Piketty cites two fundamental laws of capitalism. The first one is, truly, a law, and indeed as he says an accounting identity, but the second “law” is really more of a tendency. It suggests that the long-term path of the capital/income ratio (which he calls β but I’m going to call ב becuase we need notation diversity) is equivalent to the ratio of the trend savings rate and the trend growth rate (absolute, not per-capita, importantly). This is nothing Piketty doesn’t mention, but it’s worth stressing that this tendency can take a looooong time to manifest. A society beginning with a ב of 1 and a trend savings/growth ratio – and thus a long-term predicted ב – of 4 will take over a generation to build a ב of 3; the same society with a trend savings/growth ratio of 6 will take over 80 years to surpass a ב of 5.
Anyway, if you’re the kind of nerd I am, you’ll want to play around with the formulae too. Spreadsheet attached; enjoy.