1. r > g did not surprise any economist. It's not a radical result at all, and it's very familiar and well understood.
2. r > g does not imply that the capital stock as a share of income will go off to infinity. In fact as long as r and g (and s) remain fairly stable it implies an equilibrium capital stock level (Piketty guesses it will level off at around 700% of income).
3. The capital share of income is not the same thing as "inequality". The capital share informs us about the source of income. Inequality is about it's distribution across the population. On inequality Piketty prefers to use the 10% share and the 1% share of income.
Lewin's simple framework
4. Inequality for Piketty is not directly governed by r > g, it's primarily determined by institutional factors. Piketty does think r > g makes inheritance a more important factor, but the ultimate impact of inheritance on inequality is mediated by institutions.
I'll focus mainly on Lewin's "simple framework" in the first half of the post, although I have a few thoughts on the rest of the post as well. The framework is a straightforward national income equation, Y = rK + wL. Lewin then decomposes the growth rates of each component of national income to talk about Piketty's thinking on r > g and capital's share of income. This is all fine, until he brings Piketty into the picture. Lewin writes that "Piketty’s project is to show that the laws of capitalism imply that sK/sL rises without limit, thus destabilizing the society." In Lewin's post, sK/sL is capital's share of national income divided by labor's share of national income. This is where the problems start, of course. With a little bit of algebra we can see that Lewin is getting confused about my point #2 above. It is not Piketty's project to show that capital's share of income increases without limit. Piketty has two "fundamental laws" (a bit of an aggrandizement but the equations themselves are fine):
α = r*β, and
β = s/g
For Piketty, α is the same as Lewin's sK - it is capital's share of income. s is the savings rate and β is the capital stock divided by income. Therefore, 1- α is going to be equivalent to Lewin's sL. So Lewin is interested in α/(1-α). Substituting Piketty's second law into his first it's clear that α = rs/g, so α/(1-α) = rs/(g-rs). Piketty never puts it in these terms, of course. He's just concerned with α. But this is still the equilibrium value of the quantity Lewin thinks Piketty is concerned with. Does this "rise without limit"? No, of course not. And Piketty never says it does. In fact there's quite a bit of discussion in the book about the stability of α (and therefore the stability of α/(1-α)). Indeed the stability of α is at the very top of the list of Kaldor's facts, and the subject of quite a bit of recent discussion as labor's share has slipped a little.
Piketty spends a lot of time discussing all these issues and the steady growth of the capital share in the late 20th century (see Chapters 5 and 6), but he never claims that capital is growing without limit because r > g, which doesn't imply anything in particular about the capital share. He says there's been some increase because r has a tendency to grow somewhat with β (see pgs. 220-221), so as β climbs to its equilibrium level you're going to see some increase in r and some increase in the capital share, but only to rs/g, not an "increase without limit" as Lewin has it.
So Lewin seems to run up against my points #1 and #2 in some fashion at least. He goes on to confuse #3 as well. He writes "Piketty reasons that if the earnings of K grow more rapidly than earnings in general, this must imply that K’s share is growing, thus increasing inequality." This is where Lewin decomposes the growth rates. The problem is, the capital share is not the same thing as inequality. The capital share has to do with payments to factors of production, while inequality is a statement about the distribution of those payments across the population.
One of the strangest things about Lewin equating the two is that two of the biggest narratives that come out of Piketty's discussion of inequality directly contradict the conflation of the capital share with inequality. These are: (1.) the rise in the capital holdings of the middle class due to homeownership, and (2.) the critical role that labor income plays in the share of income held by the top 10% and the top 1%. Capital income doesn't dominate labor income until the very top of the income distribution. Piketty calls these earners of labor income the "super-managers". The capital share discussion is in Part II of the book, which deals with capital. The inequality discussion is in Part III of the book, which deals with inequality. They are not the same thing and the fundamental laws of capitalism certainly don't imply anything about inequality, at least not without a great deal of ambiguity.
The rest of the post
The rest of Lewin's post is a mixed bag. I agree with him on some of the points, and I think he agrees with Piketty more than he realizes on some of the others. After his "simple framework" Lewin explains that factor income is not the same as income inequality. Indeed, and Piketty thinks so as well which I point out above. He then criticizes Piketty for excluding human capital. I've had this concern in the past as well (as has David Weil at the AEA meeting). Lewin calls the exclusion "cavalier" which I think is extremely unfair. It makes perfect sense why Piketty would exclude human capital from this discussion. It can't be sold on capital markets, and it can't be inherited so it's not directly relevant to his discussion of physical and financial capital. I get that, but I do think it's an important part of the income distribution story which is why I'd love to hear more about it (plus I'm a labor economist so of course I'm interested).
I find the next few sections of Lewin odd. My impression is that Piketty agrees with Lewin on the rest of the post. From the very beginning of the discussion of the fundamental laws, Piketty talks about how capital is heterogeneous and how different types of capital have different rates of return (pg. 52). Lewin is also wrong when he says "It [K] is meant to be an index of the physical magnitude of the capital of the economy". No, it's not! It's the value of the capital, not a physical quantity!
Finally Lewin criticizes Piketty for allegedly equating the rate of return with the interest rate. He doesn't do this either, of course. On page 52 he writes "the rate of return on capital measures the yield on capital over the course of a year regardless of its legal form (profits, rent, dividends, interest, royalties, capital gains, etc.), expressed as a percentage of the value of capital invested. It is therefore a broader notion than the "rate of profit," and much broader than the "rate of interest," while incorporating both."
So tread carefully when reading Lewin, I think. But it is a nice illustration of some common confusions about Piketty.