It's not r>g, Piketty, it's r>3g

Piketty's main explanatory model in Capital in the 21st century is that when the rate of return to capital (r) is higher than economic growth (g), income and wealth tends to concentrate in the hands of the owners of capital. Given that the rich own most of the capital, that makes the distribution of income and wealth more unequal. Many passages could be quoted, for example the title of section "The Fundamental Force for Divergence: r>g"; or "...this leads to endless accumulation, which the inequality r>g transforms into a permanent divergence in the global capital distribution..." (462); or "...the inequality r>g implies that wealth accumulated in the past grows more rapidly than output and wages..." (571).

But r>g doesn't imply, at all, that wealth or income would tend to concentrate in the hands of capitalists.

Fortunes

Imagine you are the owner of a capital amounting to 1 million euros (sic), with a rate of return of 4%. Imagine you live in an economy growing at just 1,5%. Clearly, r>g. You'll have an annual income of 40,000 euros. Assume you consume 30,000 euros, and save 10,000. At the end of the year, your fortune will be 1,010,000 euros. (Ignore, for the sake of the argument, the effects of inflation). Your "capital" would have grown 1%. The economy has grown 1,5%. Next year, if the rate of return to capital hasn't changed, you'll have an income of 4% x 1,010,000, or 40,400, 1% higher than this year. Your income would have grown, but less than the income of the average person in the economy, which (by assumption) would have grown 1,5%. The ratio between your capital and the economy's income would have fallen.

In basic math: if r is the rate of return to capital, g economic growth and S the share that capitalists save out of their income, you need S.r>g for capitalists to increase their share of the pie. S.r is necessarily less than r. How much? More on this later.

Capital

One could argue: but capitalits aren't the only ones saving here! Workers can also save, so capital will grow more than by capitalists' savings alone. True. (Though if that were the case it would be less clear that an increase in the capital/income ratio implies a more unequal distribution of wealth). In any case: under what conditions would capital be becoming dominant (ie., increasing its share in income distribution, or increasing the capital/labor ratio)?

If K is accumulated capital, the change in the level of capital will be dK = s.Y, where Y is income and s is the savings rate in the economy.

The rate of change in the level of capital will be sY/K or s/k if we define k=K/Y.

The rate of change in income will be g. Capital will be becoming dominant (the capital-income ratio increasing) if s/k>g or s/g>k. The ratio between the savings rate and economig growth has to be higher than the capital-income ratio for the latter to increase. If the rate of interest is constant (something Piketty describes as a historical regularity) the same condition will imply that the share of income from capital (r.K/Y) is growing. So: if the interpretation of "fundamental divergence" means that capitalists (old and new) are getting a larger size of the pie, that has nothing to do with r>g but with a relation between savings, economic growth and the capital-output ratio. 

Will s>k.g? Clearly, not indefinetely. As Piketty himself shows, there's an equilibrium value for the capital-output ratio. Whenever s>k.g, k will be growing, so the condition will be harder to be met as time goes by. If there's a shock to k (for instance, a destruction to capital through war) there would be a tendency for capital to become more dominant. Until k is again so big that s=k.g. The U-shape of k along the 20th century, with the bottom occuring after the wars, is natural even with constant factors s and k.

What if capitalists are the only savers?

Let's go back to the case "Fortunes", where r does play a role to define the evolution of capitalists fortunes. Don't capitalists save most of their income? Let's assume that they do: that workers don't save and capitalists are the only savers. But Piketty shows national savings rates (s) of around 10% of GDP, and a capital share in income of around 30%. So even if capitalists are the only savers, their saving rate S is around 10/30=33%. In that case, the rule S.r > g is equivalent to r > 3g. So: even in the extreme case in which capitalists are the only savers, empirically the rule for "capital domination" is very far from r > g.

If s is the general savings rate, the highest possible saving rate of capitalists, S, would be s/(r.k) where rk is the share of capital in income. In that extreme case, capital growth would be S.r = s.r/(r.k) = s/k, and the condition for increasing domination by capital would be s/k>g. Note that (quite obviously) at the equilibrium capital-income ratio s/g=k, even if capitalists are the only savers they wouldn't be able to increase their income share. Solow, on the contrary ("Piketty is right") holds that

Suppose it has reached a “steady state” when the capital-income ratio has stabilized. Those whose income comes entirely from work can expect their wages and incomes to be rising about as fast as productivity is increasing through technological progress. That is a little less than the overall growth rate, which also includes the rate of population increase. Now imagine someone whose income comes entirely from accumulated wealth. He or she earns r percent a year. (I am ignoring taxes, but not for long.) If she is very wealthy, she is likely to consume only a small fraction of her income. The rest is saved and accumulated, and her wealth will increase by almost r percent each year, and so will her income. If you leave $100 in a bank account paying 3 percent interest, your balance will increase by 3 percent each year. This is Piketty’s main point, and his new and powerful contribution to an old topic: as long as the rate of return exceeds the rate of growth, the income and wealth of the rich will grow faster than the typical income from work. 

Again: everything we know about (net) savings rates (on the order of 10%) and capital shares (on the order of 30%) is inconsistent with capitalists saving most of their income. Note that these are *net* savings rates. In economies with high k, even a small depreciation demands large savings just to replenish capital. For example: with k=5, a typical value for rich countries, a depreciation of 3% requires 15% of GDP just to replenished obsolete capital, so that a gross savings rate of 25% implies a net savings rate of 15%.

Going back to the more empirical question. Assume capitalists are the only savers, their share of income is 30% and the general savings rate is 10%. That would imply that the "fundamental force of divergence" would be 0,33.r<g, or r>3g. Is r>3g common? It used to be -- back in the Malthusian era. But even in the very dismal growth scenario for the 21st century, Piketty's estimates show r < 3g, clearly for 2012-2050, slightly for 2050-2100:

The rate of return to capital

Many economists reacted to Piketty's argument with the elasticity question. If k really increases with time, ie., if capital becomes more abundant, r would tend to fall. That seems to be the case if one compares the "capital-labor split" (the share of capital and labor in income) with k, the capital-income ratio. The capital share seems to be much more stable than the capital-output ratio, which suggests that in fact if k falls r increases and viceversa.

It is unlikey, however, that this specific critique of Piketty would damage his argument, because there's an offsetting force to the offsetting force. If in fact r falls as k increases, then the price of capital would go up because a lower interest rate implies a higher relative price of capital. It is very unlikey that capital deepening wouldn't lead to at least some increase in the share of income going to capital.