Educating Brad DeLong

Brad DeLong writes:

Department of “Huh?!”–I Don’t Understand More and More of Piketty’s Critics: Per Krusell and Tony Smith

As time passes, it seems to me that a larger and larger fraction of Piketty’s critics are making arguments that really make no sense at all– that I really do not understand how people can believe them, or why anybody would think that anybody else would believe them. Today we have Per Krusell and Tony Smith assuming that the economy-wide capital depreciation rate δ is not 0.03 or 0.05 but 0.1–and it does make a huge difference.

Let me do my best to try to educate Brad.

Here’s your first clue. Try reading Krusell and Smith’s summary of their core criticism, and tell me where the assumption of a specific numerical value for δ makes any appearance:

Piketty’s second law is not mathematically incorrect, but it relies on assumptions– as do all economic theories. The central assumption concerns how the economy saves. Piketty assumes that the ‘net’ saving rate is constant and positive, i.e. the economy increases its capital stock from year to year by an amount that is a constant fraction of (net) national income.

This assumption may sound standard but actually it is not– precisely because it is expressed in net terms. In particular:

  • With zero growth in population or technology, the assumption that the capital stock is always growing (because net saving is positive) implies that more and more output must be diverted away from consumption towards investment.
  • Eventually, because capital needs to keep rising, it is necessary to devote 100% of GDP to capital formation!

Here’s your second clue. I made exactly the same point as Krusell and Smith in an earlier Econbrowser post in which I made no claims whatever about how big the depreciation rate has to be. It’s true that I illustrated the implications of Piketty’s assumptions using a simple numerical example. My numerical example used GDP = $100 and δ = 0.10, but those numbers were chosen just to keep the arithmetic simple. I know that GDP isn’t really $100! If you run through the numerical example instead with δ = 0.05, or 0.02, or any positive number and any numerical value for GDP, you will arrive at exactly the same necessary implication of Piketty’s “second fundamental law of capitalism” as in my numerical example. His assumption of a constant net saving rate implies that capitalists always try to increase the capital stock if they have any level of positive net income whatever. That mathematically and necessarily implies that the economy’s total depreciation expense has to be higher every year. The necessary implication of Piketty’s assumed saving behavior is that as the growth rate becomes smaller and smaller, the capital stock would tend to an arbitrarily large multiple of net income (tending to infinity as the growth rate goes to zero) and the net income of capitalists after paying the depreciation bill would become arbitrarily small (tending to zero as the growth rate goes to zero).

To summarize: Piketty’s assumption that the ratio of net saving to net income remains constant as the economy’s growth rate falls is incompatible with any coherent model of saving behavior.

Brad further asks why do Krusell and Smith

imply that this is a point that Piketty has missed, rather than a point that Piketty explicitly discusses at Kindle location 10674?

One can also write the law β = s/g with s standing for the total [gross] rather than the net rate of saving. In that case the law becomes β =s/(g + δ) (where δ now stands for the rate of depreciation of capital expressed as a percentage of the capital stock).

Kindle location 10674, for those of you with a physical copy of Piketty’s book, will be found in footnote 12 on page 594. As far as I can determine, this footnote is the only point in the book at which the alternative formulation β =s/(g + δ) gets mentioned. The two versions of Piketty’s “law”, the one given in the text on page 168, and the one given in footnote 12 on page 594, cannot both be true. Either net saving is a constant fraction of net income (as Piketty assumes throughout his text) or gross saving is a constant fraction of gross income (as Piketty assumes in footnote 12). If one assumption is true, the other must be false. And if it’s not being asserted that either the gross or net saving rates are constant, then neither equation is a law at all, but instead could only be a definition of the saving rate (net or gross) that would be associated with a particular steady-state capital/income ratio β. In other words, if s is not a constant, the “second fundamental law” has no implications or predictions whatever for what will happen to the share of capital income in the economy or anything else, and claims that Piketty has uncovered some underlying principles of capitalism are completely without substance.

Having I hope clarified that the debate is not over whether the economy-wide depreciation rate is 10% or some other number, I nevertheless cannot resist entering the discussion of what is a reasonable number to assume for δ for purposes of characterizing steady-state growth paths. Brad writes as though the only sensible number to use for δ would be “0.02 or 0.03 or 0.05.”

The fact that Brad mentions such a wide range of possible values makes it obvious that defending a particular number for δ is anything but cut and dried. If you look at the assumed depreciation rates that underlie the national income accounts to which Brad appeals, you will find that rates of 10-20% are quite common for most forms of producers’ machinery and equipment (and perhaps it would be unfair at this point to mention the fine study by DeLong and Summers (1991) which concluded that this is the category of capital to which we should pay the most attention). Brad likewise casually insists that the U.S. capital/income ratio is somewhere between 4 and 6, another number that anyone who has looked at the details behind how this particular sausage gets made would treat with some caution. For example, the Census Department estimates the total net stock of fixed assets and durable goods in 2009 to have been $48.5 T, or 3.4 times 2009 GDP of $14.4 T. Steve Parente recommends using a value for K/Y of 2.75, while Paul Evans regards the relevant number for the U.S. to be around 2.

Update: Reader Salim points out that I was misinterpreting Piketty’s use of a 10% figure in his book’s calculations of depreciation. Piketty uses 10% for depreciation as a percent of GDP, not as a percent of capital as my original post suggested. In order not to mislead, I have deleted the inaccurate paragraphs that were included in the first version of this post.

116 thoughts on “Educating Brad DeLong

  1. AS

    Professor Hamilton,
    For those who have an accounting background rather than an economics background, would you summarize the similarities or differences between depreciation related to financial statement accounting and GDP accounting? Financial statement accounting may use either straight-line or accelerated depreciation over useful lives of assets often assuming salvage value. Are there similar economic conventions concerning method and salvage value? I assume that the useful life for GDP accounting is implied by the annual depreciation rate.

    1. Steven Kopits

      AS –

      I believe Jim is referring to substantive, not accounting, depreciation. You could consider it cash-on-cash accounting if you want an analogy, ie, the percent of your revenues in cash would you have to spend to maintain your current capital stock.

      It is something of a slippery concept. For example, if you assume absolutely no maintenance capex, then pretty much everything would fail within ten years, although many residences would remain essentially fully functional. On the other hand, within a decade, almost every vehicle on the road today would need some major part replaced. If you assume some maintenance capital, then you can nurse a capital stock along for some time, to wit, the 1950s cars still roaming the streets of Havana.

      You can turn around the question and ask how many years would it take to replace our current capital stock. Probably more than a decade, but probably less than 20 years. It took Germany about 20 years to rebuild after WWII.

      This is turn has interesting implications for adaptation to climate change. It always assumed, somehow, that everything is fixed and we can’t adapt–New York will be flooded! But if we essentially have to buy all our assets again every 10-20 years anyway, then we’re really largely looking at effects no more than 20 year out. There’s not much point in mitigation for issues more than twenty years into the future–we’re going to have to “buy” those assets again in the meanwhile anyway.

      1. Nick G

        if we essentially have to buy all our assets again every 10-20 years anyway, then we’re really largely looking at effects no more than 20 year out.

        Seriously? Housing and urban infrastructure have a much greater life than that, and the war left behind much of the underground infrastructure, as well as the locational value of the land (which Climate Change would not). Ask the Germans if it would be trivial to plan on sufferering even worse than the devastation of WWII in 20 years.

        And, of course, ecosystems take much, much longer to adapt. Much of our capital is natural, not built, and irreplaceable in human timeframes.

        1. Steven Kopits

          Well, then you’re using a depreciation rate of less than 5%, Nick.

          Let’s take a look at some major ‘depreciation’ in Germany. Watch the video; it documents the fire bombings of Hamburg in 1943. I think we can consider the commercial districts of the city fully ‘depreciated’ in GDP terms.

          Now here’s a couple of panoramic shots of Hamburg in 1950. You can see most of the rubble has been cleared and some of downtown has been restored.

          Here’s Hamburg in 1960. The city looks essentially rebuilt.

          Thus, it would appear that a city can be rebuilt from scratch in about 15 years, if necessary.

          It is fair, however, to say that restoring Germany’s wealth took much longer. Our family traveled in Austria and Hungary in 1971, and Europe was still poor compared to the US. By 1980, much of the difference had disappeared.

          Thus, while buildings can last a long time–there are some in Rome which date back more than 1,000 years–a city can be built in 20 years, and the losses of total devastation repaired in 30 years or so.

          1. Nick G

            it would appear that a city can be rebuilt from scratch in about 15 years, if necessary.

            Well, sure. But why would you choose to suffer such a disaster?? And, again, Climate Change’s damages would be worse – abandoning a coastal city means a loss of underground infrastructure, much of the transportation infrastructure to that hub, the loss of the locational value of the land, uprooting and diaspora of the citizens, etc.

            And, again, ecosystems take much, much longer to adapt. Much of our capital is natural, not built, and irreplaceable in human timeframes.

            Why would you suggest that a disaster that takes place more than 20 years in the future can be ignored??

          2. Achim

            According to the economic historian Werner Abelshauser who wrote the now-standard Deutsche Wirtschaftsgeschichte seit 1945 (German economic history since 1945), the capital stock in 1945 was about as large as it was in 1936, if I recall correctly. A large part of the infrastructure was destroyed, but once that was restored, it was no wonder that the German economy was quickly on the path to “Wirtschaftswunder”.

  2. Salim

    I’m referring to Williamson’s quote of Piketty. Piketty’s saying that delta*k = 0.1 * Y. That’s consistent with delta = 0.02 and k = 5Y, for example. Likewise on p. 62, where he says, “subtract 10 percent for depreciation”, he’s referring to 10 percent of output, not 10 percent of the capital stock.

    Otherwise, I totally agree with your critique of DeLong. Minor errors like the one Williamson made will distract from the broader issue that you, Krussell & Smith have rightly pointed out, that Piketty’s doomsday relies on a model that departs unduly from any reasonable decision rule.

    1. James_Hamilton Post author

      Salim: Yikes, you’re right! I deleted the erroneous paragraphs in my original post, and hope I did not mislead too many people in the brief period they were up. Thanks for catching this.

  3. Salim

    Also, the reason Piketty’s K/Y ratios are so much larger than standard ones (and his /delta/ so much lower) is Piketty’s odd choice to define “capital” to mean “wealth” (p. 46). I think it’s an intentional sleight-of-hand, so that Piketty can apply the Solow model to wealth. If he’d labeled his variable “wealth”, economists would have caviled at him sticking wealth into a production function.

    A very important critique came from four Sciences Po profs, who point out that if you take out real estate, capital/income hasn’t been rising in Europe after all. Piketty thinks he’s found the robot revolution, but he’s actually found restricted housing supply on the Left Bank.

  4. Lord

    Sorry, but numbers matter. I don’t think what may happen in the year 10000 or 10000000 has any bearing on Piketty. Simply pointing out something can’t go on forever has little to say about it not going on a very long time. Reductio ad absurdum arguments are not stong ones, but weak ones.

  5. BC

    In a hyper-financialized economy such as those of the English-speaking world, “capital” is increasingly “financial” capital, which morphs into “savings”, which then is a vehicle for increasingly leveraged rent seeking of ~7-10%+ annualized returns WELL BEYOND the rate at which physical and human capital can be deployed under the net exergetic flow constraints at ~3% returns to “compete” with MUCH HIGHER speculative rentier returns to “financial” capital.

    Moreover, imperial corporate-states, as is the US, cannot abide with 3% aggregate returns to productive capital and labor because it does not permit inflationary funding of imperial statecraft, if you will, including supplying an imperial military around the globe to defend and protect supranational firms’ investments and operations abroad, as well as the foreign resources, cheap labor, and shipping lanes required for the global supply chain and system of distribution.

    “Savings” then flows to the higher rentier speculative vehicles, i.e., stocks, bonds, forex, levered real estate, foreign expansionism, military goods, etc., rather than domestic productive value-added activities, employment, after-tax wages, purchasing power of labor, and gov’t receipts.

    The long-term result of this hyper-financialization of land, labor, and future resource supplies by the rentier caste is that cumulative rentier claims by the owners of financial capital, i.e., the top 0.1-1% to 10%, reach a level at which they preclude growth of domestic private investment, wages, profits, and gov’t receipts after tax, debt service, and price changes.

    Wealth and income concentrates increasingly to the top 0.01-0.1% to 1% owners of financial capital and the debilitating net claims therefrom on all value-added economic activity in perpetuity, which in turn results in buying and co-opting of elected officials who write laws in favor of the rentiers, further reinforcing the non-productive, speculative rentier zeitgeist and the increasing costs to productive economic activity, precluding any growth of real final sales per capita after net flows to the financial sector owned by the rentiers.

    The US is not savings/investment short, as some claim; rather, the US is savings malinvested/misallocated and concentrated unproductively in the form of highly overvalued corporate equity and bonds of the Fortune 25-100 to the top 0.01-0.1% to 1%, who seek unsustainable returns of 7-10% when the economy is capable of only ~2-3% sustainable, non-inflationary growth of value-added output. The more the central banks and their masters, the owners of the TBTE banks, succeed in pumping up financial assets far in excess of wages and GDP, the larger the net cumulative claim on future wages, profits, gov’t receipts, the slower the growth of real final sales per capita will be, and the more the US will resemble the Third World.

    As for US capital stock, the situation is far worse than most perceive, even grim by implication. Adjusted for depreciation, population, and the declining net exergetic flow rates to the necessary rate of capital accumulation/replacement to sustain current real final sales per capita hereafter, the US is at the level of sustainable capital stock equivalent to 1929-30 to before WW II.

    The “American Dream” or the non-negotiable “American Way of Life” is over and no longer retrievable for the bottom 90%+ of American households hereafter, and a growing majority are awakening to this reality.

    1. PeakTrader

      BC, I stated before: Lower interest rates and higher asset prices induce people to spend and borrow, and reduce saving, a lower cost of capital spurs production, refinancing at lower rates increases discretionary income, lower mortgage rates makes buying a home more affordable, 401(k)s and IRAs increase in value, etc. There are massive multiplier effects throughout the economy.

      What concerns me is the continued L-shaped recovery, at best, after the severe recession. Fiscal policy “crowded-in” growth, through massive borrowing, and “crowded-out” future growth. Unfortunately, the result has been anemic growth.

