Criticisms of Piketty

There has been much discussion of Thomas Piketty’s new book, Capital in the 21st Century. Some of the criticisms I agree with, and some I do not.

The latest concern was raised by Chris Giles, who found “a series of problems and errors” in Piketty’s publicly posted data and spreadsheets. Here is Paul Krugman’s reaction:

Giles finds a few clear errors, although they don’t seem to matter much….

But is it possible that Piketty’s whole thesis of rising wealth inequality is wrong? … That can’t be right– and the fact that Giles reaches that conclusion is a strong indicator that he himself is doing something wrong.

I suspect Krugman is correct in this evaluation. An error in one spreadsheet does not refute the conclusion that inequality in income and wealth have been increasing over the last generation. We have abundant evidence for that conclusion from a good many sources besides Piketty.

Of course, Krugman would also have been correct if he replaced “Giles” in the first sentence above with “Herndon, Ash, and Pollin”. The contrast between Krugman’s defense of Piketty and the zeal with which he jumped on top of the Reinhart-Rogoff dog pile is amusing. But kudos to Justin Wolfers ([1], [2]) for even-handed summaries of both disputes.

A more telling criticism of Piketty’s thesis relates to his treatment of depreciation, an issue raised by Larry Summers and Matt Rognlie. The need to keep up with what would otherwise be a growing cost of depreciation over time is a key factor that pins down the stable growth path in most economic models and is one reason I am puzzled by Piketty’s claims of an inherent instability in capitalist economies.

On page 168 of Piketty’s book the reader is introduced to “the second fundamental law of capitalism” according to which β = s/g, where β denotes the capital/income ratio, s the economy’s saving rate, and g the overall economic growth rate. Note that a curious corollary of this “law” is the claim that if the economy is not growing (g = 0), the capital/income ratio β has to be infinite.

To arrive at such a conclusion, Piketty is defining (page 174) the saving rate s to be the ratio of net investment to net income, where “net” here refers to net of depreciation. To see why under these definitions his “second fundamental law” would require an infinite capital/income ratio in the absence of growth, consider the following modification of the numerical example presented in Robert Solow’s sympathetic review of Piketty’s book. Consider an economy with GDP = $100, a depreciation rate of 10%, a net saving rate of 10%, and a capital stock of $100. The annual depreciation would be $10, which is subtracted from the $100 GDP to arrive at $90 in net income. The economy is assumed to save 10% (or $9) of this net income, meaning gross investment is $19, or 19% of GDP. Because gross investment exceeds depreciation, the capital stock would have to grow even if there is no other source of growth in productivity or population. Thus a capital stock of $100 is too low to be consistent with a steady state for this economy.

Let’s take those same initial conditions ($100 GDP, 10% saving and depreciation rates) and now suppose that the capital stock is $500. Then annual depreciation would be $50, leaving $50 in net income, of which the economy is again supposed to save 10%, or $5, making gross investment a total of $55, or 55% of GDP. But because we have added $5 net of depreciation to the capital stock, the capital stock would still have to grow if these were the initial conditions. So $500 is still too low a number for the capital stock for this economy.

For an economy with GDP of $100 and 10% saving and depreciation rates, the steady-state capital stock turns out to be $1,000. At this level, depreciation is $100, leaving zero net income with which anybody could do any further net investing. 100% of GDP is going to gross investment, but this doesn’t increase the capital stock any further above $1,000 because it is just sufficient to cover depreciation. Unfortunately, nothing is left over from the $100 GDP for capitalists or anybody else to consume, and everybody would starve. The ratio of the capital stock to net income is infinity because we have driven net income all the way down to zero.

This is the mechanism whereby Piketty’s law would predict a value of infinity for the steady-state ratio of the capital stock to net income. When the growth rate g = 0, Piketty’s capitalists save themselves into exhaustion until, with zero net income, they still dutifully try to salt away 10% of what is now nothing.

Thinking these numbers through leads you to understand that the assumption of a constant net saving rate, regardless of the levels of income and the capital stock, could not possibly be a sensible characterization of the decisions that would be made by real people in the real world.

A more plausible and standard assumption (though one that can still easily be improved on further if one thinks of the saving decision as the outcome of some kind of rational calculation) would be that the gross saving rate s* is constant. Under this alternative specification, the relevant characterization of the steady state would be β = s*/(g + d) where d is the depreciation rate. For example, when g = 0, with GDP of $100 and s* and d both 10%, the economy would be in a steady state with a capital stock of $100. Depreciation is $10 annually, and gross investment is just sufficient to cover depreciation each year with no growth in the economy and no reason for any variables to change over time. But the good news is that $90 would be left over for people to consume on a sustained basis every year. That’s a rather boring but much more plausible notion of what an economy with no growth could look like.

But a sensible characterization of the implications of depreciation for the dynamics of wealth accumulation would not have caught the attention of as many commentators as Piketty’s book seems to have done.

47 thoughts on “Criticisms of Piketty

  1. PeakTrader

    The rate of return on capital (r) doesn’t necessarily rise at the expense of the rate of economic growth (g).

    For example, in the U.S. both r and g have been higher than in Western Europe.

    Also, at the height of the U.S. Information Revolution, from 1982-07, both r and g rose substantially.

    Moreover, when the U.S. is in a recession or depression, r is high and g is low. The high r didn’t cause the low g. The high r raises g, e.g. through monetary policy. When the economy recovers, r falls, or slows, and g rises, or accelerates.

