Duncan Williamson: Lorenz Curves 1

A Lorenz Curve has an X axis that begins at 0% and ends at 100% and a Y axis that begins at 0% and ends at 100%. The purpose of such a curve is to demonstrate relative proportions of X versus Y.

Lorenz Curves are perfect for demonstrating Pareto analysis and were first used in the latter part of the 19th century. The first LC demonstrated that a small proportion of the population owned a large proportion of the wealth: for example, 20% of the population might own 70% of the wealth.

Stock Controllers have found that Pareto analysis can be applied in the store room: here it tends to be called ABC analysis; and the following table illustrates why this is the case.

Classification % of Total Value % of Total Units

A 70% 10%

B 20% 20%

C 10% 70%

From this table we see that

Category A stock items are high in value but low in volume: there are relatively few of them but they are expensive.
Category B stock items have a medium level value and a medium level volume.
Category C stock items are low in value but high in volume: there are relatively many of them but they are cheap.

Stock Control

Applying Pareto analysis to stocks is important because it gives stock controllers and accountants a focus. Look carefully after category A items; be careful with careful B items; but category C items are unimportant and can be controlled in a relatively relaxed way.

LC and Stock Control

If we adjust table 1 and accumulate the percentages and add the "dummy" row where both percentages are set at 0%, we have the basis for a LC:

Cumulative

Classification
% of Total Value
% of Total Units

0%

0%

A
70%

10%

B
90%

30%

C
100%

100%

Please note: these figures are for illustration only. In reality, we might find the percentages to be, say, 60%, 30%, 10% for A, B and C respectively. An analysis of actual stock holdings in an organization will reveal the truth!

Plotting this on a graph gives us:

The Equality Curve

To better illustrate the situation, it is common to add what is called an equality line to a LC: this is the line that shows a 1: 1 relationship … total equality. Consequently, the equality line shows us how equal or unequal a situation is by illustrating the distance between the Pareto curve and the equality line. Figure 2 illustrates the equality line:

How to plot this graph using Excel

Go here to see how to plot Figure 2 using Microsoft Excel.

Multi Curve LC

There is no reason why we might not illustrate a LC with more than one curve. The following tables illustrates some National Income Data, both before and after tax; and we might be interested to compare these two sets of data with each other … and with the equality line. Table and Figure 3 show such an example:

Cumulative Percentage of Income

Cumulative Percentage of Households
Before Taxes and Transfers

After Taxes and Transfers

Equality

0%
0.00%

0.00%

0.00%

20%
0.90%

5.10%

20.00%

40%
8.30%

16.10%

40.00%

60%
23.70%

32.80%

60.00%

80%
49.00%

56.70%

80.00%

100%
100.00%

100.00%

100.00%

Alternative LC

There is another way to prepare a LC that we might frequently come across. Table and Figure 4 illustrate this alternative:

Classification
Cumulative % of Total Units

Cumulative % of Total Value

Equality Line

0%

0%

0%

A
10%

70%

10%

B
30%

90%

30%

C
100%

100%

100%

The interpretation of this graph is exactly the same as the interpretation of the previous graph.

The Gini Coefficient

The Gini coefficient measures the degree of inequality of a variable in a distribution of its elements. It compares the Lorenz curve with the line of perfect equality. The Gini coefficient ranges between 0, where there is perfect equality, and 1 where there is perfect inequality.

Here is the Gini Coefficient for the UK from 1979 to 2002:

The following diagram shows the elements of the Gini Coefficient:

Source of graph: http://people.hofstra.edu/geotrans/eng/ch4en/meth4en/ch4m1en.html

We are looking at trying establish just how equal or unequal a distribution is.

Formally, the function that describes the Gini Coefficient is:

Whilst the function looks horrific, we can take an example and simplify it enormously!

When deriving the Gini Coefficient:

X is the reciprocal of the number of observations or data points

Y is the proportion of the variable we are interested in

Imagine that the traffic at five airport terminals is as given in the following table, our task is to calculate the Gini coefficient from them:

Terminal
Traffic

A
25,000

B
18,000

C
9,000

D
3,000

E
2,000

Total
57,000

Developing this table now:

Source of data: http://people.hofstra.edu/geotrans/eng/ch4en/meth4en/ch4m1en.html

Remember X = 1/n where n is the number of observations

Y = traffic per terminal/total traffic eg for terminal A:

25,000/57,000 = 0.438

sigmaX is the accumulation of X

sigmaY is the accumulation of Y

sigmaXt-1 - sigmaXt is, eg, sigmaX for B less sigmaX for A,

sigmaXt-1 - sigmaXt is, eg, sigmaX for C less sigmaX for B

Similarly for sigmaYt-1 + sigmaYt: eg sigmaX for A plus sigmaX for B

The A * B is just that!

Finally the value of the Gini coefficient is found by subtracting to total in the A*B column from 1 and IGNORING any minus sign ... that's why the formula has those straight lines at the ends to tell you to ignore any minus sign. That's the Gini Coefficient.

Simply, the nearer the Gini Coefficient is to a value of 1 the more unequal the distribution. In theory, if the income of a country was distributed with a Gini Coefficient of 1, it would mean that ONE PERSON had all of the income and everyone else would have none! The statistics for real income and wealth distributions show that developed economies tend to have a Gini coefficient of around 0.400 to 0.460.

Reference

Adapted from Andrew Shephard Inequality under the Labour Government Briefing note No 33 The Institute for Fiscal Studies March 2003