Gini IndexPart I: Income Distribution The distribution of income in our society is a concept of ongoing interest to economists, politicians, public policy analysts, and other concerned individuals. In a capitalistic society such as the US, perfect equity in income distribution is neither possible nor desired. There would be no incentive to develop new products if you weren’t able to capitalize on your invention. However, there is also a limit to how much of the total income should be controlled by a small group. Some suggest that this inequity in income distribution is playing an important role in the unrest apparent in Tunisia, Egypt, Yemen, and Bahrain. In the US, are the “rich getting richer, and the poor getting poorer” and is the “middle class disappearing” as some politicians suggest? And if so, how could yo u tell? To quantify distribution of income in a country, economists consider the percent of the country’s total income that is earned by certain groups of the population. To understand how this is done, we will consider a very small society consisting of the individuals with the following jobs and salaries: Administrative Support $28,369 Public Relations Specialist $39,913 President of the country $400,000 Advertising $40,424 Mail Carrier $36,619 CEO $100,271 Electrical Engineer $62,201 Congressman $150,000 Secretary $23,311 Teacher $33,123 Pediatrician $113,510 Governor $110,346 Head Nurse $48,000 Migrant farm worker $2,500 Drafter $37,500 Farm worker $7,500 Mechanic $29,521 College Basketball Coach $260,000 Firefighter $27,976 Microbiologist $55,411 Cashier $15,184 Forensic Science Technician $32,864 Travel Agent $27,373 Librarian $42,120 Aircraft Mechanic $42,370
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6. Optional Extension: Finding the Lorenz Curve using Least Squares
There are many ways of finding the equation for the Lorenz curve based on the data from the table in #1. If you
search the Web for Gini indices for various countries, you will find that two estimates for the same year may
differ. This is due to possible differences in their sources of information and to the method used to approximate
the area A. Since we are interested in comparisons over time or among countries, as long as all indices are
computed using the same techniques, comparisons are legitimate.
The Least Squares Method: Since (0, 0) and (1, 1) are always points on the curves, a reasonable model for this
data is a power function of the form y xn
, with . We cannot use a power least squares regression on our
calculator to fit a power function to the data because a Lorenz curve must contain the point (1,1) and a power
regression curve does not necessarily contain (1,1). Also note that we only use the four points
0.2,0.036 , 0.4,0.125 , 0.6,0.273 , and 0.8,0.503 in our calculations. By using y xn
as our model, we
guarantee that (0,0) and (1,1) fall on the curve.
a. The least square method uses the fact that a log-log re-expression linearizes data that is modeled by a
power function. Since y xn
, we take the logarithm of both sides of the equation to obtain ln ln y n x .
We now can use our knowledge of calculus to find a least-squares estimate of n. Consider the linear
equation Y nX (in our case lnY y and ln X x ). Use the methods of calculus to minimize
4
2
1
i i
i
S n Y nX
(remember i X and i
Y are constants). Find a general solution for n for the Lorenz
curve y xn
, given the cumulative aggregate income percentages for the quintiles. Use the general
solution to find the equation for the Lorenz curve for the 2000 data.
b.
Another method is to fit a true least squares model to the data. In this case, we have
2
n
i iS n y x . What equation must be solved to find the value of n that minimizes S? You will not
be able to solve this equation analytically, so feel free to use a calculator or computer to find the value ofn
for 2000.
c.
Graph the Lorenz curves you created in parts a and b over a scatterplot of the data from 2000. In your
opinion, which Lorenz curve fits better, and why?
d.
For the 2000 data, calculate the Gini Index using each version of the Lorenz curves (from parts a and b) byincorporating calculus methods to find Area A. Discuss the differences between these indices, and compare
them with the index calculated with the trapezoid approximations in #4. How accurate would comparisons
among Gini indices for an economy over time (or perhaps comparisons of Gini indices for different
countries) be if the indices were computed using different methods?
Choose Income, then for example, choose Poorest Fifth. This graph shows a map of the world in which theterritory size
shows the earnings of the poorest fifth of the population living there, as a proportion of the earnings of the poorest
fifth living in all territories.
“Japan is the region with the richest poor people in the world. The average income of the poorest fifth of the population inJapan is at least 7 times more than that of the equivalent group in 8 other regions.
The regions with the lowest average incomes for the poorest fifth of the population are Central Africa, Southeastern Africaand Northern Africa. The poorest fifth of the population of South America have especially low relative incomes given theaverage incomes there. Despite being located in South America, French Guiana and the Falklands / Islas Malvinas share data
with France and the United Kingdom respectively, so are resized accordingly.”