Thanks for your detailed reply, which seems more responsive to me than Eugine_Nier's.
Every actual population differs from a parameterised mathematical function with few parameters, and for pretty much anything you can measure, if the mathematical distribution has infinite support, there will be some reason that the population cannot. But the question to ask is not, are they different, but, does the difference make a difference?
Yes, just so.
The way to answer this question is to repeat the analysis in the paper Eugine cited using a truncated power law. The bounds must be placed at the limits of what is possible, not at the accidental maximum and minimum values observed in the current population, as the point here is that the population is not fully exploring the tails.
OK, so, I think this has helped me pinpoint the root of the disagreement here: we have different beliefs about what the relevant statistical population is. Here are four possibilities, in increasing order of expansiveness.
The statistical population is whatever big sample one can get one's hands on.
The statistical population is the literal population of (working-age? adult?) people in a place of interest at a given time.
The statistical population is that implied by "a truncated power law", with its bounds "placed at the limits of what is possible".
The statistical population is that implied by a power law with only a lower bound.
(1) is what E_N apparently believes I endorse, even though it's obviously a stupid choice based on confusing sample & population. I actually contend (2). You propose (3). T&D seem to assume (4).
In other circumstances one could simply elide the differences between these distributions, as there'd be no reason to expect the precise choice to matter much; the analysis being carried out would be insensitive to it. But here T&D's analysis rests on a particular subtle, pathological feature, and the chance of that feature being present likely depends on which population one chooses.
Given that a choice should be made, I think (2) is a more natural, sensible choice than (3) & (4) for the purposes of estimating inequality at a given point in time, because (2) refers to the actually existing population of interest. (4) is a mathematical abstraction which might be a useful approximation in other circumstances, but risks spuriously introducing a pathological feature here. (3) better matches reality (having an upper bound) but is less parsimonious and harder to operationalize; how do we determine "the limits of what is possible"? And which upper limit ought one use — the maximum individual income/wealth that one'd witness if one could see into the pre-ordained future of humanity, or the maximum individual income/wealth possible under some counterfactual ensemble of futures, or...?
We might choose (3) or (4) in spite of these issues if we wished to predict the future course of inequality, because then we'd need to go beyond people who currently exist (and could simply have their incomes observed) and start modelling the distribution of incomes which haven't been earned yet. But if we'd just like an index of current (or past) inequality, our interest is in the population of people who exist now (or existed at a given time in the past).
If God handed a data file with the income of everybody on the planet to an economist, and the economist used that to calculate some inequality index, the economist wouldn't wring their hands about that index being a noisy sample statistic; they'd consider it the precise population value of the Gini coefficient (or whatever coefficient) for the world, and mark that job as done. (At least until they had to produce next year's statistics, and needed fresh data!) In T&D's notation, the number they'd come up with wouldn't be κ-hat but κ. Complaints about κ-hat systematically diverging from κ would therefore be irrelevant.
Returning to what you wrote, yes, the way to answer the question is to repeat T&D's analysis with a different distribution — but if I used a truncated power law it would be because it matched the empirical distribution of income/wealth well. (It would also be useful to see how κ-hat and κ differed under different sampling strategies; economists often deliberately try to oversample the upper end of the wealth distribution by using tax data or rich lists, and I'd expect that to lessen the small-sample bias T&D identify.)
...OK, so, I think this has helped me pinpoint the root of the disagreement here: we have different beliefs about what the relevant statistical population is. Here are four possibilities, in increasing order of expansiveness.
1. The statistical population is whatever big sample one can get one's hands on.
2. The statistical population is the literal population of (working-age? adult?) people in a place of interest at a given time.
3. The statistical population is that implied by "a truncated power law", with its bounds "placed at the limits of wh
Politics as gymnastics for rationalists. No one one Less Wrong is quite sure why politics is a taboo topic or how things got to be that way. What we do think we know is that politics is a great way to bring out the irrationality in people. So why not take advantage of that and use politics as a way to measure rationality? Since politics brings out the most irrationality, it should provide the strongest signal. Since there aren't useful objective metrics of how a political discussion went, we'd have to use subjective judgements by neutral third-party raters, kind of like they do in gymnastics. (In the comment thread for this post, feel free to find fights that you have no dog in, improvise a rationality rubric, and grade participants according to your rubric... let's see how it goes.)
Be a sheep. This is probably the exact opposite of what you were taught in your high school civics class. But if my friend Jane is more intelligent, more informed, and less ideological than I am, it seems like voting however Jane is going to vote is a strict improvement over voting however I would naively. It also saves me time, and gives Jane an incentive to put even more time in to carefully considering political issues since she now controls two votes instead of one. Done on a large scale, this could provide an interesting twist on representative democracy. Imagine a directed graph where each node represents a person and an edge is directed from person A to person B if person A is auto-copying person B's votes. There's a government computer system where you can change the person you're auto-copying votes from at any time or override an auto-copied vote with your own personal guess about what's best for society. Other than that, it's direct democracy... all bills are put before all citizens to vote on. Problems this might solve: