In the pills example in Kjellsson et al., moving from the original shortfall distribution to the blue distribution, which keeps relative inequality unchanged, involves giving both groups the same percentage increase in life expectancy.
By contrast, moving from the original shortfall distribution to the red distribution, which keeps absolute inequality unchanged, involves giving both groups the same number of years of additional life expectancy.
The aforementioned result linking the concentration indices for attainments and shortfalls reminds us that in such cases, the ordering of countries vis-à-vis relative inequality can change as we move from the attainment to the shortfall distribution simply because the means of the attainment (and hence shortfall) distributions differ.
One important result is that the concentration index for the shortfall distribution is equal to the negative of the concentration index for the attainment distribution multiplied by the ratio of the means of the attainment and shortfall distributions.
Subsequent work generalized W to nonbinary-bounded variables, and showed that the normalization W solves not only the bounds problem (like the concentration index for an unbounded variable, W lies between −1 and +1) but also the mirror problem (in the Kjellsson et al. pills example, both the initial attainment and initial shortfall distributions give the same [absolute] value of W, namely 0.133).
When the choice does matter, however, it is worth keeping in mind why the results differ, namely that that the two approaches give different pictures of inequality (at least in part) because of differences in the means of the attainment and hence shortfall distributions.
And that's the conundrum: environmental groups readily point out our water systems' failures (the dozens of unregulated contaminants, the discharge of 850 billion gallons of raw sewage into the nation's waterways each year, the $22-billion-a-year 22-billion-a-year 22-billion-a-year 22-billion-a-yearment plants).
Furthermore, the distribution of item thresholds indicate a shortfall in their distribution across the middle of the construct suggesting the potential for adding items which reflect levels of depression at the middle of the scale.
They were reminded of that a year ago when a temporary shortfall in imports combined with distribution snags to cause the price of tortillas to soar.Despite the long transition to free trade, little has been done to prepare small-scale farmers.
