Results for “piketty”
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Is Piketty’s “Second Law of Capitalism” fundamental?

Per Krusell and Tony Smith have a new paper on Piketty (pdf), which I take to be reflecting a crystallization of opinion on the theory side.  Here is one excerpt:

There are no errors in the formula Piketty uses, and it is actually consistent with the very earliest formulations of the neoclassical growth model, but it is not consistent with the textbook model as it is generally understood by macroeconomists. An important purpose of this note is precisely to relate Piketty’s theory to the textbook theory. Those of you with standard modern training have probably already noticed the difference between Piketty’s equation and the textbook version that we are used to. In the textbook model, the capital-to-income ratio is not s=g but rather s/(g+ δ), where δ is the rate at which capital depreciates. With the textbook formula, growth approaching zero would increase the capital-output ratio but only very marginally; when growth falls all the way to zero, the denominator would not go to zero but instead would go from, say 0.12—with g around 0.02 and δ = 0.1 as reasonable estimates—to 0.1. As it turns out, however, the two formulas are not inconsistent because Piketty defines his variables, such as income, y, not as the gross income (i.e., GDP) that appears in the textbook model but rather as income, i.e., income net of depreciation. Similarly, the saving rate that appears in the second law is not the gross saving rate as in the textbook model but instead what Piketty calls the “net saving rate”, i.e., the ratio of net saving to net income.

Contrary to what Piketty suggests in his book and papers, this distinction between net and gross variables is quite crucial for his interpretation of the second law when the growth rate falls towards zero. This turns out to be a subtle point, because on an economy’s balanced growth path, for any positive growth rate g, one can map any net saving rate into a gross saving rate, and vice versa, without changing the behavior of capital accumulation. The range of net saving rates constructed from gross saving rates, however, shrinks to zero as g goes to zero: at g = 0, the net saving rate has to be zero no matter what the gross rate is, as long as it is less than 100%. Conversely, if a positive net saving rate is maintained as g goes to zero, the gross rate has to be 100%. Thus, at g = 0, either the net rate is 0 or the gross rate is 100%. As a theory of saving, we maintain that the former is fully plausible whereas the latter is all but plausible.

With the upshot coming just a wee bit later:

…Moreover, whether one uses the textbook assumption of a historically plausible 30% saving rate or an optimizing rate, when growth falls drastically—say, from 2% to 1% or even all the way to zero—then the capital-to-income ratio, the centerpiece of Piketty’s analysis of capitalism, does not explode but rather increases only modestly. In conclusion, at least from the perspective of the theory that we are more used to and find more a priori plausible, the second law of capitalism turns out to be neither alarming nor worrisome, and Piketty’s argument that the capital-to-income ratio is poised to skyrocket does not seem well-founded. [emphasis added by TC]

Krusell and Smith really know their stuff on this topic and their arguments to me seem completely correct.

By the way, here is a Chris Giles follow-up post from The FT, very useful.

What do the Piketty data problems really mean?

In some ways the new FT criticisms may not matter much, although I think not in a way which is reassuring for Piketty.  There were already several major problems with Piketty’s analysis and also empirics, including what Alex has called the asset price problem.  He wrote:

According to four French economists, Piketty’s measure of the capital stock is greatly influenced by the Europe-US housing bubble that preceded the financial crisis.

Adjusting for that factor seems to make the main results go away, and that is a purely empirical problem which has not been answered, at least not yet.

Another pre-existing empirical problem is that 19th century data seem to indicate that a “Piketty world,” even if we take it on its own terms, far from being a disaster, would likely be accompanied by rising real wages and declining consumption inequality, albeit rising wealth inequality.

That hasn’t been answered either, although a few people have suggested (without serious back-up) that if wealth inequality is going up that has to lead to political problems, or problems of some kind or another, and thus it can’t be something we can approve of or accept with equanimity, because inequality is really really bad, and therefore Piketty is somehow right anyway.  That’s a weak response to begin with and furthermore it doesn’t fit the available data.

Empirically, inheritances aren’t nearly as important as Piketty seems to suggest.

On Twitter Clive Crook wrote of the:

…distance between treacherous data and super-bold conclusions an issue at the outset. This underlines the point.

