Question

3. The Gini Index Problem How is possible to measure the distribution of income among the inhabitants of a given country One such measure is the Gini index, named after the Italian economist Corrado Gini, who first devised it in 1912. We first rank all households in a country by income and then we compute the percentage of households whose income is at most a given percentage of the country's total income. We define a Lorenz Curve y L(x) on the interval [0, 1] by plotting the point (a/100, b/100) on the curve if the bottom a% of households receive at most b% of the total income. For instance, the points (0.4, 0.12) and (0.8, 0.5) are on the Lorenz curve for the United States in 2008 because the poorest 40% of the population received just 12% of the total income and likewise the bottom 80% of the population received 50% of the total income respectively. (The Lorenz curve is named after the American economist Max Lorenz.) All Lorenz curves pass through (0,0) and (11) and are concave upward. In the extreme case, L(x)-x, society is perfectly egalitarian: The poorest a% of the population receives a% of the total income and so everybody receives the same income. The area between a Lorenz curve y- L(x) and the line y-x measures how much the income distribution differs from absolute equality. The Gini index (sometimes called the Gini coefficient or the coefficient of inequality) is the area between the Lorenz curve and the line y-x divided by the area under y-x. 1 a) Show that the Gini index G is twice the area between the Lorenz curve and the line y = x, that is, b) What is the value of G for a perfectly egalitarian society (everybody has the same income)? (o po c) What is the value of G for a perfectly totalitarian society (a single person receives all the income)? Lle Pot 2. The following table (derived from data supplied by the US Census Bureau) shows values of the Lorenz function for income distribution in the United States for the year 2008. 0.0 0.000 0.2 0.034 0.4 0.120 0.6 0.267 0.500 1.000 a) What percentage of the total US income was received by the richest 20% of the b) Use a calculator or computer to fit a quadratic function to the data in the table. Graph the c) Use the quadratic model for the Lorenz function to estimate the Gini index for the United population in 2008? (IPonts data points and the quadratic function. Is the quadratic model a reasonable fit? Why? (o Ponts States in 2008 4. a) If a function is integrable, does it have to be continuous? Explain your reasoning and give examples. io Pon s b) Describe the relationships among continuity, differentiability, and integrability. Which is the strongest? Which is the weakest? Which condition imply the other conditions? Ponts

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