Piketty on the Dynamics of Inequality: Four Useful Theorems

Matt Breunig has posted three excellent new pieces on Thomas Piketty’s analysis of the dynamics of inequality. They’re at the Demos Policyshop blog, and can be found here, here and here. Since Breunig comes to many of the same interpretive conclusions I have reached myself, I will just refer the reader to these pieces without much further comment on them. But I do want to call special attention to one thing Breunig says in response to a recent critical essay on Piketty by John Aziz. In characterizing Piketty’s account of inequality, Aziz says that according to Piketty inequality will tend to increase when the rate of economic growth for the entire economy is less than the average return on capital. But Breunig notes in response:

This is fine enough as a gloss of an explanation, but is it not strictly true. Piketty’s actual point [is] that the larger the spread between r and g, the more forcefully the dynamics of capital income pushes in the direction of increasing wealth inequality.

Breunig raises an important point here, and it bears emphasis. Piketty couches most of his arguments about equality and inequality in terms of forces of divergence and forces of convergence. His approach is to identify those conditions under which the forces of divergence will predominate, and if so, how strongly they will predominate. His view is that at the beginning of the 21st century, the conditions appear to be in place for the forces of divergence to acquire renewed strength, although he also stresses that nothing is certain, and the exact course of 21st century inequality depends on a host of political, demographic, technological and economic factors. One thing Piketty routinely stresses, however, is that the forces for divergence operate very strongly when r is “significantly and durably” higher than g, and automatically lead to a very high concentration of wealth.

I want to add a bit of precision to these initial statements, and in the process shed some light on why Piketty lays very significant stress on the fact that different kinds of wealth owners earn different rates of return on their wealth, and also on the fact that the wealthy save their incomes at higher rates than those who are less wealthy. Both of these phenomena play an important role in Piketty’s analysis of the forces for divergence and the structure of inequality in Chapters 10, 11 and 12 of Capital in the Twenty-First Century.

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Let's End the Confusion over Piketty's "Second Fundamental Law"

A great deal of unnecessary confusion continues to surround Thomas Piketty’s Second Fundamental Law of Capitalism. I have already dealt with the most common source of confusion here, and refer the reader to that post. But the same misreading has reappeared frequently in the ongoing debate over Piketty’s Capital in the Twenty-First Century, so it is worth returning to the topic briefly.

James Hamilton writes:

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.

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

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The Financial Times’s Lazy Reading of Piketty

The Financial Times has sparked a major debate on Thomas Piketty’s Capital in the Twenty-First Century by releasing an investigation that purports to find important data problems and errors in Piketty’s work. A helpful summary of some of the initial contributions to this debate can be found here. My conclusion at this stage is that the only really major data issue appears to lie in the analysis of UK wealth inequality for the past four decades. While I strongly suspect that Piketty’s basic conclusions with regard to the recent UK trends will be sustained after his results are compared with other measures of wealth inequality, including the revised estimates by Saez and Zucman that incorporate wealth held abroad, including wealth held in offshore havens that Zucman has called the “missing wealth of nations,” the data issues themselves are not the topic of this post.

What I am interested in here is the bearing of the data problems the Financial Times purports to have discovered on Piketty’s central theoretical arguments, and on his assessment of the likely path of inequality in the 21st century absent policy corrections that might redirect that path. Yesterday, the Times described the impact of its investigative results on Piketty’s argument in very dramatic terms. But some of the statements they make about that impact are so odd that one can only conclude that the people at the Financial Times who are making these charges have not read the book with even minimal care. It is apparent they don’t understand its argument very well.  Thus they are in a poor position to evaluate the bearing of the data issues they have raised on the book’s central themes, and are misleading their readers about the substance of Piketty’s arguments and claims.

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Naked Piketty – with a Bonus Workbook!

As is well known by now, Thomas Piketty argues in Capital in the Twenty-First Century that the historical record shows the rate of return on capital in market economies consistently exceeding the growth rate of national income in those economies, usually by a sizeable margin. Piketty finds typical values for the pre-tax rate of return on capital during the 19th century, for example, of around 5%, with typical income growth rates of around 1.5%. He encapsulates this historical regularity with the formula r > g.

Yves Smith, commenting on a post at her blog Naked Capitalism, is convinced that that Piketty’s claim is absurd:

There’s no sound basis for Piketty’s contention that capital makes a steady 5%, or perhaps more important, a 4% premium to GDP growth. Pull out a calculator. It does not take that long before capital that consistently showed that much of a return premium would eat the entire economy. Then by definition it would be the entire economy and unable to earn a premium. His r>g as some sort of constant is absurd, tantamount to “trees grow to the sky.”

And she adds later:

Folks, this is basic compounding, and that’s where the fallacy in his reasoning lies. It’s bloomin’ obvious. That’s why it being a materially higher rates than GDP growth has to break down, and not in a very long period of time either.