Now, when you cut through the small stuff, the new empirical problem seems to be that UK revisions, combined with a population-weighted series for Europe, contradicts Piketty’s claim of rising wealth inequality for Europe.   I would call that a serious problem.  I am not impressed by the “downplaying” responses which focus on coding errors, Swedish data points, and the other small stuff.  Let’s face up to the real (new) problem, namely that robustness suddenly seems much weaker.  You can’t argue that population-weighting is “the right way to do it,” but it is an entirely plausible way to estimate the wealth inequality trend.  If Piketty’s results don’t survive population weighting (and what are apparently the superior UK numbers), that suggests the overall rise in European wealth inequality is not very robust to how the pie is carved up and also that it is not backed by dominant, “rule the roost” sorts of forces.

It should be noted that Piketty’s response to the new criticisms was quite weak.  Maybe he’s not to be blamed for what was surely a rapid and caught-off-guard response, and perhaps there is more to come, but it doesn’t reassure me either.  He also should have run it by a PR person first (for instance, don’t start your response with a sentence ending in an exclamation point.)

That said, don’t focus on Piketty.  When evaluating debates of this kind, never ever confuse a) is he right? with b) “how much should we raise/lower the relative status of the author as a result of the new exchange”?  So responses like “he made all his data freely available,” or “he admits all along how complicated this all is,” address b) but not the more important a).  And if you are seeing people focus on b) rather than a), they have a problem themselves.  On empirical grounds it does seem we have another reason for thinking Piketty’s central claim isn’t quite right, at least not for the reasons he sets out, and perhaps not quite right altogether.

Addendum: Ryan Avent has a good survey of some key issues and responses.

Piketty update

…according to a Financial Times investigation, the rock-star French economist appears to have got his sums wrong.

The data underpinning Professor Piketty’s 577-page tome, which has dominated best-seller lists in recent weeks, contain a series of errors that skew his findings. The FT found mistakes and unexplained entries in his spreadsheets, similar to those which last year undermined the work on public debt and growth of Carmen Reinhart and Kenneth Rogoff.

The central theme of Prof Piketty’s work is that wealth inequalities are heading back up to levels last seen before the first world war. The investigation undercuts this claim, indicating there is little evidence in Prof Piketty’s original sources to bear out the thesis that an increasing share of total wealth is held by the richest few.

Prof Piketty, 43, provides detailed sourcing for his estimates of wealth inequality in Europe and the US over the past 200 years. In his spreadsheets, however, there are transcription errors from the original sources and incorrect formulas. It also appears that some of the data are cherry-picked or constructed without an original source.

For example, once the FT cleaned up and simplified the data, the European numbers do not show any tendency towards rising wealth inequality after 1970. An independent specialist in measuring inequality shared the FT’s concerns.

The full FT story is here.

Addendum: Here is the in-depth discussion.  Here is Piketty’s response.

The Piketty Bubble?

Piketty’s Capital is not very clear on how to distinguish greater physical capital from higher asset prices. For the most part, Piketty discusses capital as something that builds up over time through savings. The increase in physical capital then generates large returns to rentiers and those returns increases the capital share of income. When it comes to measuring capital, however, this has to be done in money terms which means that we need the price of capital. But the price of capital can vary significantly; as a result, Piketty’s capital stock can vary significantly even without changes in physical capital or savings. When capital increases because of changes in its price, however, the implications are quite different from a physical increase in capital.

Consider an asset that pays a dividend D forever; at interest rate r the asset is worth P=D/r. As r falls, the price of the asset rises. An asset that pays $100 forever is worth $1000 at an interest rate of 10% (.1) but $2000 at an interest rate of 5%. Piketty measures a higher P as more capital but notice that P is high only because r is low. You can’t, therefore, multiply P by some fixed r and conclude that rents have increased. Indeed, in this case rents, the dividend, haven’t increased at all. For the most part, Piketty simply ignores this issue (at least in the book) by arguing that changes in the price of capital wash out over long time periods but that does not appear to be the case in his data.

According to four French economists, Piketty’s measure of the capital stock is greatly influenced by the Europe-US housing bubble that preceded the financial crisis (Tyler earlier pointed to the French version of this paper, this is the English version). Since Piketty’s theory is based on rents from physical capital, the authors suggest that measures of housing capital based on prices should be corrected using the rent to price ratio. In other words, if the rentiers aren’t getting more rents then their capital hasn’t really increased. When measured in this way, the authors find little to no increase in the capital stock in either France or the United States.


Addendum: Do note that the debate here is not about income inequality but rather the source of income inequality and the implications for the future that Piketty draws from a  (possibly not) rising capital stock.

