Economists use a cumulative distribution called a Lorenz curve to describe the distribution of income between households in a given country. Typically, a Lorenz curve is defined on [0, 1] with endpoints (0, 0) and (1, 1), and is continuous, increasing, and concave upward. The points on this curve are determined
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If the question involves interpreting the Lorenz curve or calculating properties like the Gini coefficient, here’s a general approach:
Explanation of the Lorenz Curve
- Definition:
The Lorenz curve plots the cumulative proportion of income earned by the bottom x% of the population. On the graph:
- The x-axis represents the cumulative percentage of the population (from the poorest to the richest).
- The y-axis represents the cumulative percentage of income.
- Properties:
- It always starts at (0, 0) and ends at (1, 1).
- It is monotonically increasing because adding households cannot decrease cumulative income.
- It is concave upward because wealthier segments typically contribute disproportionately to total income.
- Key Insight:
- A perfectly equal income distribution corresponds to the line of equality: ( y = x ).
- The greater the curvature below this line, the more unequal the income distribution.
- Gini Coefficient:
The Gini coefficient measures income inequality, calculated as:
[
G = \frac{A}{A + B}
]
Where ( A ) is the area between the line of equality and the Lorenz curve, and ( A + B ) is the area under the equality line.
If you describe the exact task or provide numbers/curves, I can calculate specific results or draw a Lorenz curve. Let me know how you’d like to proceed!