I find it somewhat depressing that an informed economics blogger like Yves Smith could be in this case so deeply uninformed about Piketty’s fundamental conceptual framework, to the extent that she thinks Piketty must be some kind of bungling duffer who has flunked basic accounting algebra. But her claims here can easily be shown to be false. Piketty’s framework is simple, elegant and coherent, and working within that framework one can easily see how the typical values Piketty finds for r and g can be sustained indefinitely without capital overwhelming the economy and without capital income devouring all other income.

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A Proper Pile of Piketty Posts

Over the past few weeks I have written several pieces on Thomas Piketty’s Capital in the Twenty-First Century. It is likely that there are more to come. But here is a convenient round-up of the posts so far, with permalinks:

Summers’s Review of Piketty: Underestimating the Argument for the Forces Driving Inequality

How Hassett Gets Piketty Wrong

Nothing Magical about Piketty’s Mathematics

A Flaw in The Economist’s Explanation of Piketty

Laissez-faire’s Piketty Problem

Summers's Review of Piketty: Underestimating the Argument for the Forces Driving Inequality

Lawrence Summers, in what is generally a very favorable review of Thomas Piketty’s Capital in the Twenty-First Century, pushes back on some of Piketty’s arguments in favor of the conclusion that capitalism shows an inherent tendency toward increased inequality. The review raises a number of important questions, many of which deserve careful future attention. But in this post, I just want to touch on two areas where I think Summers has misconstrued Piketty’s argument in some important ways, misconstruals that lead Summers to underestimate the argument’s overall strength. I will deal first with what I think is the less important of the two misconstruals, and then turn to the more important one.

At one point in the review, Summers says this:

His [Piketty’s] argument is that capital or wealth grows at the rate of return to capital, a rate that normally exceeds the economic growth rate. Thus, economies will tend to have ever-increasing ratios of wealth to income, barring huge disturbances like wars and depressions.

But this isn’t quite right. Piketty does, indeed, believe that the average rate of return to capital in an economy normally exceeds the economy’s growth rate. But he does not argue that the rate of capital growth is always equal to the rate of return to capital, nor does he argue that capitalistic economies will tend to have ever-increasing ratios of wealth to income. The rate of capital growth will always be equal to s/β, where s is the savings rate and β is the ratio of capital to income. If β is less than s/g, where g is the growth rate, then capital will grow more rapidly than the economy, and β will increase. But if β is greater than s/g, then capital will grow more slowly than the economy and β will decrease.

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How Hassett Gets Piketty Wrong

Gwen Ifill and PBS’s NewsHour hosted a brief debate last night on Thomas Piketty’s Capital in the Twenty-First Century. The participants in the debate were Heather Boushey of from the Washington Center for Equitable Growth and Kevin Hassett of the American Enterprise Institute. In the course of the debate, Hassett made an important representation about Piketty’s argument that Boushey effectively let pass. That’s a shame, because Hassett has misinterpreted Piketty in a rather serious way, and the error shouldn’t be allowed to stand.

At one point in the discussion, Hassett says this:

And if you look at what’s been going on in his data, then the share of income going to capital in the United States has gone up over time. And what he [Piketty] does is, he gives a theory for why that’s going to continue, and eventually capital is going to have everything, unless we have 80 percent tax rates and so on.

But the problem that he has is that his story for why that could happen, why capital will ultimately get all the income, is that we’re going to substitute capital for workers, and so we’re going to have robots making hamburgers and so on, which is a story — it could be true that that’s something that we need to think about.

So according to Hassett, Piketty has argued that eventually capital is going to “have everything” and “get all of income.” And the reason for this is that our economy can indefinitely substitute capital equipment for labor, so that all returns will eventually accrue to the owners of that capital.

But Piketty makes no such argument.

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Nothing Magical about Piketty’s Mathematics

George Cooper worries that Thomas Piketty has advanced some “magical mathematics” in Capital in the Twenty-First Century: mathematics which lead to absurd results. And Cooper argues that when one repairs these absurdities in the most logical way, Piketty’s main policy prescription – a 2% global wealth tax – is seen as a recipe for economically disastrous effects. But the mathematical fix Cooper proposes is not at all plausible given the historical trends Piketty’s research has revealed. More importantly, I think Cooper has simply misconstrued Piketty’s mathematics, and that it doesn’t require any fix at all. When we interpret the mathematical formulae in the way Piketty has indicated, and impose economically natural conditions on their range of application, Cooper’s problem disappears, and Piketty’s historical analysis and projections emerge unscathed.