Stefan Homburg on Piketty

It is a useful overview (pdf, but better link here) of some of the major problems in the argument, here is one key passage:

Piketty’s erroneous claim is due to the implicit assumption that savings are never consumed, nor spent on charitable purposes or used to exert power over others. It is only under this outlandish premise that wealth grows at the rate r. If people use their savings later on, as they do in the Diamond model as well as in reality, the growth of wealth is independent of the return on capital. This holds all the more in the presence of taxes.

Piketty’s allegation that the relationship r>g implies a rising wealth-income ratio is not only logically flawed, however, but also rebutted by his own data: On p. 354, the author reports that the return on capital has consistently exceeded the world growth rate over the last 2,000 years. According to his “central contradiction of capitalism”, this would have implied steadily increasing wealth-income ratios. Yet, over the last two centuries or so, the period for which data are available, wealth-income ratios have remained relatively stable in countries like the United States or Canada. In countries such as Britain, France, or Germany, which were heavily affected by the wars, wealth-income ratios declined at the start of World War I and recovered after the end of World War II. The book’s references to these wars and the implied destruction of capital abound. They are intended to rescue the claim that r>g implies an ever rising wealth-income ratio. The United States and Canada as obvious counter-examples remain unmentioned in this context.

For the pointer I thank David Levey.

Aaron Hedlund on Piketty

He writes me in an email:

I feel comfortable saying that his extensive documentation of inequality– it’s trends, composition, etc.– is a big contribution that should drive future research.

…the thesis that r > g is the explanation for inequality or an ominous predictor of future inequality is, to be blunt, ridiculous.

1) Consider the most basic economic growth model: the Solow model where households arbitrarily follow a constant savings rate rule. In that model, the long-run growth rate equals the rate of technological progress, and the rate of return to capital is constant and completely independent of that growth rate. Therefore, you could have r > g, r = g, or r < g, simply because there is NO relationship. In that model, the capital/output ratio is stable in the long-run, again regardless of what r is relative to g.

We could beef the model up a bit by allowing households to actively choose how much to save (rather than impose a constant savings rate rule on them). In that model, the economy will also get to a point with stable long-term growth where the growth rate is determined purely by technological progress. In that model under log utility, 1 + r = (1 + g)*(1 + rate of time preference). As long as people are at all impatient, the implication is r > g. Therefore, the economy will have r > g, stable growth, and a stable K/Y ratio.

The flaw with both of these models, of course, is that they are representative household models where there is no inequality. Therefore, we can go a step further and add uninsurable risk to the model (whether that be health risk, earnings risk, or any other important source of economic risk). In *these* models, households also engage in precautionary savings, so in equilibrium r is lower than it is in the rep agent models. In fact, the greater amount of risk, the more wealth inequality and the SMALLER the gap between r and g.

2) Another huge fallacy is to translate “r > g” as “the return to capital is greater than the rate of return to labor.” The notion “rate of return” indicates an intertemporal dimension: for example, if I invest $1, how much do I get back in return a year later? The growth rate of the economy is not the return to labor. In fact, the “return” to labor is static: I give up x units of time in exchange for y dollars.

The RELEVANT comparison would be to compare the rate of return on capital to the rate of return on investing in human capital (ie you go to college and then reap labor market rewards in the future). The rate of return on human capital is most definitely not just “g.” In fact, the college premium is at an all-time high, which suggests that the rate of return on human capital is quite high and very possibly higher than the rate of return on physical capital.

3) The r > g –> inequality thesis is also based on ignoring the fact that r and g are both determined in equilibrium. Here’s what I mean: it is bad economics to say “Look, r > g, therefore IF people behave in such and such manner, their wealth will grow at a higher rate than g indefinitely.” The reason it is bad economics is because you can’t take the “r > g” as given and THEN impose whatever behavioral assumptions you want. The fact is, people’s behavior affects r and g. In the heterogeneous models I mentioned above, r > g, inequality is STABLE, and the behavior of households is determined by their desire to maximize utility. If I were to go to the model and arbitrarily force the households to behave differently, then the equilibrium r would change.

Here is Hedlund’s home page.

