François Nielsen
Department of Sociology
University of North Carolina Chapel Hill

What Is Inequality?

Inequality is a property of the distribution in a population of some (presumably valued) resource such as income or wealth but also cattle, wives (in a polygynous society), and articles published by scholars in scientific journals.  The distribution of such quantities is typically highly skewed, with a long tail to the right.
One can conceptualize inequality with the Lorenz curve.  Take the example of income.  Imagine that all income-receiving units (IRUs) are ranked by income from the smallest to the largest, and calculate the cumulative share of income accruing to each category of the populations from poorest to richest, as in the following table.
Family Income Distribution: U.S. 1983 (from Kerbo 2000, Table 2-7)
Income Category Share of Total Income (%) p = Cumulative Share of Population (%) L = Cumulative Share of Income (%)
Top 20% 42.7 100 100.0
4th 20% 24.4 80 57.3
3rd 20% 17.1 60 32.9
2nd 20% 11.1 40 15.8
Lowest 20% 4.7 20 4.7
Total 100

The Lorenz curve is the plot of the cumulative income share L against the cumulative population share p.

Can One Measure Inequality?

Yes.  One common measure of inequality is the Gini coefficient.  The Gini coefficient (or "Gini index" or "Gini ratio") G is calculated from the Lorenz curve as the ratio
G = Area A/(Area A + Area B)
Note that (Area A + Area B) is the area of a triangle, given by 100*100/2=5000.
The Gini coefficient for the 1983 U.S. family income distribution is given by the following calculations.
Calculation of Gini Coefficient
Area A + Area B 100*100/2 =  5000
Area 1 20*4.7/2 = 47
Area 2 20*(4.7+15.8)/2 = 205
Area 3 20*(15.8+32.9)/2 = 487
Area 4 20*(32.9+57.3)/2 = 902
Area 5 20*(57.3+100)/2 = 1573
Total Area B 3214
Area A 5000 - 3214 = 1786
Gini Coefficient 1786/5000 = 0.36 or 36%

Thus the Gini coefficient in this example is 1786/5000 = 0.36 or 36%.
In the Lorenz curve the 45 degrees line represents a situation of perfect equality.  (Why?).
In general, the closer the Lorenz curve is to the line of perfect equality, the less the inequality and the smaller the Gini coefficient.

Countries differ in the inequality of their income distribution.

Note how the Lorenz curves of different countries can intersect; this means that 2 societies with the same Gini ratio can have different Lorenz curves (i.e., more inequality "at the bottom" versus more inequality "at the top").  Thus the Gini coefficient is not a complete description of the income distribution.

The following table shows the distribution of family wealth in the U.S. in 1983.  You may want to calculate the Gini coefficient as an exercise.

Family Wealth Distribution: U.S. 1983 (from Kerbo 2000, Table 2-7)
Income Category Share of Total Wealth (%) p = Cumulative Share of Population (%) L = Cumulative Share of Wealth (%)
Top 20% 78.7 100 100.0
4th 20% 14.5 80  21.4
3rd 20% 6.2 60  6.9
2nd 20% 1.1 40  0.7
Lowest 20% -0.4 20  -0.4
Total 100

Inequality tends to be greater for wealth than for income.  (Why?).

The fifteen wealthiest people in the world in 2005

Are There Other Ways to Measure Inequality?

Yes.  Other more or less sophisticated measures of inequality include the following
  1. Share of total income accruing to the top 20% of IRUs; this is 42.7 for U.S. family income distribution in 1983; share of the top 10%, or 5% are also used; the larger, the more unequal the distribution
  2. Share of total income accruing to the bottom 40% of IRUs; this is 15.8 for U.S. family income distribution in 1983; the larger, the less unequal the distribution
  3. Gap in constant $ between income of the 90th percentile and income of the 10th percentile; often used by economists; advantage is that knowledge of very top and very bottom incomes does not need to be accurate
  4. Gini coefficient calculated form individual IRU data (as opposed to grouped data).  The formula is
    1. G = (2/(μ*n2))Σi=1 to ni*xi - (n+1)/n
    where xi is income, i is the rank of income xi (i=1 for smallest income, i=n for largest), and μ is the mean income.
  5. Theil coefficient (or "information theory" or "entropy" measure) calculated from individual IRU data.  The formula is
    1. T = [(1/n)Σi=1 to n(xi*log(xi)) - μ*log(μ)]/μ
    where log denotes the natural logarithm.
  6. Varlog (variance of the logarithms of income) calculated from individual IRU data.  The formula is
    1. V = (1/n)Σi=1 to n(zi - z.)2
    where zi = log(xi) and z. denotes the mean of the zi
  7. The coefficient of variation calculated from individual IRU data.  The formula is
    1. C = sX
    where sX is the standard deviation of the xi and μ is the mean of the xi
Income inequality measures may or may not have certain desirable properties such as Only G, T, and C satisfy both properties.  For a detailed discussion see Allison (1978).
There is a substantial literature analyzing the properties of inequality measures with respect to social welfare.

