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).
The Lorenz curve diagram splits the triangle under the perfect equality line into two parts: A, the gap caused by unequal income distribution, and B, the remaining area bounded by the actual Lorenz curve. The question tests the standard formula that converts these two areas into the Gini coefficient.
$\frac{A}{A+B}$: A+B together make up the full triangular area under the equality line. Dividing the inequality gap A by this whole area gives a ratio that is 0 when there is no inequality (A = 0) and approaches 1 under extreme inequality (B close to 0). This matches the definition of the Gini coefficient and is bounded correctly between 0 and 1.
$\frac{A}{B}$: This ratio is not bounded by 1; as inequality worsens and B shrinks toward 0, this fraction grows without limit, which cannot represent a coefficient that must stay between 0 and 1.
$\frac{A}{B-A}$: This expression can become negative (when B is smaller than A) or undefined (when B equals A), so it fails to give a sensible inequality measure across all cases.
$\frac{B}{A}$: This moves in the wrong direction altogether, since it grows large as inequality falls (A close to 0) rather than as inequality rises, the opposite of what the Gini coefficient should do.
Only $\frac{A}{A+B}$ behaves correctly at both extremes, equal to 0 under perfect equality and approaching 1 under total inequality, so it is the correct formula.
Let's summarize:
A+B is the full area under the perfect equality line (a right triangle).
A alone is the area of inequality, the gap between equality and the real Lorenz curve.
Gini coefficient $= A/(A+B)$, always between 0 and 1.
