Obviously it's difficult to get a handle on the distribution of personal income even today, but I have some theoretical interest in this question for the following reason:

When people try to build models to predict how population size and gross domestic product correspond to military power, a popular index of "national strength" that's used is

for some exponent . The idea is that, all else equal, if two countries have the same GDP the one with the smaller population should be at an advantage in any conflict. This matches many historical episodes, such as the wars fought by European countries in Asia in the late 18th and 19th centuries.

However, there is a big flaw with this kind of measure: it is not monotone. As an example, suppose and we have two countries A and B. A has a GDP per capita of $100k and a population of 100 million, B has the same population but a GDP per capita of only $10k. Now, the union of A and B has GDP per capita equal to $55k and a total population of 200 million, and so for the union actually appears weaker than the country alone.

There is a natural alternative which is monotone: we take expectations after exponentiating income by rather than before. Since is a convex map, this gives us a larger answer. It's not only monotone but also extensional, because it's equal to

where the sum runs over all individuals in a country and denotes the income of individual . So this index has desirable theoretical properties and we can trust it more when trying to evaluate the strength of international alliances or coalitions.

However, computing it and fitting a model based on it to data requires knowledge of the higher moments of the income distribution across countries and across time. We have data that's at least adequate when it comes to population and mean income, but I haven't seen any data on the mean of income squared or cubed, say. I imagine someone has already looked into this problem, and I'd be happy enough just with data pertaining to the past 300 years or so.

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