Inequality "r> g" is the key to Piketty's book...."When the rate of return on capital [r] is higher than the growth rate of the economy [g], logic dictates that inherited wealth is growing faster than GDP and personal income"(p. 25).
And that is the fundamental message of the book: as the rate of return on capital is higher than the growth rate of the economy (and therefore wages), the "capitalists" obtained an increasing share of the pie, leaving inheritance to their children, who therefore, are born rich. The return of inherited wealth that they will get will also be higher than the growth rate .... And so dynasties; the rich getting richer relative to workers, and capital has become increasingly large relative to the rest of the economy.
This constant increase in inequality is a major problem for capitalism because it generates political instability: per Piketty, the mass of poor workers will end up rebelling against the rich minority, and through democracy or violence will destroy the system. That is the "central contradiction of capitalism". A contradiction that will ultimately lead to self-destruction.
To fix the problem requires taxing inheritances and capital, to break the dynasties of billionaires and their possibly dominating the economy. Everything from the finding that the rate of return on capital is higher than the growth rate of the economy. Everything from "r> g".
This logic of Piketty, however, has a little problem: ITIS FALSE!
Let the rate of return on capital, r, be greater than the growth rate of the economy, g. Far from being a "contradiction of capitalism" this is a condition that economists have dubbed "dynamic efficiency". If an economy has "r <g", it is inefficient in the sense that it has saved too much. That is, if "r <g", the current generation could increase their consumption (reducing their savings), with no future generations being forced to suffer. In this case the savings would be inefficiently large so it would be socially desirable to be reduced until "r> g." And that would be true in a capitalist economy and a planned economy. so, the inequality "r> g" isn't the "central contradiction of capitalism", it is an aberration.
Moreover, contrary to the claims of Piketty, the fact that r is greater than g implies neither the rich spend their savings on to their children, nor that wealth grows faster than GDP, or that rich dynasties are increasingly richer, or that social inequalities grow.
Imagine, for example, a world in which individuals work when young and retire when old. Knowing that someday they will retire, so when young, save money and invest. When they are old they use their savings (and the rate of return on their savings) to survive. It's no longer a dollar inheritance to their children if they die with nothing.
The children do the same as their parents and thus, generation after generation too. All economists know that in this world of "overlapping generations" the rate of return on capital, r, may be higher, exceeding the growth rate, g. If the economy is dynamically efficient, then it is true that "r> g" and yet, no one gets an inheritance!
Unlike what Piketty says, logic does not dictate in any way, that "r> g" implies that inherited wealth grows faster than GDP, partly because inherited wealth can be exactly zero in worlds where "r> g"!
In the real world, of course, the rich do not consume everything they have and leave part of their wealth in inheritance to their children. It is also true that for many of them an important part of that wealth is spent on lavish parties, boats, airplanes, luxury travel or philanthropic actions such as Bill Gates or Warren Buffet.
Furthermore, unlike what happened in ancient times, where all the wealth went to a single heir, now the property of the rich is divided between many children (often from different marriages) so that from a very rich grandfather You can have very poor grandchildren. It is well known; the saying that the grandfather created a fortune, his children extend the fortune, and the grandchildren squandered same.
The world could have "r> g" and, in turn, be filled with families whose grandparents create fortunes , the grown children and grandchildren destroy them. And contrary to what Piketty says, in that world there would not be more and more rich and powerful dynasties. However, it would be true that "r> g"!