      If we had a U-shaped or V-shaped recovery instead, government spending (including on the unemployed) would fall and tax revenue would rise to shrink budget deficits dramatically.

      There seems to be too many ineffective and counterproductive economic policies out of Washington. Basically, there’s one foot on the accelerator and the other foot on the brake causing an expensive “recovery.”

      It seems, rather than bringing down the mountain of household debt through large tax cuts (to generate a self-sustaining consumption-employment cycle to shrink budget deficits), a mountain of government debt was added to the mountain of household debt. If you owe $5,000 on your car, a $5,000 tax cut would help you substantially. However, hiring a worker for $5,000, through borrowing, to pave the road won’t help you much.

      1. spencer

        Really, lower and lower interest rates crowded out investments. Can you show me an a real life example of where higher rates lead to greater growth?

        1. PeakTrader

          There’s nothing in my statement that supports your response.

          When there’s an actual recovery, or enough potential output is destroyed, interest rates will rise on the national debt, to preempt inflation, crowding-out future growth.

          Moreover, lower interest rates induce demand and reduce saving, in the current period (somewhat similar to lower prices), to raise consumption = income – saving, through spending and borrowing, also crowding-out future growth.

  6. Patrick R. Sullivan

    Obviously, the attack on Krussell and Smith was written by Brad’s evil twin, Bad DeLong.

  7. Rick Stryker


    Indeed, Delong’s defense is pretty weak. I made some related criticisms of Delong’s post to 2slugs in the comments section of the last post. However, having thought it over some more, I think Delong has stumbled upon a real point in questioning whether d is too high. To explain that, I’ll need to comment some more on Delong’s other point about whether Piketty was aware of the standard model.

    Delong’s post does raise the question of why Piketty is using his bizarre “second law” when he seems to be well aware of the standard model. Although PIketty gives the standard model short shrift in his book, he discusses it much more in his papers. Moreover, PIketty’s “second law” does have a distinguished lineage. Solow in his 1956 paper “A Contribution to the Theory of Economic Growth” works with a growth model in which agents save from net income. Why doesn’t Piketty discuss the relationship between the standard model and the “second law” more?

    I think the answer is that Piketty mistakenly concluded that the “second law” and the standard model are actually the same. Evidence for that can be seen in his 2013 paper Capital is Back: Wealth Income Ratios in Rich Countries 1700-2010 with Zucman. In that paper, they derive the “second law” in section 3.2 and then say the following:

    “Should we use gross-of-depreciation saving rates rather than net rates, the steady-state
    formula would be B =s/(g+d) with s the gross saving rate, and d the depreciation rate expressed
    as a proportion of the wealth stock. We find it more transparent to express everything in terms
    of net saving rates and use the B =s/g formula, so as to better concentrate on the saving versus
    capital gain decomposition. Both formulations are equivalent and require the same data.”

    Now, I wondered why they might think the models are equivalent. A little algebra reveals what I think happened (with apologies to Hans)

    If we define Y(t) to be gross income at time t, K(t) to be the capital stock at time t, g the growth rate, s the gross saving rate, and d the depreciation rate, we can write the standard model as a pair of difference equations:

    Y(t) = (1+g)Y(t-1)
    K(t) = (1-d)K(t-1) + sY(t-1)

    Setting B = Y(t)/K(t) = Y(t-1)/K(t-1) for large t, then we can divide the second equation by the first to get

    K(t)/Y(t) = (1-d)K(t-1)/(1+g)Y(t-1) + s/(1+g) or

    B(1+g) = B(1-d) + s. Solving for B, we have

    B = s/(g + d). That’s clearly not the “second law” and Piketty is aware of it. However, if we calculate the net saving rate in the standard model, it’s

    (sY(t)-dK(t))/(Y(t) – dK(t)) = gs/(g + d(1-s)). So the net saving rate s(net) in terms of the gross saving rate is

    s(net) = gs/(g + d*(1-s))

    We can also calculate the capital to net income ratio in the standard model. It’s K(t)/(Y(t) – dK(t)) = B/(1-dB) = s/(g + d(1-s)

    However, if we substitute the formula for s(net) into the formula for the capital to net income ratio, we get the second law:

    B(net) = s(net)/g

    I believe that an argument like this must have led Piketty to think the standard model and his “second law” are equivalent.

    However, the conceptual mistake here can be seen in the formula for s(net). For a given gross saving rate s, s(net) is not constant but is a function of g. And yet Piketty holds s(net) constant when he lowers g.

    I agree with you completely on the crazy behavior of the “second law” as g goes to zero. However, I don’t think Piketty is driving g to zero but is just lowering it, although as I mentioned in a previous comment Piketty fully buys into the neo-Marxist implications of driving g to 0 in the second law. But how much does this matter for the argument he’s actually making?

    In the section of the book entitled “What Will the Capital/Income Ratio Be in the Twenty-First Century?” Piketty says that he expects the net savings rate to stabilize to about 10 percent and he expects growth to slow from 3% to 1.5%. So, by his second law, he thinks the capital to net income ratio will go from 10%/3% = 3.33 to 10%/1.5% = 6.66, i.e., it will double. That seems to be his prediction. However, he’s made the mistake of holding the net savings rate constant in this calculation.

    If Piketty thinks the net savings rate will stabilize at 10%, then the equation that relates s(net) to s implies, given g = 3% and d = 2%, that the gross saving rate is about 16%. So, the “second law” for the standard model would imply the capital to net income ratio is 10%/3% = 3.33. However, if we drop g to 1.5%, we should hold gross saving constant at 16% and calculate the new net saving rate, which turns out to be 7.5%. So, the new capital to net income ratio is 7.5%/1.5% = 5.

    Thus, the capital to net income ratio should have gone up by 5/3.33 = 50%. That means that Piketty overstated the percentage increase in the capital to net income ratio by a factor of 2 in the book, assuming that d = 2%

    Now let me make one point in Delong’s defense. In my analysis, I used Piketty’s preferred d to be in the 2% to 3% range–I used 2%. But if we look at the equation for s(net), we can see that for a given decrease in g and holding gross saving constant, s(net) increases more slowly as d increases. That means that if d is larger, Piketty is making an even bigger overestimate since the correct capital to net income ratio would move up less for a given drop in g. Indeed, if I repeat my analysis but use d = 10% and s(net) = 10%, gross saving would be held constant at 33% and I would get that the capital to income ratio should have gone up 18% if g dropped from 3% to 1.5% rather than Piketty’s estimate of 100%, and overestimate in the percentage increase by a factor of 5.

    So, Delong has a real point I think, though not for the right reasons. We need to know the value of the depreciation parameter if we want to understand how much Piketty is overestimating the increase of the capital to net income ratio when g slows. If d is in the low end of the range, 2%, then Piketty’s overestimate of the percentage increase is about a factor of 2, significant for sure. And if d is higher, Piketty’s overestimate is higher.

  8. Tom

    Piketty is actually making a very similar error to one made by Keynes in the General Theory.

    In a nutshell, Keynes has savings as a stable percentage of income, not as an absolute “law of capitalism” but as a generally stable “psychology of the community.” He therefore assumes that additional investment from public works stimulus will increase output by the inverse of the ratio of savings to investment, because that psychology will push people to consume as much more as it takes to keep the savings/income ratio stable.

    “It follows, therefore, that, if the consumption psychology of the community is such that they will choose to consume, e.g. nine-tenths of an increment of income, then the multiplier k is 10; and the total employment caused by (e.g.) increased public works will be ten times the primary employment provided by the public works themselves, assuming no reduction of investment in other directions.”

    To be fair Keynes adds some qualifications eg re inflation in full employment conditions.

  9. Kevin Donoghue

    “Piketty assumes that the ‘net’ saving rate is constant and positive….”

    No he doesn’t.

    1. Tom

      Indeed it’s the sole basis of his claim that if growth slows, the capital/output ratio and thus inequality must rise.

  10. Ricardo

    Thank you professor. It is amazing how much time we spend attempting to illustrate bad economics and economic error. But then it is more amazing that there are so many who spend more time trying to rationalize bad economics and economic error.

  11. Ricardo

    There are much more urgent economic concerns than imaginary problems caused by income inequality. From THE STREET:

    “Through destroying the value of one’s own currency, the wages of workers in real dollars are driven steadily toward zero, and so (supposedly) this will allow a nation to undercut its trade partners and export more goods.

    “The sick joke here is that with all nations destroying the value of their currencies (and the wages of their workers) simultaneously, no nation gains any “advantage” and the wages of workers are being destroyed for no reason whatsoever. This does, however, produce the paradigm of all currencies simultaneously falling in value. Only the rate of decline of this paper-destruction varies.

    “This is why anytime we see some talking head refer to a currency as “rising in value” it is an implicit admission that the person has no understanding of the global economy. If two people jump off the roof of a 100-story building at the same time, and while on the way down one individual climbs on top of the shoulders of the other, that person hasn’t “risen.” He will merely go “splat” on the pavement a millisecond later.”

  12. Ben

    Professor DeLong has a new post on the subject here . From what he says, Picketty is using a lot of very non-standard definitions. If you take time to understand these and his rationale for using them, all of these arguments melt away.

    1. You already noticed that depreciation is % of GDP – not capital stock.

    2. Capital is defined as wealth – not productive equipment/machinery. In the Belle Epoch, the majority of “capital” was land not plant/equipment. Today, we have a lot of housing/buildings, and in both periods some people can count the ability to summon the police to protect an income stream as “capital”.

    3. Increased K/Y now means something else, too, because K is “capital” – not the standard definition of capital.

    Your thoughts, professor?

  13. Rick Stryker

    Delong has a new attempt to rescue Piketty’s argument that the capital to net income ratio will rise from about 300% to about 700%. However, you just can’t get there using Piketty’s model and assumptions.

    Following on my previous comment, in the book Piketty claims that he expects the net savings rate to stabilize to about 10% and for g to decline from 3% to 1.5%. So, using his second law, the capital to net income ratio would increase from 10%/3% = 333% to 10%/1.5% = 666%. That’s the basis for Piketty’s claim that the ratio will get to 700%. But that claim depends very strongly on his “second law,” which JDH has shown is a nonsense model.
    If you take the view that I did in my comment above that Piketty’s second law could alternatively be based on the the textbook model (and I suspect that’s what Piketty thought he was doing), then you have to adjust for the fact that net saving is not constant but a function of g. Delong has told us that he thinks the depreciation rate is 3.33% per year. So, going through the adjusted “second law” calculation, a net saving rate of 10%, g = 3%, and d = 3.33% would imply a gross saving rate of 19%. The adjusted “second law” gives a starting capital to net income ratio of 10%/3% = 333%. Now, if g drops to 1.5% but we hold the gross saving rate and d constant, we get a new net saving rate of 6.7%. The adjusted “second law” thus would yield a predicted capital to net income ratio of 6.7%/1.5% = 452%. That’s a 37% increase in the capital to net income ratio, not Piketty’s predicted 100% increase.

    Given that Delong is not directly addressing the criticisms that have actually been made (and Krugman is silent), we can only conclude that these guys understand that there are problems with Piketty’s analysis. Delong is already showing how the rescue mission will proceed. Ignore the real criticisms and make alternative arguments for the plausibility of Piketty’s conclusions. Then claim that the critics have been shown to be wrong.

    But the fact remains that Piketty’s book is chock full of highly questionable assertions, crankish views, and conceptual mistakes, all in an attempt to get us to buy into a world of very high taxes, very high government spending, and intrusive government.

    1. Nick G

      Why does an argument about inequality say anything about *overall* levels of spending and taxation?

    2. Patrick R. Sullivan

      DeLong is a veteran of this smoke screen retreat. I know from personal experience how he will persevere with this.

      Back in 2005–on the now defunct Economic History Net–he was still trying to rescue Paul David (and Paul Krugman) from the egregious error in ‘Clio and the Economics of QWERTY’ (AER 1985!). That was another ‘market failure’ argument; essentially what Piketty is arguing. DeLong found himself baffled…baffled, at David’s critics. Which brought this response;

      Dear Brad,

      It’s my turn to be puzzled. No typing-intensive company since computers have made the change trivial has ever adopted the Dvorak board. Have I got that right? I’m no student of these matters, so maybe it’s wrong. Suppose it’s right. Changing keyboards is not hard for professional typists–they do it for example when shifting between Danish and British boards; more, between Cyrillic [you know what I mean: Russian!] and Latin.

      Clarinetists play oboe after a little practice. So it’s no big trick to retrain typists if the filling out of, say, thousands and thousands of insurance forms would be sped up by the change. Doesn’t that mean that Qwerty (boy, is that easy to type!) is not superior? So doesn’t that make Paul’s story into an urban myth? I’ve not had success in getting Paul to answer this, or similar questions about, say, economies of farm size in reapers or economies of scale in railways. But maybe you can, Brad: I mean, answer it; or get Paul to.

      Does the market “work”? Of course not, says the postmodern free market feminist, by the standard of blackboard perfection that Pigou and his followers have imposed. But we’re talking quantitatively here, about which the blackboard can say nothing at all (it’s a contradiction in Coase’s career that though he is always and everywhere talking about quantitative oomph he is not a quantitative economist).

      So does use of property rights do better than use of central planning rights, or on the contrary is the blackboard result that lighthouses are public goods of any use? Coase early and late said that the blackboard is of no use, and implicitly said that property rights do better, often (not always: back to blackboards), than more direct forms of government intervention.

      Love, Deirdre McCloskey

      Btw, in that infamous Paul David paper–which, to my knowledge, DeLong has never admitted is flat out wrong–the opening line was; ‘Cicero demands of historians, first, that we tell true stories.’

  14. Peter K.

    “For example, the Census Department estimates the total net stock of fixed assets and durable goods in 2009 to have been $48.5 T, or 3.4 times 2009 GDP of $14.4 T.”

    As with what Ben is saying, does Piketty include liquid assets and financial claims in his measures of wealth? I mean the famous r in r > g is the return on all wealth.

    And what about tax shelters, offshore accounts and Bitcoins?

  15. Nick G

    Does anyone have any comments about the assumption that overall economic growth will fall sharply?

    1. Steven Kopits

      There has been some tendency for a lowering of growth expectations compared to the pre-Recessionary period.