    Furthermore, the U.S. offshored old industries, imported those goods at lower prices and higher profits, and shifted resources into high-end manufacturing and emerging industries. Increasingly larger trade deficits subtract from g and globalization adds to r, although U.S. living standards improved faster. The U.S. has been a “Black Hole” in the global economy attracting imports and capital, and even attracting the owners of that capital themselves.

  2. 2slugbaits

    First, I haven’t had a chance to read the book yet. I only got it a couple of days ago. It was on backorder for quite awhile…long enough that my initial order was canceled and I had to reorder when it was in stock. But I have read at least a dozen reviews. That said, I’m a bit stumped why so many economists (both those who agree and disagree with Piketty) work from a framework in which savings is constant and exogenous. Just about every neo-classical growth model has savings as exogenous in the first chapter, but then moves to a Ramsey endogenous savings model with the elasticity of intertemporal substitution being the exogenous factor. I don’t think it’s too much of a stretch to go from there to an endogenous intertemporal substitution parameter that depends on wealth. Put another way, no one seriously believes that we understand the dynamics of growth well enough that we can really lean too hard on any neo-classical inspired model. I wouldn’t want to build a critique of advanced market economies on the basis of a neo-classical growth model, but then I again, I wouldn’t want to critique a critique using that model either. The truth is that no one actually understands the determinants of long term growth. That’s just the hard truth. The gist of Piketty’s argument seems to be “to hell with a model, let’s just look at the long run empirical data.” And that strikes me as quite reasonable…and it also puts more pressure on Piketty to rework his numbers without the summation errors in his spreadsheet. I’m also a little bit puzzled by Solow’s comments on depreciation since a very long time ago he was the one who wrote a rather famous paper that said NIPA basically got things all wrong because it did not correctly account for the depreciation of exhaustible natural resources. And if you look at the sources of wealth for many of the elite, it comes from the extraction of natural resources. In other words, we are not correctly measuring depreciation.

    Finally, I don’t believe Krugman was all that harsh with R&R’s spreadsheet error. He found it a little amusing, but didn’t see it as a deep criticism. The heart of Krugman’s critique of the R&R paper was their weighting technique.

    1. Anonymous

      Also at the heart of Krugman’s criticism of R&R was their assumption of the 90% cliff and, crucially, directionality of causation, from high debt levels to low GDP growth. Several papers since Herndon et al. have looked at this and find that even R&R’s data show a tilt to the opposite directionality, from low GDP growth to high debt levels. It’s really a basic error to assume correlation is causation. As a geneticist I too have noted that economics relies far too heavily on models and not enough on empirical data. Some models that show inflation exploding with increased money supply have obviously had a real world test these last 6 years and failed badly. Still the theorists appear wedded to an errant thesis unable to see that money locked in Excess Reserves does not affect the real economy. The same is true of the theoretical assumption from modeling that indicate increasing the minimum wage must perforce lead to higher unemployment. Arindrajit Dube’s empirical data show this theory doesn’t hold in the real world. For me as an economics dilettante it seems Piketty’s contribution is the wealth of data he and his team have assembled. Would that the economics profession could learn this lesson and lean more in the direction of empiricism.

      1. Lilguy

        A big AMEN to your last sentence! The notion that the behavior of billions of people can be reduced to the Greek alphabet is a little ridiculous.

      2. Anonymous

        You make a good point generally regarding the importance of empiricism; however, I want to point out that your critique of monetary models is wrong. Not only does the most widely taught monetary model–the quantitative theory of money– hold following the Fed’s unconventional policies over the past 6 years (it accounts for money “velocity” and looks at money supply as opposed to monetary base–I don’t know of any model that would look at monetary base in the context of inflation), but any model that shows that increasing the monetary base automatically leads to inflation is clearly wrong and I would be very surprised if such a model has been used in any serious analysis.

  3. jonathan

    I had clipped Summers’ review and highlighted this bit: “I think he misreads the literature by conflating gross and net returns to capital. It is plausible that as the capital stock grows, the increment of output produced declines slowly, but there can be no question that depreciation increases proportionally. And it is the return net of depreciation that is relevant for capital accumulation. I know of no study suggesting that measuring output in net terms, the elasticity of substitution is greater than 1, and I know of quite a few suggesting the contrary.” So it was very interesting to read your expansion of the point. Thanks. It helped make his point a lot clearer for me.

    My reaction is that people love to make grand statements and that these statements tend to attract supporters (and naysayers). The problem with R&R was not any spreadsheet error or the clearly correct idea that too much debt is bad for growth but that there is a cliff at a specific level. People love that kind of grand statement: hit this point and you fall off the earth and they ran with it. That cliff became a substantive political point. It wasn’t the first nor will it be the last “hard” fact that isn’t. (As for who defends whom, it’s simply too much to expect absolute or objective consistency. We’re all people. We all have our perspectives. We are far more likely to be subjectively consistent. And we get angry with subjective, not objective consistency.) When I think about how we see “fact” and how we use those, I think of how we used to bleed the sick. That wasn’t as stupid as dunking “witches” to see if they float but rather a completely mistaken “fact” about disease that science used to think was right. An extreme example, yes, but think about it.

