Tuesday, February 3, 2015

Peter Lewin on Thomas Piketty

I've pointed out several times now that there are a few basic points that seem to commonly trip people up on Piketty. A confusing new critique of Piketty by Peter Lewin illustrates the trouble people often get into with at least a few of these points (HT Bob Murphy). I'd summarize the basic points I have in mind as:

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. 

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.
Lewin's simple framework

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.


  1. Out of curiosity, Daniel Kuehn, have you read Nassim Nicholas Taleb's critique of Thomas Piketty's research?

    1. Well, Suvy Boyina is more familiar with Nassim Nicholas Taleb's research than I am, but this is the paper I'm referring to.


    2. First, he starts off by saying anything measured in a centile way in any domain with winner-take-all effects is useless. In other words, Gini is a useless way of measuring inequality. Then, he shows how aggregating inequality statistics across countries will lead to a measurement error in the amount of inequality and will measure inequality higher than it otherwise would be. He then goes on to show how you can get an increase in greater wealth that will usually show up as higher inequality. In other words, it's stupid to assume that wealth will accrue equally AND when wealth does accrue, it's likely to be with winner-take-all effects.

      The particular paper Blue Aurora cited isn't a direct critique of Piketty BTW.

  2. Ok, I think I've followed some of your arguments here - but if r> g drives inequality or is a significant determinant, then wouldn't it's share of all income also go up? As people earn returns on their capital beyond requirements they re-invest earning yet more - what forces that process to equilibrate at 7x approx.?

    1. I would have to run through the derivation to be sure, but I think you're being thrown by the fact that both quantities are going towards infinity. It's a L'Hopital's rule thing, in other words. Remember Y is growing at a smaller rate (g) from a larger base and K is growing at a higher rate (r) from a smaller base.

      Anyway, I really would have to derive it to make it exactly clear.

      Now remember Piketty is keeping the production function and savings in the background through all of this. The MP of capital is going to adjust to the amount of capital (Piketty regularly refers to "too much capital killing the returns to capital". In what's referred to as a decentralized model they'll pin down the result too. Whether it's centralized (Solow) or decentralized (RCK), the equations beta=s/g all have the same basic form in the end. There are little differences depending on where optimizing behavior comes in but it's the same idea.

    2. There has been good criticism of Piketty on how he treats depreciation in all this. That would lead to a slightly different set of "laws" (again same basic form, though). But it wouldn't change at all the points I'm trying to make here - that given a positive but low g and r >g, we still have an equilibrium value of alpha and beta - alpha doesn't veer off to 1 and beta doesn't veer off to infinity.

    3. Look to Hamilton, Rognile, and some others on that if I recall correctly.

    4. How can capital even be measured? In my definition of capital, I include things like social, political, and entrepreneurial capital, but how can you measure the efficacy of political institutions? Capital isn't some number that has a static growth rate that's equivalent to the rate of interest. Capital is an economic input that can go across a wide range of areas.

      For example, having a society of well-educated people with technical skills is a form of capital (human capital), but is it's return r? Of course not.

  3. OK, fair enough - I'm gonna take your word for it - it does make sense. Still, not to derail or anything - can't we take more seriously the idea to get more capital in to the hands of the poor so that they can start to get a piece of that sweet r>g thing working for them. What's the classical economics view of systems like Singapore where you have these forced savings and it's essentially a replacement for taxes that you would normally pay - and this goes towards health, education, welfare, social security, even housing, etc. Is it seen as something that can only work in a small state or something? I mean that's real savings, that's capital, esp. when you're older or for your kids or whatever. I feel that Piketty should have a toolbox of pet plans to deal with poverty rather than just tax the rich more - shouldn't we ask what the govt. is doing now with the taxes it already takes?

  4. I know this will be deleted. I'm leaving it here for lack of an alternative. You might think I'm an asshole, but your buddy advocated creating a separate class of citizens because the consensus (which, mind you, is usually uniformed) disagrees with them so that they can then either be persecuted and intimidated into following that consensus or drive into ghettos or something if they don't comply. I think we've seen this kind of thing several times in the past, and we usually end up feeling guilty about it as a species... until we forget we did and do it again, because "this time it's different." This type of thought is a virus far more dangerous than measles, which parents used to intentionally expose their children to, b/c its not so scary when you get it as a child. I humbly ask that you introspect and see if this sort of thought process is also yours and if you, as an educator, should be friends with such people. If this is your thought process, then you are dangerous, just like your friend. He might be "nice," but that doesn't mean that those sorts of ideas are straight up evil.

  5. Piketty thinks r>g leads to greater inequality in the distribution of wealth. He has a Pareto distribution with random multiplicative shocks to wealth; the Pareto coefficient is r-g, as it turns out. I wish he had been clearer about this in the book - you have to go the on-line appendices.

    1. So my understanding is that it contributes to inequality but since people don't mechanically inherit things at a constant rate it's not determinant in the way that the wealth share is determined by these parameters. So when he talked about shocks in the book, that's a way of modeling the vagaries of inheritance in the real world. So yes, when shocks hit randomly the functional relationship with r-g will remain. Of course in the real world things are a little messier than that.

      Is that your understanding as well?

      Do you know off hand where in the online appendix he works this out (don't worry if it's too much work to dig up). If I recall this also comes up in his lecture notes, but that's harder to tie directly to the book.

  6. Daniel: I am working through the lecture notes, too. He cites those and a co-authored paper - which I haven't had a chance to read yet. So I think we're thinking of the same part of the appendices.

  7. Starting at page 60-61 in technical appendix


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