      If you look at slide 57 of my Columbia presentation (, you can see some analysis. The forecasts come from Statoil (not sure if they’re actually internally developed there).

      As some of you may know, I have been commissioned to write a book on supply-constrained oil markets analysis, which is in process. Part of my recent analysis looks at increasing efficiency of oil consumption as a share of GDP. The sustainable, non-recessionary rate is around 2.8%. (Jim had postulated 2.5%, I had stated 3.0%.) Thus, weak economic growth–if we accept this model–is essentially explained by oil rationing.

      I think at this point we can see another oil crisis coming at the edge of the radar. Bernstein yesterday reported that the marginal cost of shale oil in the US is $111/barrel. WTI spot today is under $103. Bernstein sees Bakken and Eagle Ford production peaking around 2016/2017, with uncertain upside potential from the Permian, and that’s all we have in terms of big plays.

      More importantly, a document is circulating that Statoil intends to cut capex by up to 25%. If this occurs, similar cuts could be assumed throughout the industry, with frankly disastrous implications for IOC oil production at the 3-5 year horizon. Many of the NOCs are also struggling with similar cost issues. I had earlier stated that I felt that a hard landing for the IOCs was likely, and the data are lining up that way.

      And I have earlier commented that Aramco’s intention to drill for gas in deepwater Red Sea suggests that marginal barrel costs in Saudi Arabia are approaching IOC levels. This is also disturbing.

      So, a constrained oil supply, in my opinion, provides a good explanation for continued weak economic performance. I think it interesting that GDP has been consistently coming in about 1.1% (percentage points) below earlier consensus expectations. (see here: This seems to be the unaccounted for “surprise” which economists for some reason cannot anticipate. As I stated before, I think a constrained oil supply is knocking 0.7-1.2 percentage points off of OECD growth rates, and thereby provides an explanation. If we are headed into another oil shock (supply-side, this time) around 2016/2017, then we need to be prepared to see anemic growth, followed by a distinct downturn, followed by…well, we’d better hope that technology or increased global access finds a way to offset increasing exploration and production costs. Otherwise, we will see OECD oil consumption decline rates fall to -2% to -3% per annum, such that efficiency gains will essentially only be able to offset consumption declines. In such a case, OECD GDP growth would average about zero.

      1. Steven Kopits

        OPEC this morning reporting they might have challenges in meeting demand requirements in the second half of the year.

      2. Nick G


        I don’t think you’re addressing Pikkety’s claims. He argues that growth will start to decline around 2030, and decline more quickly around 2050. He argues that the US will have a long-term growth rate of about 1.2% per capita (which isn’t dramatically different from now) but that developing country growth will decline dramatically, apparently because they’ll catch up to developed countries. Here’s a discussion at the World Bank, which suggests that his dates are much too early for such convergence:

        I’d agree that Peak Oil can reduce economic growth in the short term: high oil prices transfer income & wealth to oil exporters, which reduces growth for oil importers (though not for the world). Peak Oil shocks scare consumers, so they stop investing in new cars and other capex – but that’s temporary. Finally, there is some cost to developing alternatives: for instance, GM spent $1B developing the Chevy Volt (which reduces oil consumption by 90%). Of course, such costs are pretty small compared to the overall car industry, or the overall economy.

        Will the US continue to transfer it’s income and wealth to countries like Saudi Arabia? The US did manage to reduce it’s net oil imports by 55% in the last several years. On the other hand, if we continue to listen to arguments that oil is essential to “our way of life”, then it will take longer than necessary to free ourselves from this unreliable, dirty and expensive energy source…

        1. Steven Kopits

          Nick –

          Oil is not essential to our life, at least for transportation. However, mobility probably is essential. So the question is how we provide affordable mobility.

          The issue with oil is not that we’re transferring more money to the Arab Gulf. Rather, it’s that the cost of exploration and production is rising so quickly. That is, the productivity of the oil business, as measured in oil output per dollar invested, is plummeting. This is the core of the problem.

          How we adapt as an economy is an interesting, and potentially problematic issue. The track record since 2005 is not comforting. At least 1/3 of the adjustment has come from reduced economic activity–that’s about how much oil looks to be knocking off of GDP growth. Now, if we face a hard peak in 2017, and I think that’s looking increasingly likely, then I would not be surprised to see the oil supply either flat or out-and-out declining. This translates into -2.5% annual oil consumption change for the US, implying essentially GDP growth goes to zero at a 2.8% annual efficiency gain.

          Personally, I think self-driving EV’s are the answer, and Google has created the “dork car”, which essentially conforms to the specifications I laid down more than a year ago.

          But it comes down to feasible rates of change and whether society thinks it needs to respond. For example, Noah Smith writes, moronically:

          First, the good news. President Barack Obama’s proposed rules on coal-fired power plants aren’t a last-ditch, desperate measure. They’re designed to keep a good trend going — the U.S. is already decreasing its carbon emissions, thanks to cheap natural gas and to the fact that Americans are driving less.

          So, if you believe we have a problem on the oil front, and if you think that matters for GDP, then you have a reason to act. If you’re like Noah Smith, and your little brain can’t comprehend that a society losing mobility is losing fundamentally important capabilities, then there’s no call to act.

          So which is it

          1. Nick G


            Let’s celebrate that we’re agreeing on some things:

            1) Oil is expensive and unreliable, and we need to transition away from it ASAP.

            2) EVs are a large part of the answer, and

            3) Our society doesn’t sufficiently recognize the importance of points 1 and 2.

            Now, if you want, we can discuss some less important points of disagreement:

            The issue with oil is not that we’re transferring more money to the Arab Gulf. Rather, it’s that the cost of exploration and production is rising so quickly.

            The US oil trade deficit isn’t new. It’s a significant part of our long-term debt to the rest of the world. Higher prices just make it that much worse. Further, the recent increase in prices may arguable be due to increasing *marginal* costs, but average costs haven’t risen nearly as much, meaning that the cost of raising oil hasn’t risen nearly as much as oil prices, and therefore oil exporters are receiving enormous windfall profits. Those profits are coming out of the wallets of oil consumers.

            At least 1/3 of the adjustment has come from reduced economic activity–that’s about how much oil looks to be knocking off of GDP growth.

            But how is oil reducing GDP growth? Inflation and interest rates are still low, so the price of oil isn’t capping economic growth. No, it’s simply the increased drain on consumers’ wallets.

            This translates into -2.5% annual oil consumption change for the US, implying essentially GDP growth goes to zero at a 2.8% annual efficiency gain.

            This suggests that reducing oil consumption is difficult. It’s really not – we just haven’t tried hard. SUV drivers can cut their fuel expenses by 90% overnight by buying a Chevy Volt – the average SUV purchase prices is much higher than the price of a Volt, and the Volt’s 5 year cost is about the lowest of any vehicle on the market. So, why aren’t Volts selling out? Well, Fox News attacking Volts every chance they get isn’t helping.

            self-driving EV’s are the answer

            I love the idea of self-driving cars, and I like EVs, but why suggest that self-driving is a necessary part of EVs?

            Finally, I do think we’re making progress. The recent growth of EVs is encouraging. The revised CAFE rules are encouraging, even if they’re too timid.

            Just as you and I seem to be converging on agreement, the country is getting there, albeit too slowly.

          2. Steven Kopits

            Regarding oil and EV’s, Nick –

            “1) Oil is expensive and unreliable, and we need to transition away from it ASAP.”\
            Oil remains the cheapest transportation fuel bar CNG, and we don’t have the infrastructure for the latter. It is and has been a very reliable fuel, but we cannot grow our mobility easily using just oil, and there’s a good chance the oil supply will falter, and sooner rather than later.

            “2) EVs are a large part of the answer,”
            Agreed. Put it another way, as I look out at the ten year horizon, I don’t see much else with which to provide incremental mobility on a large scale (ie, suitable to motorize China).

            “3) Our society doesn’t sufficiently recognize the importance of points 1 and 2.”
            I don’t think people realize the fragility of the oil supply, agreed. Because of that, I don’t think the public sector understands the importance of looking for alternatives, yes.

            “The US oil trade deficit isn’t new. It’s a significant part of our long-term debt to the rest of the world. Higher prices just make it that much worse. Further, the recent increase in prices may arguable be due to increasing *marginal* costs, but average costs haven’t risen nearly as much, meaning that the cost of raising oil hasn’t risen nearly as much as oil prices, and therefore oil exporters are receiving enormous windfall profits. Those profits are coming out of the wallets of oil consumers.”
            Low cost oil producers are making a lot of money, but less than earlier, and they will make less next year. Costs are rising faster than revenues. At the IOCs, average costs are rising 3 percentage points of sales faster than revenues.

            “But how is oil reducing GDP growth? Inflation and interest rates are still low, so the price of oil isn’t capping economic growth. No, it’s simply the increased drain on consumers’ wallets.”
            I disagree. To make your case, you have to show that oil efficiency can increase at a pace faster than, say, 4.5% per year outside a recessionary environment. If it’s true, you can’t see it in the data.

            “This suggests that reducing oil consumption is difficult. It’s really not – we just haven’t tried hard. SUV drivers can cut their fuel expenses by 90% overnight by buying a Chevy Volt – the average SUV purchase prices is much higher than the price of a Volt, and the Volt’s 5 year cost is about the lowest of any vehicle on the market. So, why aren’t Volts selling out? Well, Fox News attacking Volts every chance they get isn’t helping.”

            Economy-wide, reducing oil consumption is difficult and time consuming. You can see efficiency gains of 3-4% per year in the historical record–all during recessions.

            “I love the idea of self-driving cars, and I like EVs, but why suggest that self-driving is a necessary part of EVs?”
            Battery-powered cars are not cost competitive. This includes the Chevy Volt, which is more or less at $40,000 Chevy Cruze. Assuming no material improvements in battery technology, you can make EV’s competitive if

            i) you can reduce their requirements (ie, low speed, only two passengers, low range requirement)
            ii) reduce their cost (see i. above)
            iii) increase (double) average utilization
            iv) reduce the need to have them fully recharged on average (see i above)

            Do we have a vehicle that meets these specs? Indeed, we do. The Google Dork Car.

            But no one’s going to buy that car. It’s not intended for sale to individuals. It’s intended to provide a stream of transportation services for specific (ie local) applications only. So how does it get around? Easy, self-driving, which is what Google’s been working on. And it depends intrinsically on smart phones and location-based services.

            So, Google’s vision–and mine–is not that EV’s get cheaper on an apples-to-apples basis with today’s ICE vehicles (although they certainly could). Rather, we use technology to change the business model and create a way to employ EV’s in a way which makes economic sense.

          3. Nick G

            Again, it’s great that we’re agreed on the need to find alternatives to oil, the need for our society to make that a higher priority, and the value of EVs for that.

            Now, back to points of disagreement:

            Oil remains the cheapest transportation fuel bar CNG

            Electricity is far cheaper. At an average of $.11/kWh, and 3 miles per kWh, an EV costs less than 4 cents per mile to “fuel”. The average US vehicle, at 22 MPG and $3.75/gallon, costs 17 cents to fuel. A Chevy Cruze at 35 MPG costs 11 cents per mile. Over a 150,000 mile lifetime the EV saves $10k over the Cruze and $20k over the average vehicle.

            Take a look at Edmunds’ 5 year costs: you’ll find that a Chevy Volt is $4k cheaper own than a Cruze. And, of course, a Volt is a much better car than a Cruze – they really aren’t comparable.

            That includes the credit, which partially adjusts for the cost being an early adopter, before economies of scale reduce prices. Don’t forget, GM originally expected the Volt to be priced at $30k – that’s why it’s a Chevy!

            It ignores external costs. Here’s just one discussion:

            In 2006, the cost of US military expenditures related to oil (like protecting the supply route from the Persian Gulf) was estimated at $133 billion a year. Add in squishier externalized costs, like environmental damage, carbon emissions and pollution-related health care costs related to oil, and you’re talking at least another $400 billion a year.”


            We can’t ignore external costs, right??

            Oil…is and has been a very reliable fuel

            Every oil-related recession says otherwise. Look at the cost of oil-shock induced recessions. I think you’ve argued that the 2008 recession was in large part caused by oil. Certainly Prof Hamilton has. Well, that recession vaporized roughly $6T in wealth around the world, and created a long-term $1T annual output gap. Many analysts argue that other oil-related recessions include those of : 1973,1980,1991,2001.

            how is oil reducing GDP growth? Inflation and interest rates are still low, so the price of oil isn’t capping economic growth. No, it’s simply the increased drain on consumers’ wallets.

            To make your case, you have to show that oil efficiency can increase at a pace faster than, say, 4.5% per year outside a recessionary environment.

            No, all we have to observe is that inflation is low. That means that a growing economy isn’t being held back by the cost of commodities.

            Economy-wide, reducing oil consumption is difficult and time consuming.

            We have the means to reduce oil consumption very quickly. Passenger transportation accounts for more than 50% of oil consumption. We have passenger vehicles that use between 50% and 10% as much fuel, available right now.

            Battery-powered cars are not cost competitive. This includes the Chevy Volt, which is more or less at $40,000 Chevy Cruze.

            See the discussion above. First, the Volt is not at all a Cruze: it has much better handling and acceleration, and better options and amenities. 2nd, the Volt is cheaper to own than a Cruze, and that will only increase with time, even without the tax credit. 3rd, that doesn’t include external costs!

            We can’t ignore external costs, right??

  16. anon2

    I still get a kick out of these Picketty conversations. The bottom line for me is what it says about the state of macroeconomics. The commentators on the left agree, generally, with Picketty. The commentators on the right disagree, generally, with Picketty. As far as I can tell there is no academic discipline for macro economics. It just does not exist. What does exist is a lot of preexisting beliefs that continually need to be restated to keep the faithful in line.

    I have no idea if Picketty is right or wrong. What I do see is lots of crony capitalism, entrenched interests and lots of preferences for already rich people. What I do see is little different from the King and his courtiers of past times.

    Government of the rich, for the rich and by the rich.

