It seems that Piketty offers one stylized fact about inequality, which is that it rose during the long 19th century, fell during the period of the World Wars and their aftermath, and has risen since. What might explain this?
I would offer a theory in the spirit of Adam Smith: the division of labor is limited by the extent of the market.
My theory is that the more division of labor, the greater the inequality within a nation. Of course, from a world perspective, inequality may go up or down, and in recent decades it has gone way down.
The long 19th century was an era of globalization. World War I interrupted that, and there was little recovery of international trade between the wars. After World War II, trade also was limited. We had the cold war, which isolated the East from the West. We had economic policies in many countries that were anti-trade (remember import substitution?). Finally, in the 1980s, we began to see liberalization in the West, and then greater liberalization in China, India, and some of the former Soviet bloc. Then we had the Internet, which opened up new opportunities for specialization and trade.
This unleashed a new round of globalization, and the “extent of the market” became greater. The business opportunities this created helped to increase inequality within nations. At the same time, incomes increased in India, China, and several other poor countries, so that world inequality fell.
This theory won’t give me a best-selling book. But I think it has merit.