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Does a low Gini index correspond to an equitable or inequitable distribution of wealth? Explain. - for a general L(x), explain why G=2A

Does a low Gini index correspond to an equitable or inequitable distribution of wealth? Explain. - for a general L(x), explain why G=2A

Economics Today and Tomorrow, Student Edition
Economics Today and Tomorrow, Student Edition
1st Edition
ISBN:9780078747663
Author:McGraw-Hill
Publisher:McGraw-Hill
Chapter11: Marketing And Distribution
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Problem 16AA
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Lp = x^p for p is greater than or equal to 1

- Does a low Gini index correspond to an equitable or inequitable distribution of wealth? Explain.

- for a general L(x), explain why G=2A

Scientists studying the economics of a country, community, or group of people are often interested in the distribution of wealth, where wealth is understood to mean the income or the personal assets of individual households. A powerful tool for illustrating how wealth is distributed across a society is the Lorenz curve (invented by the American economist Max Otto Lorenz in 1905). A typical Lorenz curve is given by y = L(x), where 0 < x < 1 and 0 < y < 1. The variable I represents the fraction of all households in the society and y = L(x) represents the fraction of the total wealth that is owned by the fraction x of the society. For example, for the Lorenz curve shown in Figure 1, we see that L(0.5) = 0.2, which means that 0.5 (50%) of the society owns 0.2 (20%) of the wealth. bottom botom y A 1- Line of perfect equality 0.8 of households control 0.6 of the wealth. Lorenz curve 0.5 of households control 0.2 of the wealth. 0.2 B 0.5 0.8 a1 1 Fraction of Households Figure I Fraction of Total Wealth
Transcribed Image Text:Scientists studying the economics of a country, community, or group of people are often interested in the distribution of wealth, where wealth is understood to mean the income or the personal assets of individual households. A powerful tool for illustrating how wealth is distributed across a society is the Lorenz curve (invented by the American economist Max Otto Lorenz in 1905). A typical Lorenz curve is given by y = L(x), where 0 < x < 1 and 0 < y < 1. The variable I represents the fraction of all households in the society and y = L(x) represents the fraction of the total wealth that is owned by the fraction x of the society. For example, for the Lorenz curve shown in Figure 1, we see that L(0.5) = 0.2, which means that 0.5 (50%) of the society owns 0.2 (20%) of the wealth. bottom botom y A 1- Line of perfect equality 0.8 of households control 0.6 of the wealth. Lorenz curve 0.5 of households control 0.2 of the wealth. 0.2 B 0.5 0.8 a1 1 Fraction of Households Figure I Fraction of Total Wealth
7. The information in the Lorenz curve is often summarized in a single measure called the Gini index (proposed in a 1912 paper by the Italian statistician Corrado Gini). The Gini index is defined as follows. Let A be the area of the region between y = x and y = L(x) and let B be the area of the region between y = L(x) and the x-axis (see Figure 1). Then the Gini index A is G = A+B Compute the Gini index for L, when p = 1.1 and when p= 4. (A+B) A = (x - L(w) dx 6= 2 L-S Llo)dx A = G = 2A S LIx) dx %3D |-2 A = A- i-S Lx) dx A = (1-0)= (0.0476 2.1 o.6 %3D
Transcribed Image Text:7. The information in the Lorenz curve is often summarized in a single measure called the Gini index (proposed in a 1912 paper by the Italian statistician Corrado Gini). The Gini index is defined as follows. Let A be the area of the region between y = x and y = L(x) and let B be the area of the region between y = L(x) and the x-axis (see Figure 1). Then the Gini index A is G = A+B Compute the Gini index for L, when p = 1.1 and when p= 4. (A+B) A = (x - L(w) dx 6= 2 L-S Llo)dx A = G = 2A S LIx) dx %3D |-2 A = A- i-S Lx) dx A = (1-0)= (0.0476 2.1 o.6 %3D
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Gini coefficient is a way to measure income inequality. The Gini-coefficient is calculated by the ratio of the area between the Lorenz curve and the perfect equality line, and the area below the perfect equality line. Its value measures the extent of income inequality and its value lies always between 0 to 1.

1.

The lower value of the Gini coefficient indicates the equitable distribution of wealth in comparison to a higher value which indicates the inequitable distribution of wealth. If the value of the coefficient is near 1, there is greater income inequality and if the value if close to 0, there is lower income inequality.  G = 0.0476 indicated lower inequality.

 

 

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