The problem Cooper believes he has detected emerges from what Piketty calls the Second Fundamental Law of Capitalism. The law relates an economy’s long-term capital-to-income ratio β to its savings rate and growth rate. For convenience, Piketty states this law in the form “β = s/g”.  Since β is the ratio of a nation’s capital K to its annual national income Y, it might seem that the law can be restated in the form of the identity:

K/Y = s/g

And if we do interpret it as an identity – or at least an approximate identity – the Second Fundamental Law would lead to some rather absurd results for hypothetical low-growth economies. For example, using one of Cooper’s examples of an economy with a 10% savings rate and 0.25% growth rate, we get an s/g of 40. If K/Y is approximately equal to s/g, and if the rate of return on capital is between 4% and 5%, then the capital share of income rK/Y falls approximately between 160% and 200%. Obviously income from capital cannot exceed total income, so something has gone wrong here.

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A Flaw in The Economist's Explanation of Piketty

The Economist, in its “The Economist Explains” column, offers a terse four-paragraph summary of Thomas Piketty’s Capital in the Twenty-First Century. The summary is useful on the whole. However, on two important points, I believe, the benefit of terseness has been purchased at the cost of accuracy.

The Economist begins by outlining the historical elements of Piketty’s monumental study:

“Capital” is built on more than a decade of research by Mr Piketty and a handful of other economists, detailing historical changes in the concentration of income and wealth. This pile of data allows Mr Piketty to sketch out the evolution of inequality since the beginning of the industrial revolution. In the 18th and 19th centuries western European society was highly unequal. Private wealth dwarfed national income and was concentrated in the hands of the rich families who sat atop a relatively rigid class structure. This system persisted even as industrialisation slowly contributed to rising wages for workers. Only the chaos of the first and second world wars and the Depression disrupted this pattern. High taxes, inflation, bankruptcies, and the growth of sprawling welfare states caused wealth to shrink dramatically, and ushered in a period in which both income and wealth were distributed in relatively egalitarian fashion. But the shocks of the early 20th century have faded and wealth is now reasserting itself. On many measures, Mr Piketty reckons, the importance of wealth in modern economies is approaching levels last seen before the first world war.

The statement that “high taxes, inflation, bankruptcies, and the growth of sprawling welfare states caused wealth to shrink dramatically, and ushered in a period in which both income and wealth were distributed in relatively egalitarian fashion” is very misleading, and does not represent Piketty’s account very well, since it creates the mistaken impression that all four of the cited factors were responsible for both a reduction in wealth and its more egalitarian redistribution. But high taxes, inflation and welfare states caused a redistribution of wealth, not its destruction. In some cases the redistribution went from private hands to other private hands; in other cases, the redistribution took the form of a conversion of private wealth to public wealth. But since the emergence of the social state in the mid-20th century, national wealth has grown consistently in both Europe and the US. Whether the social state was responsible for a slower rate of wealth accumulation, or in fact supported higher rates of wealth accumulation, is a matter about which people may disagree, but there was no aggregate wealth destruction during this period. The outright destruction of wealth was caused earlier in the 20th century by war and depression. The other factors, combined with strong postwar growth, were responsible for slowing the return of pre-20th century levels of inequality in the distribution of wealth and income as wealth continued to grow.

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Laissez-faire's Piketty Problem

Noah Smith remarks that Tyler Cowan’s portion of the Marginal Revolution blog, since the release of Thomas Piketty’s Capital in the Twenty-First Century, seems to have become “your one-stop-shop for anti-Piketty links and analysis.” Smith goes on to say:

This is interesting, because Cowen himself recently wrote a book called Average is Over, which makes predictions somewhat similar to those made by Piketty. From the Wikipedia article about Average is Over:

Cowen forecasts that modern economies are delaminating into two groups: a small minority of highly educated and capable of working collaboratively with automated systems will become a wealthy aristocracy; the vast majority will earn little or nothing, surviving on low-priced goods created by the first group, living in shantytowns working with highly automated production systems.

There is a difference, of course. Piketty forecasts that wealth inequality will go up because of an increase in capital’s share of income, while Cowen forecasts that wealth inequality will go up because of increased inequality in labor income. But the basic future foretold by the two is the same.

Speculating on why Cowan has turned into “an anti-Piketty crusader,” Smith continues:

But it also seems possible that Piketty has deeply worried economists and pundits who thought that concern over inequality was a thing of the past, and that laissez-faire had basically won the battle of ideas. Piketty’s immense popularity might seem, to these folks, to threaten to drag us back into a dark age when radical wealth redistribution was taken seriously, not only by large segments of the public, but by a number of prominent economists as well. Piketty might seem like the vanguard of an onrushing wave of socialist thought that could succeed in turning back the tide of neoliberalism that had been advancing for at least 40 years. So Cowen – and the numerous anti-Piketty writers he links to – may simply be worried about Piketty and what he represents.

And I think that account is quite plausible. But I also think more can be said about the specific nature of the challenge Piketty poses to contemporary neoliberal and laissez faire attitudes. Smith touches on this challenge in the passages quoted above, but doesn’t really pick it up and develop it.

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