I’m not sure why Broda and Romalis, or Levitt and Wilkinson, think this should alter our assessment of the trend in inequality. Do they mean to suggest that the revealed preference of the poor for cheap goods is exogenous to their income? In other words, people with low incomes simply like buying inexpensive lower-quality goods, and they would continue to do so even if they had the same income as the rich. Likewise, the rich simply have a taste for better-quality but pricier goods, and they would continue to purchase them even if they suddenly became income-poor. If this is the assumption, I guess the conclusion follows. But I can’t imagine the authors, or anyone else, really believe that.
Maybe I'm missing something, but it seems as though Kenworthy's response might be based on some kind of conceptual misunderstanding. I'm not sure that this is it, but is the idea here that there is a single, standard, uniform price index, perhaps kept in a vault in Paris next to that famous platinum-iridium bar, the standard meter? But there is no standard index with which to determine the one true rate of inflation, or one true rate of change in real wages, because there is no one true standard consumption basket.
It seems that Kenworthy thinks there is something suspect about looking at the typical consumption basket of people at one part of the income distribution, looking at the typical consumption basket of people at another part of the income distribution, and then determining separately the change in rate of actually experienced inflation for people at those points in the distribution. I don't see how this requires any weird assumptions about the exogeneity of preferences to income. All it requires is that we take seriously what different kinds of people tend to buy.
Think about it this way. Suppose you've got a country with only poor people and a country with only rich people. In each country, their version of the BLS creates something like the CPI. We find that price inflation is lower in the poor country. Then the rich country annexes the poor country. Does calculating separate CPIs suddenly become a kind of mistake?
As I noted in my first post on this paper, when I talked very, very briefly to Sachs about my paper on inequality, looking at the change in price of the typical consumption baskets of the rich and poor was the one thing he suggested one might try to do to get a better sense of what's happening in terms of the trend in real consumption inequality. I said I didn't have the technical wherewithal to do that. But Broda and Romalis do. I'm not convinced that they or Sachs or Levitt is confused.