    And the problem with Piketty – to me – is that he seems in my historical mind to be following in the footsteps of summarizers like Marx who look at the past and extrapolate … and the problems with that are that any look at the past is subjective and any application of a grand idea is going to come out wrong because the future is too hard to predict. In other words, I think there’s something really good in some of the big ideas in Piketty’s work but that it follows in the model of people trying to put the future in a box. The future refuses to be boxed. With that caveat, I enjoy the book as an exploration of the concepts and don’t expect it to be entirely accurate (how could it be when it treats the past?) or entirely reasonable (how could it be when it needs to make so many choices?). This stuff isn’t physics.

  4. IS

    As I recall, Krugman’s piling-on of R-R tended to stress that the spreadsheet error was embarrassing but ultimately not the point, noting instead that the 90% cliff was driven by the needlessly broad aggregation function used and how the high-growth low-debt data points were primarily the immediate postwar years of the former Axis powers, characterized by rebuilding from a low base and low debt as a result of repudiation, while a large part of the high-debt low-growth data points had the causality backwards since they came from Japan, where low growth led to high debt.

  5. Lord

    RR didn’t have much other data backing them. Even their own conclusions were largely the result of one data point, New Zealand.

    Piketty doesn’t consider the steady state of a far distant future, only the direction the system moves in over long periods and the results that accumulate. I agree there is another tale to be told about where we will end up, but that is one to be written by the institutions and policies we adopt.

  6. PeakTrader

    Paul Krugman wrote a related article:

    Profits Without Production
    June 20, 2013

    “Economies do change over time, and sometimes in fundamental ways.

    “…the growing importance of monopoly rents: profits that don’t represent returns on investment, but instead reflect the value of market dominance.

    …consider the differences between the iconic companies of two different eras: General Motors in the 1950s and 1960s, and Apple today.

    G.M. in its heyday had a lot of market power. Nonetheless, the company’s value came largely from its productive capacity: it owned hundreds of factories and employed around 1 percent of the total nonfarm work force.

    Apple, by contrast…employs less than 0.05 percent of our workers. To some extent, that’s because it has outsourced almost all its production overseas. But the truth is that the Chinese aren’t making that much money from Apple sales either. To a large extent, the price you pay for an iWhatever is disconnected from the cost of producing the gadget. Apple simply charges what the traffic will bear, and given the strength of its market position, the traffic will bear a lot.

    …the economy is affected…when profits increasingly reflect market power rather than production.

    Since around 2000, the big story has been one of a sharp shift in the distribution of income away from wages in general, and toward profits. But here’s the puzzle: Since profits are high while borrowing costs are low, why aren’t we seeing a boom in business investment?

    Well, there’s no puzzle here if rising profits reflect rents, not returns on investment. A monopolist can, after all, be highly profitable yet see no good reason to expand its productive capacity.

    And Apple again provides a case in point: It is hugely profitable, yet it’s sitting on a giant pile of cash, which it evidently sees no need to reinvest in its business.

    Or to put it differently, rising monopoly rents can and arguably have had the effect of simultaneously depressing both wages and the perceived return on investment.

    If household income and hence household spending is held down because labor gets an ever-smaller share of national income, while corporations, despite soaring profits, have little incentive to invest, you have a recipe for persistently depressed demand. I don’t think this is the only reason our recovery has been so weak — but it’s probably a contributory factor.”

  7. Tom

    In defense of Giles, I believe the mistakes he found related to whether inequality has risen in Europe in recent decades. Giles wasn’t challenging the obvious increase in disparity in the US.

    At least a hundred debunkings like the one you’ve done on savings ratios could be written about Piketty’s book. The most basic is that comparing returns on capital to growth tells you nothing about whether disparity is rising or falling. (Comparing them isn’t really possible at all: r is a pace of income relative to a stock of savings, g is an acceleration of a pace of income).

    But by far the most important priblem with Piketty is that he attempts to study disparity in advanced economies in isolation. The recent decades that he claims have seen a return of widening disparity have in fact been an era of rapidly reducing disparity, once you broaden your horizon and look at the whole world. The income gaps between the West and the East and the North and the South have narrowed dramatically. Disparity has widened within the US mainly because our lesser skilled were exposed to lower income competition while our capitalists expanded from the US to global markets.

    1. David

      This is actually a controversial topic and has been a long standing one amongst people in development studies. Your view has been the World Bank view (until recently, I think even they now do not push it). A lot of the reduced income gaps between rich and poor countries is explained by the rise of emerging markets in Asia, China in particular. Take the latter out in particular and the picture is a very different one. The gaps between the very poorest countries – and there are a lot of them, basically the whole of Africa, and richest is wider than ever.

      So really it depends on how you define inequality. It is true that China’s entry into the international economy, and the careful unorthodox way it managed it – ie it did not rush into privatisations or liberalisations of international capital flows – unlike the former Soviet bloc, has brought a lot of people out of poverty.

      But the disparity between the haves and have nots is bigger than ever.

      1. Tom

        Well, if you’re trying to make a claim that China’s development wasn’t capitalist, I disagree but fine. It was undoubtedly built on US and European foreign investment in factories for export to the US and Europe.

        The gap between the world’s poorest and richest must always grow so long as there is somebody somewhere still starving and somebody somewhere getting richer. That gap has been growing since the paleolithic era.

        If we are to say anything sensible about global disparity we have to look at the distribution of income and wealth globally and we have to of course include China. But the picture is not all that different if you exclude China. Incomes have been rising in South and Southeast Asia, Latin America and some of subsaharan Africa.

