Many seem to use the rate of return on capital to calculate the next year's wealth. I made the same mistake until I came to page 351 in Piketty. From Piketty and Zucman's paper, it is calculated as follows.

Wealth next year W_(t+1) is calculated from wealth this year W_t by the formula W_(t+1)=W_t (1+s/β) assuming that there is no capital gain or loss, where s is the savings rate and β the capital to income ratio. So wealth does not grow exponentially as some have been saying. Suppose β=7 (sometimes Piketty writes this as 700%), current income 200 units so that , W_t is 1400, s is 10%, and growth rate g=2%. We have the current income which is 200 which goes to 204, the next year. Savings 20 is added so that W_(t+1) is 1420=1400(1+10/100times1/7)=1400(1+1/70)=1420 as it should be. So the next year β goes down. On the other hand, if β=3 with others s,g the same, then income still goes from 200 to 204, W_t goes from 600 to 620, so now β is slightly bigger than 3. In both cases, income grows at the rate of 1/50, but in the first case wealth grows at the rate 1/70 where as in the second case, it grows at the rate of 1/30. If s, g remain the same, in the long run β converges to 5, assuming various conditions are satisfied through out. In any case, it is only a rule of thumb to see the direction of β under various restrictions. see also Dan Kervick's post (he says that he will have an update along the lines of his comment here. See also Seth Ackerman's comments on aggregate production functions and elasticity of substitution)

http://ruggedegalitarianism.wordpress.com/2014/05/28/lets-end-the-confusion-over-pikettys-second-fundamental-law/

P.S. I made this comment (upto the links) in Naked Capitalism. Here is a response from one Paul Boisvert which explains better.

Wealth next year W_(t+1) is calculated from wealth this year W_t by the formula W_(t+1)=W_t (1+s/β) assuming that there is no capital gain or loss, where s is the savings rate and β the capital to income ratio. So wealth does not grow exponentially as some have been saying. Suppose β=7 (sometimes Piketty writes this as 700%), current income 200 units so that , W_t is 1400, s is 10%, and growth rate g=2%. We have the current income which is 200 which goes to 204, the next year. Savings 20 is added so that W_(t+1) is 1420=1400(1+10/100times1/7)=1400(1+1/70)=1420 as it should be. So the next year β goes down. On the other hand, if β=3 with others s,g the same, then income still goes from 200 to 204, W_t goes from 600 to 620, so now β is slightly bigger than 3. In both cases, income grows at the rate of 1/50, but in the first case wealth grows at the rate 1/70 where as in the second case, it grows at the rate of 1/30. If s, g remain the same, in the long run β converges to 5, assuming various conditions are satisfied through out. In any case, it is only a rule of thumb to see the direction of β under various restrictions. see also Dan Kervick's post (he says that he will have an update along the lines of his comment here. See also Seth Ackerman's comments on aggregate production functions and elasticity of substitution)

http://ruggedegalitarianism.wordpress.com/2014/05/28/lets-end-the-confusion-over-pikettys-second-fundamental-law/

P.S. I made this comment (upto the links) in Naked Capitalism. Here is a response from one Paul Boisvert which explains better.

Hi, Gaddeswarup,

No need for corrections, you have it precisely correct!

The mistake you refer to (that of confusing the rate of return to capital with the rate of growth of capital), from which you no happily longer suffer, is (obviously) easy to make, since some very bright people have made it. To avoid it, one need only remember that owners of capital spend much of their return from that capital on consumption–they don’t save all of it as new capital. Moreover, those with no capital in January may save some of their labor income during the year, and thus own some (new) capital when December rolls around.

Thus, capital increases each year due to various people (owners of capital and owners of labor power alike) saving some (not all) of their income–but “r”, which refers only to the rate of return on previous capital, doesn’t indicate by itself how capital is growing. To know the latter fact, one also needs to know how much new income was created by labor and how much of all income was saved (became new capital.) These two additional factors, represented by g and s, together with r, determine the trend (towards some stable ratio of income from capital to total income) that Piketty analyzes mathematically, and which you have correctly captured in your comment.

Again, this doesn’t mean Piketty’s overall take is valid, or even relevant–it just means that there is no reason to believe that he made an elementary math mistake in his model, or that the fact that r could be greater than g for a long time leads to absurdities or infinities. Neither belief is warranted in the slightest. If one wants to critique Piketty, one has to do it on grounds other than that r > g leads trivially to mathematical fallacies.

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