  17. Ben

    From page 46 in the Kindle, “Capital is defined as the sum total of nonhuman assets that can be owned and exchanged on some market. Capital iincludes all forms of real property (including residential real estate) as well as financial and professional capital (plants, infrastructure, machinery, patents, and so on) used by firms and government agencies. ”

    I have heard (though I have not the book) that one of Picketty’s breakthrough s was identifying how much wealth owned by people in one country (i.e France) was hidden off shore in shell corporations (i.e. Swiss bank sets up a Panamanian corporation for its French client) by examining why international flows didn’t balance. But yes, all wealth that can be exchanged is included in Picketty’s definition of capital.

  18. 2slugbaits

    Rick Stryker Piketty’s “second law” does have a distinguished lineage. Solow in his 1956 paper.

    Indeed. That was exactly my point when I said the his second law was “fundamental” in the dictionary sense of the term, meaning “base or foundation.” The Harrod/Domar/Solow approach was the base or foundational approach.

    Let me make yet another attempt to clarify why the magnitude of the magnitude of the capital consumption (hereinafter referred to as “d”) matters, and why a pedagogical value of 10% tends to mask that reason. This goes back to JDH’s first point about the infinity problem if “g” goes to zero. If “d” is 10%, then there is no plausible scenario under which the denominator could be anything but a positive value. And this matters because Krussel and Smith insist that it must be positive even if “g” is zero. But if “d” is something much lower…say 1% or 2%, then there is nothing at all implausible about “g” being negative at -1% or -2%. With that you have exactly the same infinity problem. The point is that this zero denominator problem applies irrespective of whether or not you include “d”. Also, in the real world these things tend to be discrete, so there is no particular reason negative growth could not pass through zero and be stronger than “d”, which would flip the sign of beta.

    One of the problems I have with your difference equations is that Y(t) and Y(t-1) are strictly flow variables in your formulation because they represent income in the NIPA sense. As Piketty says many times, that is not his definition. You should think in terms of net wealth, which includes stock variables as well as new income flows. And one of those stock variables will be non-productive financial claims equal to the value of the capital depreciation. If you don’t add those financial claims into wealth then you are committing the crime of Immaculate Capital Consumption.

    I think the constancy of the savings rate is being overplayed. Piketty says that this second “law” only holds under some fairly severe conditions. It is a long run tendency. As Piketty says, “…it is important to be clear that the second fundamental law of capitalism, b = s/g, is applicable only if certain crucial assumptions are satisfied.” Piketty goes even further and says that predictions of the savings rate are “extremely uncertain.” The savings rate can and will change over time. His argument is not that the savings rate is fixed, constant and predictable. His argument is that the second law is “an asymptotic law, meaning that it is valid only in the long run: if a country saves a proportion s of its income indefinitely, and if the rate of growth of its national income is g permanently, then its capital/income ratio will tend closer and closer to B=s/g and stabilize at that level.” Note that Piketty is not saying a country’s savings level must be constant; he is saying that if its savings level is constant and if national income is permanently fixed at “g”, then and only then will the second fundamental law hold over the very long run. That’s my idea of a very qualified and conditional statement.

  19. Ben

    Also, from pp 43-44 of the Kindle edition: “In order to calculate National Income, one must first subtract from GDP the depreciation of capital that made this production possible: In other words, one must deduct wear and tear on buildings, infrastructure, machinery, vehicles, computers, and other items during the year in question…

    …Then one must add net income received from abroad (or subtract net income paid to foreigners, depending on each country’s situation).

    Once these non-standard definitions of income and capital are established, I cannot see any reason to quibble with Picketty’s 2nd law. Depreciation is already removed from GDP before the ratio is taken and saving rates are defined vis-a-vis this definition of income after depreciation has been removed.

    1. James_Hamilton Post author

      Ben and others: The question is not whether Piketty accurately defined his s as the ratio of net saving to net income– indeed he did. The question is whether it makes any sense to assert that a magnitude so defined would not change as the growth rate declines.

      1. Nathanael

        It stays high as the growth rate declines. This is observed behavior, and I explain in other comments why it is observed behavior. In short, rentiers will change the laws in order to *keep* the savings rate high.

        This is asymmetric with respect to time; when the growth rate rises, the ratio does not stay constant. Rentier behavior is different when there is inreased growth than when there is slowing growth.

        Yes, there is a limit to the ability of rentiers to keep this ratio high when growth stops, but by that point we aren’t living in a market system any more.

        Some examples of changes which kept the savings rate high: The 2005 bankruptcy bill, and the changes to make student debt non-dischargeable. These eliminated the “negative savings” caused by defaulted bonds. These also created “debt slaves”, people who labor and hand their wages over to creditors immediately and forever.

        Study your economic history, and you’ll realize that debt slaves often eventually get converted into chattel slaves. Sometime after that is the point at which the ability of the rentiers to keep the savings rate high stops. It’s happened a few times in history.

        The alternative is a massive transfer of wealth from the rich to the poor. This is generally done by a jubilee — abolishing debts, and wiping out the wealth associated with the debts.

  20. Ben

    2 days ago, I had not yet purchased Capital in the 21st Century , by Thomas Picketty. I was most impressed with Professor Hamilton’s critiques as well as critiques by Krussel & Smith, Galbraith, and Summers. Yesterday, I bought the book and have only gotten to the definitions section. I must say that DeLong is correct. All of these critiques depend fundamentally on misreading Picketty. Picketty is very clear and up front about the fact that he is using non-standard definitions of Income, Capital, and Savings. He goes into a great deal of depth to explain why he departs from the standards. You can argue about the appropriateness of the definitions, but what you cannot do is impose your own definition on his analysis. It is pretty clear that all of these critiques did exactly that. Did the critics even read the book?

    Income = output is fairly standard economics. GDP is at once all output produced in a country and all income of that country. Not to Picketty. Picketty wants a measure of income that subtracts all depreciation and accounts for net domestic claims on foreign income. To Picketty, income is not equal to output.

    Capital is the stock of plant, equipment, and machinery used to produce output according to standard economics. Not according to Picketty. By capital, Picketty means wealth, and he explicitly says he will use these terms interchangeably throughout the book.

    S is the rate of income savings using Picketty’s definition of income, while g is the growth rate of the same – which is not the same as output growth and output savings.

    Using Picketty’s definitions, K/Y = s/g merely describes the steady state. Since s is the increase in the capital stock and g is the increase in income, the steady state will occur only when their ratio equals the ratio of wealth to income. This is the second fundamental law. The first is just as simple.

    I must say I am most disappointed with Hamilton et al. I expect a critic of this caliber to have read the book being critiqued and to have retained it. You have better reading comprehension skills than you have exhibited in these critiques.

    1. Patrick R. Sullivan

      ‘Picketty is very clear and up front about the fact that he is using non-standard definitions of Income, Capital, and Savings. ‘

      So your argument is that Piketty is talking about something different than the rest of the world, when he says ‘capitalism’?

      If so, then of what use is the book?

    2. Patrick R. Sullivan

      ‘By capital, Picketty means wealth,….’

      Then why didn’t he title the book ‘Wealth in the Twenty-First Century’.

  21. Rick Stryker


    I would agree that the magnitude of d matters for the empirical question concerning how much Piketty is overestimating the capital to net income ratio. I offered some calculations on that in my comment above. But I don’t agree that the magnitude of d has any bearing on the theoretical point that JDH made on Piketty’s model. In fact, I think that’s a major conclusion of JDH’s current post, that d doesn’t matter for the theoretical point.

    As I understand you, I think you are suggesting that both models have a problem with the capital to income ratio going to infinity for some parameter choice and that d very big obscures that problem. Both models have limiting cases it’s true, but the problem with Piketty’s model is not that the capital to net income ratio goes to infinity at g = 0. The problem is how it goes to infinity. JDH I think looked at the limiting case to clarify what’s actually going on in the model. It might be worthwhile to look at both the standard model and the Piketty model limiting cases in more detail to understand the differences between the models.

    In the Piketty model, capital K is always increasing, since depreciation is always covered and some saving is additionally added to K. However, net income Ynet is not necessarily always increasing. However, in the non-singular case, net income will eventually catch up to the growth rate of K for large t. At that point, the ratio of K/Ynet will converge to s/g. But the details of how Ynet catches up is the problem with Piketty’s model.

    If growth is normal–say g = 3%, d = 2%, Y0= K0 = 100, and s = 10%, Ynet just starts increasing along with K and eventually has the same growth rate as K when the ratio of K/Ynet is approximately s/g = 3.33. But things get more interesting as g drops. For example, if g = 0.25%, Ynet will start to drop at first, but then it will eventually turn around catch up to the growth rate of K, so that in the limit K/Ynet = s/g = 10%/0.5% = 40. But what happens if g = 0.01%? In that case, Ynet will continue to drop to a very low point close to 0, over thousands of years, before it eventually stabilizes at a very low point and then starts to climb at the same rate as capital K, so that the ratio K/Ynet = 1000 in the long run.

    That’s the problem with Piketty’s model. As g declines, agents starve themselves in order to keep replenishing capital. In the limiting case when g = 0, netY just keeps dropping to zero. Since there is no g, Ynet never stabilizes and then turns up to catch up with the growth rate of capital. In Piketty’s model, agents just keep replenishing depreciation of K while they starve, despite the fact that they produce Y0 income per year. In the limit, all the income goes to K and none to themselves. And, amazingly enough, in the book Piketty actually compares this crazy behavior of the model with Marx’s view.

    The standard model behaves very differently. K is not necessarily always increasing as in Piketty’s model. Looking at the limiting cases in the standard model is pretty interesting too. If for example we take g = -1%, d = 2%, s = 20% (gross saving rate), and Y0 = K0 = 100, we would see that K starts out by rising, since the saving that is being added to K is much greater than the depreciation that is being lost. But Y is always declining. So eventually, the saving that is being added from a smaller Y is not enough to compensate for the depreciation of K and K starts to decline. As t becomes large, both Y and K decline at the same rate, such that there ratio approaches s/(d + g) = 20%/(2% – 1%) = 20. Both go to zero at the same rate as t goes to infinity, with a ratio of 20 in the levels.

    In the limiting case, when g = -d, the negative growth rate of K never catches Y. Both are going to zero over time, but Y is always going to zero faster, and so the ratio of K/Y blows up. But in all these limiting cases in the standard model, consumers always save sY and consume (1-s)Y. Contrast that behavior with the Piketty model limiting case. In the Piketty model, even though there is the same amount of presumably ample income Y0 each period with g = 0, the economy saves itself into starvation in order to keep adding to K.

    None of this theoretical argument has anything to do with the value of d.

    You also mentioned that if g is negative and has absolute value greater than d, then the ratio K/Y can go negative. But that can’t really happen in the standard model. With g < 0, Y goes to zero but not below. In the case that g d, K and Y both decline eventually, but the ratio of K/Y just blows up faster than the case g = -d. However, you could get a negative K/Ynet ratio in Piketty’s model unless you constrain it in some way. The reason is that since the depreciation of K is always paid for from Y, with Y declining Y will eventually go negative to pay the depreciation cost. When that happens its growth rate will eventually stabilize such that K/Ynet = -s/g, another oddity of the Piketty model.

    1. Rick Stryker

      typo alert in comment above

      “In the case that g d, K and Y both decline eventually” should read “In the case that g d, K and Y both decline eventually”

      1. Rick Stryker

        Well that’s interesting–I got the same typo again. I had it right the first time with no typo but the blogging software is confused for some reason. Let me write the sentence out in all English.

        “In the case that g d, K and Y both decline eventually” should read “In the case that g is less than 0 and the absolute value of g is greater than d, K and Y both decline eventually”

    2. Ben

      “In Piketty’s model, agents just keep replenishing depreciation of K while they starve, despite the fact that they produce Y0 income per year.”

      No they do not. Picketty does not say they will replenish the capital stock. He says to the extent they do not, wealth falls. By Picketty’s definition of income, the drop in wealth must be accounted for in the flow variable income.

      Read the book, people!!!

      1. James_Hamilton Post author

        Ben: If the ratio of net saving to net income is positive, it means that gross investment has to be bigger than total depreciation. This is simply the definition of what it means to have a positive value for net saving. Net saving positive by definition means that saving is bigger than depreciation.

        Having gross investment bigger than total depreciation means that you not only replenish the capital stock, you make it grow every year. That means it will cost you every more just to replenish the stuff that wears out every year, because if the capital stock is bigger this year than last, then the total depreciation expense will be bigger this year than last.

        Please understand that the core issue we are challenging is Piketty’s assumption that the net saving rate would not decline if the growth rate of income declines. The basis for our challenge is the observation that if you try to maintain a constant net saving rate as the growth rate went to zero, you would indeed be forced, as an unavoidable implication of elementary mathematics, to starve yourself as you try to keep up with a growing cost every year of trying to replenish the capital stock.

        Piketty’s calculation of what would happen to the distribution of wealth when the growth rate declines is based on a fundamentally erroneous assumption about saving behavior.

        1. Kevin Donoghue

          James Hamilton,
          If it were indeed the case that Piketty made the assumption that you attribute to him then he would certainly be making a mistake. Many of your readers clearly despise Piketty and they will take your word for it. But some of us have read the book. AFAICT he does not in fact make the constant-saving assumption you keep referring to.

          As an example of that sort of thing he does say, consider this: “Despite wide variations in individual behavior, we find that savings rates increase with income and initial endowment, but variations by age group are much smaller….” Are you really trying to say that Piketty believes that the savings rate is increasing in the level of income, but he also believes it is independent of the rate of growth?

          I suppose it’s possible that he may believe such a thing, but you present no evidence.

          1. James_Hamilton Post author

            Kevin Donahue: Please tell me how you interpret (and how you are suggesting that Piketty interprets) the equation β = s/g, where remember β is the ratio of capital to net income, s is net saving as a fraction of net income, and g is the growth rate. Does this equation determine s as a function of β and g, or does it determine β as a function of s and g? I am claiming that Piketty uses it in the latter sense, in particular using it to calculate the implication for future β of changing g while holding s constant. That is what I am objecting to. When g changes, s has to change. If you use the equation, as I claim Piketty does, to predict a value for β by changing g and holding s constant, then you are assuming nonsensical behavior on the part of the savers.

          2. Tom

            Kevin, instead of repeatedly denying the point with nothing more than an accusation that others haven’t read, respond to the point. What have you read that in Piketty that disagrees with either of the following points:

            – Piketty believes capital/output inevitably rises when growth falls
            – Piketty derives that conclusion from holding the savings rate steady as growth slows in the formula,
            capital/output = savings/output/growth
            – Piketty nowhere in the book acknowledges that savings rates respond to changes in growth rates.
            – Piketty nowhere in the book puts forward any alternate theoretical argument why capital/output should rise with slowing growth.