    2. 2slugbaits

      Tom Disparity has widened within the US mainly because our lesser skilled were exposed to lower income competition while our capitalists expanded from the US to global markets.

      There might be some truth to this, but it doesn’t really explain why virtually all of the gains in GDP have gone not just to the top 10%, or the top 1%, but the top 0.1% and 0.01%. My understanding is that Piketty is not just thinking of returns to capital in the narrow sense of some neo-classical model. My understanding is that he has in mind returns to the ownership of the income flows from wealth broadly understood. So this would include royalties from copyrights and patents, rents from land and mineral extractions, etc. So this is a much broader definition than simple profits from capital formation. Indeed, talking about returns to capital in the context of a neo-classical growth model wouldn’t make a lot of sense in the context of the Ancien Regime. Piketty isn’t just saying that advanced capitalist economies tend towards wealth concentrations; he is arguing that all economies tend towards wealth concentration. The brief period between the end of the Great War and 1980 was an historical anomaly. I think his equations of motion have to be understood as driving toward wealth concentration. Also, I do not understand him to be saying that this equation of motion is necessarily leading to a stable equilibrium.

      My biggest concern with some of the critiques of Piketty’s book is that they are ultimately based on neo-classical growth models that are elegant on the blackboard, but no one seriously believes they are at all up to the job of explaining economic growth. The one thing that all of the neo-classical models agree upon is that capital deepening does not explain changes in growth rates. Growth levels, yes; growth rates, no. What really determines growth rates is some fuzzy thing called “technology.” If the wealthy can capture, own and restrict the expansion of technical knowledge, then this would slow growth rates and lead to wealth concentration.

      1. Tom

        Disparity has undoubtedly decreased, not increased since 1980, once you widen your horizon and look at the whole world. So then do we have to consider that another anomaly that must soon reverse? I’m not trying to make an argument that capitalism inherently reduces disparity. I’m just pointing out that Piketty’s claim that capitalism inherently increases disparity is based on a blinkered look at a tiny subset of the world. That subset was not evolving independently of the rest of the world, and much of the increased disparity he finds within that subset is obviously related to decreased overall disparity as emerging world incomes gained at the expense of some lesser skilled advanced economy workers.

        Your point about the 1% and 0.1% and 0.01% seems to be your own not drawn from Piketty. I have a similar impression about the 0.01% but surely you must agree that globalization of business is a big part of that. My impression is the rise of hedge funds and the extremely rapid growth of tech sector companies seem to have something to do with it also. I’m less sure that there’s been much concentration in the 1% but not 0.01% range.

        I don’t get your comments on the neo-classical growth model which seem to have nothing to do with anything I wrote.

        1. Tom

          PS Be very careful with net worth disparity, which doesn’t differentiate among the very large portion of the population that hasn’t (yet?) inherited and doesn’t (yet?) save. A lot of quite well off people have no net worth. Net worth disparity is also more subject to misleading fluctuations. Housing prices carry enormous weight in middle class net worth, greatly amplified by leverage. Since domestically held public debt is a household asset, war debt inflates net worth and post-war consolidation shrinks it.

  8. DeDude

    “The contrast between Krugman’s defense of Piketty and the zeal with which he jumped on top of the Reinhart-Rogoff dog pile is amusing”

    Actually, I can see why someone would go much harder at R&R than at Piketty for essentially the same type of mistake. R&R did not publish their raw data and methods for anybody to look at and confirm – to his credit Piketty immediately allowed anybody to repeat his work and find the error. Furthermore, Piketty’s claims of increased inequality was supported by many other studies – R&R made a brand new claim of a cliff at 90% that nobody else had ever claimed. If you make a brand new claim you have a much larger responsibility to ensure that your data supports it.

  9. J J Flattery

    Last time I checked, the notion of “the second fundamental law of capitalism” where β = s/g, where β denotes the capital/income ratio, s the economy’s saving rate, and g the overall economic growth rate does not lead to the curious corollary: that if the economy is not growing (g = 0),capital/income ratio β has to be infinite – but rather it is NOT DEFINED.

    1. James_Hamilton Post author

      J J Flattery: Then check again. For s a fixed positive number, the limit of s/g goes to infinity as g goes to zero.

      1. Arun

        When the economy is in a steady state, the growth rate g is zero, net investment (net of depreciation) is zero and so the net savings rate s (with net investment in the numerator) is zero; s/g is undefined.

        However, the capital/income ratio β is some definite number, since neither capital nor net income is zero.

        But think of a purely agricultural society, where land is the main capital asset, and where all cultivable land is being exploited (so there is no new capital possible). To keep land productive some small fixed amount is necessary per year, regardless of the price of the land, i.e., as the value of capital (land) goes up, the depreciation rate (the expenditure needed to keep land productive divided by the value of the land) goes down. In such a zero growth agricultural society, Piketty’s law is saying, I think, that the value of land keeps rising, even as it produces a fixed net income. Is this correct?

        1. James_Hamilton Post author

          Arun: Since you won’t read my response when Dan Kervick and others had the same confusion you had, let me copy and past my answer to Dan here for you.

          If the use of the term “infinity” upsets you, then just substitute “very small” for “zero” and “ridiculously large” for “infinity” and everything still goes through exactly as I said.

