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Lorenz curves

What is a Lorenz curve?

The statistician M.O. Lorenz introduced a new method of thinking about inequality in a very influential paper published in 1905. This method is now called the Lorenz curve.

We can construct a society’s Lorenz curve using the following approach:

  1. Sort the population in order of increasing incomes.
  2. At each income level, calculate the cumulative share of the population with an income of that level or lower. Then, calculate these individuals’ collective income share by finding the sum of all their incomes and dividing this sum by the society’s total income.
  3. Starting from zero, plot the cumulative share of total income on the vertical axis against the cumulative share of total population on the horizontal axis. Note that 0% of the population has 0% of the income. This means that \((0, 0)\) should always be the first point on the Lorenz curve.

Exercise

Construct a Lorenz curve for a society of 3 people whose incomes are 10, 60, and 30.

  1. Sort the population in increasing order of incomes:

    \[10, 30, 60\]

  2. Calculate cumulative population and incomes shares.

    Each person in this society is \(\frac{1}{3}\) of the total population.

    Person 1 has an income of 10. At this point, the cumulative population share is \(\frac{1}{3}\) because we are only counting one person. The cumulative income share is just their income share of \(\frac{10}{100}\).

    Person 2 has an income of 30. They are the second person, so now the cumulative population share is \(\frac{1}{3} + \frac{1}{3} = \frac{2}{3}\) and the cumulative income share is \(\frac{10}{100} + \frac{30}{100} = \frac{40}{100}\).

    Person 3 has an income of 60. They are the final person, so now the cumulative population share is \(\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3}\) and the cumulative income share is \(\frac{10}{100} + \frac{30}{100} + \frac{60}{100} = \frac{100}{100}\).

    Therefore, the cumulative population shares are:

    \[\frac{1}{3}, \frac{2}{3}, \frac{3}{3} = (0.33, 0.67, 1.00)\]

    And the cumulative income shares are:

    \[\frac{10}{100}, \frac{40}{100}, \frac{100}{100} = (0.10, 0.40, 1.00)\]

  3. Plot the cumulative share of total income against cumulative share of total population.

    Draw the Lorenz curve by plotting the points \((0, 0)\), \((0.33, 0.10)\), \((0.67, 0.40)\), and \((1, 1)\) and connecting them with a line. See the figure below.

     

Exercise

If everyone in a society has identical income, what does the Lorenz curve look like?

It is a 45-degree line. This represents a society in which everyone has the same share of total income. It is also called the line of perfect equality.

Lorenz curve examples

In the starting example, see how the society with incomes \((25, 75)\) is closer to the line of perfect equality than the society with incomes \((20, 80)\). We can conclude that the society with income \((25, 75)\) has less inequality than the other society by the Lorenz criterion.

Lorenz curves

The Lorenz criterion and the transfer principle

The Lorenz criterion and the transfer principle yield the same ranking of income distributions. If the Lorenz curves cross, societies cannot be ranked by the transfer principle. If the Lorenz curves don’t cross, the curve closer to the line of perfect equality represents the more equal society.

Exercise

Does the Lorenz criterion respect income neutrality? Use the tool above to determine.

If you double the income of Society 1 in the input box to \((40, 160)\), notice how its line is identical to the \((20, 80)\) line. This illustrates the fact that the Lorenz criterion respects income neutrality.

Exercise

Does the Lorenz criterion respect population neutrality? Use the tool above to determine.

If you clone Society 1 in the input box above to \((40, 40, 160, 160)\), notice how its line is still identical to the \((20, 80)\) line (although with more points). This illustrates the fact that the Lorenz criterion respects population neutrality.

Exercise

Input the four previous examples to see how they look graphically. Can you identify the examples in which the societies cannot be ranked? If an example can be ranked with the help of the Lorenz curves, which society is more equal?

  1. Let \(X = (15, 16, 69)\) and \(Y = (17, 34, 49)\)

They can be ranked since the Lorenz curves do not cross. \(Y\) is more equal because its Lorenz curve is closer to the line of perfect equality.

  1. Let \(X = (15, 40, 45)\) and \(Y = (17, 34, 49)\)

They cannot be ranked since the Lorenz curves cross.

  1. Let \(X = (10, 30, 60)\) and \(Y = (5, 16, 29)\)

They can be ranked since the Lorenz curves do not cross. \(Y\) is more equal because its Lorenz curve is closer to the line of perfect equality.

  1. Let \(X = (24, 76)\) and \(Y = (17, 34, 49)\)

They can be ranked since the Lorenz curves do not cross. \(Y\) is more equal because its Lorenz curve is closer to the line of perfect equality.

Notice that these are the same answers we got when applying the three principles.