            The one example you have cited is a description of how individual savings behavior varies by income over a population. That has nothing to do with whether national savings rates respond to changes in national growth rates over time.

            If in your opinion Piketty implicitly acknowledges that national savings rates adapt to changes in national growth rates, how does that not invalidate his conclusion that capital/output must rise when growth slows?

          3. Kevin Donoghue

            James Hamilton: “Does this equation [β = s/g] determine s as a function of β and g, or does it determine β as a function of s and g?”

            Neither. It’s an asymptotic condition. Think of sequences {β}, {s} and {g}. Under certain assumptions, which Piketty spells out quite carefully, the equation β = s/g must hold in the limit. But in general it won’t hold for every term of the sequence.

            The determinants of g are population growth and technology, whereas the savings rate s is determined by the desire of households to provide for their heirs. Obviously there are variables which constrain the evolution of both g and s; they are not independent variables. But to say that we must write either s = f(β,g) or β = f(s,g) is too simplistic.

            The crucial point is that even in societies where growth is very slow the desire to accumulate can be very strong. There are many historical examples of this. A very low value for g doesn’t ensure a very low value for s. Ancient Egypt wasn’t a fast-growing society, on average, but a lot of monuments got built all the same.

          4. Tom

            Kevin, your reply is another dodge. The point of Jim’s post is to demonstrate what’s wrong with Piketty’s conclusion that capital/output must rise when growth falls based on the behavior of his capital/output = savings/income/growth model when the savings rate is assumed stable as growth slows. You apparently interpret Piketty as leaving some out by which the model doesn’t apply when growth is zero. That’s not the point. The point is that the savings rate always responds to changes in growth rates and when you admit that Piketty’s theory that capital/output must rise when growth slows falls apart.

          5. Nathanael

            The “savings rate” among capitalists stays high when growth slows. Empirical fact. Deal with it. Piketty is right.

            How does it stay high? Manipulation of the legal system, mostly. Stuff like the 2005 Bankruptcy Bill, making it harder for debtors to get rid of their debts, *also* increase the calculated “savings rate” of the creditors who hold the notes of those debtors.

            Once you understand the mechanism, you begin to reallly see what’s going on.

            And yes, Piketty is correct.

          6. Nathanael

            And yes, there does *eventually* come a point where the rich are unable to keep their savings rates up even by massive changes to the legal system. By that point capitalism is long gone and we’re back to feudalism.

            Piketty does not lay this out in detail, but that is the point at which his model stops working, and it’s a very depressing point.

    3. 2slugbaits

      Rick Stryker I blame part of the confusion on Piketty and part of it on JDH. In JDH’s example he got to the infinity problem by sneaking in an additional increment of capital replenishment stock without also having enough savings to create the stock. For example, in the first period after the initial position savings is only $10, but the total amount of new capital is $19, which is what’s needed to fully replenish the capital stock. There is still an infinity problem, but there is a better way to explain it. But doing it that better way involves a slight correction to Piketty’s version of the second fundamental law. When Piketty refers to the second fundamental law, he is appealing to the old Harrod warranted growth model. Under that model G = [S / (K/Y) – S]. So if G = zero and S is fixed, then obviously the (K/Y) in the denominator has to go to infinity. That does not mean net income goes to zero. It means capital endlessly accumulates each period. With a little algebra you can rearrange terms and express the relationship in terms of (K/Y) on the left hand side. So we get (K/Y) = S[(1/G) +1] Note that this is slightly different from Piketty’s version. Piketty left off that little “+1” at the end. It turns out not to make much practical difference, as I will show. So instead of calling it the second fundament law, Piketty should have called it an approximation to the second fundamental law. Let’s work a couple of examples.

      Let S = 0.10, and G = 0.05. In that case (K/Y) = 2.01.

      Let S = 0.10 and G = 0.05. In that case (K/Y) = 3.43

      Let S = 0.12 and G = 0.06. In that case (K/Y) = 2.12.

      Let S = 0.20 and G = 0.06. In that case (K/Y) = 3.53.

      Let S = 0.20 and G = 0.03. In that case (K/Y) = 6.87.

      If you run the first example without the “+1” the answer is 2.00 rather than 2.01. If you run the second example without the “+1” the answer is 3.33 rather than 3.43. So as a practical matter Piketty’s formulation is good enough for his purposes. The virtue of Piketty’s approximation is that it is easier for the reader to keep the fundamental relationship in mind, and in this case I’m saying that the “+1” is not fundamental enough to clutter the reader’s mind.

      If you assume depreciation with zero growth, then you have to set savings exactly equal to capital consumption or you get a logical impossibility. If there is depreciation without saving, then growth must be negative and by definition growth cannot therefore be zero.

      The more important question is how to interpret Piketty’s assumption of a constant savings rate. My reading of Piketty is that he is saying there is a tendency for capitalist economies to settle into a constant savings rate. That’s an empirical question. His interpretation of his data suggests that over the long run capitalist economies do settle into a long run savings rate. This conflicts with endogenous savings models, but that might just mean those endogenous savings models are not properly microfounded. I would agree that for 99.999976% of the world’s population, the reason people save is for retirement and lifetime consumption smoothing. The problem is that 0.000026% of the world’s population controls ~10% of global GDP and many of them are octogenarian cranks. Is it plausible that these people are saving because they want to smooth out consumption over their lifetimes? For them the act of saving is a kind of consumption even if it means pouring their wealth into that ocean. Or buying the Clippers for $2B. A famous economist once illiterated that capital accumulation is all about “the perpetual postponement of pleasure.” Saving for the sake of saving. Ever read Max Weber’s “Protestant Ethic”? I first read it in high school and by chance reread it a few months ago.

      A second issue in Piketty’s thesis is whether or not “r” can be greater than “g”. Piketty’s argument is that wealth is cumulative whereas income is a flow variable. Wealth also includes assets that do not depreciate. It also includes assets that earn an overseas return, which can easily exceed “g” at home. In short, there are plenty of reasons to believe that “r” can exceed “g” provided we recognize that “r” is not just the return to productive, physical capital.

      Finally, the biggest moral question that lurks around Piketty’s thesis is whether or not the improvements in the absolute level of material well-being that advanced capitalism can provide are worth the cost of extreme inequality. I think this is a real issue and there is no simple answer. There is no question that capitalism can and has improved the material well-being of 3rd world countries. And if we look at inter-country comparisons inequality is falling. But if we look at intra-country data income and wealth inequalities are on the rise almost everywhere. There are also reasons to be concerned that extreme inequality won’t just present us with moral and political issues, but it at some point it might hurt economic growth. The value of Piketty’s book is that he provides important data sets and reminds us that wealth is a cumulative process and the wealthy will get wealthier long before the poor will become middle class. The usual explanations for rising inequality (e.g., skill levels, education, superstar, etc.) go a long way towards explaining inequality within the bottom 99%, but those explanationos do not come close to explaining the extreme inequality of the super-plutocrats that Piketty has in mind.

      1. Rick Stryker


        JDH didn’t sneak anything into his example. His example is correct and it motivated me to read the book and write down Piketty’s model for myself. I verified JDH’s example by solving the difference equation for capital in Piketty’s model for g = 0.

        Piketty’s statement of the second law is correct given his assumptions. We can easily verify this fact.

        Let Y(t) = gross income at time t, netY(t) = income net of depreciation at time t, K(t) = the level of capital at time t, s = the net saving rate, and g = the growth rate of Y. Then Piketty’s model is simply

        K(t) = K(t-1) + s*netY(t-1)
        Y(t) = (1+g)Y(t-1)

        In the long run, everything will grow at rate g. In particular, netY(t) will grow at rate g for large t so that in the long run

        netY(t) = (1+g)netY(t-1).

        Since K(t) will grow at rate g in the long run, the ratio K(t)/netY(t) = K(t-1)/netY(t-1) = B, a constant for large t
        If we divide the capital equation by the netY(t) equation, we get for large t

        K(t)/netY(t) = K(t-1)/((1+g)netY(t-1)) + snetY(t-1)/[(1+g)netY(t-1)]


        (1+g)B = B + s

        Therefore, B = s/g

        Piketty’s formulation of the second law is right, given his assumptions. There is no extra 1 in there.

    1. Rick Stryker


      That the capital stock gets always replenished is implicit in Piketty’s second law. Unfortunately, that’s not so obvious given Piketty’s imprecise and misleading explanation of where the second law comes from. Let me try to explain how to derive the law intuitively. The real explanation is lot more intricate I’m afraid.

      The first thing to realize about the second law is that net income is changing at some rate in the long run. Since that’s true, it must also be the case that in the long run, K must also be changing at the same rate. If they are both changing at the same rate in the long run, then there is some level of K and netY such that the ratio is constant as time gets longer and longer. Piketty’s second law is asserting that that constant ratio is equal to s/g. The common growth rate in the long run is a critical point to understand about the second law.

      Now let’s think intuitively about how we get to that long run in a particular case. Let’s take the case when g is very low, say g = .01%, so that Y is barely increasing in the long run. To get to the long run of the second law, K must be increasing at some rate that we need to figure out. But we know that every period, K starts life decreasing because of depreciation. So in every period, we must add back the depreciation of K to itself just to keep K constant. And then we need to add a little more to make sure that K grows. The depreciation that we add back to K must come from income Y. And the little bit more that we add back comes from what we save from income after we paid out the depreciation cost, i.e., we save from net income. For Piketty’s second law to be true, K must be growing in the long run and the only way that can happen is if depreciation of K is always paid for from Y and we always save a little more from net Y.

      How do we get to equilibrium? If you think about it, since K continues to grow, the depreciation cost continues to grow as well and in the long run we’ll need to subtract more and more from our barely growing Y just to pay the depreciation bill. So, that means that net income will continue to fall. But since net income falls, we’ll save less and less to contribute to the growth of K and the growth of K will have to slow down. Eventually, K will grow at rate g–in the very long run.

      Why is that? The reason is that K and netY have to grow at the same rate in the long run, and netY will grow at rate g in the long run. netY = Y – d*K, or in english, it’s equal to income Y (which is growing at rate g) minus depreciation times the level of K. But K is slowing and converging to something close to initial income divided by d, or Y0/d, so d*K is converging to Y0, initial income. Since we are subtracting something that is almost a constant from Y in the long run, netY must be growing at the same rate as Y, which is g. And that means that K is also growing at rate g.

      We are almost there. In long run equilibrium, the incremental growth to K is g*K, since K is growing at rate g. And where does that incremental growth in K come from? It comes from savings from net income, i.e., s*netY.

      Thus, g*K = s*netY, or as Piketty writes the second law

      K/netY = s/g

      There are a couple crucial takeaways from this:

      1) In long run equilibrium, what makes Piketty’s second law true is that K and netY have to eventually grow at a common rate g
      2) For low g, the only way that will happen is if net income declines toward zero and then grows at rate g from a low level.

      Point 2 is what JDH has been hammering home.

      You’ll note that the only point I glossed over was how in the long run the depreciation from capital, d*K, becomes approximately constant at Y0/d (for small g). I left that out so as not to have too much complexity in the explanation.

      Hope this helps.

      1. Rick Stryker

        I noticed that I omitted an important detail in my explanation above. d*K does go to a constant for small g as I mentioned. But it’s important to realize that for g > 0, d*K grows at rate g in the long run, and therefore netY grows at rate g in the long run. Since Y is growing at rate g and is the source for growth in K, in the long run both the replacement of the depreciation and the savings from net income must be growing at the same rate as Y. And K must grow at rate g in the long run.

  22. Ben

    Professor Hamilton,

    Picketty is using non-standard definitions for income, capital, and savings. You are imposing standard definitions on his analysis. To understand what he is saying start with GDP. Subtract depreciation (but note Picketty is not saying the worn out capital is being repaired/replaced. If it isn’t, this is a drop in wealth which is a realized income loss). Next add/subject net domestic/foreign claims on foreign/domestic output. This result is what Picketty calls income. Savings is likewise defined by Picketty as the amount of this non-standard income definition that is not consumed. Note that this savings is neither net nor gross savings according to the standard definition. Capital is wealth according to Picketty. Finally g is not the growth rate of GDP. It is the growth rate of this odd income as defined by Picketty.

    When Picketty says K/Y = s/g he is merely describing the steady state. Y, after all is not GDP in his model. In the special case of g=0, Y=0, by Picketty’s definition of Y. To claim that s must be sufficient to cover all depreciation so people starve is to misunderstand Picketty. S is merely the fraction of income (as defined by Picketty) that is not consumed. As I pointed out above, the savings don’t have to overcome the depreciation. A nation doesn’t have to save at all, but any depreciation not covered by savings will, by Picketty’s definition of income, be deducted from this flow as a realized capital loss.

    1. Peter K.

      I have not read the book, but does how income flows becoming capital stock enter into the equations under discussion about savings? At his blog, DeLong recently asked “But if the savings rate necessarily falls as the wealth-to-annual-net-income ratio rises, why was the (gross) savings rate half again as high back before World War I when the economy was wealth-dominated as it is today? ”

      It could be that back before World War I, more income flows became capital stock so that the gross savings rate was half again as high as today. Progressive taxation of income and of inherited wealth happened in the 1910s and ’20s.

    2. Tom

      Ben, the important point is that Piketty derives his conclusion that capital/output rises when growth slows by artificially holding the savings rate steady as growth changes in his model of the relationship of national aggregates, capital/output = savings/income/growth.

      That is the key error – assuming savings rates won’t respond to changes in growth rates. Don’t get lost in the other minor details. If savings rates are allowed to respond to changes in growth rates, Piketty’s conclusion that capital/output must rise when growth slows disappears into thin air. Piketty’s whole theoretical basis for concluding that inequality will rise as growth slows disappears.

      As for the minor details, I accept your point that Piketty doesn’t say net savings must be positive. Jim might need to revise his elaboration of what happens with Piketty’s theoretical models if taken to extremes by allowing net savings rates to go negative. That’s not important.

