Morgan Bishop
MA 1723
Gini Index Investigation
3 October 2013
1(a) By definition, the Gini Index is the quotient of the area between the Lorenz curve and the line y=x and the area under the line y=x.
G=area between y=x and L(x)area under y=x
G=01[x-Lx]01xdx.
If one was to solve the bottom integral, the area would be 122-022, or 0.5.
Therefore the area between the Lorenz curve and the line y=x is essentially being multiplied by 2. Properties of integrals allow one to factor constants out of the integrand. This gives one another way to express the Gini Index:
G=201[x-Lx]dx.
1(b) In a perfectly egalitarian society, Lx=x. Therefore, there is no area between y=x and L(x), because they are the same line. The Gini Index would equal zero in this case. In a perfectly totalitarian society, Lx is the vertical line x=1. Therefore, the area between y=x and L(x) is represented by a triangle with the area of ½. The Gini Index would equal 1 in this case.
x | L(x) |
0 | 0.000 |
0.2 | 0.034 |
0.4 | 0.120 |
0.6 | 0.267 |
0.8 | 0.500 |
1 | 1.000 |
2(a) Based on the 2008 census, the richest 20% of the American population received 50% of the total income.
2(b) A quadratic function that best models this data is L(x) = 1.3027x2 - 0.3677x + 0.0264. It seems to be a good fit since it has a R2 value of 0.98825. Looking at the graph though, one can see why it is not a perfect fit. For example, the y-intercept is not zero.
2(c) Using this model of the Lorenz curve, one could estimate the Gini Index from 2008. Using the equation found in Part 1:
G=201[x-Lx] dx
G=201[x-(1.3027x2-0.3677x+0.0264)] dx
G=201(x)dx-01(1.3027x2-0.3677x+0.0264) dx
G=2(0.5-[1.3027x33-0.3677x22+0.0264x]01)
G=2(0.2232)
G=0.4464
3. Using Census data from other years, the Gini Index leading to the approximant index found based on the 2008 census.
x | 1970 | 1980 | 1990 | 2000 |
0 | 0 | 0 | 0 | 0 |
0.2 | 0.041 | 0.042 | 0.038 | 0.036 |
0.4 | 0.149 | 0.144 | 0.134 | 0.125...