Why Piketty’s book is a bigger deal in America than in France

On The Upshot I have a new piece, co-authored with Veronique de Rugy, here is an excerpt:

…the book’s timing may be behind the state of French debate. Had it been released in the halcyon days of Mr. Hollande’s 2011 presidential campaign, when many French considered soak-the-rich talk and 75 percent marginal tax rates to be practical fiscal strategies, Mr. Piketty’s book might have made a bigger splash in France. Today, with the economy still struggling, Mr. Hollande is talking about tax cuts rather tax increases. The 75 percent rate has suffered constitutional challenges, and even celebrity backlashes, such as when Gérard Depardieu pursued and received Russian citizenship to lower his tax rate. Mr. Hollande seems to be steering France away from its traditional role as a defender of high taxes and toward some structural reforms, albeit at a slow pace. During his New Year address, Hollande even turned into a rhetorical supply-sider, making the case for cutting taxes and public spending, improving competitiveness, and creating a more investor-friendly climate. In any case, the French appetite for stiff tax increases has diminished.

…Finally, some other French economists have taken the lead in challenging Mr. Piketty’s empirical claims. One recent paper by four economists at l’Institut d’Etudes Politiques de Paris challenges Mr. Piketty’s view that inequality has increased because the return to capital has been greater than general growth in the economy. The current shorthand is “r > g.”

The paper argues that the higher growth of capital rests entirely on returns to housing, and takes technical issues with the book’s treatment of housing, too. If Mr. Piketty’s argument depends on housing, it hardly seems to match his basic story about the ongoing ascendancy of capitalists.

There is much more at the link.

Updated Priors (Ryan Decker) reviews Piketty

Here is one good part of a consistently good and interesting review:

Most of the analysis in the book is more about accounting than economics. Piketty takes nearly everything as exogenous then divides things arithmetically. His ubiquitous r > g heuristic takes both sides of the inequality as given for almost the entire book. Lines like “the richest 10 percent appropriate three-quarters of the growth” (297) enable lazy readers to avoid thinking about what actually determines income. Language about “appropriation” suggests that we live in an endowment economy, as does the claim that post-World War I wealth inequality fell “so low that nearly half the population were able to acquire some measure of wealth” (350). Endogeneity, anyone? Taking income as exogenous leads to other large problems with inference, such as the claim that “meritocratic extremism can thus lead to a race between supermanagers and rentiers, to the detriment of those who are neither” (417). Piketty does not consider the possibility that this race results in more income than otherwise, nor does he consider the notion that an increase in the bargaining power of elite executives could actually come at the expense of capital owners rather than workers. I’m not making an argument for either here; I’m simply suggesting that Piketty’s ideological quips don’t deserve the certainty with which he delivers them. Models with endowment economies have their purposes, but a 600-page book should be able to relax such strict assumptions. His criticisms of mathematical economics (32, 574) are not surprising given that he relies so heavily on assumptions and mechanisms that would be highly vulnerable to criticism if they were forced into the transparency of a formal model.

Hat tip goes to Angus.

Garett Jones reviews Piketty

Here is one bit:

Market-oriented economies that learn to live with inequality will reap the rewards: More domestic capital for workers to use on their jobs, more foreign capital flowing in to a country perceived as a safe investment, and a political and cultural system that can spend its time on topics other than the 1 percent. Market-oriented economies that instead follow Piketty’s preferred path—taxing capital heavily, preferably through international consortiums so the taxes are harder to evade—will end up with less domestic and foreign capital, fewer lenders willing to fund new housing projects, fewer new office buildings, and a cultural system focused on who has more and who has less.

…The Boston University economist Christophe Chamley and the Stanford economist Kenneth Judd came up independently with what we might call the Chamley-Judd Redistribution Impossibility Theorem: Any tax on capital is a bad idea in the long run, and that the overwhelming effect of a capital tax is to lower wages. A capital tax is such a bad idea that even if workers and capitalists really were two entirely separate groups of people—if workers could only eat their wages and capitalists just lived off of their interest like a bunch of trust-funders—it would still be impossible to permanently tax capitalists, hand the tax revenues to workers, and make the workers better off.


…One lesson of this story is that it’s good to be patient. So let’s start training ourselves and our children to delay gratification, to forego that great sound system on the new car, to eat at home a little more often.

The full review is here.

The policy proposals Thomas Piketty forgot to mention

Suresh Naidu writes:

…let me suggest that if we’re aiming for politically hopeless ideas, open migration is as least as good as the global wealth tax in the short run, and perhaps complementary.One weakness of the book is its focus on the large core economies (the data obviously is better and the wealth is obviously larger). But liberalizing immigration, while not solving the ultimate problem the book diagnoses, can go some of the way by raising growth of both income and population.

