Economists use a cumulative distribution called a Lorenz curve to describe the

Chapter 5, Problem 63

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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 with endpoints and , and is continuous, increasing, and concave upward. The points on this curve are determined by ranking all households by income and then computing the percentage of households whose income is less than or equal to a given percentage of the total income of the country. For example, the point is on the Lorenz curve if the bottom of the households receive less than or equal to of the total income. Absolute equality of income distribution would occur if the bottom of the households receive ofthe income, in which case the Lorenz curve would be the line. The area between the Lorenz curve and the linemeasures how much the income distribution differs fromabsolute equality. The coefficient of inequality is the ratio ofthe area between the Lorenz curve and the line to thearea under .(a) Show that the coefficient of inequality is twice the areabetween the Lorenz curve and the line , that is, showthat(b) The income distribution for a certain country is representedby the Lorenz curve defined by the equationLx 512 x 2 712 xc What is the percentage of total income received by the a%bottom of the households? Find the coefficient ofinequality.