      Piketty’s not making up a non-standard definition of income. What you have described is national income, a standard aggregate of national accounts defined in UN conventions. What’s non-standard is Piketty’s model of the interaction of national aggregates, capital/output = savings/income/growth. This model looks similar to other popular models, but Piketty’s model is crucially different in that it relates net aggregates, rather than gross aggregates as those other similar models do. That’s not important either.

      One key difference of Piketty’s model from others is that it leads to more absurdly wrong conclusions when its net savings rate is assumed to remain constant as growth changes, whereas those other models lead to less absurdly wrong conclusions. That’s not important. All models lead to absurdly wrong conclusions when net or gross savings rates are assumed to remain constant when net or gross growth rates change, the differences in how absurdly wrong are only a matter of degree, and so not important either.

      This is about whether or not the model is crucially flawed, though no such models are perfect and some are less perfect than others. This is about whether it’s correct to assume that capital/output will rise as growth slows based on an assumption that savings rates won’t respond to changes in growth rates.

      Piketty does make that assumption, and it’s a wrong assumption.

      1. Tom

        Sorry, I meant to write: “This isn’t about whether or not the model is crucially flawed, …”

      2. Nathanael

        Piketty does not make that assumption. Read it again.

        Perhaps you’re having trouble with Piketty’s use of “savings rates”. What Piketty means by “savings rates” (again, not the standard economics meaning) do, in fact, consistently stay high when growth drops, although they may change when growth increases. (There’s an asymmetry with respect to time here.)

  23. Ben

    Okay, I made a mistake. When g=0, Picketty’s model doesn’t imply Y=0. It is clear, though, that if s > 0 that wealth (i.e. K) is increasing while Y is not. This scenario would not be a steady state. A steady state only occurs if Y (as defined by Picketty) is increasing/decreasing at the same rate as K. S negative implies that the economy is not keeping up with depreciation and net foreign claims, which means g is negative, too. For a steady state to exist when g=0, s must be zero, too.

  24. Ben


    From your earlier post:

    “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.”

    I object to your claim, “… making gross investment a total of $55…” That is not what Picketty says. Picketty does not say what gross investment will be. It could be $5. If it is $5, this implies deterioration of the capital stock to $455. Consumption is $95 as we consume out of wealth, so savings are $50 – $95 = -$45 and s = -$45/$50 =-90%. With less wealth, we will have less productive capacity and GDP will fall, ergo g < 0. Note that g is not the growth rate of GDP. The capital stock and GDP will continue to fall until depreciation can be sustained by the savings. The 2nd fundamental law only applies in the long run.

  25. James_Hamilton Post author

    Ben: Net investment is defined as gross investment minus depreciation. If you agree that Piketty’s assumption means net investment would be $5, and if you agree that under the stated conditions depreciation is $50, then you cannot possibly argue with the conclusion that under these conditions, gross investment would have to be $55.

    If net saving is 10% of net income, and if depreciation is 10% of the capital stock, and if the capital stock is $500, then the numbers must be exactly as I describe– gross investment has to be 55% of GDP. If you perceive that it makes no sense to think that gross investment would be equal to 55% of GDP under those conditions and would be predicted to rise even further as a percent of GDP from those already ridiculous levels, then you are beginning to understand what I have been saying from the beginning.

  26. Ben

    “If you agree that Piketty’s assumption means net investment would be $5, …”.

    No I don’t think Picketty is assuming any particular level of either net or gross investment. While under standard definitions S=I holds as an accounting identity, Picketty’s definitions are so different that both S and I mean something different in the context of his model. Can you explain why you think Picketty postulates $5 net investment?

    I believe I understand your critique, professor, and before I began reading the book I fully agreed with your analysis. But Picketty defines his terms in a way that turns everything on its head. I was rather perplexed as to why he gave standard terms new definitions, but he does explain the difficulties in separating the standard terms from aggregated data sets available.

  27. Tom

    Though as I said the debate here is about whether savings rates can be assumed steady as growth slows, and not about the quality of Piketty’s model or what he calls “law”, it’s worth thinking a minute about what the model means.

    The model is, capital/output = savings/income/growth.

    Capital is the net monetary value of assets, private and public, or the sum of household and public sector net worth.

    Savings is net savings, growth is net national output growth.

    Output and income are both net national and thus equal, so the formula reduces to:

    The value of assets equals the net savings of the income earned from them times the inverse of that income’s growth rate.

    Chew on that for a while. There’s no end to its bizarre implications.

  28. Ben


    In response to Kevin Donahue you said:

    “Please tell me how you interpret (and how you are suggesting that Piketty interprets) the equation β = s/g, where remember β is the ratio of capital to net income, s is net saving as a fraction of net income, and g is the growth rate. Does this equation determine s as a function of β and g, or does it determine β as a function of s and g? I am claiming that Piketty uses it in the latter sense, in particular using it to calculate the implication for future β of changing g while holding s constant. That is what I am objecting to. When g changes, s has to change. ”

    I don’t think Picketty is claiming this equation holds true at all times – only that the two sides converge to equality in the long run. Imagine wealth growing at a faster rate than income. Savings must be high enough to make this possible. But next year, the same savings rate will not increase wealth by the same percentage even if g remains the same. β will fall until it reaches s/g. The same convergence holds if wealth grows slower than income. With savings growing larger each year until β converges to s/g from below. Nothing in Picketty’s assumptions requires that s or g be determined by β.

  29. 2slugbaits

    I think the problem here is that people are misunderstanding what Piketty means by savings. In PIketty’s argument the saving rate is the rate at which new capital accumulates above and beyond the gross capital relative to net income. Remember, most of Piketty’s book is not about 21st century America with all of our fancy NIPA tables. His project is to use 18th and 19th century records to reconstruct that net rate at which new capital is formed. So if the initial capital stock is 1000, and depreciation is 1%, or $10, and gross income is $100, and net income is $90 and gross investment is $10, then net investment is zero. See why? Gross investment exactly offsets depreciation losses of $10, meaning there is no net addition to the capital stock. In this example the saving rate is not the ($100 – $90) / $100 = 10%. The saving rate is defined as the ratio of net investment to net income. In this case it would be the difference between gross investment of $10 and the amount needed to cover depreciation, which is also $10. So it’s zero. The saving rate is therefore $0 / $90 = 0%. Now let’s assume a 5% growth rate and plug all of this into Piketty’s simplified version of the fundamental law: (K/Y) = s/g = 0%/5% = 0%. Yes, the capital/output ratio asymptotes to zero. Why? Because the growth rate is 5% and the capital stock is unchanging. That means output (Y) will increase (e.g., increasing labor). At the limit (K/Y) will go to zero. But suppose total saving equaled $15 rather than $10, all else equal. In that case the new saving rate would be $5/$90 = 5.55%. The new (K/Y) target would be 5.55%/5% = 111%. If the growth rate falls to 3%, then the new (K/Y) would be 185%.

    I suspect that many reviewers are looking at Piketty’s model through the eyes of a 21st century economist used to 21st century economic models and terminology. Piketty’s work is primarily a work of history rather than economics, so what’s needed is an historian’s view. The key thing to keep in mind is that for Piketty savings means the net addition to capital…it’s the change in investment relative to net income. Saving represents the rate at which the capital stock increases relative to net income. And that is why all of these discussions about depreciation not entering into his formulation have been red herrings. People are simply talking past one another.

    1. Rick Stryker


      No, we are not talking past each other. Piketty is using the model that I and others have claimed he’s using to obtain his second law.

      Please consult Piketty’s online appendix to the book and look at page 28. You will see that Piketty explains the second law by writing down the same difference equation model I’ve been using in my comments. But that model necessarily implies JDH’s critique.

    2. Rick Stryker

      And just to generalize the comment to others who saying, “You are misreading Piketty, he really means x by wealth and he really means y by net income.” Whatever he means about those terms, Piketty has written down in his technical appendix the difference equation that governs wealth and net income. That difference equation implies mathematically that

      1) the second law holds in the long run
      2) if g goes to 0
      a) the wealth to net income ratio goes to infinity
      b) net income goes to zero
      c) wealth goes to a fixed constant, which is (initial gross income/depreciation rate)

      These are mathematical facts about wealth and net income if they satisfy the model that implies Piketty’s second law

      You can’t really argue against JDH’s point by claiming that Piketty is using non-standard definitions for wealth and net income. Whatever definitions he’s using JDH’s critique applies.

  30. Peter K.

    My question is why the focus on the savings rate? In an interview Piketty says “”My bottom line is that the average rate of return for all assets combined is not going to zero. It has been going down a little bit over the past 20 to 30 years because of the rise in the capital-to-income ratio, but it has declined less than the increase in the capital–income ratio, so that the capital share has actually increased.”

    So he admits that the return for all assets has come down somewhat because of the rise in the capital-to-income ratio. Does this have to with g slowing and s remaining constant or changing little? How does s relate to r?

    So β has risen. But r – which includes s? – has declined less than the rise in β, so the capital share (?) has actually increased. Even if s has declined? Or rather the savings rate could have declined for the 99 percent as growth and income slows, but it could increase for the 1 percent as their return on wealth is higher than most.

  31. 2slugbaits

    Rick Stryker I have looked at the appendix. Sorry, but you have completely misunderstood what each of the variables means. Savings does not mean the percent of GDP not otherwise consumed. That is a modern definition, but that is not Piketty’s definition. He explicitly says that savings is the ratio of NET investment to NET income. NET investment is the addition to productive capital and other unproductive wealth (e.g., gold, silver, jewelry, bonds, etc.) that represent claims against net national income. He uses the ratio of net investment to net income because he is trying to arrive at the rate at which productive capital and unproductive wealth accumulate relative to net national income.

    Look, the problem here is that Piketty has gone outside of the usual tools and models in most econ departments. Back in yesteryear I remember taking a few seminars under people like Fogel and McCloskey and Fenoltea. The “econometric” tools used to study pre-modern economies are the tools of what was then known as “cliometrics.” Defining savings as the ratio of net investment to net income is a common tool within cliometrics to estimate the the trajectory of the capital/output ratio when you don’t have convenient NIPA tables around that might help you figure out the capital stocks of 13th century granaries, or the relative impacts of the Plague and Hundred Years War on French vineyard capital & labor costs. Stuff like that. If you’re unfamiliar with cliometric approaches to economic history you might want to start out with the foundational book by Robert Fogel, “Time on the Cross.” Fogel won a Nobel with that book. His thesis is hotly debated today, but my point is that it’s a good introduction to the tools of cliometrics. And that’s exactly what Pikkety is doing here.

    1. Rick Stryker


      Pikety’s peculiar definition of net saving is precisely what’s causing the problem. Yes, you are correct; Piketty is defining net saving s as the ratio of net investment to net income, i.e,

      s = [I(t)-dK(t)]/[Y(t) – dK(t)]

      where I(t) is investment at time t. It’s also the case that

      K(t) = (1-d)*K(t-1) + I(t-1)]

      Solving the net saving equation for I(t-1) we have
      I(t-1) = d*K(t-1) + s*(Y(t-1) – dK(t-1)

      Now put this into the capital equation for K(t) and we get

      K(t) = (1-d)*K(t-1) + d*K(t-1) + s*[Y(t-1) -d*K(t-1)] = K(t-1) + s*netY(t-1)

      This is exactly the model I wrote down. If you set g = 0 so that Y(t) = Y0, the model can be written

      K(t) = (1-s*d)*K(t-1) + sY0

      Now, lets solve the difference equation. We get:

      K(t) = [(1-s*d)^t]*[K0 – Y0/d] + Y0/d

      Thus, as t goes to infinity, we have the three facts I mentioned

      a) K(t) goes to the constant Y0/d as t goes to infinity
      b) Net income goes to zero as t goes to infinity since netY(t) = Y0 – d*K(t) which goes to Y0 – d*Y0/d = 0
      c) Since K goes to a constant and net income goes to zero, the ratio of K to net income goes to infinity

      There is really no way to escape this since it follows from Piketty’s definitions. JDH’s point is precisely correct. At some point, you really have to stop denying mathematical reality.

  32. Ben

    Rick Stryker,

    You and Professor Hamilton are reading way more into Picketty’s 2nd law than Picketty intended. Picketty is not saying that s is determined by g and β. β is just the wealth to income ratio, s is just the savings rate, and g is the rate of growth of Y. Given s and g, wealth may be growing faster than income or income growing faster than wealth. If s and g remain the same indefinitely (which Picketty says is unlikely) then β will change each year, resulting in a sequence βi with a different value for β each year. The 2nd fundamental law says no more than the sequence β1, β2, … βn, where βi = Ki/Yi converges to β = s/g as long as s and g remain the same. Notice that Picketty never says s is a function of g. If g changes, all bets are off as s may change, too. Read from Kindle location 2885 to see that β =s/g is only an equilibrium condition for an economy taking s and g as fixed without going into any reasons why people have made the choices they have or any assertion that these choices will continue for any length of time. He is certainly not claiming that if g drops, s will remain the same no matter how little income is left over for consumption.

  33. Tom

    Rick, thanks for pointing to the note in the online appendix. I hadn’t seen that. It helps explain where Piketty went wrong deriving his β = s/g model or “law”.

    His fundamental error is that he derives a model of the relationships of nominal aggregates from the relationships of real aggregates. His relationships might hold true for real net savings and real national income growth and real wealth. But “real wealth” is an impractical concept. Nobody tries to measure it. Nobody uses it for anything.

    Even for real aggregates, it’s a somewhat dubious derivation. What he’s arguing in a nutshell is that the real net savings to real wealth ratio must hover around or “converge to” the real growth rate, or else the real wealth to real income ratio would continually grow or shrink. That’s conceptually probably usually true but since the real growth and real net savings rates fluctuate all the time you’d struggle to see any such relationship in the data. Also since nobody knows or cares what “real wealth” is, arguably nobody might care if the real wealth to real income ratio continually grew or shrank.

    1. Tom

      PS The crucial problem with the derivation, even in real terms, is that β is a stock relative to a flow. Real flows (eg income) are practical to measure relative to a previous period. Real stocks are inherently impractical and immeasurable.

    2. Nathanael

      The ratio of nominal income to nominal wealth is always equal to the ratio of real income to real wealth, for a given time period. I don’t see what your problem is with this. Everyone knows what real wealth is. “Real” in economics is merely a synonym for “inflation-adjusted”.