Maybe, that will likely improve welfare but in an Alvin Hansen model it can make Piketty-like phenomena (I won’t call them problems) more extreme.  In any case there are more straightforward remedies.  Social Security privatization is another option, if r > g is truly such a likelihood.  Yet Piketty and his boosters won’t mention this.  By the way, I am opposed to social security privatization — scroll down in that link — but I probably would favor it if it my views were closer to Piketty’s.

Here is a somewhat biting paragraph on Piketty and policy from my Foreign Affairs review (use “open private window” in Firefox, if need be):

Piketty also ignores other problems that would surely stem from so much wealth redistribution and political control of the economy, and the book suffers from Piketty’s disconnection from practical politics — a condition that might not hinder his standing in the left-wing intellectual circles of Paris but that seems naive when confronted with broader global economic and political realities. In perhaps the most revealing line of the book, the 42-year-old Piketty writes that since the age of 25, he has not left Paris, “except for a few brief trips.” Maybe it is that lack of exposure to conditions and politics elsewhere that allows Piketty to write the following words with a straight face: “Before we can learn to efficiently organize public financing equivalent to two-thirds to three-quarters of national income” — which would be the practical effect of his tax plan — “it would be good to improve the organization and operation of the existing public sector.” It would indeed. But Piketty makes such a massive reform project sound like a mere engineering problem, comparable to setting up a public register of vaccinated children or expanding the dog catcher’s office.

Here is another:

Worse, Piketty fails to grapple with the actual history of the kind of wealth tax he supports, a subject that has been studied in great detail by the economist Barry Eichengreen, among others. Historically, such taxes have been implemented slowly, with a high level of political opposition, and with only modestly successful results in terms of generating revenue, since potentially taxable resources are often stashed in offshore havens or disguised in shell companies and trusts. And when governments have imposed significant wealth taxes quickly — as opposed to, say, the slow evolution of local, consent-based property taxes — those policies have been accompanied by crumbling economies and political instability.

The simple fact is that large wealth taxes do not mesh well with the norms and practices required by a successful and prosperous capitalist democracy. It is hard to find well-functioning societies based on anything other than strong legal, political, and institutional respect and support for their most successful citizens. Therein lies the most fundamental problem with Piketty’s policy proposals: the best parts of his book argue that, left unchecked, capital and capitalists inevitably accrue too much power — and yet Piketty seems to believe that governments and politicians are somehow exempt from the same dynamic.

And finally the review closes with this:

A more sensible and practicable policy agenda for reducing inequality would include calls for establishing more sovereign wealth funds, which Piketty discusses but does not embrace; for limiting the tax deductions that noncharitable nonprofits can claim; for deregulating urban development and loosening zoning laws, which would encourage more housing construction and make it easier and cheaper to live in cities such as San Francisco and, yes, Paris; for offering more opportunity grants for young people; and for improving education. Creating more value in an economy would do more than wealth redistribution to combat the harmful effects of inequality.

The original pointer to the first link was from Henry Farrell.  And my previous criticisms of Piketty on the analytics are here.  Barry Eichengreen on wealth taxes is here.

Why I am not persuaded by Thomas Piketty’s argument

My Foreign Affairs review is here.  (Open up “New private window” in Firefox, if need be.)  I won’t attempt to cover all of the review, but rather will rephrase a few of my points for MR readers, in slightly different terminology:

1. If the rate of return remains higher than the growth rate of the economy, wages are likely to rise and quite a bit.  You can find a wonky version of that idea here from Matt Rognlie.   But it suffices to apply common sense, namely that capital accumulation bids up wages.  Piketty suggests we are headed back to something resembling the 19th century.  Well, that was a pretty good time for the average working person in Western Europe, especially once we get past the first part of that century, which had lots of war and a still-incomplete industrial revolution.

Since we today have had some wage stagnation, perhaps it does not feel that kind of favorable outcome is what we will get and many commentators are trading off this mood.  But also realize the (risk-adjusted) return on capital hasn’t been that high lately and it has been falling for decades.  This combination of variables — low returns and stagnant wages — does not refute Piketty but it doesn’t exactly fit into his mold either.