          For example, if the growth rate is g = 0.01% per year, then for the numerical example I presented, the steady-state capital would be $990.0990099… (a little less than the $1,000 for the zero-growth case), depreciation would be $99.00990099… leaving a tiny net income of 99.00990099…. cents (instead of literally zero net income in the zero-growth example) associated with a value of β = s/g = 1000. So about 99 cents in net income (= $100.00 – $99.01) is left over from the original $100 annual GDP after replacing the machines that wore out during the year. If Piketty’s capitalists dutifully use about a dime of their pitiful net income to increase the capital stock, that will add about another cent to next year’s depreciation bill. But fortunately next year’s GDP will be $100.01 as a result of the assumed growth, so they will still have about 99 cents left over after covering what is now $99.02 in depreciation costs to try to add about another dime to the capital stock for the following year. And so we continue, year after year along an absurd Piketty steady-state path associated with a 0.01% growth rate.

          People are saving more than $99 out of the annual GDP of $100 , leaving themselves less than a dollar to try to live on, and all that saving accomplishes is to produce an even larger sum that will be wasted on depreciation the following year. If that sounds like a sensible model of economic behavior to you, then bless your soul.

      1. Wes

        Perhaps a silly question on my part but why are you considering the land resource as capital?

  10. nottrampis

    I am with 2 slugs baits.

    Kruggers criticisms was always about the article not the book and not so much the error on the spreadsheet.

    Me thinks James has a Kruggers problem which proved to be embarrassing last time.

    Apples and oranges to be honest.

  11. Rick Stryker

    JDH referred to the amusing contrast between Krugman’s take on the questions raised about R&R and Piketty, given their similarity. However, a number of commenters continue to whitewash Krugman’s behavior in the R&R affair. It might be helpful to spell out in detail the differences between Krugman’s treatment of R&R and Piketty.

    Fortunately, the internet will not let us forget that Krugman wrote a NYT article called The Excel Depression. In that article, Krugman started by comparing R&R’s excel error to the error that caused the Mars Orbiter to crash as well as to the error made in the London Whale incident. Krugman then asks, “So, did an Excel coding error destroy the economies of the Western world? ”

    After that bit of amazing hyperbole, Krugman gets to his main accusations. First, he says, “Finally, Ms. Reinhart and Mr. Rogoff allowed researchers at the University of Massachusetts to look at their original spreadsheet — and the mystery of the irreproducible results was solved,” thereby strongly implying that R&R somehow withheld their data from public scrutiny. That’s completely false of course. R&R had made their data available on a public website.

    Then Krugman goes on to say, “First, they omitted some data; second, they used unusual and highly questionable statistical procedures; and finally, yes, they made an Excel coding error. Correct these oddities and errors, and you get what other researchers have found: some correlation between high debt and slow growth, with no indication of which is causing which, but no sign at all of that 90 percent “threshold.”

    Again, these are false or misleading charges. R&R did not omit data and their statistical technique was not “unusual and highly questionable.” R&R did make an excel coding error and they acknowledged that but that error didn’t really change the results. To get a few more licks in, Krugman went on to refer to the Reinhart and Rogoff “fiasco” and asked “will toppling Reinhart-Rogoff from its pedestal change anything?”

    It’s hard to see this article as anything else other than a hit piece. Later on Krugman moderated his attacks somewhat but this article was how he responded initially.

    Now contrast with Piketty’s case, which has a number of close parallels. The FT article did not claim that Piketty omitted data but it did say that he judgmentally altered or filled in data and questioned that. It also raised questions about his averaging technique. And it found some coding errors. What was Krugman’s response?

    1. On Piketty’s judgmental altering or filling in of data, Krugman does not say that Piketty made up data, as he surely would have described this situation had it been R&R. Instead, he very evenhandedly says that Giles “questions some of the assumptions and imputations Piketty uses to deal with gaps in the data and the way he switches sources.”
    2. On Piketty’s averaging technique, Krugman does not describe it as an ” unusual and highly questionable statistical procedure.” Rather, he passes over the point in silence.
    3. On the excel error, Krugman does not compare it to the error that caused the crash of the Mars Orbiter, or to the error involved with the London whale. He does not refer to an excel depression. He does not ask whether this coding error destroyed the economies of the Western world. Instead, Krugman very evenhandedly says, “Giles finds a few clear errors, although they don’t seem to matter much.”
    4. Finally, Krugman nowhere implies in the article that Piketty withheld his data from public scrutiny.

    What to make of this contrast in the way Krugman handles the two cases? Yes, it is amusing. But it’s also revealing. It reveals a former economist who is now engaged in pure political partisanship rather than economic analysis.

    Interestingly, Krugman hedges his defense of Piketty by saying: “Piketty will have to answer these questions in detail, and we’ll see how well he does it.” He does not say “Piketty will have to answer these questions in detail, and I’m confident that his answers will satisfy the critics.” No, Krugman says, “and we’ll see how well he does it.”

    I for one find it very amusing that Krugman is already hedging on Piketty after he recently exclaimed on his blog: “And let me add something here: if you think you’ve found an obvious hole, empirical or logical, in Piketty, you’re very probably wrong. He’s done his homework!”

    1. baffling

      krugman went after R&R to a large degree because the paper was used as an excuse to encourage policy which was extremely detrimental to the world pulling out of this last major recession-depression if you want to call it that. conservatives used the paper as a “fact” that certain debt levels would lead to calamity. the paper did not explicitly state this, but when the conservative commentators ran with the idea, R&R stood by and did not correct this “fact”. this was dishonest behavior by the pundits and unethical behavior by R&R. since people continued to project this paper as important in forming policy, krugman felt it necessary to clearly identify the fallacy of the paper and any policy that used it as support. not sure why you want to defend R&R against the arguments laid out by krugman, unless you are ok using misleading data to perpetuate an ideological policy.

