# Is Piketty’s “Second Law of Capitalism” fundamental?

Per Krusell and Tony Smith have a new paper on Piketty (pdf), which I take to be reflecting a crystallization of opinion on the theory side. Here is one excerpt:

There are no errors in the formula Piketty uses, and it is actually consistent with the very earliest formulations of the neoclassical growth model, but it is not consistent with the textbook model as it is generally understood by macroeconomists. An important purpose of this note is precisely to relate Piketty’s theory to the textbook theory. Those of you with standard modern training have probably already noticed the difference between Piketty’s equation and the textbook version that we are used to. In the textbook model, the capital-to-income ratio is not s=g but rather s/(g+ δ), where δ is the rate at which capital depreciates. With the textbook formula, growth approaching zero would increase the capital-output ratio but only very marginally; when growth falls all the way to zero, the denominator would not go to zero but instead would go from, say 0.12—with g around 0.02 and δ = 0.1 as reasonable estimates—to 0.1. As it turns out, however, the two formulas are not inconsistent because Piketty defines his variables, such as income, y, not as the gross income (i.e., GDP) that appears in the textbook model but rather as income, i.e., income net of depreciation. Similarly, the saving rate that appears in the second law is not the gross saving rate as in the textbook model but instead what Piketty calls the “net saving rate”, i.e., the ratio of net saving to net income.

Contrary to what Piketty suggests in his book and papers, this distinction between net and gross variables is quite crucial for his interpretation of the second law when the growth rate falls towards zero. This turns out to be a subtle point, because on an economy’s balanced growth path, for any positive growth rate g, one can map any net saving rate into a gross saving rate, and vice versa, without changing the behavior of capital accumulation. The range of net saving rates constructed from gross saving rates, however, shrinks to zero as g goes to zero: at g = 0, the net saving rate has to be zero no matter what the gross rate is, as long as it is less than 100%. Conversely, if a positive net saving rate is maintained as g goes to zero, the gross rate has to be 100%. Thus, at g = 0, either the net rate is 0 or the gross rate is 100%. As a theory of saving, we maintain that the former is fully plausible whereas the latter is all but plausible.

With the upshot coming just a wee bit later:

…Moreover, whether one uses the textbook assumption of a historically plausible 30% saving rate or an optimizing rate, when growth falls drastically—say, from 2% to 1% or even all the way to zero—then the capital-to-income ratio, the centerpiece of Piketty’s analysis of capitalism, does not explode but rather increases only modestly. In conclusion, at least from the perspective of the theory that we are more used to and find more a priori plausible, the second law of capitalism turns out to be neither alarming nor worrisome, and

Piketty’s argument that the capital-to-income ratio is poised to skyrocket does not seem well-founded. [emphasis added by TC]

Krusell and Smith really know their stuff on this topic and their arguments to me seem completely correct.

By the way, here is a Chris Giles follow-up post from The FT, very useful.