2. The crude seven-word version of Piketty’s argument is  “rates of return on capital won’t diminish.”  Is that really such a powerful forecast?  I say over the next fifty or one hundred years we don’t have a very good sense of which factors will show diminishing returns and which will not.  It is hard enough to make predictions of trend over a twenty-year time horizon.  NB: At many points in the Piketty book he seeks to have it both ways: loads of caveats, but then he falls back into the basic model, and he and his defenders cite the caveats when it is convenient.

3. Piketty’s reasons why rates of return on capital won’t diminish are fairly specific and restricted to only a small share of capital.  He cites advanced financial management techniques of the very wealthy and also investing abroad in emerging economies.  Neither of these covers most capital, and thus capital returns as a whole may not be so robust.  Nor is it obvious that either technique will prove especially successful over the next few decades or longer.  Again, is there any particular reason to think either of these factors will outrace the basic logic of diminishing returns, or for that matter EMH, relative to other factor returns that is?  They might, to be sure.  They also might underperform.  In any case this is pure speculation and Piketty’s entire argument depends upon it.

4. The actual increases in income inequality we observe are mostly about labor income, not capital income.  They don’t fit easily into Piketty’s story and arguably they don’t fit into the story at all.

5. Piketty converts the entrepreneur into the rentier.  To the extent capital reaps high returns, it is by assuming risk (over the broad sweep of history real rates on T-Bills are hardly impressive).  Yet the concept of risk hardly plays a role in the major arguments of this book.  Once you introduce risk, the long-run fate of capital returns again becomes far from certain.  In fact the entire book ought to be about risk but instead we get the rentier.

Overall, the main argument is based on two (false) claims.  First, that capital returns will be high and non-diminishing, relative to other factors, and sufficiently certain to support the r > g story as a dominant account of economic history looking forward.  Second, that this can happen without significant increases in real wages.

Addendum: Still, it is a very important book and you should read and study it!  But I’m not convinced by the main arguments, and the positive reviews I  have read worsen rather than alleviate my anxieties.  I’ll cover the policy and politics of this book in a separate post.  Do read my review itself, which has much more than what is in this blog post.

Clive Crook reviews Piketty

Here is his bottom line and it is correct:

Over the course of history, capital accumulation has yielded growth in living standards that people in earlier centuries could not have imagined, let alone predicted — and it wasn’t just the owners of capital who benefited. Future capital accumulation may or may not increase the capital share of output; it may or may not widen inequality. If it does, that’s a bad thing, and governments should act. But even if it does, it won’t matter as much as whether and how quickly wages and living standards rise.

That is, or ought to be, the defining issue of our era, and it’s one on which “Capital in the 21st Century” has almost nothing to say.

The full review is here.

More Matt Rognlie on Piketty

From the comments:

Krugman correctly highlights the importance of the elasticity of substitution between capital and labor, but like everyone else (including, apparently, Piketty himself) he misses a subtle but absolutely crucial point.

When economists discuss this elasticity, they generally do so in the context of a gross production function (*not* net of depreciation). In this setting, the elasticity of substitution gives the relationship between the capital-output ratio K/Y and the user cost of capital, which is r+delta, the sum of the relevant real rate of return and the depreciation rate. For instance, if this elasticity is 1.5 and r+delta decreases by a factor of 2, then (moving along the demand curve) K/Y will increase by a factor of 2^(1.5) = 2.8.

Piketty, on the other hand, uses only net concepts, as they are relevant for understanding net income. When he talks about the critical importance of an elasticity of substitution greater than one, he means an elasticity of substitution in the *net* production function. This is a very different concept. In particular, this elasticity gives us the relationship between the capital-output ratio K/Y and the real rate of return r, rather than the full user cost r+delta. This elasticity is lower, by a fraction of r/(r+delta), than the relevant elasticity in the gross production function.

This is no mere quibble. For the US capital stock, the average depreciation rate is a little above delta=5%. Suppose that we take Piketty’s starting point of r=5%. Then r/(r+delta) = 1/2, and the net production function elasticities that matter to Piketty’s argument are only 1/2 of the corresponding elasticities for the gross production function!

Piketty notes in his book that Cobb-Douglas, with an elasticity of one, is the usual benchmark – and then he tries to argue that the actual elasticity is somewhat higher than this benchmark. But the benchmark elasticity of one, as generally understood, is a benchmark for the elasticity in the gross production function – translating into Piketty’s units instead, that’s only 0.5, making Piketty’s proposed >1 elasticity a much more dramatic departure from the benchmark. (Keep in mind that a Cobb-Douglas *net* production function would be a very strange choice of functional form – implying, for instance, that no matter how much capital is used, its gross marginal product is always higher than the depreciation rate. I’ve never seen anyone use it, for good reason.)