What is Dualism?

In this context dualism is the amount of income inequality generated by the difference in average income between two distinguishable categories of IRUs in a society.
Examples are

How is Dualism Measured?

One measure of dualism is simply a special case of the Gini coefficient, in which the number of IRUs is reduced to 2.  Then the Gini coefficient can be calculated as
D = |p - L|
where p is the percentage of IRUs in the poorest category and L is the percent income share of IRUs in the poorest category.  The symbol |  | denotes the absolute value.
The relationship of dualism to the Lorenz curve and the Gini coefficient is shown in the following exhibit.

Historically the concept of dualism was emphasized in a famous article by economist Simon Kuznets (1955) that initiated a stream of research on the relationship of income inequality with economic development.  Kuznets thought that dualism generated by income differences between traditional/agricultural and modern/urban sectors was a principal reason for the high level of income inequality in developing countries.  The following exhibit shows sector dualism figures for countries of the world around 1970.

The Big Picture: Inequality Trends and Sociocultural Evolution

Is U.S. society today more or less unequal than it was 200 years ago?  How do modern industrial societies compare with simple societies, such as hunters and gatherers?
Sociologist Gerhard Lenski has proposed a typology of human societies based on subsistence technology, and the nature of the environment of the society.  The basic typology (simplified) is
Simplified Typology of Human Societies after Gerhard Lenski (Nolan and Lenski 1999)
Type of Society Main Technological Innovation Approximate Date of Appearance
Hunting & Gathering (primordial)
Horticultural Plant Cultivation 8,000 BC
Agrarian Use of the Plow 3,000 BC
Industrial Use of Machines Powered by Inanimate Forms of Energy 1,750 AD

Inequality has evolved in a systematic way during sociocultural evolution according to Lenski.

Patterns of Inequality in the Modern World

During industrialization, inequality of the distribution of income has been characterized by 2 historical trends: the Kuznets Curve and the Great U-Turn.

1.  The Kuznets Curve

The Kuznets Curve was named after economist Simon Kuznets (1955).  Kuznets conjectured that during industrial development in the long run, income inequality at first rises and then declines, tracing an inverted-U-shaped trajectory.  As a result, industrial societies are more equal than non-industrial societies The inverted-U shape of the Kuznets curve is due to

2.  The Great U-Turn in the U.S. and a Few Advanced Industrial Societies

The phrase "The Great U-Turn" was coined by Harrison & Bluestone (1988).
The term refers to the inequality upswing that took place since the early 1970s in some advanced industrial societies (especially the U.S. and the U.K.)
Why these inequality trends?
Research suggests that the inequality upswing is the result of a combination of factors including
Additional reference: the following article from the Census Bureau looks at income inequality trends in the US from 1947 to 1998. Two salient features of the recent trend discussed in the article are

3. Current World Trends

Hans Rosling's Gapminder site provides graphic depictions of trends in income inequality in countries of the world.

Major trends:

4.  Regional Inequality Patterns in North Carolina

Race dualism (= inequality associated with the average income difference between black and white families) varies among counties of North carolina.  The following exhibits show the geographical distribution at the county level of race dualism, overall income inequality (distribution of family income), and the relationship between income inequality and dualism. It appears that regions of North Carolina are characterized by distinct patterns of inequality.
Regional Inequality Patterns in North Carolina
Region Race dualism Income inequality
Mountain Low High
Piedmont Low to High Low to High
Coastal plain High High
Tidewater High High

Discussion: Is Bill Gates Hazardous to Your Health?

How does the degree of inequality in the social environment affect individuals in that environment?  There are many issues involved here, including

One study of the effect of income inequality on health outcomes at the county level finds a "modest independent effect [of income inequality] on the level of depressive symptoms, and on baseline and follow-up self-rated health, but no independent effect on biomedical morbidity or subsequent mortality"; these effects are substantially smaller than the effects of individual income.



Last modified 21 Feb 2008