      “On Piketty’s judgmental altering or filling in of data, Krugman does not say that Piketty made up data, as he surely would have described this situation had it been R&R.”
      stryker, this is a straw man argument. you do not know what krugman would say on the topic, but you choose to describe an action krugman did not take in misleading terms to try and strengthen your argument. this is dishonest commentary.

  12. Johnny

    Yes of course, inequality has risen. Sure, the wealthy will not give away a cent, and their paid political leaders will do everything that inequality will rise even more (in their favor).

    And this is why the right wing groups are marching again in France. And Denmark. And Britain. And Austria.

    Well James, keep on discussing, until your house is burning.

  13. Jordan

    This is okay as far as it goes. The only problem is that it’s simplistic to the point of uselessness. The first giant problem is that as time has passed in the global economy, an increasing share of wealth and capital have become virtual wealth and intellectual capital, which has a depreciation rate of more or less zero. This phenomenon is well known, and is referred to as the financialization of the global economy. So it’s very reasonable to surmise that the net depreciation rate of the total capital stock is falling as the gross stock gets larger.

    Secondly, to the extent that real depreciation – rather than things made up to create tax write-offs – actually occurs, that represents physical obscolescence and deterioration. Thus the capital stock needs to be replaced, which means g does not equal zero. Which means the math changes completely.

    It’s an interesting point, but far from a convincing counterargument.

    1. James_Hamilton Post author

      Jordan: You are missing the point. The need to replace deteriorated and obsolete capital is what is counted in d, and is never included in g, not in Piketty’s calculations of g, not in anybody else’s. Because Piketty is not including d in his calculation of the appropriate steady-state capital/labor ratio, he is incorrectly predicting how much the capital/labor ratio should change when g declines. You are further missing the point that his formulation of the net saving rate s as a fundamental and stable parameter makes no economic sense. His “second fundamental law of capitalism” is total bunk.

  14. Johnny

    @James/Jordan :

    Agreed, his “second fundamental law of capitalism” is total bunk.
    But in general, Piketty is right indeed. And the question remains : the times- they are chang’in – but how will it be ?

  15. A Jones

    This is just so sickening. You just won’t be reasoned with, and will do about anything to throw yourself on the R-R cross. The R-R paper was terrible. It was terrible long, long, long before the spreadsheet error came out. It has obvious correlation-causation issues and was clearly being used well beyond anything useful it had to say, with the tacit acceptance of its authors, who enjoyed the attention.

    I don’t want to go in the other stuff–that the R-R spreadsheet was much simpler than the Piketty spreadsheet; that the errors in the Piketty spreadsheet have not yet been shown to consistently go against Piketty’s claims; that other studies that tend to confirm Piketty’s conclusions (and the absence of such in the R-R case). It’s just embarrassing that you are trying to make yourself look less foolish by comparing two very dissimilar things when you are only making yourself look more foolish (it’s like a house of mirrors with a fool in the middle). Whatever, you had so little reputational capital after you lit it in a bonfire over the R-R catastrophe, I guess it doesn’t matter anymore.

    If the Piketty data are wrong, and his conclusions incorrect, I will reject them (and you will still be defending R-R).

    1. Rick Stryker

      A Jones,

      You and others keep missing the point. No one is objecting to legitimate criticism of R&R. The objection is to illegitimate criticism of R&R. The criticism leveled by Herndon, Ash, and Pollen and Krugman was largely illegitimate. As has been documented here ad infinitum, HAP and Krugman made demonstrably false, misleading, and unfair charges against R&R which impugned their professional reputations.

      Fine, claim that R&R confused correlation and causation. I wouldn’t object to that criticism. Go ahead, claim that R&R should have corrected the record about the threshold. I wouldn’t object to that either. We can argue about that but those points are not out of bounds. But don’t insinuate that R&R withheld data from scrutiny when that is demonstrably false. Don’t impugn their reputations by implying that they omitted data in order to get the result they wanted. Don’t compare trivial excel coding errors to errors that crashed a spaceship. Don’t call perfectly defensible averaging techniques “unusual and highly questionable.” And don’t go on a comedy show to mock R&R as one of the authors of HAP did.

      And don’t call JDH a fool for standing up to bullying.

      1. baffling

        if you want to find blame for why R&R was so ridiculed, simply look to the conservative movement who rallied around the paper. the paper was wrong and this group of conservatives new it, but continued to push the paper to support their ideology of austerity. now if those folks had acknowledged this paper did not provide support for their ideological agenda, nobody would have gone after R&R. but that did not happen. but conservatives continued to use the paper for their ideology, R&R did not acknowledge the fallacy of the 90% debt point, and so the issue was pointed out quite loudly so that the myth would not be perpetuated. lesson learned: do not promote flawed research to support ideological policy.

  16. Rick Stryker


    Your post stimulated me to read the book. I looked at the “second law of capitalism” and came to the same conclusion that you do–it seems to be nonsense.

    If g = 0, the capital to net income ratio is indeed infinite. And if g >0, the capital to net income ratio converges to s/g as Piketty asserts. But this is hardly a law. It follows from Piketty’s definitions and the behavioral assumption that depreciation of capital will be replenished from income and that net saving will be added to capital. But does this “law” describe anything that makes sense?

