Wthin the last week, the Census Bureau released new data on the 2009 American Community Survey. The most publicized piece of information was the growing inequality of incomes in the US. The link below is a census piece on income from the ACS.
www.census.gov/prod/2010pubs/acsbr09-2.pdf
In figure 2, there are 3 states with a Gini index higher than the national average: Texas, Connecticut, and New York. There are only this many, largely because the margin of error is fairly large in comparison to the actual difference between state and the national Gini index numbers. Now, all three of these states are among those with the most unequal distribution of income. But are they really the worst three?
Texas vs. Louisiana
Texas has a slightly higher Gini index than Louisiana based on census data. The difference is well within the margin of error. For this exercise, I will treat the census numbers as actuals instead of estimates and presume there is no margin of error.
One method of using income data to estimate the Gini index is to look at means of various income quintiles. The census bureau also provides a number for mean income of the top 5%. Since these figures are aggregated, the Gini index calcluated based on these numbers will not be as precise as those produced by the Census Bureau. Indeed, more aggregated inputs result in lower inequality. Caluculating the Gini index shows .457 for Louisiana and .458 for Texas. The Census calculations are .473 and .474, higher for both but still separated by the same amount (.001).
Proportionally measuring each household income with all others however shows inequality to be greater in Louisiana. The means of the quintiles are given below, along with the mean for the top 5%, and a calcualted mean of those in the top quintile but OUTSIDE the top 5%.
Lowest: LA-$9,315; TX-$11,021
2nd: LA-$24,623; TX-$28,627
3rd: LA-$42,879; TX-$48,384
4th: LA-$69,492; TX-$76,793
Highest: LA-$148,787; TX-$170,844
Top 5%: LA-$254,657; TX-$302,285
Top quintile, not top 5%: LA-$113,497; $127,030
Raw numbers mean little, since income is higher across the board in Texas. Since we can divide the highest quintile into 2 subgroups, we have 6 distinct cohorts. Calculating the proportional gap between each cohort illustrates the relative gap at each step of the income ladder. For these 2 states, the numbers are listed below.
Lowest-2nd: LA: 2.64; TX-2.60
2nd-3rd: LA: 1.74; TX-1.69
3rd-4th: LA: 1.62; TX-1.59
4th-Highest outside top 5: LA: 1.63; TX: 1.65
Highest outside top 5-top 5: LA: 2.24; TX: 2.38
So in 3 cases Louisiana had a higher proportional gap, and in 2 Texas had the larger gap. In comparing each household against one another, some of these will apply and some won't - depending from which cohort each household is included. Given all the possible combinations of households, we know that these gaps will apply as follows:
Lowest-2nd: in 32% of cases
2nd-3rd: 48%
3rd-4th: 48%
4th-Highest outside top 5: 32%
Highest outside top 5-top 5: 9.5%
For example 20% of the population will fall below the break from Lowest-2nd, namely the lowest quintile. The other 80% will be above. If selecting a random combination of households, we'd have the following odds for combinations:
Lowest 20-Lowest 20: 20% x 20% = 4%
Lowest 20-Highest 80: 20% x 80% = 16%
Highest 80 - Lowest 20: 80% x 20% = 16%
Highest 80 - Highest 80: 80% x 80% = 64%
In the two middle cases, the selected households fall on opposite sides of the Lowest-2nd gap, and thus that proportional gap applies. The mean (geometric, since this involves proportions)gap between any 2 households in Louisiana is 2.84. In Texas it is 2.78. The actual mean difference in relative terms between household incomes is higher in Louisiana.
The Gini index does much the same thing, though it deal with ABSOLUTE values. Of course, absolute difference increases as all incomes increase proportionally, and so the Gini value is indexed to the mean. The consequence of this is to put much greater emphasis where absolute difference is highest-at the top of the income ladder. And that's place where we find larger gaps in Texas than Louisiana.
Cutting the Louisiana lowest quintile income in half would increase the mean relative difference significantly - to 3.54. The calculated Gini index would increase to .469, that is, it barely moved. What about doubling the top 5% incomes? Mean relative difference moves up to 3.03 since only 5% was affected instead of 20% as in the first case. The Gini jumps to .662!
What are the ramifications? A geography can look pretty equal even with a very poor underclass, with incomes way below the median, since this is less of a factor in calualting the Gini index. Overall, the Gini pretty closely shows income disparity. But there are better measures, no more difficult to calculate that could be used instead.
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