  34. Rick Stryker


    A major claim of the book is that the capital (or wealth) to net income ratio will approach 700%. That’s the whole point and why Piketty gets into the policy discussion of what to do about it. Piketty arrives at that conclusion by fixing s at 10% and dropping g from 3% to 1.5%. That takes the ratio from 333% to 666%. If you want to claim that Piketty never said that s won’t change if g goes down, then you are also saying that Piketty can’t justify a major conclusion of the book.

    So, pick your poison. Either you can believe JDH that Piketty’s model is flawed, in which case Piketty can’t justify a major conclusion of the book.

    Or you can trust your own reading that Piketty never claimed that s is fixed if g goes down, in which case Piketty can’t justify a major conclusion of the book.

    Works for me either way.

    1. Ben


      Where in the book does Picketty claim that g is 3% now? At Kindle position 1676 he makes extensive claims that g is 1.5% now and has been over an extended time period. He says 3% growth is quite unusual and only took place historically in economies that were playing catch up.

      Are you assuming g is growth rate of GDP? Picketty is using a different measure of income.

  35. Tom

    PPS Piketty sure is sloppy. That formula he derives,

    β [t+1] = β[t] * (1 + s[t]/ β[t] / (1 + g[t] )

    Should be

    β [t+1] = β[t] * (1 + s[t+1]/ β[t] / (1 + g[t+1] )

    Not an important mistake in this case. Just saying.

    1. Tom

      Oh dear, I didn’t find all the errors in Piketty’s formula the first time round. At first look I thought it was just a matter of mismatched time periods, of wrongly using beginning period wealth as a ratio of period income and weirdly writing g[t] for g[t+1]/g[t] when everybody else calls that g[t].

      But there’s a bigger problem that when you adjust the time reference of the wealth/income to end period wealth over period income, it changes the whole structure of the formula. The correct formula for real aggregates is:

      β [t] = β[t-1]/ (1+g[t]) + s[t]

      or β [t+1] = β[t]/ (1+g[t+1]) + s[t+1]

      compare that to Piketty’s

      β [t+1] = β[t] * (1 + s[t]/ β[t] / (1 + g[t] )

      That is a substantial difference. But note that since β is a ratio of W/Y and W is an unmeasurable real stock, the formula is purely theoretical and has no practical application.

      To construct a similar formula applicable in the real world, you have to change to nominal aggregates and introduce a revaluation component, which I’ll call R:

      β [t] = β[t-1] * ( 1 + R[t]/W[t-1] ) / ( 1+ g[t] ) + s[t]

      Anyone care to try to derive any “laws” from that formula? At least you’d be starting with an accurate accounting identity whose components can be measured.

        1. Tom

          Oh blast, I really did garble that explanation of Piketty’s mistake with his g[t] notation. I guess I’m one to talk about sloppy, but then these are blog comments not a book.

          Just to clarify, the reason Piketty’s formula is wrong is his wealth to income ratio in the formula is beginning of period wealth to period income. Fixing that to end of period wealth substantially changes the formula.

          The other non-standard quirk of Piketty’s formula is he makes his growth g[t] = Y[t+1] / Y[t], where everybody else would call that g[t+1]. That’s not as important but worth pointing out. Sorry I garbled that part of my explanation.

  36. Ben


    That says global output, which is not the same as income in Picketty’s definition. The drop in g isn’t nearly as big. I note also that he assumes s will drop to the lower end of the range he has estimated. Earlier he estimated global s to be 10 – 15 percent. Finally he says, “Obviously, this is just one possibility among others. As noted, these growth predictions are extremely uncertain, as is the prediction of the savings rate. ”

    Note that he is not making growth and savings predictions based on the 2nd law – only suggesting plausible values for the parameters.

  37. Nathanael

    “His assumption of a constant net saving rate implies that capitalists always try to increase the capital stock if they have any level of positive net income whatever. ”

    Um, that’s what they do.

    Look at Piketty’s definition of “capital” — you may know it better as “wealth”.

    If capitalists have any level of positive net income whatsoever, they try to increase their wealth, a.k.a. “the capital stock”.

    That’s what makes them capitalists.

    They may eventually find this impossible, but they will try forever, and will go to such lengths as changing the bankruptcy laws (in 2005), rewriting copyright and patent laws (several times), and other means of creating new “property” through government fiat.

    In short, Piketty’s right. Your analysis starts out right, and then you idiotically declare that this is not how capitalists behave. This IS how capitalists behave.

  38. Rick Stryker


    Like many economists, Piketty is using the terms “income” and “output” interchangeably. He does mean g = 3% and g = 1.5% in his second law formula.

    This calculation is a central claim in the book. The book’s title is “Capital in the Twenty-First Century” and Piketty is predicting that the capital to net income ratio will rise to almost 700%, based on a second law calculation, in a section called “What Will the Capital to Income Ratio Be in the Twenty-First Century?” Of course, like every economist he issues the obligatory caveats about uncertainty of the parameters–but the fact remains that this is a central claim of the book.

    Piketty repeats the claim at kindle location 4058 and then adds a new claim– that at the predicted capital to net income levels the return on capital will also stay fairly stable at 4-5 percent, implying that capital’s share of global income will be in the 30-40 percent range.

    JDH’s serious of posts and all these comments have been about the first claim–using the second law to predict that the ratio will go up to almost 700% if growth cuts in half. That prediction only happens because Piketty is using an underlying model that JDH has shown is nonsense.

    The claim that the return on capital will stay relatively stable at 4-5% is another highly problematic assertion as well, by the way.

  39. Rick Stryker

    Interestingly enough, Piketty seems to be running away from the implications of his own law. In response to a business week article on the Krusell and Smith critique, Piketty apparently sent this email:

    “We’ve never written that the capital income ratio beta=s/g should go to infinity if g goes to zero: presumably people would stop saving (i.e. s would go to zero) much before that! We’re just saying that the simplest way to explain the rise in capital-income ratios that we observe in the data in recent decades is that saving rates did not fall as much as growth rates, so that mechanically the capital-income ratio tends to rise to relatively high levels, just like in the 19th century. I don’t think they are disputing this. Also note that the rise of capital-income ratio is certainly not bad per se, and does not necessarily imply high inequality. Tell me if I missed something!”

    Except that Piketty did discuss just that case in the section of the book entitled “Back to Marx and the Falling Profit.” In that section, he explicitly considers the case when g = 0 or is close to zero. He notes that the law illustrates the logical contradiction of capitalism: with any motive to save from capitalists, whether to increase their own power or because their standard of living is already high, g going to zero implies that the capital to income ratio will go to infinity. Piketty says that explicitly.

    Piketty is now claiming that saving would go to zero as income goes to zero. Of course it would. That’s been the whole point of JDH’s and Krusell and Smith’s critique: the underlying model behind the second law does not allow saving to go to zero. And by writing down the second law as he did, PIketty did most certainly claim that saving would not go to zero as g goes to zero but would rather go to 100%.

    Piketty’s also claiming in the email that he doesn’t think that Krusell and Smith are disputing that if savings rates didn’t fall as much as g, then capital to income ratios will rise to relatively high levels. But that’s precisely what Krusell and Smith are disputing.

    A very weak response from Piketty that shows that he really didn’t have any basis for the second law.

  40. Tom

    All right, since nobody’s giving any ground and almost nobody’s responding to any important points, let’s review what we’ve found about Piketty’s β = s/g model or “law”.

    1) He derived it partly from a false accounting identity. He correctly started with the true observation that real wealth grows over a period by the volume of that period’s real net savings, Wt+1 = Wt +St, but in the process of transforming that equation, he mistakenly made his real wealth/ real income ratio, β, equal to the beginning of the period’s wealth over the period’s income. As a result he substantially misstated how real wealth / real income grows.

    Whereas he incorrectly derived: β [t+1] = β[t] * ( 1 + s[t]/ β[t] ) / ( 1 + g[t] )

    Where g is the growth rate of real national income and s is real net savings over real national income,

    Correct math derives: β [t+1] = β[t] * (1 + ( s[t+1] * ( 1 +g[t+1] ) ) / β[t] ) / ( 1 + g[t+1] )

    That is, his formula leaves out that the real net savings rate, s, needs to be multiplied by ( 1 + real national income growth, g). He also uses the wrong time periods for s and g.

    ( Reduced to its simplest form, the correct formula for the development of the real wealth to real income ratio in terms of real net savings and real national income growth rates is β is: β [t] = β[t-1]/ (1+g[t]) + s[t] ) )

    2) Piketty implicitly assumes a model that applies to real aggregates will apply to nominal aggregates. That’s a far bigger mistake.

    3) If Piletty’s boiled-down β = s/g model were true and applied to nominal aggregates, the following formulas would also be true:

    W/Y = S/Y/g , since β = W/Y and s = S/Y

    W = S/g

    Where W is nominal national wealth (private and public net worth), S is some period’s nominal net savings, and g is that period’s nominal national income growth rate.

    That is, the aggregate value of assets, W, would always be inversely proportional to the nominal national income growth rate, With a growth rate of 4%, assets would be worth a quarter as much as they would with a growth rate of 1%.

    4) If you interpret Piketty’s application of what he calls a “law” as JDH does, as something that’s always true, then wealth approaches infinity when national income growth is zero (and btw, wealth turns negative when national income contracts). Personally I’m willing to concede that Piketty may simply have used misleading language, and his “law” is only meant to be an approximation of long-term trends.

    5) If, as JDH (and I) do, you interpret Piketty as deriving his conclusion that wealth/income must rise when growth slows from the behavior of β = s/g when s is assumed constant, than besides having a dubious model, Piketty has made another grave error assuming that savings rates don’t respond to changes in growth rates. If, like Ben and Kevin, you don’t see any such contention by Piketty, then Piketty has no theoretical justification for predicting that wealth/income will rise as growth slows.

  41. Tom

    Just one more comment on this topic. If Piketty’s model, β = s/g, even very loosely predicts the long-run relationships of wealth, income, savings and growth, it’s only really important lesson is that s and g will tend to move proportionally with each over the long run.

    Go back to my reduction of the formula, W = S/g, which just multiplies both sides times income. I made the point that if it were true, wealth would be inversely proportional to the growth rate. Of course it isn’t. So how can you salvage any sense from the formula, even only as an approximation of a long-run relationship? Only if S and g (or s and g) tend to change in step and proportionally.

    Which again invalidates the conclusion that wealth/income must rise when growth slows. If β = s/g tells us anything, it’s that s will over the long run move in step with g.

  42. dilbert dogbert

    Found this over at Delong’s in a comment:

    “Jan Milch said…

    Excellens Brad! Robert Solow described pretty well what this anti-Piketty nittpicking is all about:
    ” Suppose someone sits down where you are sitting right now and announces to me that he is Napoleon Bonaparte. The last thing I want to do with him is to get involved in a technical discussion of cavalry tactics at the battle of Austerlitz. If I do that, I’m getting tacitly drawn into the game that he is Napoleon. Now, Bob Lucas and Tom Sargent like nothing better than to get drawn into technical discussions, because then you have tacitly gone along with their fundamental assumptions; your attention is attracted away from the basic weakness of the whole story. Since I find that fundamental framework ludicrous, I respond by treating it as ludicrous – that is, by laughing at it – so as not to fall into the trap of taking it seriously and passing on to matters of technique.”

    Gave me a laugh.

  43. joan

    Piketty´s saving rate is not the gross saving rate S but the “net saving rate” = S-(K/Y)d, because it is K that depreciates not Y. This gives K/Y=(S-(K/Y)d)/g for Piketty´s model and solving for K/Y we get K/Y=S/(g+d)

      1. joan

        In my previous post I showed that there is no difference between piketty´s model and the textbook Solow model which means Krusell and Smith were wrong in their claim that there is. Since they are just 2 way to write the same equation, they must predict the same S for g=0. Setting g=0 in K/Y=S/(g+d) gives S= d K/Y. When the equation is written K/Y=(S-(K/Y)d)/g to find S we must first mutiply both sides by g getting gK/Y=(S-(K/Y)d) and when we set g to zero we again obtain S= d K/Y not infinity.

        The value from Evens for K=2 does not include housing which accounts for almost half of K for the US. However he does agree with piketty about what the fact that K/Y being mean stationary while d is increasing by a factor of 2.

        ¨The previous section provided evidence that the following stylized facts have
        characterized the postwar US economy, (his data set ends in 1998)
        (a) The growth rate of per capita output is mean-stationary;
        (b) The capital-output ratio is mean-stationary;
        (c) The net rate of return paid on capital is mean-stationary;
        (d) The net share of output paid to capital is mean-stationary;
        (e) The net investment rate is mean-stationary;
        (f) The depreciation rate trends upward;
        (g) The equipment-structures ratio trends upward;
        (h) The gross rate of return paid on capital trends upward;
        (i) The gross share of output paid to capital trends upward; and
        (j) The gross investment rate trends upward.¨

        1. James_Hamilton Post author

          Sorry, Joan, but you have missed the whole point. Let s denote the net saving rate and s* the gross saving rate. Given any values for K, Y, and d, one could calculate either the net saving rate s or one could calculate the gross saving rate s*. The correctly calculated value of s would be perfectly consistent with the correctly calculated value of s*. None of this is in dispute.

          So, for example, if the value of s* happened to satisfy K/Y = s*/(g + d), then the value of s would certainly satisfy K/Y = s/g. Again, this is not in dispute.

          Instead the question at hand is, what would happen if g were to change to a new lower value g*? One possibility, if you assume that s* does not change, is that K/Y would, over time, tend toward a new value of K/Y = s*/(g* + d). A second possibility, if you assume instead that s does not change, is that K/Y would tend toward s/g*. Both predictions cannot be correct. If one is true, the other is false, because if s/g = s*/(g+d), it cannot also be the case that s/g* = s*/(g*+d). K/Y cannot trend to both s/g* and also to s*/(g*+d), because these are two different numbers.

          And this is the core of the dispute. Piketty’s calculations assume that, if g falls to g*, it is s that would stay the same, and s* that would adjust. The point of my original post, which I would still encourage you to try to read and understand, is that Piketty’s assumption that s would stay the same as g declines is quite nonsensical.