    For g = 0, it certainly does not for the reasons you’ve stated. But even for g > 0, you can get very strange behavior. Here’s an example.

    Suppose that income Y = 100, capital K = 100, saving rate s = 10%, depreciation rate d = 10%, and growth rate of Y, g = 3%. Then the “law” says that the capital to net income ratio should converge in the long run to s/g = 3.333. Assuming a time step of 1 year for saving and depreciation contributions to capital from income, the ratio will have largely converged in about 100 years to 3.333. At that point, the ratio of Y/K is about 0.4.

    So far so good. But what happens if at that point growth g decelerates to 0.5%? What is the new steady state? In this case, Y = 2000, K = 5000, s = 10%, d = 10%, and g = 0.5%. In about 500 years, the dynamics will converge to the theoretical value s/g = 20. However, the path to get there is quite strange. In the first 60 years, net income will continually decline from its starting value of 2,000, reaching a nadir of 1,363. Then Y will start to increase, returning to its starting value of 2000 at about year 210. At that point, the capital to net income ratio will be a bit over 18. And in about 300 more years, the capital to net income ratio will converge to 20.

    Obviously this makes no sense and can’t be describing how real economic actors make real economic decisions. The problem is that this “law” isn’t a law and not even economics. Piketty claims that s and g are “macrosocial parameters” that are largely independent of each other, variables which he then links into a “law” by making some very simple behavioral assumptions. From this law, he draws broad conclusions.

    Given all the hype about the book, I expected a lot more. I am very surprised to see such poor analysis.

  17. Dan Kervick

    “On page 168 of Piketty’s book the reader is introduced to “the second fundamental law of capitalism” according to which β = s/g, where β denotes the capital/income ratio, s the economy’s saving rate, and g the overall economic growth rate. Note that a curious corollary of this “law” is the claim that if the economy is not growing (g = 0), the capital/income ratio β has to be infinite.”

    That is a misreading. Piketty’s Second Fundamental Law is not an identity or an approximate identity. It is, as Piketty makes clear at some length on pages 166-170, a long-term asymptotic law. For the benefit of the general reader, Piketty writes it in the form “β = s/g”. But it might more carefully be stated this way: “For a fixed savings rate s and growth rate g, the capital-to-income ratio β converges over time to s/g.” He might have written it in more conventional fashion in the form β(t) –> s/g. He sketches an elementary proof of this convergence theorem in his technical appendix. Like any limit theorem true in the real numbers, it requires an implicit restriction on the range of the variables to avoid singularities.

    However, if you the augment the real numbers with a point-at-infinity, the theorem also says that in the case where g = 0, then β –> infinity, or in other words, the function β(t) fails to converge over time, where t ranges over years starting with some arbitrarily chosen year 0. That makes perfect sense in this case. In a society which maintains a constant annual savings rate with a zero rate of national income growth, then the capital-to-income ratio would rise without bound rather than converging to a limit. So the law doesn’t say that if g = 0, β is infinite; rater it says that if g is zero, then β increases without bound.

    I think the best way to understand Piketty’s concept of the national savings rate is to use the wealth growth rule

    W(i+1) = W(i) + s(i)*Y(i)

    as an implicit definition of s. So s(i) is by definition equal to [W(i+1) – W(i)]/Y(i). It’s just the change in wealth as a percentage of national income . As noted already, Piketty’s concept of national income already includes the subtraction of depreciation. He defines national income as (national output + income from abroad – depreciation). And he gives a definition of national wealth as well. So the concept of national savings is well-defined.

    1. James_Hamilton Post author

      Dan Kervick: You are the one who has misread both Piketty and me.

      If the use of the term “infinity” upsets you, then just substitute “very small” for “zero” and “ridiculously large” for “infinity” and everything still goes through exactly as I said.

      For example, if the growth rate is g = 0.01% per year, then for the numerical example I presented, the steady-state capital would be $990.0990099… (a little less than the $1,000 for the zero-growth case), depreciation would be $99.00990099… leaving a tiny net income of 99.00990099…. cents (instead of literally zero net income in the zero-growth example) associated with a value of β = s/g = 1000. So about 99 cents in net income (= $100.00 – $99.01) is left over from the original $100 annual GDP after replacing the machines that wore out during the year. If Piketty’s capitalists dutifully use about a dime of their pitiful net income to increase the capital stock, that will add about another cent to next year’s depreciation bill. But fortunately next year’s GDP will be $100.01 as a result of the assumed growth, so they will still have about 99 cents left over after covering what is now $99.02 in depreciation costs to try to add about another dime to the capital stock for the following year. And so we continue, year after year along an absurd Piketty steady-state path associated with a 0.01% growth rate.

      People are saving more than $99 out of the annual GDP of $100 , leaving themselves less than a dollar to try to live on, and all that saving accomplishes is to produce an even larger sum that will be wasted on depreciation the following year. If that sounds like a sensible model of economic behavior to you, then bless your soul.

      The issue is not whether the ratio s of net saving to net income is “well-defined.” The issue is whether it makes any sense at all to assume that the value of s so defined should be treated as a constant that doesn’t change when you lower the growth rate of the economy.

      I submit to you that it most clearly does not.

