This is a linkpost for /r/discussion/lw/ov1/putanumonit_contra_noah_smiths_bad_argument_for_a/
I don't know if it's a fatal objection, it's the main thing differentiating Logini from regular Gini. Logini isn't a "pure" measure of inequality, but I argue that measuring pure inequality is not very useful.
First of all, there's no such thing as "pure inequality". Gini measures just "reported income, within country, household aggregated" inequality or something equally full of caveats. Asking for inequality without qualifications or context gets you nowhere, so why not combine absolute prosperity with income distribution? I don't think Logini is a better econometric formula in some abstract sense, just that it correlates closer with what we should care about (life quality).
I think you may be misunderstanding my objection, which has nothing to do with any notion of purity.
Suppose you measure the "logini coefficient" in some country. Immediately afterwards, they decide that instead of denominating their currency in Foobars, which is beginning to produce slightly silly results after many years of inflation, they will switch so that the New Foobar equals 1000 Old Foobars.
Suddenly everyone's income is nominally 1000x less. All your logs will decrease by whatever log-to-your-chosen-base of 1000 is. And this will complete...