Exercises 57 and 58 use the notation and results of Exercises $49-51$ of Section $3.4 .$ For a given country, $F(r)$ is the fraction of total income that goes to the bottom rth fraction of households. The graph of $y=F(r)$ is called the Lorenz curve.

Let $A$ be the area between $y=r$ and $y=F(r)$ over the interval $[0,1]($ Figure 22$) .$ The Gini index is the ratio $G=A / B,$ where $B$ is the area under $y=r$ over $[0,1]$ .

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\begin{array}{l}{\text { (a) Show that } G=2 \int_{0}^{1}(r-F(r)) d r} \\ {\text { (b) Calculate } G \text { if }}\end{array}

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F(r)=\left\{\begin{array}{ll}{\frac{1}{3} r} & {\text { for } 0 \leq r \leq \frac{1}{2}} \\ {\frac{5}{3} r-\frac{2}{3}} & {\text { for } \frac{1}{2} \leq r \leq 1}\end{array}\right.

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\begin{array}{l}{\text { (c) The Gini index is a measure of income distribution, with a lower }} \\ {\text { value indicating a more equal distribution. Calculate } G \text { if } F(r)=r \text { (in }} \\ {\text { this case, all households have the same income by Exercise } 51(b) \text { of }} \\ {\text { Section } 3.4 \text { ). }} \\ {\text { (d) What is } G \text { if all of the income goes to one household? Hint: In this }} \\ {\text { extreme case, } F(r)=0 \text { for } 0 \leq r<1}\end{array}

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