Indeed, with this point in mind, the sources cited in support of high elasticities do not necessarily support Piketty’s argument. For instance, in their closely related forthcoming QJE paper, Piketty and Zucman cite Karabarbounis and Neiman (2014) as an example of a paper with an elasticity above 1. But K&N estimate an elasticity in standard units, and their baseline estimate is 1.25! In Piketty’s units, this is just 0.625.

And this:

What does this all mean for the Piketty’s central points – that total capital income rK/Y will increase, and that r-g will grow? His model imposes a constant, exogenous net savings rate ‘s’, which brings him to the “second fundamental law of capitalism”, which is that asymptotically K/Y = s/g. The worry is that as g decreases due to demographics and (possibly) slower per capita growth, this will lead to a very large increase in K/Y. But, of course, this only means an increase in net capital income rK/Y if Piketty’s elasticity of substitution is above 1, or if equivalently the usual elasticity of substitution is above 2. This is already a very high value, and frankly one to be treated with skepticism.

Meanwhile, it is even harder to get growth in r-g, which most readers take to be Piketty’s central point. Suppose that in recent decades, r has been roughly 5% while g has been 2.5%, and suppose that g will ultimately fall to around 1%. In Piketty’s framework, this implies an increase in steady-state K/Y of 2.5. If there is an elasticity of 1 (in Piketty’s units), this implies a decrease in r from 5% to 2%, and thus a *decrease* in the gap r-g from 2.5% to 1%. The point is that with this unit demand elasticity and the exogenous net savings assumption, it is the ratio r/g rather than the difference r-g that is constant, which means that a decline in g leads to a proportionate decline in r-g. (Note that Krugman’s review is ambiguous about this distinction.)

What would we need to obtain even a tiny increase in r-g in this setting – say, of half a percentage point? We would need r to fall from 5% to only 4% while g fell from 2.5% to 1%, increasing r-g from 2.5% to 3%. But given the 2.5-fold increase in K/Y, a decline in r by a factor of only 1/5th implies an elasticity of substitution (in Piketty’s sense) of nearly 4. This implies an elasticity of substitution in the *usual* gross production function sense of nearly 8, not plausible by any stretch of the imagination.

Unless I’m missing something, the formal apparatus in Piketty’s book simply is not capable of generating the results he touts. There are two very simple issues that break it quantitatively – first, the distinction between elasticities of substitution in the gross and net production functions; and second, the fact that as g falls, an extraordinarily high elasticity of substitution is necessary to prevent r from falling along with it and actually compressing the arithmetic gap between r and g. Perhaps there are modifications to the framework that can redeem it, but as it currently stands I’m baffled.

I believe Matt is correct.  I would simply note that diminishing returns to capital — relative to other factors of production — are likely to hold in the long run.  See also these earlier MR comments by Rognlie and Harless.  And here are Piketty’s lecture notes.

Krugman’s review of Piketty

You will find it here.  Excerpt:

Just about all economic models tell us that if g falls—which it has since 1970, a decline that is likely to continue due to slower growth in the working-age population and slower technological progress—r will fall too. But Piketty asserts that r will fall less than g. This doesn’t have to be true. However, if it’s sufficiently easy to replace workers with machines—if, to use the technical jargon, the elasticity of substitution between capital and labor is greater than one—slow growth, and the resulting rise in the ratio of capital to income, will indeed widen the gap between r and g. And Piketty argues that this is what the historical record shows will happen.

Krugman calls the book “awesome,” but here are his critical remarks:

I don’t think Capital in the Twenty-First Century adequately answers the most telling criticism of the executive power hypothesis: the concentration of very high incomes in finance, where performance actually can, after a fashion, be evaluated. I didn’t mention hedge fund managers idly: such people are paid based on their ability to attract clients and achieve investment returns. You can question the social value of modern finance, but the Gordon Gekkos out there are clearly good at something, and their rise can’t be attributed solely to power relations, although I guess you could argue that willingness to engage in morally dubious wheeling and dealing, like willingness to flout pay norms, is encouraged by low marginal tax rates.

My own review is still due out in about a week’s time.