The graph below shows cumulative population (in %) on the X-axis and cumulative income (in %) on the Y-axis. If A and B are areas as shown in the graph, then the Gini coefficient is measured as .
Show Hint
Gini coefficient = area between the equality line and the Lorenz curve (A), divided by the full area under the equality line (A+B).
Step 1: Understanding the Question:
The graph is a Lorenz curve: cumulative population (%) on the X-axis against cumulative income (%) on the Y-axis. The straight diagonal from O to the top-right corner is the perfect equality line, where the bottom x% of the population always earns exactly x% of income. The bowed curve below it is the actual Lorenz curve for the real, unequal income distribution. Area A lies between the equality line and the Lorenz curve; area B lies between the Lorenz curve and the axes below it. We need the formula for the Gini coefficient in terms of A and B.
Step 2: Key Formula or Approach:
The Gini coefficient is a standard measure of income inequality, defined as the ratio of the area between the line of perfect equality and the actual Lorenz curve (area A), to the total area under the line of perfect equality (the full triangle below the diagonal, which is A plus B together):
\[ \text{Gini coefficient} = \frac{\text{Area between equality line and Lorenz curve}}{\text{Total area under equality line}} = \frac{A}{A+B} \]
Step 3: Detailed Explanation:
The triangle formed by the equality line, the X-axis, and the vertical line at 100% cumulative population is the full area under perfect equality; this triangle is exactly area A plus area B together, since the Lorenz curve splits that triangle into the upper sliver (A, the gap caused by inequality) and the lower region (B, bounded by the Lorenz curve and the axes).
When income is perfectly equal, the Lorenz curve coincides with the diagonal, so A becomes 0, and Gini \(= \frac{0}{0+B} = 0\), correctly showing no inequality.
When inequality is extreme, one person holding all the income, the Lorenz curve hugs the axes and B shrinks to nearly 0, so Gini \(= \frac{A}{A+0} \to 1\), correctly showing maximum inequality. This behaviour confirms \(\frac{A}{A+B}\) as the right ratio, since it always stays between 0 and 1 and moves the correct direction with inequality.
Option (B), \(A/B\), is wrong since it does not stay bounded between 0 and 1, it can exceed 1 as inequality grows, which a coefficient meant to range from 0 to 1 cannot do.
Option (C), \(A/(B-A)\), is not a recognized definition and can turn negative or blow up depending on the relative sizes of A and B, which is not meaningful for an inequality measure.
Option (D), \(B/A\), is the inverse of the wrong ratio in (B), and would rise as inequality falls, the opposite of how the Gini coefficient is defined to behave.
Step 4: Final Answer:
The Gini coefficient is defined as \(\frac{A}{A+B}\), option (A).
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