more striking when stratified by racial group.51 “Black”versus “White” inequality, expressly, measured via di↵erences in income and wealth, is enor-mous. It has been estimated that 85% of black and Latino households have a net worth lessthan the median of white households; the median net worth of black households is less thana tenth of white households.52 Further, income inequality within blacks appears even greaterthan income inequality among whites only. Figure 11 plots Gini coe�cients for inequalityfor ‘White’ versus ‘Black’ households by state: most states have higher income stratificationamong blacks than among whites.

These inequities can be measured in very real outcomes: black men and women are over-proportionally arrested, convicted and sentenced; black men are particularly overrepresentedin low-wage jobs and underrepresented in high-wage jobs; black professionals are discrimi-nated against before and after hiring at all levels of employment; and most directly, African-Americans are more likely to die younger, sooner, and from disease.53

In general black Americans even more appear to be more economically unequal, within racialgrouping, than whites; consider figures 2 and 3 — the income distribution for blacks onlyis closer towards singularity than the distribution for whites only. Greater within groupinequality exists at the same time as relative inequality between black and whites remainshigh: the distribution of black incomes is more stratified though at lower levels of incomethan for whites.54

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US INEQUALITY

Figure 13 illustrates the association of within versus across group inequality for US blackand white Americans. It may be imprudent to infer an overall functional relationship be-tween within racial group and across racial group inequality. Nevertheless, the geographicdistribution of inequalities is interesting. Using the across to within group decomposition ofthe Theil index, equation 7 above. Figure 13 suggests the Southern states — in conjunctionwith New York, Illinois and California — have the greatest ratios of across to within racialgroup inequality. This can be interpreted as relatively large di↵erences between the incomesof whites versus blacks to at commensurate levels of income concentration within blacks andwhites separately.55

Figure 13: Heatmap of US via ratio of decomposed Theil index (equation 7), state by state.Darker states have higher levels of across ‘Black’ and ‘White’ inequality at commensuratelevels of within race income inequality. The scatterplot is overlaid with a lowess curve, see[23]. Data from [53].

Finally, while the focus here on black versus white inequalities is not the only narrative onAmerican inequality it is perhaps the dominant one. Income and wealth di↵erences betweengroups identified as wholly white or black tend to be the largest. In addition, intra-racialdi↵erences among people who are or do identify as black are persistent and measurableon things as depressingly meaningful as prison time received as an increasing function ofpigmentation.56

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FINAL WORDS

[One cannot derive ought from is.]...Since vice and virtue are not discoverable merely by reason, or the comparison of ideas, it must be by means of some impression or sentiment they occasion, that we are able to mark the difference betwixt them. -David Hume