The Lorenz curve was described in 1905 by Max O. Lorenz to display the distribution of wealth or assets in a society. A Lorenz curve is obtained by plotting on the x-axis the percentage of people and on the y-axis the corresponding percentage of wealth. Thus, a point at(x, y)on the Lorenz curve indicates the percentage y of total wealth that the bottom x% of people have. Figure A1 shows the approximate Lorenz curve for wealth in the U.S. in 2001. The thin line with slope 1 is the line of perfect equality, which corresponds to an equal distribution of wealth among individuals in a society.
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Proportion of Population
ProportionofWealth
Figure A1. A Lorenz curve for wealth in the U.S. (bold) approximated using data from 2001, along with the line of perfect equality (thin).
A Lorenz curve has properties useful in approximating it:
• It begins at(0, 0)and ends at(1, 1),
• cannot rise above the line of perfect equality,
• is increasing, and
• is convex.
The Lorenz Curve for the U.S.
We approximate the Lorenz curve for the U.S. using data from 2001. We display in Table A1 the data from Wolff [2004].
Table A1.
Financial wealth distribution by household in the U.S. in 2001, according to Wolff [2004, 30].
Top 1% Next 19% Bottom 80%
% wealth 39.7% 51.5% 8.8%
We also know that 0% percent of the population have 0% of the wealth, and the collective population has all the wealth. This give us the boundary conditions(0, 0)and(1, 1).
We approximate the Lorenz curve using a B´ezier spline fit algorithm because of its ability to generate a smooth curve with relatively few data points. The B´ezier fit also guarantees that the curve will be convex, as we would expect a Lorenz curve to be. The disadvantage is that the curve does not pass through all the data points.
Computation of C
∗and C
∗We compute the weightsC∗ andC∗ using the Lorenz curve, which we denote byL(x). We defineC∗ to be the ratio of the cumulative wealth of the median person in the lowest quartile to the cumulative wealth of the average person. Thus,C∗ is given by:
C∗ = R.125
0 L(x) dx R.50
0 L(x) dx ≈ .17
Similarly, defineC∗ to be the ratio of the cumulative wealth of the median person of the upper quartile to total wealth. SoC∗ is given by:
C∗ = R.875
0 L(x) dx R1
0 L(x) dx ≈ .63 The Gini Index of L(x)
The Gini index is a numerical measure of the distribution of wealth in a country, defined as
G = 2 Z 1
0 [x − L(x)] dx = 1 − 2 Z 1
0
L(x) dx
where L(x)is a Lorenz curve. Thus, the Gini index is 1 minus twice the area below the Lorenz curve. Perfect equality in wealth corresponds to G = 0, perfect inequality toG = 1. Numerically integration of our function L(x)givesR1
0 L(x) dx ≈ .21and henceGUSA,2004 ≈ .579.
Limitations of our Approximation
• Although we use data from 2001, the distribution of wealth does not change dramatically from year to year.
• We use only five data points, including the boundary conditions (0, 0) and(1, 1).
• The B´ezier curve passes through the boundary points but not through the data points.
The Gini index for financial wealth of households in the U.S. in 2001 was .888 [Wolff 2004, 30], while our approximation is .579. We used scant data; moreover, the bottom 40% of households combined have negative financial wealth (“net worth minus net equity in owner-occupied housing”
[Wolff 2004, 5]). Davies et al. [2008] have different data (Table A2) for household wealth, which they take more conventionally to include “non-financial assets [presumably including home equity], “non-financial assets and liabilities” [2008, 2].
Table A2.
Wealth distribution for families in the U.S. in 2001, according to Davies et al. [2008, 4, Table 1].
Top 1% Top 5% Top 10% Bottom 50%
% wealth 32.7% 57.7% 69.8% 2.8%
Editor’s Note: Calculation of the Gini Index from Available Data The U.S. Census Bureau publishes wealth and income data by quintiles.
The income data are published separately for families and for households [2005a; 2005b], while the wealth data are published for households only [2008a]. A household includes related family members plus any unrelated people who share the housing unit. The Bureau also publishes Gini indexes for income [2008b; 2007b; 2007a] calculated from the full Lorenz curves, together with other measures of inequality [n.d.].
The Gini index cannot be approximated from quintile data by using Simpson’s rule for an integral, since Simpson’s rule requires an even num-ber of intervals. Using the trapezoid rule would underestimate the Gini coefficient because of the concavity of the Lorenz curve.
Gerber [2007] gives a simple method suitable to quintile data. For U.S.
family income in 2000, the method gives a Gini index of .422, while the index given by the Census Bureau (based on the full Lorenz curve) is .433.
Further information about both the Lorenz curve and Further details about the Lorenz curve and the Gini index are given in a series of UMAP Modules by Schey [1979].
Editor’s References
Gerber, Leon. 2007. A quintile rule for the Gini coefficient. Mathematics Magazine 80 (2) (April 2007): 133–135. Note: The URL for reference 3 of this paper should read http://www.census.gov/hhes/www/income/
histinc/f02.html.
Schey, Harry M. 1979. The distribution of resources. UMAP Modules in Undergraduate Mathematics and Its Applications: Modules 60–62.
Newton, MA: COMAP.
U.S. Census Bureau. 2005a. Historical income tables—Families. Table F-2.
Share of aggregate income received by each fifth and top 5 perc [sic] of households (all races): 1967 to 2001. http://www.census.gov/hhes/
www/income/histinc/f02.html.
. 2005b. Historical income tables—Families. Table F-2. Share of aggregate income received by each fifth and top 5 percent of fam-ilies (all races): 1947 to 2001. http://www.census.gov/hhes/www/
income/histinc/f02.html.
. 2007a. Historical income inequality tables. http://www.census.
gov/hhes/www/income/histinc/ineqtoc.html.
. 2007b. Historical income tables—Households. Table H-4. Gini ratios for households, by race and Hispanic origin of householder: 1967 to 2006. http://www.census.gov/hhes/www/income/histinc/h04.
html.
. 2008a. Net worth and the assets of households: 2002.
http://www.census.gov/prod/2008pubs/p70-115.pdf.
. 2008b. Historical income tables—Families. Table F-4. Gini ratios for families, by race and Hispanic origin of householder: 1947 to 2006.
http://www.census.gov/hhes/www/income/histinc/f04.html. . n.d. Selected measures of household income dispersion: 1967 to 2005. http://www.census.gov/hhes/www/income/histinc/p60no231_
tablea3.pdf.
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