Picturing inequality: the Lorenz curve and the Gini coefficient

Overview

Two very different distributions produce the same Gini coefficient.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   Describe the axes of a Lorenz curve and what a single point on the curve tells you.

   Model answer

   The horizontal axis shows the cumulative percentage of the population, ranked from the poorest to the richest, running from 0 to 100. The vertical axis shows the cumulative percentage of total income received, also running from 0 to 100. A point on the curve reads as a sentence: at a horizontal value of 40 and a vertical value of 17, the poorest 40 per cent of the population between them receive 17 per cent of all income.

2. 2

   Why is the line of perfect equality on a Lorenz curve diagram a straight line at 45 degrees?

   Model answer

   Under perfect equality every person receives the same income, so any given percentage of the population receives exactly that percentage of total income. The poorest 20 per cent would receive 20 per cent, the poorest 60 per cent would receive 60 per cent, and so on. Plotting those points gives a straight line from the origin to the top right corner, which has a gradient of one and therefore sits at 45 degrees when both axes are scaled identically.

3. 3

   What does it mean when one Lorenz curve lies further below the diagonal than another throughout its length?

   Model answer

   It means that at every point in the distribution the poorer group receives a smaller share of total income than in the comparison distribution, so the distribution is unambiguously more unequal. The curve must always start at the origin and finish at the top right corner, because 0 per cent of people receive 0 per cent of income and 100 per cent receive 100 per cent. It must also be convex when viewed from above, since the population is ranked from poorest to richest and each successive group adds more income than the one before.

4. 4

   Two countries have Lorenz curves that cross. What does that tell you about ranking their inequality?

   Model answer

   Crossing curves mean that one country is more equal at one part of the distribution and less equal at another, for instance with a smaller gap at the bottom but a more concentrated top. There is then no unambiguous ranking from the curves alone: the answer depends on which part of the distribution is weighted more heavily. A summary statistic such as the Gini will still produce a single number for each, but that number is imposing a particular weighting rather than resolving the ambiguity.

5. 5

   Define the Gini coefficient in terms of the areas on a Lorenz curve diagram.

   Model answer

   Let A be the area between the line of perfect equality and the Lorenz curve, and let B be the area beneath the Lorenz curve down to the horizontal axis. The Gini coefficient is A divided by the sum of A and B. Because A and B together make up the triangle below the 45-degree line, that sum is always 0.5 when both axes are measured as proportions, so the Gini is equivalently A divided by 0.5, or twice A.

6. 6

   What are the limiting values of the Gini coefficient and what distribution does each describe?

   Model answer

   The Gini runs from 0 to 1. A value of 0 means the Lorenz curve lies exactly on the line of perfect equality, so area A is zero and everyone receives an identical income. A value of 1 means the curve runs along the horizontal axis and then straight up the right-hand edge, so one person receives all the income and everyone else receives nothing. Real economies sit well inside these limits.

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The same content, four other ways.

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