  18. Gerald Silverberg

    Two observations:

    1. Piketty is actually interested in private sector wealth, not the (real accumulated productive) capital stock in the sense of growth theory. As Bob Solow points out in his review of the book (New Republic), these are not at all the same thing, neither in trend nor levels. Private wealth includes non-producible forms of capital like land and natural resources, and monetary forms like the credit end of debt relationships (government and corporate bonds) and equities that may be market valued higher than book value. Thus the Harrod-Domar steady-state golden rule is largely irrelevant to wealth dynamics. If the government just issues more bonds to the private sector the private wealth/GDP ratio will go up even though the capital/GDP ratio is unchanged (as will changing land prices or a stock market boom). We’re talking a 100-200% GDP discrepancy for realistic values of just national debt (50-150% GDP) and land values (20-50% GDP) in OECD countries.

    2. The Reinhart-Rogoff vs. Piketty comparison is misleading for reasons many have already mentions: Piketty made all his data and methods available in exhaustive detail from the start, while RR hid data cherrypicking (New Zealand 1951!) and a questionable weighting scheme on which their main result crucially depended (the infamous Excel error was the least of their worries). See my post “Reinhart-Rogoff vs. New Zealand: Final Round?” at

  19. Swedish

    Piketty and Zucman never argued that net savings would remain constant as growth goes to zero. Their argument is that if g goes from 2% to 1%, then K/Y doubles unless the net savings rate is divided by 2. The question now is: is the net saving rate this elastic to growth? Cross-country regressions made by PZ suggest not, Krusell and Smith suggest a large time-series elasticity in the US postwar case. It would be interesting to replicate KS time series analysis over longer periods and more countries than the US (I can see many reasons why the US savings rate declined at the same time as growth, in particular their ever greater ability to tap into other countries’ savings). It remains that the large K/Y ratios in the 19th century remain to be explained, and the formula s/g seems to do a good job there. Do you have other theories for that time period?

  20. 2slugbaits

    JDH I think I’m following our numerical example, but don’t you get the same “infinity” problem if negative growth equals the depreciation rate? In your example, what happens if d=10% and g=(-10%). Don’t you get Beta = s/(10% – 10%) = s/0? What am I missing here?

    1. James_Hamilton Post author

      2slugbaits: That’s an interesting example! If the economy starts out with K = Y = $100, d = 0.1, g = -0.1, and a gross saving out of GDP given by s* = 0.1, then in year 1 it saves $10, just enough to replace the depreciated capital. So in year 2 we still have K = $100 but output has shrunk to about $91, so the capital/GDP ratio has gone up from 1 to 1.1. In year 2 the economy saves $9.10, so capital shrinks to $99.10, but Y has shrunk to about $82.60, and K/Y has gone up again to about 1.2. Thus Y and K are shrinking, but under the assumed saving behavior Y always shrinks faster than K so that K/Y goes to infinity.

      Your example illustrates a situation in which an assumption of a constant gross saving rate s* would not be consistent with rational decisions by the savers (though Piketty’s constant net saving s assumption in such an economy would be even more ludicrous). I would say that a constant s* specification is not an appropriate model for an economy where output is shrinking rapidly, just as a constant s specification is not an appropriate model for an economy in which output growth is very low (which is the situation to which Piketty wants to apply it). Your example underscores the point I made above, that better models than the constant s* assumption are models in which the saving behavior is derived from some kind of consideration of what savers are trying to accomplish and how they respond to incentives to save and consume. In an economy with zero or moderately positive growth, a constant s* model behaves in many ways like a model of rational investors, but in an economy with very strongly negative growth, such as you propose we consider, it clearly does not.

  21. zeeba neighba

    ” Note that a curious corollary of this “law” is the claim that if the economy is not growing (g = 0), the capital/income ratio β has to be infinite.”

    Wrong. If the economy is not growing, the values are undefined. If g=0, the value for s/g is undefined, and therefore the capital/income ratio β is undefined as well. You cannot divide by zero.

    1. James_Hamilton Post author

      zeeba neighba: You are mistaken. As I explained to J J Flatery above, s is claimed by Piketty to be a fixed positive constant. The limit of s/g as g goes to zero is therefore perfectly well defined, and is infinity. Please also read my answer to Dan Kervick above.

    2. Rick Stryker

      zeeba neighba,

      As Piketty makes clear in the book, the capital to net income ratio is not literally equal to s/g. Rather, it tends to s/g asymptotically, in the long run. If you look at the model that produces this asymptotic behavior, you will see that as g goes to 0, the capital to net income ratio goes to infinity asymptotically. And if g is exactly 0, it’s still true that the ratio goes to infinity, since asymptotically the capital converges to a limit while net income goes to zero.

    3. Johnny

      A divided by 0.1 equals 10A
      A divided by 0.01 equals 100A
      A divided by 0.001 equals 1000A
      A divided by 0.0001 equals 10000A
      A divided by zero equals perfectly well defined positive unlimited (but your pocket calculator says “error”, I know. Gosh, shall ya keep trusting your machines ?).

  22. Roger McKinney

    Nice take down of Piketty’s “fundamental” law. There are so many problems with Piketty’s reasoning, not the least of which is the Marxist habit of defining terms in such a way that his argument automatically wins the debate. I have my own, more philosophical, criticisms of Piketty on my blog at

    On the other hand, I don’t think it’s helpful for good economists to defend the status quo. The West has grown increasingly socialist since 1929, with a small reduction in the regression in the 1980’s, after which the process resumed full speed. We need to point out the real reasons for growing inequality in the West, if not in the world, which have to do with increasing fascism.

Comments are closed.