QUESTION THREE (30 marks) Consider the following diagram which shows the Lorenz curve and line of Equality for a particular country. Line of Equality (45 Degree) Lorenz Curve B Cumulative share of people from lowest to highest incomes A 100% 100% Cumulative share of income earned 3a- What is Lorenz curve? Explain briefly at least 5 of its main advantages and its benefit in measuring economic inequality. (20 marks) 3b- The above diagram shows two areas A and B, what information could be obtained through them? (10 marks)
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It plots the cumulative share of income earned by the population from lowest to highest incomes on the horizontal axis, and the cumulative share of people from lowest to highest incomes on the vertical axis. Advantages of the Lorenz curve: Show more…
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Economists use Lorenz curves to illustrate the distribution of income in a country. A Lorenz curve, $y=f(x)$, represents the actual income distribution in the country. In this model, $x$ represents percents of families in the country and $y$ represents percents of total income. The model $y=x$ represents a country in which each family has the same income. The area between these two models, where $0 \leq x \leq 100$, indicates a country's "income inequality." The table lists percents of income $y$ for selected percents of families $x$ in a country. $$ \begin{aligned} &\begin{array}{|c|c|c|c|c|c|} \hline x & 10 & 20 & 30 & 40 & 50 \\ \hline y & 3.35 & 6.07 & 9.17 & 13.39 & 19.45 \\ \hline \end{array}\\ &\begin{array}{|c|c|c|c|c|} \hline x & 60 & 70 & 80 & 90 \\ \hline y & 28.03 & 39.77 & 55.28 & 75.12 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to find a quadratic model for the Lorenz curve. (b) Plot the data and graph the model. (c) Graph the model $y=x .$ How does this model compare with the model in part (a)? (d) Use the integration capabilities of a graphing utility to approximate the "income inequality."
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Refer to the diagram, where curves (a) through (e) are Lorenz curves for five different countries. Income is most unequally distributed in: Country a. Country b. Country c. Country d. Country e.
Rashmi S.
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 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 $(a / 100, b / 100)$ is on the Lorenz curve if the bottom $a \%$ of the households receive less than or equal to $b \%$ of the total income. Absolute equality of income distribution would occur if the bottom $a \%$ of the households receive a $\mathscr{H}$ of the income, in which case the Lorenz curve would be the line $y=x .$ The area between the Lorenz curve and the line $y=x$ measures how much the income distribution differs from absolute equality. The coefficient of inequality is the ratio of the area between the Lorenz curve and the line $y=x$ to the area $x \cos y e^{x} x$. (Check your book to see figure) (a) Show that the coefficient of inequality is twice the area between the Lorenz curve and the line $y=x,$ that is, show that $$\text { coefficient of inequality }=2 \int_{0}^{1}[x-L(x)] d x$$ (b) The income distribution for a certain country is represented by the Lorenz curve defined by the equation $$L(x)=\frac{5}{12} x^{2}+\frac{7}{12} x$$ What is the percentage of total income received by the bottom $50 \%$ of the households? Find the coefficient of inequality.
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