   Chapter 13.3, Problem 45E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Income distribution With U.S. Census Bureau data, the Lorenz curves for the income distribution in the United States for Black households and for Hispanic households (of any race) are given below. Find the Gini coefficient of income for both groups and compare the distribution of income for these groups.Black: y =   2.292 x 4 −   3.127 x 3 +   2.006 x 2   −   0.1710 x   +   0.0005556 Hispanic : y =   1.830 x 4 −   2.564 x 3 + 1.806 x 2 −   0.1137 x +   0.0003512

To determine

To calculate: The Gini coefficient of income for black household and Hispanic household if income distribution for black household and Hispanic households are y=2.292x43.127x3+2.006x20.1710x+0.0005556 and y=1.830x42.564x3+1.806x20.1137x+0.0003512 respectively.

Explanation

Given Information:

The income distribution for black household and Hispanic households are y=2.292x43.127x3+2.006x20.1710x+0.0005556 and y=1.830x42.564x3+1.806x20.1137x+0.0003512 respectively.

Formula used:

The power rule of integration is;

The integration of function f(x)=xn is given by;

xndx=xn+1n+1+c n1

Here c is any arbitrary constant.

The sum rule is;

f(x)±g(x)dx=f(x)dx±g(x)dx

Calculation:

Consider the given income distribution for 2012 in United States for Black households and Hispanic households as below

For Black households,

y=2.292x43.127x3+2.006x20.1710x+0.0005556

For Hispanic households,

y=1.830x42.564x3+1.806x20.1137x+0.0003512

The formula for Gini coefficient of income is;

Gini coefficient=01xy(x)12

Thus, to find the Gini coefficient of income for Black households first solve the integration as;

(xy(x))dx=x(2.292x43.127x3+2.006x20.1710x+0.0005556)=x222.292x55+3.127x442.006x33+0.1710x220.0005556x

Evaluate the limits as;

01(xy(x))dx=(122

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