It's a good thing I wasn't eating when I read this passage, or might have choked when I came to the sentence, "Marx did not use mathematical models, and his prose was not always limpid, so it is difficult to be sure what he had in mind." The irony is that the section of Capital Volume 1 where he introduces the concept is one of Marx's more limpid passages. And while one needs to delve into the wilds of the Grundrisse to find the mathematical model Marx was trying to explicate, it is there in quite clear terms, both as a formula and as inequality. As a formula, it is S / (C + V), the formula for calculating the rate of profit, where S equals surplus-value, C equals constant capital (the value of the capital equipment, facilities, materials and supplies used up in the production process) and V equals variable capital (the value of the capital paid out as wages for the purchase of labor power). The inequality compares the rate of profit to the rate of exploitation S / V, i.e., S / (C + V) < S / V, for the straightforward mathematical reason that in no branch of production, at no stage of economic history, can C be equal to zero.
What follows Piketty's dismissal of Marx's prose and mathematics is an attempt to restate Marx's arguments in Piketty's terms that bears no resemblance to what Marx actually wrote anywhere. The presence of this chapter seems to be an attempt to maintain respectability by stressing that, whatever the implications of his statistical analyses, surely it is not Marxism.
The problem for Piketty is that, for a reader with an understanding of what Marx actually wrote, its mathematical basis, and the economic phenomena those formulas describe, there is a clear homeomorphism between several of Piketty's key concepts and those of Marx. His α (share of the national income represented by earnings on capital) corresponds to Marx's S [or more precisely, S / (S + V)], albeit on a nationally aggregated scale. His r is indistinguishable from Marx's rate of profit, expressed as S / (C + V). And while his β (ratio of the national capital to the national income) is not identical to what Marx terms "the organic composition of Capital" (C / V), since Piketty's definition of the national income includes earnings from capital (S) within it, then to the extent however that the rate of exploitation (S / V) remains fairly constant, there will be a linear relationship between Marx's "organic composition" and Piketty's β.
If anything, Piketty's formulas show why it is that, contrary to the Straw Marx, the actual Marx recognized that the end of capital was never going to be an automatic result of the declining rate of profit. If α = r × β, then even if there is a low r (rate of return, or rate of profit) capital can still appropriate for itself a significant share of the national income as long as there is a high β, that is, as long as the accumulated value of constant capital (Marx's C) is significantly greater than the value of earned wages (Marx's V).
Piketty thinks his assiduous collection of historical data in his technical appendix has disproven Marx. Rather, it has provided Marxists with the datasets necessary to do what they have not been able to do in the past, either for lack of tenured sinecures, or due to energies directed otherwise: to test the truth or falsity of specific predictions of the 19th century Capital.
There may be a methodological limit to this, however, insofar as Piketty uses the market valuation of capitalist enterprises, rather than their book value, to estimate the aggregate value of national capitals. This comes out in his discussion of Germany, where he mentions that the market value of German firms is consistently lower than their book value, in contrast to Britain, France and the U.S., where on average it is the reverse. (Elsewhere in the book, it turns out that this ratio between market value and book value has a name in mainstream economics, "Tobin's Q". I did not know that.) Neither market value nor book value correspond precisely to Marx's C, but of the two book value comes closer. The problem with market value is the potential for "irrational exuberance" in asset valuation, or in Marxist terms, fictitious capital, as can be seen in those countries where Tobin's Q routinely exceeds 1. The problem with book value is that it is basically an accounting term defined at the level of the individual firm: It takes account of one form of depreciation, material depreciation, but not moral depreciation, i.e., when a capital asset is revealed to no longer be as valuable as it once was thought to be, either because it has been superseded by a more productive set of technologies for making its commodity (see the example in my last post of Bob's Widget Company) or because that commodity is shown to no longer be socially necessary, or no longer socially necessary in the quantities that can be produced.
There's more that can be said: About the weakness of his grasp of the significance of foreign asset positions in the U.S. economy, past and present. About his ideological treatment of slavery in its relation to capital, and his complete erasure of the Haitian revolution. But that's enough, I think, for now.