  44. joan

    In your post you says the saving rate will go to infinity when g=0 using Piketty´s equation but simple algebra shows that the gross savings rate will =d K/Y and the net saving rate will be 0 no matter which model you use so I can not agree with this part of your post

    ¨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.¨

    1. James_Hamilton Post author

      Joan: Let me try to use numbers to help you understand this. Let Y* denote GDP, say Y* = $100. Let K = $250 denote the capital stock, d = 0.05 the depreciation rate, g = 0.03 the growth rate, and s* = 0.20 the gross saving rate. I’ve picked numbers so that we’re starting out at the steady state, namely K/Y* = 2.5 = s*/(g + d) = 0.20/(0.03 + 0.05) = 2.5. To confirm that this is indeed a steady state, note that what happens in the current period is gross saving = (0.2)(100) = 20, depreciation = (0.05)(250) = 12.5, leaving 20 – 12.5 = 7.5 for net investment. So next period the capital stock would be 257.5, but GDP would be 103 as a result of the assumed 3% growth, so next year’s ratio K/Y* = 257.5/103 = 2.5, the same as it was this year.

      We could alternatively perfectly well characterize the economy I just described in terms of a net saving rate. Let Y denote net income; here Y = Y* – dK = 100 – (0.05)(250) = 87.5. For the numbers above, net saving is 7.5, so the net saving rate s = 7.5/87.5 = 3/35. The ratio of capital to net income is K/Y = 250/87.5 = 2.857, and this is correctly predicted by the formula β = s/g = 3/(35)(.03) = 2.857. So far, so good.

      Now, the question is, what would the new steady state for this economy look like if the growth rate were to decline from 0.03 to 0.015? One assumption you might make is that the gross saving rate s* holds constant at its old value of 0.20. In this case, the new steady state is K/Y* = s*/(g* + d) = 0.20/(0.015 + 0.05) = 3.0769. In other words, the steady-state capital stock associated with GDP of $100 would be K = $307.69 rather than K = $250 if the growth rate g = 0.015 rather than 0.03.

      Suppose alternatively that we assumed that the net saving rate s holds constant at its old value of 3/35. Then we predict a new value for K/Y = s/g* = 3/(35)(0.015) = 5.7143. If K/(Y* – dK) = 5.7143, that requires a value for K of 444.44 when Y* = 100.

      Under the first assumption (constant s*), the capital stock would increase from 250 to 307.69, a 23% increase. Under the second assumption (constant s), the capital stock would increase from 250 to 444.44, a 78% increase. You cannot claim that the second assumption is consistent with the first. Either the capital stock would increase 23% when the growth rate falls, or it would increase by 78%. It cannot do both.

      And this is the core debate. Piketty assumed that the net saving rate s would not fall when the growth rate falls, and as a consequence predicted that the ratio of K/Y will rise dramatically in this century. This is the key conclusion from his book.

      But look at what Piketty’s savers are doing in the above example. By always trying to increase the capital stock faster than the economy is growing, they continue to pile up capital until the only thing that stops them is the higher depreciation rate begins to eat up so much of their net income they cannot go any farther. Depreciation went from 12.5% of GDP in the initial configuration to 22.2% of GDP after the growth rate falls, and the level of net income associated with the new steady state fell from $87.50 to $77.78. What principle of saving would lead people to want to throw away money in this way, accomplishing nothing for themselves?

      1. joan

        ‘I think and I think you agree that when g= 0 the net saving rate is zero and and the gross saving rate is d K/Y. From data we know that when g=.03 it is about .075-.01 and gross saving is about .2-.3 so it is obvious that neither the net savings rate nor the gross saving rate is constant and the Piketty´s assumption that it is constant, is only an approximation for moderate changes in g. This does not bother me since all models in economics are only approximately true. Even an approximate theory of why ratio of K/Y changed so little during the the 19th century and helps us understand how we got from the the world of Jane Austen to the world of Henry Ford should be more than enough.

  45. Rick Stryker


    You are not writing Piketty’s model down correctly. Please see my comment of June 7 at 2:03 PM above. I write down Piketty’s definition of net saving and thus derive the difference equation that Piketty claims in his technical appendix and his paper is the model that underlies his second law. I then solve that difference equation for the case g = 0 and show that JDH’s critique necessarily follows.

  46. dsquared

    Here’s a fairly simple analogy that might explain what’s going on.

    Say I’m considering going on a diet. But I want to be sure that my weight stays above 90kg (I want to compete in the next Olympics as a heavyweight boxer or something).

    Doctor P says “The amount of calories that you burn every day is a function of your body weight B. So in the very long run, your weight will settle down at a level where the amount of calories that you burn is equal to the amount you ingest”

    Immediately I respond “But that model would suggest that if I were to go on a fast, I would disappear entirely!”

    Doctor P says “Yes, in the real world your behaviour would also change as your weight fluctuated. And I suppose in your limiting case, at some point you would either stop fasting or die, although you would have lost a lot of weight quickly. But for most reasonable levels of calories intake, which I thought we were talking about, my model makes sense, and this is my reason for telling you that your 1000 calorie-a-day diet is highly unlikely to leave you competing as a heavyweight in four years’ time”.

  47. dilbert dogbert

    I loves me some food fights among economists. Esp economists I find interesting. I really like dsquarded as he was the guy who said a decade or so ago that good ideas don’t need to be sold by lies. This was about the excellent Iraq adventure. The dsquared digest is no longer available to scum like me. Sad.

  48. Rick Stryker


    I saw that Brad Delong continues to ignore the real criticism of Piketty and would rather focus attention on depreciation rates. Delong is implicitly arguing that the Hamilton/Krusell/Smith critique doesn’t matter since the effect of Piketty’s bizarre model isn’t that important for low d, although Delong won’t acknowledge that the model is bizarre. Based on my understanding of Piketty’s data, it seems to me that Piketty’s claim of 2% for the depreciation rate is reasonable. But that doesn’t mean that Delong is correct. I took a closer look at Piketty’s data and, as a result of the use of his bizarre model, Piketty has dramatically overestimated the capital to net income ratio going forward even if d = 2%. The depreciation rate argument is just a red herring.

    Although Piketty’s second law is based on a nonsensical model, that doesn’t invalidate the analysis he did in Capital is Back or in the book as far as explaining history goes. Piketty remarks in the paper that his model and the textbook model are equivalent, and for the historical purposes to which he puts the second law, they are equivalent.

    As we know, the relationship between net saving snet and gross saving s in the textbook model is given by

    snet = gs/(g + d*(1-s)) and equivalently s = snet*(g + d)/(g + snet*d).

    And since K/netY = s/(g + d*(1-s)), Piketty’s second law, K/netY = snet/g, holds in the textbook model. As long as g, s, and d don’t change, Piketty is free to define snet and netY and use the second law to explain long run capital to income ratios historically. For the purpose of explaining the past, Piketty has not necessarily committed to his bizarre model–as long as g, s, and d don’t change

    As you have pointed out, the problem arises when Piketty uses the second law to project into the future, since in that case he changes both snet, which implicitly changes s, as well as g. Once he changes those values, it matters what the underlying model actually is. If the underlying model is the more sensible textbook model, then Piketty must make adjustments if he uses the second law. If we look at Piketty’s actual numbers and make the adjustments he should have made, we can see that his argument collapses even if d = 2%

    Figure 5.8 in the book shows that the world capital to net income ratio is currently about 450%, having gotten to that point in long run equilibrium. Table 2.1 in the book shows that world g as been 3% over most of the last 100 years and Piketty notes that g = 3% in his discussion, implying that the world net saving rate was 3% X 4.5 = 13.5% over that long period.

    Now, Piketty wants to change the net saving rate to 10% going forward. However, if he does that, then he needs to change the gross saving rate in the textbook model. We can infer what the gross saving rate over the long period was since we know the net saving rate was 13.5% and we know g = 3% and d = 2%. The gross saving rate was

    s = 13.5%*(3% + 2%)/(3% + 13.5%*2%) = 20.6%

    over the past 100 years. Now if Piketty expects the net saving rate over the coming 100 years or so to be about 10% then we can calculate the associated new gross saving rate to be

    s = 10%*(3% + 2%)/(3% + 10%*2%) = 15.6%

    That’s the gross saving rate that would be expected to prevail until 2100 in Piketty’s projection. Now, if Piketty also wants to cut g from 3% to 1.5%, then he must also change the net saving rate, since it is a function of g. Thus, the new net saving rate would be 1.5%*15.6%/(1.5% + 2%*(1-15.6%)) = 7.4%.

    Now, applying the second law, Piketty should get 7.4%/1.5% = 490%, which is a 9% increase in the ratio from its current value of 450%. Thus, Piketty has dramatically overestimated the change in the capital to net income ratio to be 666%, even when d equals his preferred value of 2%.

    As a result, Delong is wrong to claim that a low depreciation rate somehow rescues Piketty. Interestingly enough, if d is actually higher than 2%, a run through this calculation will show that the ratio could actually decline somewhat. For example, if d = 5%, then the capital to net income ratio would actually decline about 5% to 427% in Piketty’s projected scenario.

    1. joan

      Everyone is wrong if they treat the depreciation rate as a constant. Since land does not depreciate, it was very small ( probably less than .01) when land was a major component of K. and rose to over .02 when when railroads and factories became more important and Evens estimates that it has than doubled in the last 50 years. This means net savings rate has changed less over time than the gross savings rate so is a better choice for the model that is trying to explain long term behaivor.

  49. gaddeswarup

    A question for Rick Stryker. Piketty and Zucman say in section 4.4 that the residual capital gains induced growth q in US is around .8 percent on the average from 1970 to 2010. How are the results affected if q is taken into account?


    All of you are correct in some parallel universe. But remember that on their own, mathematical models, especially so trivial, yield neither interpretations nor ontological conclusions. A way to save the economies of the West is to impose a 80% tax on automation product use, including any type of software of machine that replaces humans doing simple repetitive tasks. Otherwise, in 10 years the economies of the West will be owned by the robot/automation product makers. Only one machine I once designed as part of a group and installed in an assembly factory somewhere put out of work more than 10,000 good men and women who were just trying to make a living. The result: more dividends in the pockets of the 1% and more money to make more such machines. The rest are red herrings and straw man arguments. The 1%, soon, 0.01%, will accumulate all capital and then issue credit on will, a form of slavery. The root of all evil is man making machines to put man out of work and I did that and I feel sorry and I promised to myself not to do it again, I quit my job and I chose the hard way instead of making a living by putting others out of work. Most economists do not have a clue of what is going on, they are detached from reality and they play with trivial math like kids playing with toys. Piketty is right even if he is wrong. He tries to find an intellectual way to present a problem to avoid being called a crackpot. We are in the final stage of digital economics. See for more details < a href =""title = "this"

  51. Rick Stryker


    For the first couple of calculations I did at the top of this chain, I abstracted from the issue of capital gains in the capital to net income ratio and just computed the percentage overestimation due to second law error with respect to the textbook model. I did that to keep the comment and calculations simple. In the calculation you asked about, I also ignored the capital gains, but for a different reason–because Piketty ignored capital gains in his forecast.

    As you are aware, Piketty claims that perhaps 20% of the capital to net income ratio is accounted for by capital gains. He comes to that conclusion by measuring the discrepancy between the prediction of the second law and the actual ratio and forms a residual. The residual could represent measurement error in the saving rate or it could represent capital gains. Piketty argues for capital gains.

    However, in his prediction in figure 5.8 in the book, he ignored capital gains and assumed the second law holds exactly. You can see that in his prediction for 2100, where the ratio is about 666%, or 10%/1.5%. But if we are assuming that the second law holds exactly in the prediction, it’s not clear why we don’t also assume that it held exactly historically, implying that the residual is measurement error and that the savings rate is actually larger. I calculated that implied savings rate by assuming that the second law held exactly, and with a long run equilibrium of 450% in 2010 and a growth rate of 3%, the net saving rate would be 13.5%. I then just calculated the change in the net saving rate and showed that under the assumptions implicit in Piketty’s forecast–the second law holds exactly–the error he makes with respect to the textbook model is not very dependent on a small depreciation rate d.

    Alternatively, if we want to include capital gains, we need to start by assuming they are included consistently in the forecast. The second law gives 666% by 2100, but that would need to be scaled up by 20% to 800% if we include capital gains. Then I’d suggest proceeding as follows: we start with a ratio of 450% and then scale it down to 360% which represents the second law component. Then just go through the same logic as the original calculation. The net saving rate would be 10.8% and the gross would be 16.8%. Then, drop the net saving rate to 10%, yielding a new gross saving rate of 15.6%. Then drop g to 1.5% and recalculate the net saving rate to be 7.3%, producing a new capital to net income ratio of 7.3%/1.5% = 490%. That’s the second law component. Then scale that up by 20% to add capital gains back in, to get 613% for an all-in forecast.

    So, to compare, the second law unadjusted with capital gains would change the ratio from 450% to 800%, a 78% increase. But the adjusted second law would go from 450% to 613%, a 36% increase. Thus, when d = 2%, the second law overstates the capital to net income ratio by about a factor of 2 with respect to the textbook model. This result is consistent with the calculation I did at first in which I abstracted from the capital gains issue to keep things simple

    For d = 3.33%, Delong’s preferred number, the second law overstates the percentage increase by about a factor of 3 when you include capital gains.

    Hope this helps.

  52. gaddeswarup

    Rick Stryker,
    Thanks. That is very kind of you. It will probably take me several days to understand this. I do not have economics background but some mathematics background. I was using W_(t+1)=W_t (1+s/β) to deduce the second law. I will try to understand it off and on over the next several days.

  53. Steve Blitz

    Growth, net saving, additions to capital stock presume a closed system, particularly that domestic firms only produce domestically. Net income and saving can rise, production (ie capital investment) occurs offshore. And voila! Skewed income, lots of cash accruing to the firm, no capex growth. Income and capital stock grow globally just not necessarily domestically. Just follow income skew and trade balance net of oil since 1982, add to it capex net of depreciation as a percent of GDP. Look today at equity market % of GDP (high!), % of corp profits (more than reasonable), capex % of profits (on the floor). This isn’t hard to understand once everyone stops looking for only domestic domiciled responses to domestic earnings. It really is that simple. Fixing it is hard. In case you want a domestic historic answer, follow capex & income skew in New England and Midwest vs Southern States 1940-80. Same thing.

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